FINAL REPORT NASA GRANT NSG-2112 for TRANSONIC FLOW STUDIES from the University of Arizona Tucson, Arizona 85721 Principal Investigator: Co-principal Investigator: Technical Officer: Dr. A. Richard Seebass Dr. K.-Y. Fung Dr. William F. Ballhaus NASA Ames Research Center Period: i November 1975 - 31 October 1980 28 February 1981 https://ntrs.nasa.gov/search.jsp?R=19810009506 2020-03-21T14:57:20+00:00Z FINAL REPORT N_-_II2 i NTRODU(]TI ON This einal report delineates research activities on NASA grant NSO-._II2 Srom 1976 to 1980. These research activities were also supported in part bg the AFOSRand the ONR. This res,_arch result_d in _i_teen pub] ications to October 31, 1980; _our addition_l publications are planned as a consequence o_ this research. ]he publications are listed in chronological order in the Appendix. [he list is annotated to indicate the cited source o_ _unding _or the research reported upon in the publication. The Appendix also lists the talks given bg vat. ioos investigators supported b,._ this grant, _s well as the eleven Transonic Fluid Dgnamic Rel)orts we have issued during the grant period. SHOCK-FREEAIRFOIL AND WINg DESIGN For most uS the grant period we had an active research program on the direct and indirect design oe shock-_ree airSoLls and wings. We report ot_ this research here, noting that while snme minor activities continue, the major thrust o_ this rese,_T'ch has be(_n completed. We limit the main part uS our discussion to results theft halve not get been published. fransonic Air_oils with a Oiven Pressure Distribution Out' original collaboration with Dr. Helmut Sobieczkg resulted _t'om a plan to adopt his rheograph design method tha% used analog computations to a comp]etelg digital approach. As a result of this collaboration, we have developed a procedure that uses a map o_ the hodograph p_Lane to _ind tT'ansonic air_oils with a given pressut'e distribution. This work complements related studies that provide air_oil modiFications to achieve shock-_ree _low. The method is limited to _ pT'escription o? the subsonic portion o? the pt'essure distribution and to shock-_ree supersonic ?lows. We have no plans to extend this capabilitg _urther. The procedure we have developed is ?airlg e1-_?icient ?or subc_itical ?lows; we Seel that this will remain true _or supercritical ?lows, but we have onig pt-eliminarg T'._SU].ts ?Or this case. The procedure we use is the _ollowing: We eirst note that i_ we use the Prandti-Meger _unction, v , and the _Low deflection angle, 0 , _s independent variables, then the equal;ions _or the stream Sunction and velocitg potential are not onlg l_near, as th(_ _ould be with the hodograph transformation, but theg are also in canonical _orm. That J s, the second-o,-der derivatives either _orm the Laplaci_n or the wave operator depending on whethe, r or' not the !:iow is subsonic or- supersonic. Thus the equations _:or the subsonic ?lo,_s are invariant in ?orm under a con?ormal t'ransFo'r'mation. We thus assu_,e _hat a con_ormal map ?tom the two--s_eeted Hiemann sur-Face o? the subsonic portion o_ the v , 8-plane into the unit circle. Part o? the boundarg og the unit circle corresponds to the air#_oil surface ,_etted bg subsonic _iow_ the other pa'.-t corresponds to the sonic line. 'These portions are then chosen and the pressure on the subsonic part oS the unit circle is prescribed On the sonic line. segment the pressure takes its critical va [ue. pre__sure given on the boundarg oS the unit circle, and subsonic portion o? the ?Io,_ inside the unit cicle, advantage o r- the ?act that the mapping to this plane is Thus the PT'_ndtl-Meger _unction _Jnd the ?]ow de_lectLon conJugate harmonics. Because the pressure is given on circle we kno,_ the Prandt]-Meger ?unction there. We Laplace's e(luation ?or the Prandtl-Meyer ?unction inside circle using FouT'ier series_ this immediatelg determknes de?!ection _ngle to within a constant. With the with the we take con_ormal. ang le are the unit then solve the unit t h e _ Io tLJ On the portion o? the air_oil sur?ace that corresponds to sub__onic ?loci the stream _unction is zero. We prescribe a distribution ?or the stream function on the sonic line to complete the boundarg value problem ?or the stream ?unction° This boundarg valu_e problem is then solved using a ?ast PoLsson solver, determining the stream _unction and therebg the velocLtg potential, as functions o_ v and 8 in the subsonic portion o.P the ?low al_d on the sonic line. These results are then mapped back to the phgsical plane to see i? the corresponding air?oil is a reasonable one. le it :i.5 not, then the input must be modi?ied. When a suitable air?oii is :-'ound, the data on the sonic line is integrated in the v, 0 -plane to _ind the zero streamline. This, th,._n, determines that portion o? the air_oi] wetted bg supersonic f:low. This calculation mag _ail due to a singu_laritg in the mapping back to the phgsical plane. This indicates that the sonic lin_._ data is not consistent uith shock-?ree ?low and req.uires that the input be c h an g e d. There are several minor points that cause great di_Sicultg and h_ve reo_ui_ed some imagination to circumvent. These include ti_e logarithmic singula_itg in v at _ = O, a -I/3 power singularitg in the normal derivative o_ the stream Sunction at the .sonic line in the v , 0-plane, and the need to choose the circulation oe th_ _:ar-?ield singularitg, which occurs at the origin in the v, 0-plane, so that the stagnation streamlines are normal to the bodg. The latter two are especiallg diSSicult because theg must be resolved verg accurately in orde_ to achieve satis_actorg results. Fictitous gas Techniques The concept o_ the eic titious gas has been described in eariieT, grant reports. Thus, we only give a brie? sgnopsis o? this research here. in two dimensions we have extended our studies to ip.clude v_.s(:ous e_ects. 0'._,_"results are reported in ,_ rec_nt A¢_ARD paper. As we suggested in last gea_''s brie_ propos_l, shock-Free designs can be ?ound with little computational e?_ort even when globally uJeak and loca].]g strong viscous - invisciJ interactions are included. This is s:imp]g a consequence o.F the fic:titious pressure gradient in the supersonic domain being c]_se enough to the actual pressure gradient _or the design process to essentiallg duplicate the boundarg lager displacement thickness o_: the renl shock-free .Flow. In a collaborative effort with Dr. Sobieczkg of the DFVI_R in gottingen w,÷ have examined adaptive airfoils and, wings in order to achieve shock-_ree _Iow over an extended range of Macn numbers and lift coe_Si,:ients. These results are reported in ,_ recent ICAS preprint. During the past gear we have implemented the Sictitous gas procedure with the full potential code FLO 22. Par,_llel work at Lockheed bg Drs. Miranda and Raj has also been successful. CONTROL ALO(]RITHM FOR ADAPTIVE WALL. TUNNELS As part oF our NASA supported research u,._ have beer, investigating a control algorithm i_or three-dimensional adaptive wall transonic wind tunnels. This research was interrupted last gear when the student carrging out the investigation, was forced to •retL_rn to Poland because his visa expired. These political problems now seem to be resolved and he is back with us. We have developed a relativelg fast ADI algorithm for the subsonic flow past li_ting wing-bodg in a circular or rectangular wind tunnel. The boundarg conditions on the wind tunne] wa].l corresponding to variable porc sitg and plenum pressure are modeled bg _r(X,R,O) + a(x,O) _bx (x,R,O) = 8(x,O) f_or a cglindrical wind tunnel ol_ radius R. Here a and B aro directlg, but empiricallg, related to the porosi_g and plenum pressure. A similar algorithm is used to compute the external ("low. Thus, we mag simulate adaptive wall strategies numericailg. The algorithm is now being extended to the tran_.onic range. When this is complete our studies can begin in earnest. Tl_e .Fundamental question ,_e wish to address is this: given a measurement o@ _r and _x on some surface near, but not at, the wind tunnel walls, what strategy is to be invoked to modifg _ and B so that the errors in, sag li._t and drag, due to the wind tunnel walls _3re minimized7 ]'here are, ol_ course, also constraints on the values that e and B can assume since theg model to some degree the ei_=ects o_ wail porositg and plenum pressure. We are thus ?aced with an optimal control problem which involves a nonlinear partial difi_erential equation. There is not much literature on optimal control with partial differential equation constraints and we mag have to invent our theorg for the case at hand. To do so we _,i_.l begin ,.J_ith the l:.near subsonic equation. We can set up certazn simple model situations and write down formulas £or analgtJcal solutions using green's _unctions. We pT'esume that we will be ab]e to discern an appropriat._ strategy _:or determining the changes to be made in _ and B in ._rde'r to Sind their optim,_m values. With this bodg oS theorv in han_, the results will be veriSied bg numerical experiments For the pr_totgpe line_r problem. 'We will next extend these results to the _eneral !inea_r pT'oblem. Wh,_nwe are convinced that we have a uJorkable strategy _oi' subsonic Slow we will then attempt to generalize it to transonic T_low. No doubt changes will be required to account Sot the possibility that part oS the tunnel supersonic Slow will the wind tunnel wall. It is, o_ course, not clear that we have posed has a unique mathematical solution. adaptive wall wind tunnels are to be develope_ three-dimensional Slows, then some practical strateg# is essential. extend to the problem Still, iF Sot tru.1 _ Sot control UNS]'EADY TRANSONIC FLOW In the course o_ resining our time-linearized a_gortihm eor unsteadg transonic _lows it became clear that acoustic waves were being re_lected Srom the boundarg and contaminatin_ the re.suits computed _°or indicial motions. This would not necessarilg be noticed in a harmonic anaigsis, nor is it especia]Ig easg to discern in the indicial motion. It is clear, however, that serious errors in pilase lag occur il_ the boundarg conditions i_or the Sar-_ield reSlect acoustic waves as the common _ = O boundaT'tl conoition does. In order to remedg this situation !)r. K-Y. Fung derived the correct unsteadg _ar-_ield and this is. n_w implemented in our algo'--ithm. The results oS this research have been accepted _or publication in the AIAA Journal. OuT" studg oS two-dimensional time-linearized Slows is essentiallg complete except Sot the studies oS the unsteadg behavior o_ shock-_ree aireoils to see how their bueeet boundaries migDt diS_e," Srom supercritical airSoils with _ell developed shock waves. A paper is being prepared bg Dr. Fung that will report the conclusions oS this studg. We note several here: I) For determining _lutter boundaries a time-linearized algorithm that _aptures shock motions oS arbitrarilg small size is probably more e_icient than a nonlinear algorithm_ _or more moderate amplitudes either a nonlinear or a time-linearized algorithm will do_ So_' larger amplitudes a nonlinear treatment is essential. _) One must be concerned about the accuracg o_ wind tunnel data Sot unsteadg trarsonic Slows as wall reelections mag lead to serious errors in the phase lags. Numerical simulations give strong evidence oS su(:h phase lag errors due to boundarg reSlections. 3) Shock-Sree airfoils have markedlg diS_erent phase lags than the_' counterparts with well eormed shock waves. C(]MPUTATIONALFACILITIES This N,_SAgrant was, at least in part, responsible for 'the development o? an advanced remoted job entrg capbiiitg in tile Universit9 of Arizona's Computational Mechanics Laboratoru. We now have an Eclipse S/140 computer with 256 kilobgtes o_ core memoru, two CRT terminals with local intelligence, i_our standard CR] terminals, and a TI 745 portable terminal. Funds from the College of E:ngineering have been used for some of these Sacilities_ th(_.g have also been used to provide work space ?or eight graduate students. These facilities provide us with remote _ccess to the NASA Ames CDC 7600 via a UT _200 interface. The same interface is used to go _lirectlg to the Universitg's Cgber 175. There is also a direct line to the Interactive graphics Engineering Laboratorg's Eclipse S/_2:30, and therebg to its other facilities such as the Ramtek color displag. PAPERS AND TALKS 1980 We list here the papers and talks ?or our last _rant theu have not previouslg been reported. As noted earlier, _or the complete grant peT'iOd is to be found in Appendix A. gear as the list Ti,I_eman, H. and R. Seebass. "Transonic Flow Past Oscillating A_r?oils, ' Annual Revie_ of Fluid Mechanics, Vol. i;_, 1980. Fung, K.-Y.. Sobieczkg, H., and R. Seebass. "Shock-Free Wing Design, " AIAA d. , Vol. 18, 1980. F ung, K.-Y. "Far-Field Boundar4 Conditions ?or Unstea_g Transonic Flows," (to appear AIAA J. ). Fung, K.-Y. "Shock Wave Formation at a Caustic," SlAM d. APPL. MATH., Vol. 39, 1980. Nebeck, H. E., Seebass, A. R. and H. Sobieczkg. "Inviscid - Viscous Interactions in the Nearlg Direct Design of Shock-Free Supercritical Air@oils, " (to appear AgARD Conference Proceedings) Sobieczkg, H. and A. R. Seebass. "Adaptive Air?oils and Wings ?or Efficient Transonic Flight," ICAS preprint, October 1980. R. Seebass: Unsteadg Transonic Flow, NASA Langleg, Februarg 1980. R. Seebass: Shock-Free Airfoil and Wing Design, Caltech Colloq.uium April 1980. R. Seebass: as above, Universitg of Colorado, April 1980. R. Seebass: as above, Universitg of Minnesota, April 1980. R. Seebass: DFVLR Seminars on Unsteadg Transonic Flows, Shock-Free Design, and Nonlinear Waves, Mag 1980. R. Seebass: as above at Hai?a Universtig, Tel Aviv Universitg, Me_thematics and Fluid Mechanics Departments, Mag 1980. R. Seebass: general Aviation Appplications o? Shock-F_ee Design, general Aviation Technolggfest, Wichita, November 1980. APPENDIX PAPERS, REPORTS, TAIKS 1977 - 1980 Papers ***i. Seebass, A. R. and Fung, K.-Y., "Unsteady Transonic Flows: Time-Linearized Calculations," Numerical and Physical Aspects of Aerodyrmmic Flows (to appear). ** 2. Fung, K.-Y., "Far Field Boundary Conditions for Unsteady Transonic Flows," (accepted AIAA J. ). * 3. Fung, K.-Y., "Shock Wave Fornmtion at a Caustic," SIAM J. Appl. Math., Vol. 39, No. 2, pp. 355-371, October 1980. ";_ 4. Fung, K.-Y., Sobieczky, H. and Seebass, R., "Shock-Free Wing Design," AIAA J., Vol. 18, No. i0, pp. 1153-1158, October 1980. 5. Sobieczky, H. and Seebass, A. R., "Adaptive Airfoils and Wings for Efficient Transonic Flight," ICAS Preprint, Munich, October 1980. **6. Nebeck, H., Seebass, A. R. and Sobieczky, H., "Inviscid-Viscous Interactions in the Nearly Direct Design of Shock-Free Supercritical Airfoils," AGARD Fluid Dynamics Panel Symposium, Computation of Viscous-lnviscid Interactions, Colorado Spry, September 1980. *7. M_ran, J., Cole, K. and Wahl, D., "Analysis of TWo-_ional Incompressible Flows by a Subsurface Panel Method," AIAA J., Vol. 18, No. 5, pp. 526-533., May 1980. 8. Tijdeman, H. and Seebass, R., "Transonic Flow Past Oscillating Airfoils," Ann. Rev. Fluid Mech., 12, pp. 181-222, 1980. ****9. Cramer, I_. S., "Lifting Three-lYimensional Wings in Transonic Flow," J. Fluid Mech., Vol. 95, Part 2, pp. 223-240, 1979. *** I0. ** II. Sobieczky, H., Yu, N. J., Fung, K.-Y. and Seebass, A. R., "A New Method for Designing Shock-Free Transonic Configurations," AIAA J., Vol. 17, No. 7, pp. 722- 729, July 1979. Sobieczky, H., "Related AnalyticalAnalogand l_mmricalMethods in Transonic Airfoil Design," AIAA 12th Fluid & Plasma Dynamics Conference, Paper 79-1556, Williamsburg, Virginia, 23-25 July 1979. **ONR funded ,__R and ONR joint funding ._,AFOSR, NASA, and 0NR joint funding ,****AFOSR and NASA joint funding NASA funded ***' 12. 13 ** 14. **** 15. Seebass, A. R., Yu, N. J. and Fung, K.-Y., "Unsteady Transonic Flow Computations," AGARDConference on Unsteady Aerodynamics, CPNo. 227, pp. II, 1-17, 1978. Fung, K.-Y., Yu, N. J. and Seebass, R., "Small Unsteady Perturbations in Transonic Flc_cs," AIAA J., Vol. 16, No. 8, pp. 815-822, August 1978. Cranmr, M. S. and Seebass, A. R., "Focusing of Weak Shock Waves at an Ar_te," J. Fluid Mech., Vol. 88, Part 2, pp. 209-222, 1978. Yu, N. J., Seebass, A. R. and Ballhaus, W. F., "Implicit Shock-Fitting Scheme for Unsteady Transonic F10w Computations," AIAA J., Vol. 16, No. 7, pp. 673-378, July 1978. "' Talks 7th _nnual General Aviation Technologyfest, Wichita, November 1980. (Seebass) International Council of the Aeronautical Sciences Meeting, Munich, October 1980. (Sobieczky) AGARD Fluid Dynamics Panel Symposium on Viscous-lnviscid Interactions, Colorado Springs, September 1980. (Seebass) University of Haifa Colloquium, May 1980. (Seebass) lhiversity of Tel Aviv Mathemmtics Seminar and Fluid Mechanics Colloquium, May 1980. (Seebass) Israel Aviation Industry, Tel Aviv, May 1980. (Seebass) DFVLR, Gottingen Colloquia (3), May 1980. (Seebass) lhiversity of Minnesota Colloquium, Minneapolis, April 1980. (Seebass) University of Colorado Colloquium, Boulder, April 1980. (Seebass) California Institute of Technology Colloquium, Pasadena, April 1980. (Seebass) Unsteady Transonic Flow, NASA langley Workshop, February 1980. (Seebass) Shock-Free Flc_s, NASA Lewis, November 1979. (Seebass) AIAA 12th Fluid & Plasma Dynamic Conference, Williamsbu_g, July 1979. AIAA 12th Fluid & Plasma Dynamic Conference, Williamsburg, July 1979. lockheed-Georgia Company Seminar, July 1979. (Seebass) AFFDL/AFDSR Review, Dayton, June 1979. (Seebass) lockheed-California Cc_y Seminar, April 1979. (Seebass) (Fung) (Sobieczky) University of Washington Colloqui_n, Seattle, February 1979. (Seebass) Ohio State Meeting on Unsteady Transonic Flow, Columbus, February 1979. University of Southern California ColloquitTn, Los Angeles, October 1978. (Seebass) (Seebass) AIAA llth Fluid & Plasma Dynamics Conference, Seattle, July 1978. AGARDFluid Dyr_cs Panel Symposiumon Unsteady Aerodynamics, Ottawa, September 1977. (Seebass) AIAA 10th Fluid & Plasma Dynamic Conference, Albuquerque, June 1977. AIAA 10th Fluid & Plasma Dynamic Conference, Albuquerque, June 1977. University of Arizona Colloqui_n, Tucson, April 1977. (Plmg) University of Arizona Colloqui_n, Tucson, Arpil 1977. (Yu) (Sobieczky) Ju) (Fung) Dr. H. Sobieczky of the DFVLR in C_ttingen was a Visiting Adjtmct Professor at the University of Arizona in 1977 - 1978 with ONR and AFOSR support. Dr. Sobieczky gave a series of lectures on Tr_ansm%ic Fluid Dynamics at the University of Arizorm. He also gave the seminars listed below. California Institute of Technology, Pasadena, January 1978. Stanford lhiversity, Stanford, February 1978. University of California, Berkeley, February 1978. Naval Pos _tgraduate School, Mmnterey, February 1978. lhiversity of Southern California, los Angeles, February 1978. NASA Research Center, Hanpton, Panel P_mber, Airfoil Conference, P_rch 1978. Virginia Polytechnic Institute, Blacksburg, March 1978. lhiversity of California, San Diego, Fsrc_h 1978. University of California, Los Angeles, May 1978. Transonic Fluid Dynamics Reports - Engineering Experiment Station Report TF9 80-03, Seebass, A. R. and Fung, K.-Y., "Unsteady Transonic Flows: Time-I/nearized Calculations ," October 1980 (same as Paper I). Report TFD 80-02, Nebeck, H., Seebass, A. R., and Sobieczky, H., "Inviscid- Viscous Interactions in the Nearly Direct Design of Shock-Free Supercritical Airfoils," October 1980 (same as Paper 6). Report TFD 80-01, Sobieczky, H., "Computational Algorithms for Wing Geometry Generation, Transonic Analysis and Design," July 1980. Report TFD 79-02, Sobieczky, H. and Seebass, A. R., "Shock-Free Supercritical Aerodynamic Structure and Method for Designing Same," August 1979. Report TFD 79-01, Fung, K.-Y., Sobieczky, H. and Seebass, A. R., "Nume_-ical Aspects of the Design of Shock-Free Wings," July 1979 (same as Paper 4). Report TFD 78-05, Ballhaus, W. F., Craner, M. S., Yu, N. J., Fung, K.-Y., and Seebass, A. R., "Unsteady Transonic Flow Studies,"_ocusing of Weak Shock Wave at an Arete (same as Paper 14); Unsteady Transonic Flow Computa- tions (same as Paper 12); Stall Unsteady Perturbations in Transonic Flow (same as Paper 13); Implicit Shock-Fitting Scheme for Unsteady Transonic Flow Computations (same as Paper 15)3 . Report TFD 78-04, Sobieczky, H., Fung, K.-Y., Seebass, A. R. and Yu, N. J., "A New Method for Designing Shock-Free Transonic Configurations," July 1978 (same as Paper I0). Report TFD 78-03, Sobieczky, H., "A Computational Algorithm for Embedded Supersonic Flow Ik_," July 1978. Report TFD 78-02, Sobieczky, H. and Seebass, A. R., "Adaptive Airfoils and Wings for Shock-Free Supercritical Flight," May 1978. Report TFD 78-01, Sobieczky, H., "A Computer Program for Analysis of Transonic Flow Past a Wall Ramp," January 1978. Report TFD 77-01, Sobieczky, H., "Transonic Fluid Dynamics lecture Notes," October 1977. Patent Application "Adaptive Shock-Free Supercritical Aerodynamic Structures and Method for Designing Same," Serial No. 06/049,846 (with Sobieczky), 1979. q& UNSTEADY TRANSONIC FLOWS: TYHE-LINEARIZED CALCULATIONS A. Richard Seebass and K.-Y. Fung Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona An accurate and efficient method of computing unsteady transonic flow is described. The flow is linearized about an experimentally measured or numeric- ally calculated steady state, as represented by a given pressure distribution. For a given mode of motion, the amplitudes and phase lags of the llft and moment coefficients at a given reduced frequency are found by superposition from an indiclal response. The computational effort is reduced by treating shock waves as discontinuities, and by applying the correct linear far-fleld behavior. A novel method of modeling the indicial response provides an analytical formulas for the dependence of the amplitude and phase lag on the reduced frequency. INTRODUCTION A combination of technical advances should improve the fuel efficiency of transport aircraft by fifty percent in the next decade. Analogous improvements in the transonic performance of mill- tary aircraft should also be realized. These large gains will come from a combination of improvements in engine, structural, and aerodynamic efficiency. More than half will come from improvements in the aerodynamic efficiency, including active control, and the use of composite materials in the primary structure. Part of the improvement in aerodynamic efficiency will result from flight at supercritical Math numbers with subcritlcal levels of lift to drag ratio and high lift coefficients at near sonic flight conditions. This improved transonic performance mandates an accurate prediction of aeroelastic behavior at transonic M.ach numbers. Of special concern are flutter boundaries. In 1976 Farmer and Hanson (i) reported that the flutter boundaries of two dynam- ically identical wings were markedly different at transonic ._ach numbers due to very minor differ- ences in wing profile thickness. The resul_s of their measurements are shown in Figure I, indica- ting _he reduced flutter boundary for the wing with a "supercritical" profile. Today we understand well the qualitative behavior of inviseid steady and unsteady transonic flows, and we have rudimentary understanding of viscous effects. For flight regimes that involve unseparated flows the main ingredient in the calcu- lation of flutter boundaries is an accurate deter- mination of the steady pressure field. This may be determined either by experiment or by calculation. But once it is known, the response of the wing to pitching, plunging or aileron motion may be found by numerical means. The most essential ingredient in predicting this response is an accurate predic- tion of the motion of any shock waves present in the flow (see, e.g., (2)). Numerical algorithms that capture shock waves must use relatively fine grid spacing near the shock wave if they are to predict its motion due to the small changes of interest in flutter studies. But this shock motion may be predicted accurately bya time-llnear- ized algorithm using a relatively coarse grid if the calculations are done correctly. This has not generally been the case, with other investigators ignoring this essential effect (3,4,5). We report here on our two-dimensional, time- llnearized computations, which not only properly account for shock wave motion, but are able to resolve them even though the grid used to calculate the flow is relatively coarse. In order to avoid the reflection of the unsteady disturbances from the grid system, only a moderate amount of grid stretch- ing is employed away from the alrfoil. To avoid unnecessarily large computational domains, the linearized far-fleld for an unsteady vortex with a circulation determined by the airfoil's llft is used to evaluate the potential there (6). The alrfoil's response to a given mode of motion is determined by superposition from that for an indicial motion. In many cases this indlcial response can be modeled in a simple way, providing an analytic result for the dependence of the lift and moment coefficlent's amplitudes and phase lags on reduced frequency. A novel feature of this modeling is that of a sequence of harmonic oscillators, each of which improves the previous simulation of the indiclal response. This provides an analytical formula for the dependence of the amplitudes and phase lags on the reduced frequency (7). The computationa ! efficiency of the time- linearlzed calculation of the amplitudes and phase lags for a range of reduced frequencies is compared with nonlinear and frequency domain computations. The tlme-llnearlzed calculation of an indicial response can result in a factor of ten or more reduction in computational effort; this is especially significant in three dimensions. The modeling of the indicial response by a sequence of two harmonic oscillators can also reduce computational effort. GOVERNING EQUATIONS As noted above, tlme-linearization about a known steady state is an effective mechanism for determining the unsteady response to a given mode of motion. For flows that are unseparated, and not at incipient separation, an inviscid treatment of the unsteady flow should be adequate for flutter studies. It is important, however, that the nonlinear and viscous aspects of the underlying steady state be determined accurately. This may either be done by experiment, or by a reliable computational algorithm such as Grumfoil (8). In any case, we assume that an accurate steady state pressure distribution has been used to provide the input fcr the inverse cal- culation of the airfoil shape that will provide this pressure distribution when the steady state flow is computedusingthesmallperturbationapproximation. AsLinet el. (9)observedmore than thirty years ago, within the context of the small perturba- tlon approximation the basic equation governing the unsteady motions is linear unless the reduced frequency, k, is 0(60). Here 6 O is the measure of the perturbation potential and k = _c/U, where _ is the frequency, c the chord and U the freestream speed. Thus, if we write the velocity potential,. #, as ¢ - Oc{x + %¢} where # is the perturbation potential and _ is o some measure of the disturbance, the governing equation is -(2k/_o)_xt + {(i- _1_)/_0- (' + l)M-2_x}¢xx (i) + cyy - O, where terms O(k2¢tt/6o ) and O(kCtcxx) have been neglected because k = 0(5 ). (Here the time has o been nondlmenslonalized by the circular frequency and the spatial coordinates by the airfoil chord, and the y coordinate is scaled by 601/2. ) The second is of little consequence. Neglecting the first is equivalent to disregarding one of the characteristics and assuming disturbances propagate downstream at infinite speed, and has important computational advantages. The boundary condition at the body is 3/29y(X,O,t) = 6Y'(x) + _Yx(x,t) + k6Y (x,t) (2)6 o t where the body is given by y - 6Y(x) + _Y(x,t). Across the airfoil wake the jump in the pressure coefficient must vanish. Thus, [ _ Cp(X,O,t)_ =- [,x(X,O,t) + k_t(X,o,_) _ = O, where E("')_ indicates the jump across the wake. This implies that in the wake a_(x,o,t) = r(x - kt)/Uc_ . (3) O In both these boundary conditions we have retained terms of O(k), which is not consistent with the approximation made In Eq. (i). But as Refs. (10) and (Ii) demonstrate, this gives good agreement with the results of linear theory for values of k up to, and even above, 1.0. This term is, of course, also retained in the evaluation of the pressure coefficients. As noted in Ref. (2), the appropriate measure of _ is max(63/2,_3/2,(k_)3/2) o and normally 53/2 . Far from the airfoil (see (6)), ¢(x,ylt') - ---1:t'f(x'y:t'2_'o - t')dF(t_)o_ dt_ (4) where f(x,y:t') - H(t' + x o /(x 2 _ y,2))× (tan -I /(t '2 + 2xt' -v'2_ + t' y' - tan-i /(t '2 + 2xt _y,2) - t',S., y' and Here H is the Heav_side unit step function. [n addition to Eq. (I) and the boundary condi- tions (2)-(4), a shock jump condition needs to be imposed if the shock wave is to be treated as a discontinuity rather than "captured" by the numeri- cal calculations. Because the former is the intent here, we need to note that - (2k.,M_/_o)[*x_2(dx/dt)s - H')I6 (7 + - , on (dy/dx) s where _('")_ and (.'') indicate the jump in and average of (-..) across the shock wave. Equation (ba) insures the conservation of mass. The conservation of momentum is replaced by the irrotationallty con- dition Eq. (bb) or its equivalent, _ - O. Time-linearlzation where We use the ADI technique introduced by Ballhaus and Steger (12) to compute the steady state solution of Eq. (I), _o(X,y), subject to the steady boundary conditions implied by Eqs. (i)-(4), using the coordinate stretching of Ref. (13). Aside from the far-field condition (4) and the inclusion of terms of 0(k) relative to O(I) in (2) and (3), this is equivalent to NASA Ames computer code LTRAN2 of Ballhaus and Goorjian (14). We next llnearlze about this steady state by assuming that _i_3/2 = o(l). This gives, with 5 - 5 . O O (2 /%)*xt + {[_1 - M°_)/_° + + " 0, :6a) wi th and - (x,t)+ kY (x,t) (6b);y(:,,o,t)_'x t !_xC_,o,t) + k_t(x,o,t) _- 0. _ith the linearization of the solution about a steady state at time t = O, we use (4) for the potential far from the airfoil as the circulation departs from its steady state value. (6c) As noted earlier, the proper account shock muttons are of prime importance in unsteady transo- nic flow. We thus use the procedure of Ref. (13) to account for shock motions. Because the shock waves are nearly normal to the freestream we assume that this is the case and satisfy on the normal shock approximation to Eq. (5), viz., dx = (y + I)M_ +÷ ;x Again we linearize about the steady state, writing in this approximation Xs(t) = Xos + (_/s_/mIx(t) which gives = y + 1 Sx(Xos,O,t) (7) dt 2k as the equation that keeps track of the shock wave osition. Straightforward linearizatlon of + (6/6_/2)_---- = 0 gives the expression that _o _etermines _ behind the shock from its value ahead of the shock (13): l;(x ,t ,y)Z" !; - 2k*1 (a> This must be integrated in conjunction with (6a). The ADI procedure is again adopted as outlined in Ref. (13), to effect a solution of Eq. (6) in conjunction with Eq. (8), subject to the time- linearlzed boundary conditions at Eqs. (6a) and (6_. INDICIAL RESPONSE One of the major advantages of time-lineariz- ation is that, for a given mode of motion, the amplitude and phase lag of the llft or moment coefficient for a given reduced frequency may he computed by a linear superposition of the results obtained for a step change. For example, if the change in llft coefficient as a function of time for a step change in angle of attack, C£ , is that a sketched in Fig. 2, then the lift coefficient for an angle of attack variation, a(t), is _C£ " .da(t - T)C£ (t) = C Z (t)s(o) + I tr)' dz dT. (9a) a " i=t Thus, for a periodic motion =(t) = _ • ae , o + C_ (=)a o. _e low frequency approximation made in Eq. (i), viz., that k2$t t was negligible, is, of course, not valid for the high-frequency components of the indicial response calculation. It is, however, perfectly satisfactory for the computation of the indicial response, provided this response is only used to compute motions for which k = o(i). In order to calculate the response for low reduced frequencies, however, we must accurately resolve the indicial response as the motion approaches its asymptotic state. Typically, this requires the computation of the indicial response for 300 chord lengths of airfoil motion. In this time the un- steady perturbations have travelled a little more than 300 chord lengths normal to the freestream. As a consequence, any boundary condition imposed on a IYl " constant boundary that is less than 150 chord lengths away can contaminate _he indicial response through a reflection from a boundary. Our experience has been that an erroneous boundary condition such as ¢ = 0 has to be imposed at lYi greater than 80 chord lengths in order to avoid errors in the phase lag determined _rom an indicial response. The same must be true for the computa_bn of a harmonic motion, although it would be more difficult to determine that the phase lag was in error in such a computation. This same observation should also be applied to unsteady wind tunnel tests. If there are significant acoustic reflect- ions from the wind tunnel walls, the observed phase lags may be in error. This experimental difficulty warrants further investigation, especially in two- dimensional studies. On the other hand, we know that with the appropriate steady state value of the potential applied at about twenty chord lengths, the steady state solution is perfectly adequate. With the imposition of the unsteady boundary condition (4), or its time-linearlzed analog, we find that once again 20 chord lengths will suffice. For low to moderate reduced frequencies, viz., k - 0.1 to 1.0, the acoustic wavelengths associated with the motion are about 1.0 to I0 chords. The grid spacing employed may be stretched, but grid spacing compar- able to or larger than the acoustic wavelength will result in acoustic reflections from the grid itself. Thus, while a grid stretching is employed in the calculations, the largest grid spacing used remains a fraction of a chord length. Harmonic Oscillator Mcdeling Typically, the indicial response of the lift coefficient to a step change in angle of attack, flap angle, or imposition of plunging velocity, is llke that as shown in Fig. 2. The same is also approximately true for the moment coefficient taken about the airfoil's leading edge (Fig.3)._ssuggests that, to a first approximation, the response is nearly exponential and governed by a simple first- order differential equation. And, further, that to a second approximation, the difference between this response and an exponential function can be modeled by a damped harmonic oscillator. Thus, if we let u(t) be a normalized indicial response such that u(0) = -i, u(m) = O, we should write where u(t) = I + :*o(t) + Ul(t) + ".- (i0) L u -= _ +lu = 0 0 o o 0 and, in general, Liui - ui + 2PiU + qiui = O. The constants _ ,pi,qi are determined to best model the indicial response. That is, the solutions to these equations, viz., -At u (t) = e (!2a) o and Ul(t) = ui(o)e-Pitsin(_t)/_ , (125) where n - /(qi - Pi 2)' are combined to best approximate the indicial response. To be specific, if we let _o " u(t) - u (t), then we choose X such that o I . S oS 2dt is minimum. Setting _ I /2k - O, we find o l-I = 2$={i- u(t)}2dt. o (13) (14) . (t) In an analogous manner we let ¢I u(t) - i - u ° -ul(t) and choose Pl and ql so that ii(Pl,ql) _ r_l%32dt (is) is minimized. This gives Pl= (_(o)®. - _ )2 and q__ = IQ(_ - u°)2dt. (16) 2$ (u - _ )2dr $ (u - Uo)dt o o o The extent to which the simple first approxi- mation is justified for selected examples is shown in Figs. (4) and (5). In many instances an accept- able determination of the phase lag requires the second approximation. This will be discussed more • (1) of the _ully in (7) The constant k and the I ° first approximation can be determined immediately from Eqs. (13) and (14) as the indicial response is being calculated. If I is not sufficiently small, =hen the constants Pl a_d ql can be calculated from Eqs. (16). If the second approximation is not judged sufficiently accurate because I 1 is not acceptably small, the modeling is abandoned as the computational expense of computing Pl and ql is com0;_rabLe co that required for three reduced frequencies. Some time ago it was noted by Tijdeman (15) and the authors (13) that the amplitude of a harmonic response decays llke k -I with increasing k. A somewhat more general result is implied by Eqs. (12). If F(t) is a harmonic response, e.g., CZ (t), then a with F(t) - Fo(t) + Fl(t) + "'', (17) we find chat the first approximation gives F /A = (i * k'2) ½ sin(kt - _o ) (ISa) o where sin@ * k'/(l + k'2) I/2 "(18b) o Here A is the amplitude of the indicial response to a unit change and k' - k/_. The second approxima- tion gives FI/A - 61(o)k'_'sin(kt - _I ) ÷ _¢I + (k' + n'32}{I + (k' - 2')2_1/2 (19a) where sin81 , (k,2 _ _,2 _ i) ÷ _ + 0) with - 2_xt + _,_ + 4"yr + _'u = 0 (3) ¢(x,O,z,t) = ½AI'(z, to)H(t-to)H(x ) Here, the dependent and independent variables are properly scaled. The solution we seek is, of course, antisymmetric in y. The boundary condition in Eq. (3) models a vortex sheet that originates at x=0. producing a jump in potential, Al'(z), instantaneously at time t= to. Far Field Solutions We employ the standard techniques of Fourier and Laplace transforms to solve Eq. (3) subject to the boundary condition for a change in the circulation, AI" (z,t o), at time t o for both two-dimensional and three-dimensional flow. We outline the steps briefly here. Three Dimensions To solve Eq. (3), we let _(x,y,z,s) be the Laplace trans- form of 4" +. +'i'm + _'=,-2s_ = 0 (4) with dimension, that b/2 AI'( _.,to)g(x,y,z- £.,t-to)d$ 4r (7) where b is the wing span and g(x,y,z,t) = H[ t + x- (x z + y2 +z 2 ) v, ] X [l+x(x2+yZ+z 2) -_]l(Z2+y 2) Because the derivation is for an incremental change AI" (Z, to ) at t= to, a more suitable form of Eq. (7) for 4" would be Y_)I b/2 • (x,y,z,t) = -_x -b/2 AI'(Lt°)g(x'Y'Z-Z't-t°)dz (8) i'(to} Equation (7) is essentially the steady-state result modified by the Heaviside function which switches on the value given by Eq. (8) when t_to = (x 2 +y2 +z z) . -x This implies the far field phase lag is simply (x z +y2 +z _) ,-x. Such simple behavior is found only in one- and three-dimensional wave propagation (see, e.g.. Lighthillt2). Equation (8) simply retards the value of the potential 4, so that it is the potential produced by the cir- culation at an earlier time corresponding to the time for a wave front to travel from the origin, to the location where 4, is being evaluated. Thus, by letting y _ b12 AF (_.,to) [ 1 +x(x 2 +y:d4"r (x,Y,Z, to) = -4x J-o/z + (Z__)2) -v_ ] [y,, + (Z__)2] -td_ " Equation (8) becomes t H(t - t o + x- "f_ + y2 + Z: ) d_r4"(x,y,z,t) = _o=o 4,(x,O,z,s) = ½AI'(z) ( l /s)H (x) The substitution _,=eSX_b, reduces Eq. (4) to the standard Helmholtz equation, viz., _, +_k,_ +¢_ -s_¢=O (5) with ¢(x,O,z,s) = ½A]" (z) (e-'_/s)H(x) We let _(E,y,z,s) denote the Fourier transform of ,_ with respect to x and _(E,y,_os) the Fourier transform of _, with respect to z. Applying such transforms to Eq. O). we find _._ (_-2+_2 +s 2),}=o with _(Si,O,_,s)= ½AP(D s-i_ s(s2 +_ _) which has the solution _(_,y,t.s)=½Ai _ s-i_ exp[_(_2+_2+s2)_y i (6) s(s 2 +_2) After performing the inverse transforms to Eq. (6) with respect to _',_,s correspondingly, we have, then, in three-space = d4"r =4"r [x,y,z,t- (x 2 +y2 +z2 ) '/..,+X] Two Dimensions In two dimensions we have with - 24,x_ + _I,,._+ 4,yy = 0 4"(x,O,t) = [AI'(to)12]H(x)H(t-t o) (9) instead of Eq. (3). Following the same procedure used in three dimensions, we have the result in two dimensions that 4"= [AI" ( t o )/2x ]f(x,y, t - to ) (I0) where f(x,y,t) =H(t+x-_--4-_) [tan -t x/t2 + 2xt-y_ + t Y _lt 2 + 2xt-y2 _ t ] +tan-/ y a This result is more complex with the time appearing not only in the unit step function, but also in the argument of the/_ arctangent function. In the limit, as we approach steady state, 182 Eq. (10) becomes f(x,y, oo) = y x-- - + tan -/x_ lyl 2 y which is the steady result of Kiunker. tl For an arbitrary circulation change AF(t), we must superimpose the results Eq. (10) to find the general two-dimensional far field 1 O(x,y,t) = -_ I_ dI'(t°)f(x,y,t- t o ) _ dt o In the case of harmonic motion, e.g., dI'(t) dt = Fmaxe'a K.-Y. FUNG Cf (t)/.x we simply change the lower limit in Eq. (l 1) to find e'_' I:f(x,y,r)e_,_, d rO(x.y.t)/I'm. , = (11) (12) which gives the phase lag at each location of the far field boundary. The condition given by Krupp and Cole 7 may be viewed as an attempt to approximate the result given by Eq. (12). We may also use Eq. (lO) to examine other far field boundary conditions. For example, the one given in Ref. 6 states that for iargey 03)Oy 4-O x =0 while analytically we see that (x_y) (X+ t) +X 2 +y2 Oy 4- O x = -- (X2 + y2 ) ( 12 +2_ '/g -- (x2 + y2 ) - v_ which shows that Eq. (13), while asymptotically correct, is not a suitable replacement for Eq. (10). Nonlinear Effect We examine here, following one reviewer's suggestion, the nonlinear effect on the far field by adding to the right-hand side of Eq. (9) a term proportional to ($°Ox) x [see Eq. (2)]. Denoting the solution of Eq. (10) as 4,n, we may write an integral equation for the solution • that 4_ O (x,y,t) =O h - 1o F(x'Y + _I'_'t) drl where + S_ F(x,Y-_,_,t)d_+ f;F(x,_-Y,_,t)d_ (14) F(x,_,_.t)= I7 f** " "('" +x-x'-xf°t-'4"'-(xS-x'_) .... 4[t'+(x-x')]:-[a2 +(x-x')2] 0 Ox" leo°" (x',_)O_. (x',_,t-t') Idt'cLr'X b We only need to examine Eq. (14) for y>0, since ¢,(x,O,t) =O,(x,0,t). For large x, or large c_, we may ap- proximate F(x,a, rl, t ) by evaluating the time integral near its singular point, i.e., t' = - (x-x') + x/aT¥ (x-x')2. Thus f_: --:v 0F(x, ct,_,t)= ®&vr_+(x-x , ) _-_x. [daoO.(x,,_,t+x 2O i0 AIAA JOURNAL - • -s ¢. /. / t _ * 0 (vma x = * III ¢'.) nonreflecti_e = + I_(v / / -• time acctlrat,. ;,re_,',_t _vml x * I i_ _large grill reS_llt fx : + _7"_ max Fig. 1 50 Ion ISO 20n indicial pitch response for an NACA 64A006 at Mo_ = 0.88. or I? X--X'F(x, ct,_,t)= ® [c& +(x_x,)2] ¢_o.ox.(x' _,t+x- x' -,_ + (x-x')r)dx' Since both _b°, and O_. decay as (x' + rl_ ) -"_, we may further approximate F(x, ce,_,t) by too F(x, ot,_,l) __ _o, ox,(x" _,l+x- x"C_ 2 +X 2 oo _ ,,/_2-+ (x _ x, ) 2)dx , (15) We recognize Eq. (I 5) as a doublet with strength in proportion to the compressibility effect. At large distances from the airfoil this term is negligible compared to the term we retain in Eq. (lO). Thus rb(x,y,t) approaches Oh (x,y,t) asymptotically. Example We illustrate the effectiveness of our results by applying them to the time development of the lift on an airfoil sub- jected to a step change in angle of attack. The time-linearized small perturbation algorithm of Ref. 9 is used as the test bed for the comparison. Figure l compares the lift response of an NACA 64A006 airfoil at M®=0.88, computed using different boundary conditions. Results obtained in a stretched grid with the outer boundary at a Ym_, of about 100 chord lengths away from the airfoil is compared with results obtained in a grid with a y_, of about 13 chord lengths. For the large grid we find no significant difference between solutions using different boundary conditions for the times indicated in Fig. 1. In the case of the small grid, we compare solutions obtained by setting 6_ =0 at y= +y=_,, by using the nonreflecting boundary condition _/-K-K6_+ _b,.= 0 on y = ±y_,, and by using the result of Eq. (!1) On Y=4-Yma,. The solutions are rather insensitive to upstream boundary con- ditions and are subject to the same downstream conditions. viz., _b_=0. The nonreflecting boundary condition of Ref. 6 achieves about 91°70 of the steady-state lift and gives a sub- stantial improvement over the conventional calculation. Results from the time-accurate boundary condition we have derived here are in very good agreement with those found using the large grid; these are uncontaminated by reflections from the boundary for ,'<200. Conclusion We have derived the far field unsteady solutions for a step change in the lift of an airfoil and a wing. These results can be used to reduce the size of the computational domain required for either time accurate or frequency domain calculations. We-x'-4_2 + (x-x'):)dx' I/ FEBRUARY 1981 BOUNDARY CONDITIONS FOR UNSTEADY TRANSONIC FLOWS 183 have illustrated their application with a time-linearized computation of a step change in the angle of attack of a two- dimensional airfoil. Acknowledgment The author thanks Dr. H. Atassi for his technical advice on formulating this problem and Dr. A. R. Seebass for many helpful discussions. The MAC's symbolic manipulation system located at MIT is responsible for some of the in- tegrations and for checking the results. This research was supported by the AFOSR Grant 76-2954G and the ONR Grant N00014-76-C-0182. References ZTijdeman, H. and Seebass, R., "Transonic Flow Past Oscillating Airfoils," Annual Review of Fluid Mechanics, Vol. 12, 1980, pp. 181- 222. 2Farmer, M. G. and Hanson, P. N., "Comparison of Supercritical and Conventiona| Wing Flutter Characteristics," Proceedings AIAA/ASME/SAE 17th Structures, Structural Dynamics and Materials Conference, King of Prussia, Penn., April 1976, pp. 608- 61 l; see also NASA TMX-72837, May 1976. 3Ashley, H., "On the Role of Shocks in the 'Sub-Transonic' Flutter Phenomenon," Journal of Aircraft, Vol. 17, March 1980, pp. 187-197. 4Ballhaus, W. F. and Goorjian, P. M., "Implicit Finite-Difference Computations of Unsteady Transonic Flows About Airfoils," AIAA Journal, Vol. 15, Dec. 1977, pp. 1728-1735. "_Magnus, R. J., "Computational Research on lnviscid, Unsteady. Transonic Flow Over Airfoils," Office of National Research, ONR CASD/LVP 77-010, 1977. 6Engquist, B. and Majda, A., "Numerical Radiation Boundary Conditions for Unsteady Transonic Flow," Journal of Com- putational Physics, to appear. _Krupp, J. A. and Cole, J. D., "Studies in Transonic Flow IV. Unsteady Transonic Flow," UCLA Eng. Rept. 76104, Oct. 1976. aTijdeman, H., "Investigations of the Transonic Flow Around Oscillating Airfoils," Doctoral Thesis, Technische Hogeschool Delft. The Netherlands, 1977. _Fung, K.-Y., Yu, N. J,, and Seebass, R., "Small Unsteady Perturbations in Transonic Flows," AIAA Journal, Vol. 16, Aug. |978, pp. 815-822. J°Houwink, R. and van der Vooren, J., "Results of an Improved Version of LTRAN-2 for Computing Unsteady Airloads on Airfoils Oscillating in Transonic Flow," A1AA Paper 79-1553, July 1979. J tKlunker, E. B., "Contribution to Methods for Calculating the Flows About Thin Lifting Wings at Transonic Speeds," NASA TN D- 6530, Nov. 1971. 12Lighthill, J., Waves in Fluids, Cambridge University Press, 1978, pp. 17-21. From the AIAA Progress in Astronautics and Aeronautics Series SPACE SYSTEMS AND THEIR INTERACTIONS WITH EARTH'S SPACE ENVIRONMENT--v. "/1 Edited by Henry B. Garrett and Charles P. Pike, Air Force Geophysics Laboratory This volume presents a wide-ranging scientific examination of the many aspects of the interaction between space systems and the space environment, a subject of growing importance in view of the ever more complicated missions to be performed in space and in view of the ever growing intricacy of spacecraft systems. Among the many fascinating topics are such matters as: the changes in the upper atmosphere, in the ionosphere, in the plasmasphere, and in the magnetosphere, due to vapor or gas releases from large space vehicles; electrical charging of the spacecraft by action of solar radiation and by interaction with the ionosphere, and the subsequent effects of such accumulation; the effects of microwave beams on the ionosphere, including not only radiative heating but also electric breakdown of the surrounding gas; the creation of ionosphere "holes" and wakes by rapidly moving spacecraft; the occurrence of arcs and the effects of such arcing in orbital spacecraft; the effects on space systems of the radiation environment, etc. Included are discussions of the details of the space environment itself, e.g., the characteristics of the upper atmosphere and of the outer atmosphere at great distances from the Earth; and the diverse physical radiations prevalent in outer space, especially in Earth's magnetosphere. A subject as diverse as this necessarily is an interdisciplinary one. It is therefore expected that this volume, based mainly on invited papers, will prove of value. 737 pp., 6 x 9, illus., $.30.00 Mere., $55.00 List TO ORDER WRITE: Publications Dept., AIAA, 1290 Avenue of the Americas. New York. N,Y. 10104 /..'Z. SIAM J APPI-. _dATH _ ] 980 SocieTy Ior Induslr_a) and AppIJed Malhcrna6cs 'Vol. 39, No 2. October 19811 tg)36-1399 80/3902-,)014 $01.00'O SHOCK WAVE FORMATION AT A CAUSTIC* K-Y. FUNGt Abstract. The behavior of a weak acoustic compression near a caustic surface has been determined for a special class of specified incoming signals. This problem arises in various contexts, including the propagation of sonic booms generated by supersonic aircraft. The solution derives from a mapping of hodograph-like solutions to the physical plane. The maximum amplification for this class of signals of fixed amplitude depends on the width of the incoming signal. The solutions contain reflected shock waves that satisfy the appropriate shock jump conditions, provided the width of the incoming signal is greater than a certain critical value. 1, Introduction. There are many sources of weak shock waves. These include the commonly experienced phenomena of thunder and the sonic boom of supersonic aircraft. At supersonic speeds, aircraft generate a nearly conical wave pattern. These wave fronts propagate along their normais at the local sound speed, i.e., along acoustic rays. Variations in the sound speed (due to temperature changes), aircraft maneuvers, and winds can lead to ray crossing and a focusing of the wavefront. In the case of the sonic boom this results in a so-called "'superboom" if the rays form an envelope, or a "super-superboom" if they meet at a point. In the case of a sonic boom, the pressure signature is, nominally, an N-shaped wave. More specifically, a weak shock wave provides an essentially instantaneous rise in pressure; subsequently the pressure falls linearly (in space or time) and is returned to nearly the ambient pressure through a second weak shock wave. The focusing of this weak shock wave is examined here. For simplicity, we phrase the problem in the context of the behavior of a weak shock wave generated by an aircraft in slightly supersonic flight in a flow with a Mach number gradient. Such gradients occur naturally in the troposphere where the ambient temperature decreases nearly linearly with altitude. Sonic booms as well as weak shock waves from other sources are, under normal circumstances, adequately described by a nonlinear adaptation of geometric acoustics [1], [2], [3]. In geometric acoustics, as in geometric optics, a ray is the path of a signal point on a wave. Neighboring rays form an infinitesimal ray tube. The area of a ray tube will, in some situations, vanish at a point in space-time, which we shall call a focal point. Such focal points may form a hyper-surface in space and time that, in most cases, is an envelope of the rays in physical space. Such a hypersurface is an acoustic caustic, and will be referred to hereafter as a caustic or caustic surface. The nonlinear adaptation of geometric acoustics needed for signals of finite (but small) strength corrects the phase of a given acoustic signal by an amount proportional to the amplitude of the local signal strength. Except near surfaces, lines, or points of focusing, such as a caustic, this is the only nonlinear correction needed for small but finite amplitude signals. In the absence of winds, the theory of geometric acoustics states that the Rayleigh acoustic energy, p':A/(pa), is constant along a ray tube. Here p' is the perturbation pressure, A is the differential ray tube area and pa, the acoustic impedance, is the product of the local density and sound speed. This, and the more general conservation law of Biokhintsev [4] applicable in the presence of winds, require that the perturbed quantities such as p' become infinite when * Received by the editors August 6, 1979. -t Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721. This research was sponsored by the Office of Naval Research under Grant N00014-76-C-0182. 355 t_ 356 K-Y. Ft:N(; (1) where the differential ray tube area A becomes zero. In these cases, geometric acoustics fails to describe the behavior of a signal and a local treatment is required. Guiraud [5] and Hayes [6] have given the equation that describes the behavior of the pressure signature near a caustic. Guiraud's derivation is quite lengthy; Hayes" is intuitive and short. Recently, Pechuzal and Kevorkian [7] obtained the same equation as a limiting result of an inner expansion governing the behavior near the caustic. In a coordinate system that is fixed with a wave front, the local behavior at a caustic is not difficult to delineate, and the nonlinear effect needed to provide the correct equation is easily deduced for normal situations. Cramer and Seebass [8] have derived the equation for a focus point, i.e., an ar_te. Despite a fairly large number of experimental, numerical and analytical investiga- tions, there has not yet been any truly adequate description of the local behavior of a nonlinear acoustic signal at a caustic. Because the nonlinear behavior at a caustic is a very local one, experimental measurements have proved difficult. Numerical cal- culations are not very reliable because some of the local details, such as the Guderley patch, if one occurs, are difficult if not impossible to resolve [9]. Analytical studies are difficult because of the inherent nonlinearity of the problem. Perhaps the most informative results are the measurements from the French flight tests discussed by Wanner et al. [ 10]. Laboratory investigations have used spark generated N-waves [ 1 1 ], [12] artificially stratified steady and unsteady flows [13], [14] and, more recently, weak shock waves generated in shock tubes [15], but have failed to resolve the local details of the flow. The results presented here derive from a hodograph technique first used by Seebass 116] and later extended by Gill [17]. Their analysis resulted in functions which satisfied the governing partial differential equation and one of the shock jump condi- tions exactly; but the other jump condition was not satisfied. This paper extends this earlier work by considering a broader class of incoming signals. This provides solutions for compressions of sufficient breadth that the reflected shock wave is never so strong that the flow behind it becomes subsonic. In this case we are able to satisfy both the partial differential equation and the shock jump relations. More severe incoming compressions are not dealt with as this would require a double-valued hodograph plane, a situation not considered here. 2. Governing equations and boundary conditions. For simplicity we consider the model situation of slightly supersonic flight in an idealized stratified atmosphere with an adiabatic lapse rate and hydrostatic equilibrium, as sketched in Fig. 1. The governing equation is M 22 8 M_=M2.[1-(_,-1)6(y/l)] -' and [32=M_-1. Here ¢ is the usual perturbation velocity potential, e is a small parameter that characterizes the perturbations produced by an aircraft of characteristic length l, 8 is the small parameter characterizing the ratio of this length to that determined by the temperature statification, and M_ is the operating Mach number of the aircraft and Me is the local Mach number at infinity. The first two terms of (1) are the usual ones for steady supersonic flow. The third term arises because of the gravitational force and should be retained in a consistent approximation. The last term is the lowest-order nonlinear term and must be retained in transonic flows. i¢ SHOCK WAVE FORMATION AT A CAUSTIC 357 At y= 0 we give boundary data that is representative of an incoming signal, say _. =f(x) for 00, then we have the following table: X Vo(X) SI =X + V) V(S) X<0 0 X X>0 1 l_-X The function V(S) is undefined in the interval 0 < S < 1. Alternatively, if Vo(X) = 0 for X < 0 and -1 for X > O, then the function V(S) would be multivalued in the interval -l0, -1, I _7 (-1 < c-a -b). Another bound on k comes from specifying the asymptotic behavior of Uk as T tends to infinity. For the caustic problem, k =-_ gives the required T -1/4 decay (_b_ - y-1/4) and k < -1_ an allowable more rapid decay; terms with k =-i_ are included to match the outer solution. If boundary conditions are prescribed at a finite distance, then the second bound for k is relaxed, but the vanishing far-field condition in the subsonic domain is still essential. To determine the coefficients A and B in (12), we need to study the behavior of the hypergeometric series near the regular singular points _ = 0, 1, oo. After putting the appropriate values of a, b, c in (12) we have Fk = A ! 1 B61/22F_(-k,3-k;2;6)+ 2F_(½-k,_-k;_;_). The behavior at the three singular points is given by an arbitrary linear combination of the two functions listed in the following table: _-_I 6=I 6_oo (1 ex1/6+2k F A+k2 1 2F_(_-k, -k; _-2k; 1 -_:) 1 6k-1/3 1 52Fl(3-k, 6-k;4; l/g) 0 61/:(1 - 6) 1/6+:k 2F_(] + k_ 1 + k; _; 6) 0 {1-_)a/6+2k2F,(½+k,_+k;V + 2k; 1-6) O(1 - _)1/6+2k 6*2F1(-k, ½-k; _; 1/6) 0 /q 362 K-Y. FUNG We see that, within the range of k allowed, the only singularity occurs at _E= 1 _i.e., S = +_T3"21. The singular behavior of Fk there is (1-_)1'6-2k for -_< k <-_ and In (1 -_:) for k = -_(T>O). There are two loci in the $,T-plane where _£= 1, namely S = w_T 3 z, corresponding to the incoming and the outgoing characteristics. By setting B -F(c) F(a-c*l)FIb-c-_l) A Fta)F(b) F(2-c) ' we can remove the singularity on the positive side of the incoming characteristics (]s¢[< 1). Using this result, we write I12) in the form F=#*[Gm, b; c" ()+_'-¢G(a -c + 1, b-c + 1; 2-c; £t], where F(a)F_b) G_a, b: c; E_)=-- 2Fl(a, b, c; _) F(c ) and/a* is a complex constant. Because _ = 1 is a singular point, in order to evaluate 2F_ for arguments greater than one we need the analytical continuation of the function F throughout the complex sO-plane, insisting on smooth behavior in the elliptic domain, viz., T < 0 (Re _E< 0). We introduce a cut in the E-plane from _"= 1 to infinity on the real axis and choose the branch that makes F continuous everywhere except on the branch line. By this means we can extend our solution to 1_1> 1. By setting # =/_[1 +i tan (21r(a +b))], we can eliminate the other singularity on the negative side of the incoming characteristic q¢J> 1). Thus solutions of equations (10), with a behavior that we anticipate will prove acceptable in the physical plane, take the form U= f Ukdk=Re[f l.t*T3kFkdk], (13) V= I Vkdk=Re[-I lx *T3k'l"2Gkdk ], where and 'rl/2G'_ k _-Fk=G(-k,_-k;½;,f)+g ,6-,. k;_;_ ¢) I 1 . 3GI, =G(_,-k,-16-k;½;sc)+_t/"G(1-k, 3-k _). Representations of Uk, Vk appropriate to different domains of the S,T-plane are tabulated in [19]. $. Construction ot solutions. Were we able to do so, we would now proceed to determine the appropriate equations needed to determine the values of lz* in (13_ that would satisfy certain prescribed boundary conditions, although it is unlikely we could solve them analytically. This formidable task is not resolved here. Alternatively, we may try to combine the Fk's and Gk "s in such a way that they represent a meaningful solution in the physical plane that corresponds to physically interesting boundary data and satisfies the jump conditions (6) for any discontinuity that may arise. The results are instructive in that their physical interpretation is believed to be more meaningful than any numerical results might be. Furthermore, they provide the basis for further analytical advances. _.,,, SHOCK WAVE FORMATION AT A CAUSTIC 363 9If The function Uk, discussed in § 3, has a jump in S across the incoming characteristic and a singularity of (1-_I 2k÷1/6 or In (1-_) on the reflected charac- teristic. The jump behaves like the Heaviside unit function multiplied by a decay factor T 3k ; the function Vk has similar properties with a decay factor T 3k÷112 Suppose we know the shape and the leading power by which the incoming signal amplifies, then k is determined and signal shapes can be constructed as described below. With only the range of k specified, we can examine a finite set of k's and try to find useful combinations of the functions Uk and Vk. If, for example, we have an incoming step wave at T0 with k specified, 0, S+So<0, = 1, S +So<_,,U(S, To) 0 < 0, ;_ -½. This follows because UB is an integral of Ut and B # 0. Examining Uk(S, T) at its singular points for k = -_, the case of interest here, U_l/12_S -1/6 forS--,0, T=0, and 3 TS__) forS..13T3/2, T>O"U-t/12 ~ln (1 -_ 366 K-Y. FUNG Therefore the behavior of UB is given by 1 fs÷n-- U-ill2 dX_B -1/6 B JS and S_O, T=O S_T 3/2, T>0. Thus UB is finite and continuous everywhere, with first derivatives that are dis- continuous at S =-2T3/2. Y B =. 1 .06 .025 0 V-based shock U-based shock no V-based shock possible T r r FIG. 3. Location of the reflected shock wave for [our B's ; k = -_, v- = 0. l, ,_ = 20. X For small B, U(S, T; B) and limB.,0 U(S, T; B) are not significantly different in the S,T-plane except where lims-,o LIB is singular and where it has a jump. Here, changes in B significantly alter the behavior of the solutions U and V in the physical plane. These are regions where shock waves form. The transformation from the solutions U(S, T) and V(S, T) to the physical plane requires a high degree of familiarity with these functions. The transformation that gives U(X, Y) and V(X, Y) for fixed values of Y and at specified intervals of X, is accomplished by a digital computer, primarily through Newton's method. Thus we solve the implicit relations Y = T- U(S, T), (18) X = S- V(S, T) numerically and to a high degree of accuracy; it is important that we can evaluate U and V to any desired degree of accuracy throughout the S, T-plane. Notice that U and V have discontinuities in their first derivatives because Uk and Vk are discontinuous. It is essential to know in advance from which region of the ;el SHOCK _A'AVE FORMATION AT A CAUSTIC" 367 4 S,T-plane a point X, Y derives; otherwise Newton's method will fail to converge to the correct values, if it converges at all. Constant YIS, T) contours in the $,T-plane are particularly helpful in designing the numerical code to effect this evaluation. For the results we discuss later, equations / 18).are satisfied to one part in 10 -8; this limit is set by the accuracy we specify in evaluating the functions. Above a certain value of Y, UIX, Y) and VtX, Y) become multivalued, indicating the formation of a shock wave. The solution is then made single-valued by introducing a discontinuity so that both the integrals x x (19a, b) [ U_l, Y)dt, f Wtt, Y)dt are continuous and single-valued. As noted by Seebass [16], this implies that the ju.mp relations _6) are satisfied if the position of the shock jumps calculated from both integrals coincide. The shock determined by (19a) implies conservation of momentum while that determined by (19b) implies conservation of mass. We carried the integra- tions in (19) using a pointwise trapezoidal rule with varying intervals. The values of {tU]] and ITV]] were determined by linear interpolation. We have calculated the shock position by both methods and the difference between the two shock positions becomes negligible for values of B greater than a critical value, Be, when no subsonic zone can be TABLE 1 lnput signal at Y = 1 B = 0.1 Signal width ,XX = .29480 )t = 20 Signal strength z$U = -. 10870 vt = 0.1 V X, f/tr_ _VD R:L 0.14 -.2046231 -.321434 × 10 -z .314971 x 10 -3 .998263 -.2046225 -.321199 × 10 -2 .314570 × 10 3 .998468 -.200442 -.011451 .196838 × 10 -2 997718 0.17 -.200436 -.011442 .196547 ×10 2 .998058 -.194564 -.018516 .402860 × 10-: .997680 0.20 -.194551 -.018501 .402236 × 10 -2 .998071 -.167396 -.037737 .011986 .997954 0.30 -.167352 -.037704 .011968 .998294 -.122108 -.056853 .022892 .998227 0.425 -.122019 -.056791 .022855 .998549 -.089789 -.062301 .029384 .999673 0.5 -.089691 -.062219 .029344 1.00020 -.037158 -.062501 .036056 .9999020.6 -.037069 -.062421 .036010 1.00015 .095230 -.061519 .045499 .9999820.8 .095302 -.061463 .045458 1.00008 .256577 -.060413 .052557 1.000011.0 .256643 -.060364 .052515 1.00006 .759148 -.057630 .065034 1.00001 1.5 .759198 -.057601 .065001 1.00003 _ indicates the jump across the shock v,a_e. R/L is the ratio of the right to left-hand side of equation (6al. 368 K-¥. FUN_ found behind the shock wave. We conjecture that the solution is an exact one. We know it satisfies the partial differential equations, and, to the accuracy with which we can check them, the shock jump relations. We have not tried to establish the value of Be, precisely; we do know however that Bc =0.10. With B = 0.06 the maximum error in satisfying the shock jump relations, as measured by the ratio of the right-hand to the left-hand side of (6a), is 6%; with B = 0.10 it is 0.3%. Detailed numerical calculations have been carried out for five values of B: 0.01, 0.025, 0.04, 0.06 and 0.10. The other parameters,/.t and A, were taken to be 0.1 and 20, respectively, for the convenience of comparing our results with those of Gill and Seebass [17]. Shock locations, and the extent of the subsonic zone based on (19a), called U-based or (19b), called V-based, are depicted in Fig. 3. Note that as the extent of the subsonic region behind the shock wave diminishes the two shock locations become more nearly one curve. This, as was noted earlier, is to be expected because we don't expect our single-valued solution (in the S, T-plane) to be valid if the flow behind the shock is subsonic. Y shock wave X FtG. 4. An incoming step wave of finite width terminated by the reflected shock wave using the area balance rule in the multivalued region B --- .06. SHOCKWAVEFORMATIONATA CAUSTIC 369 Table 1 delineates the shock positions _,nd shock jumps L" and _, determined using equations (19a) and (19b) for selected values of Y when B = u = 0.1 and _. = 20. Also tabulated are the ratios of the right- to left-hand side, R/L, of equation i6a) obtained using (19a) and (19b). The general agreement between the results for the shock position computed both ways and the fact that R/L is 1.0, to the accuracy with which we can calculate the shock jumps using tl9a) and (19b), supports the claim that the solution satisfies both shock jump relations. Figure 4 depicts the value of U as a function of X for fixed values oI Y for B = 0.06. We see that the distortion of the sonic line gives rise to weak compressions which become much stronger as they meet the incoming compression and then propagate away from the interaction, slowly decreasing in strength. We now anticipate, but did not do so originally, that if a shock wave arises, and if it has a subsonic portion as sketched in Fig. 5, then a saddle point in speed lor Mach number) will occur. Such behavior occurs on the physical plane when the solution in the hodograph plane is multivalued. Because we have limited our solutions to single-valued functions of S and T we can only expect to find solutions for large enough B, that is, for waves thick enough that the shock wave remains embedded in the supersonic flow. Y _ - = • . -. /___ _Z.9_ -_ - Ipossible local structare _f I .1 j 0.6 15 0.4 (J.3 0.2 [I. 1 FIG. 5. Reflected shock wave, sonic line, characteristics. A shock will always occur even for large B; it will form further above the sonic line and decrease in strength with increasing B. This follows because the value of V is not zero at the sonic line ( T = 0), and its maximum values increase in the supersonic domain as y_/4 along the outgoing characteristics. The location of a given U and V in the physical plane is always stretched by V, i.e, V(X = S- V, Y), U(X = $- V, Y). The larger the Y, the more they are stretched. Nevertheless, they weaken because the region of stretching is confined to a thinner and thinner strip, which eventually corresponds to the immediate neighborhood of the singular characteristic. Starting from the lowest point where the solution becomes multiple-valued, the reflected shock strength increases almost linearly with Y (this behavior is most distinct g.7 370 K-Y. FUNG Y !.0 0.8. 0.6- 0.4- 0.2 10 12s 2'0 2:5 31o FIG. 6. Maximum reflected shock strength, divided by the incoming wa t,e strength at Y = 1.0, as a functton of Y for B=O.l. for large B) to its maximum value at a Y of about 0.4. The shock strength then decreases with Y increasing, as shown in Fig. 6 for B = 0.10. This case was used for Table 1. 7. Conclusion. The role of nonlinear effects in the amplification of certain weak acoustic signals at a caustic has been studied through a hodograph-like transformation, Nonlinear distortion of the boundary data plays an important role in determining the appropriate boundary data in the hodograph plane. We have reformulated the equation governing the nonlinear behavior of an acoustic signal at a caustic in a way that avoids supplying asymptotic boundary data. In the new coordinates, the solutions for the inner caustic region are represented by similar solutions whose behavior depends upon a single parameter k. Boundary conditions in the hodograph plane are then related to physically prescribed data. Unfortunately the relation is nonlinear. We have not resolved the problem of determining the corresponding boundary data in the hodo- graph plane. Through a study of finite width step waves, with k = -_, for a small range of step width, we have shown that small changes in asymptotic boundary data near dis- :.9 SHOCK WAVE FORMATION AT A CAUSTIC 371 continuities in derivatives in the hodograph plane have a substantial influence on the solution's behavior in the physical plane. The maximum strength of the reflected shock wave decreases monotonically with increasing signal width. When the width is larger than a given critical value, the shock that forms satisfies the jump conditions with a high degree of accuracy. Solutions for other values of k, discussed to some extent here, should prove useful in obtaining acceptable results for more general incoming signals. It seems unlikely, however, that the parameterization in k provides a complete set of solutions. Still, solutions for other incoming signals should be possible. But a single-sheeted Riemann sheet in the hodograph plane won't suffice for these more interesting signals. For signals whose widths are larger than some critical value, solutions with a discontinuity representing a reflected shock are obtained that are essentially exact. These results represent an advance over previous results and should prove valuable in testing numerical methods designed to solve mixed nonlinear equations. Acknowledgment. The author is indebted to Professor A. R. Seebass for advice and counsel during the course of this investigation and to Dr. Helmut Sobieczky for stimulating discussions and important suggestions. REFERENCES [1] W. D. HAYES, Sonic boom, Ann. Rev. Fluid Mech., 3 (1971), pp. 269-288. [2] A. R. SEESASS, Sonic Boom Theory, J. Aircraft, 6 (1969), pp. 177-184. [3] R. N. BUCHAL AND J. B. KELLER. Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), pp. 85-114. [4] D. I. BLOKFIINTSEV. Acoustics of a Non homogeneous Moving Medium, Gustekhizdat, Moscow; USSR 1946. Translated as NACA TM- 1399, 1956. [5] J. P. GUIRAUD, Acoustique g_ometrique, bruit ballistique des atwns supersoniques et focalisation, J. Mdcanique, 4 (1965h pp. 215-267. [6] W. D. HAYES, Similarity Rules [or .Nonlinear Acoustic Propagation Through a Caustic, Second Con/erence on Sonic Boom Research, NASA SP-180 (1968), pp. 165-171. [7] G. PECHUZAL AND J. KEVORKIAN, Supersonic-transonic flow generated by a thin airfoil in _1stratified atmosphere, this Journal, 33 (1977), pp. 8-33. [8] M. S. CRAMER AND A. R. SEEBASS, Focusing of weak shock wat, es at an ar_te, Part 2, J. Fluid Mech., 88, (1978), pp. 209-222. [9] N.J. iu AND A. R. SEEBASS, Computational procedure for mixed equations _ith shock ,'ares, Comput. Methods in Nonlinear Mech., (1974), pp. 499-508. [10] J.-C. WANNER, J. VALLEE, C. VIVIER AND C. THERY, Theoretical and experimental studies of the focus of sonic boom, J. Acoust. Soc. Amer., 52 (1972), pp. 13-32. Ill] E. P. CORNET, Focusing of an N-wat'e by a spherical mirror, University of Texas at Austin Report, ARL-TR-72-40, 1972. [12] W. D. BEASLEY, J. D. BROOKS AND R. L. ROGERS, A laboratory int'estigation of N-ware focusing, NASA TN D-5306 (1969). [13] C. THERY, Atomospheric Refraction and Reflection in Sonic Booms, AGARD CP No. 42 (1962), pp. 13-I to 13-14. [14] M. SANAI, T.-Y. TOONG AND A. D. PIERCE, Ballistic range experiments on superbooms generated by refraction, J. Acoust. Soc. Amer., 59 {1976), pp. 513-524. [ 15] B. STURTEVANT AND V. A. KULKARNY, The focusing of weak shock waves, J. Fluid Mech., 73 (1976), pp. 651--671. [ 16] A. R. SEEaASS, Nonlinear acoustic behavior at a caustic, NASA SP-255, {1971 }, pp. 87-120. [17] P. M. GILL AND R. SEEBASS, Nonlinear acoustic behavior at a caustic, Progr. Astronautics and Aeronautics, 38 (1975), pp. 353-386. [18] M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1968. [19] K-Y. FUr/G, Shock wave formation at a caustic, Ph.D. Thesis, Cornell University, Ithaca, NY, 1976. ;-q a 1m. d a Lm AIAA 79-1557R Shock-FreeWing Design K.-Y. Fung, H. Sobieczky, R. Seebass i i t I 41 Reprinted from AIAAJournalVolume 18, Number 10, October 1980, Page 1153Copyright American Institute of Aeronautics and Astronautics, Inc., 1979. All rights reserved _ERICANINSTITUTEOFAERONAUTICSANDASTRONAUTICS• 1290AVENUEOF THE AMERICAS=NEWYORK,NEW YORK,N.Y. 10104 4k ! I I I I I i I I I I i VOL. 18, NO. I0 AIAA JOURNAL OCTOBER 1980 ARTICLE NO. 79-1557R Shock-Free Wing Design K-Y. Fung," H. Sobieczky,t and R. Seebass:[ University of Arizona, Tucson, Ariz. A _ple •umerleal method for generaltng wing sh•p_ that will be shock free st s specified supercritlcal Much a•mber is descrlhed. The method invol_es •sing • fictitious gM law for the supersonic domain to make Ihe governing equations elliptic. Requirements on this gas law Ire detailed and • method for computing the ¢ml flow in the lupenmnic domain, given it_itiul data on the embedded sonic •uflace, is described. The failure of the method to yield u shock-free flow when • limit surface occurs in the supersonic flow, and the difficlllies that ari_ becxttse the initial-value problem for the _pcrsonic domain is ill-posed, are delineated. Finally, u small perturbation algorithm is used to illustrate the procedure nod mules are give• for a simple bascqine wing. Introduction NCREASED fuel efficiency, and in the case of commercial aircraft, productivity, can be achieved by operating aircraft at supercritical Much numbers, provided that shock waves can be avoided or made acceptably weak. Two-dimensional procedures for prescribing airfoil sections that are shock free have alrea(_y provided improvements in aircraft efficiency by employing these airfoils on swept wings. Three-dimensional effects have compromised such designs to some extent, and extensive wind tunnel development tests have been required to recapture the benefits of these "supcrcritical airfoils." Sobieczky et el. j demonstrated a method of modifying baseline configurations so that they would be shock free at a prescribeo Mach number and lift coefficient. This procedure provides ,, special opportunity for improving aircraft per- formanc¢ :hrough a careful selection of the baseline con- figuration in order to provide wings and wing-body com- binations hat are shock free at supercritical Mach numbers, and that _ave acceptable off-design performance. Yu 2 and Yu and R,;bbett 3 have also documented that this procedure is possible a_:ddemonstrated its application. As v,.., first described by Sobieczky, 4 a numerical algorithr., is used to solve a fictitious set of equations for the flow p_-_ the baseline configuration. These equations are identical ,o the correct equations for subsonic portions of the flow, but :hey are modified when the flow becomes super- sonic, so tltat even though the flow speed is larger than the local spee_ of sound the equations themselves remain elliptic. This p_3ceJure generates a numerical solution that satisfies the approp:iate equations where the flow is subsonic, and the appropria°e boundary conditions on the configuration outside of the sul_ ersonic zone. The results of this calculation provide the fiowfi :ld at the sonic surface. This surface and flowfield define an ill-posed initial value problem for the supersonic domain It at is to be solved using the correct equations. Because ti is problem is ill-posed in three dimensions any numerical method must, in principle, be unstable. This in- stability, however, is of no consequence for moderate to high aspect ra_ios. However, if the detailed definition of the spanwise modifications required to make the wing shock free Present. J as Paper 79-1557 at the AIAA 12th Fluid and Plasma Dynamic., =onferenc¢, Williamsburg, Va., July 23-25. 1979; sub- mitted Au_ 13, 1979; revision received March 23, 1980. Copyright _) American lastitute of Aeronautics and Astronautics, Inc., 1979. All rightsreser_:d. Index catt.,gorics: Aerodynamics; Transonic Flow; Configuration Ikaitn. *Research Assistant Professor. M_mber AIAA. __[ rAdio•el'Professor, also Research Scientist, DFVLR, GOttingen, West Germ, ny. Member AIAA. _Pgofessor. Associate Fellow AIAA. .... , "_'1 1153 are comparable to those for the streamwise direction, as they will be for low aspect ratios, then the instability may com- promise the calculations. Fictitious Equations The flows we _¢k are to be shock free. As a consequence, they will be irrotational and the governing equation will be the conservation of mass, viz., v. (ovo) --o (I) where olp. = II + V--_--_[ l- ( V _ ) 21az ] l n/t_-n) (2) or where (v_) a GZ=aZ. +_ -j-[aZ.--(V@) _] (4) Here (), refers to the critical flow conditions where q=a. While the conservative formulation, Eq. (I),is to be p_efcrred over its no•conservative analog. Eq. (3). numerical com- putations using Eq. (3) should be satisfactory proGded that the flow being computed has no shock waves. In order to generate smooth data on embedded sonic surfaces that are potentially consistent with shock-f-¢e flow. we elect to modify the gas laws (2) and (4) so that Ec_. (I) and (3) remain elliptic when q_a,. Thus we require a(p_) >0 if q>a. (St) pPq or el> q if q>a. ($b) where D! and a/are a fictitious density and a fictitiou._ sound speed and the partial derivative in Eq. (5a) is taken along a streamline. If we restrict Pl and a! to be functions of q alone, then The choice of the fictitious equation or gas law is a tool or technique available to produce a range of designs that are shock free; the initial data found with one gas law may lead to 4t II$4 FUNG, SOBIECZYY, AND SEEBASS AIAA JOURNAL a limit surface above the wing. while that with another gas law will not. The main consideration in choosing the fictitious equations is that the equation must imply a conservation law and that the conserved quantity must be identical to the mast flux at sonic flow conditions. This insures that the initial data for the supersonic domain is consistent with the conservation of mass in the subsonic flow. Examples of fictitious gas laws which lead to an elliptic equation include: pf/p. =(a,/q)Po or a/=P-_qo P0 t - a_. o, #>0 8) These gac laws are all of the simple form p:o(q). Itmay sometim._ be of value to consider an equation that has an explicit spatial dependence in order to alter the shape of the sonic domain. If this is done, care must be taken to insure that a conservation law is implied. Thus we may use p/_p/(x,y,z) and be sure a mass flux is conserved but not, in general, n.f = Of (X, YoZ). Supersonic Domain Given a suitable numerical algorithm for solving F.qs. (!) or (3), with the fictitious density or sound speed used for supercritical speeds, we may then locate the embedded surface where q=a, and evaluate the velocity components there. Those may be the physical components, viz., u, v, w in the Cartesian coordinates x, y, z or the components U, F, W, in some mapped space X, Y. Z. Because the equations are, hyperbolic we choose to work with a first-order system (a _-uz)u,+ (a 2-v_)v,+ (a_-w_)w z-2uww. -2utmx-2uwu z =0 (6) w_-u,=O, w,-v_=O. (u_-v,=O) f'l) with one of the three irrotationality conditions bans redundant. We must then set up a suitable numerical algorithm for the computation of the supersonic flow, marching inward in some fashion toward higher Macb numbers until the stream surface upon which the supersonic surface rests can be continued. In the process two difficulties may arise. The first is that the computation may indicate that the solution has become muhivalued because a limit surface intervenes between the sonic surface and the body; then no physically acceptable solution is possible with the initial data supplied. The second is that the inherent instability of the algorithm may become manifest, providing an unacceptable solution. We discuss this problem further shortly. A nonsubstantive difficulty that may arise with an ap- proach using rectangular coordinates is due to the topology of the supersonic domain. We can expect the supesonic region to wrap around; or more picturesquely, "grab," the wing leading edge, as shown in Fig. la. When this occurs, one of the derivatives in the two equations selected from Eqs. (7) may vanish, leading to a singular system of equations. This is mo_t easily avoided by mapping the solution domain to a coordinate system in which this does not occur, as discussed in Ref. 5. The coordinate systems of the computational algorithms to be used may provide the essentials of the mapping. Thus the coordinate systems used in the com- putational algorithms of Jameson and Caughey 6 and Caughey and Jameson _ provide natural coordinate systems M Fit. I Local mptnonk ttllion is the phr_'tl nu4 ¢oupwtatio_ domalnm. for the computation of the supersonic domain. In the first of these the wing surface is mapped to the plane Z' =0, and we may envision the sonic surface to be as depicted in Fig. lb. A subsequent mapping z=z" /z;(x,Y) where Z" (X, Y) is the sonic surface, then I-',,ds to a com- putational domain like that sketched in Fig. lc. In this domain, with U. V and W the X, Y and Z com- ponents of the velocity derived from some appropriate potential, we have a system of equations of the form AUx+BUy+CUz=O which we use to advance the solution from one Z level k to the next: U, = U,_, + [ (C-_A)t_. (Ux),_ . +(C-IB)k_v. (Uv)t_.]AZ Here we use the subscript k- ½ to indicate a suitably iterated average value of the subscripted quantity and AZ is the decrement in f.he Z coordinate. At each Z station the X and Y derivatives of U are calculated using three-dimensional cubic splines to specify U. Presumably the spiine used should be one that avoids introducing, or perhaps even filters out, oscillations in the numerical results. 1 L i pJ OCTOBER 1980 SHOCK-FREE WING DESIGN 1155 I _m ,am I I Z w z" Fig. 3 Sonic surface with a limit surface embedded in the supersonic region. aa I a= a al a da a N tN /¢ Fig. 2 Change in the flow inclination on the original wing surface that is used to define the new wing surface. The calculation of U proceeds from one Z level to the next until the original wing surface, in this example Z=O, is reached. A new stream surface is then extrapolated from the values of U there, or the computation may be pushed further, to negative Z, and the new stream surface interpolated from the additional values so calculated. If we simply wish to extrapolate the new body surface we may do so in the original physical coordinates x, y, z, or in the mapped coordinates X, Y,Z. In the original coordinates we may use the new velocity field on the original body (u b, vo, w b ) to define the slope in the stream direction, tanO b = wb/,_, + vzb= wb/q' As sketched in Fig. 2, the local stream direction is then determined in order to integrate the angular difference A0 (_) along the arc length o from A to B. The new body surface may be constructed by marching in the downstream (or upstream) direction and using a cubic spline in the spanwise coordinate to define the body at the computational nodes. This procedure assumes small surface deviations so that the velocity field's initial surface is also that of the new surface. Extrapolation of the results on the initial surface, and an iterative correction of the new surface found, may be carried out if higher accuracy is required. Limit Surfaces Shock-free designs are not always possible for a given baseline configuration, Much number, and lift coefficient. The initial data generated by given fictitious equations may imply a multivalued solution before the body stream surface intervenes. When this occurs a limit surface will be found in the flowfield. The first occurrence of a limit surface is along a line where Uz, Vz, W z = oo; the algorithm used to compute the supersonic domain should be constructed so that it can recognize when this occurs, otherwise results may be obtained that have no meaning. Such a surface might look like that Fig. 4 Sketch of Math conoids for a two-dimensional flow. Initial Value Problem As we mentioned earlier, the initial value problem for the supersonic domain is ill-posed in three dimensions. That is, small changes in the initial data will cause large changes in the solution in the domain of the problem. If we return to the well-posed two-dimensional problem and consider it to he a three-dimensional problem with no variations in the third direction, then we may sketch the Mach conoids, as shown in Fig. 4. The fore and aft Mach conoids define the influence and dependence domains of P. Because we find shock-free solutions the flow is reversible and we may consider the time- like direction to be in either the ±q direction. When we calculate the solution at P using data from the sonic surface we are effectively replacing the data along AA' by that along BB'. Alternatively, in two dimensions, the normal to the streamline may also be considered time-like and we calculate the solution at P using the data on CB. In three dimensions this alternative approach fails because the solution at P now depends upon the infinite domain CC' B' B. An informative simple example is that of the linear wave equation -_,_ +¢yy +$=z = 0 with data given on the z = 0 plane as sketched in Fig. 5; viz., 1156 FUNG,SOBIECZKY,AND$EEBASS AIAAJOURNAL z Fig. 5 Mach cones of the linear wave equation and their intersection with the plane where the initial values are given. We can construct the solution by Fourier superposition of the modal solutions O=exp[i(klx+k2y+k_z)] in the x and y directions. But then k 3 = ± _ leads to exponential growth in the z direction when the wavenumber in the y direction, k 2, is larger than that in the x direction• In this simple model, then, we may expect the ill- posed nature of the problem to manifest itself when the y variation of the initial data is comparable to or larger than the x variation. Translating this to practical terms, we can expect the inherent instability to cause difficulty for small aspect ratio wings. Payne Io has used energy arguments to show that exponential growth in the z direction must eventually occur for a large class of functions f and g. We demonstrate this instability in a subsequent section. Small Disturbance Equations: An Illustration We illustrate the procedure for computing the supersonic domain, as well as the difficulties that may arise, with the small disturbance equations. The small disturbance ap- proximation introduces difficulties unique to this ap- proximation; these we do not discuss. We first compute the elliptic flowfield using fictitious equations in supersonic regions to maintain elliptic behavior. While we use the Ballhaus-Bailey-Frick line relaxation algorithm as im- plemented by Mason et al.S for our computations, we use a simpler equation in this discussion, viz., - [½(K-cbx)2]x+ I_bj,]y + [q_zlz =0 (8) When (K-_x)<0 we make a change in the difference algorithm that corresponds to changing the first term to [IO,,-KlP/P]_,, P>--1 (9) The sonic surface on an AR=6 rectangular wing corresponding to P= 2 is shown in Fig. 6. Figure 7 shows the corresponding vertical velocity component of the redesigned airfoil for selected spanwise stations. We note that the more elliptic we make the fictitious gas (i.e., smaller P) the broader ....... "z -- z .3 • .2.6_ 1 .5 1. • Fig. 6 Sonic surfaces for a rectangular wing with /R=6, P=2, M** =0.87. w o \ \ \ \ Fig. 7 New surface slopes for the rectangular wing with AR= 6, P-- 2, M** :0.87. P D D III D IP that the body will be thinner; this is reflected in the vertical velocity component. For the original parabolic arc section the body slope, and hence the vertical velocity, decrease linearly with x. The numerical solution then provides values of O, and hence its derivatives, on an embedded sonic surface, _, z = Zo (x,y), as sketched in Fig. 6. The hyperbolic problem for the supersonic domain is solved using the simplified system of equations corresponding to Eq. (8). -½[(K--u)2]x-FVy-l-wz=O, wx-uz=O, wy-vz=O, (uy-vx=O) We may define a new variable _ (x,y) to replace the coor- dinate z, and facilitate the computations, e.g., (10) J! li (x,y) = Z/Z. (x,y) In this variable the equations become u_=D -I (K-u)_ x 0 1+_ 2 (K-u)_y 0 - (K-U)_x_y -(K-u) 0 (K-u)_ • U x i _X --_X_y + _._ l+(K-u)_ -I • Uy OCTOBER 1980 SHOCK-FREE WING DESIGN 1157 m m_ al i O g I m I I t G O o G where u has the components u, v, w and D= 1 + _2y+ (K- u)_. Our choice of the irrotationality conditions was dic- tated by the requirement that the determinant of the system D be nonsingular. Any other choice of equations gives D' = _yD and the system is singular when _y(x,y)=0, which always occurs. Limit surfaces occur only if D changes sign. For the system considered here this occurs when (K-u)_(<0) becomes larger in magnitude than 1 +/_y:. But D may also vanish because the coordinate system used for the set of equations chosen is not the appropriate one. This failure can be remedied by another choice for the coordinate system and must not be confused with that which occurs when a true limit line is present. As discussed earlier, the initial value problem we solve is ill- posed. The implied numerical instability becomes more serious when the spanwise gradients are large. Indeed, as we can see from Eq. (10), u_ =D-I( (K-u)_xux+ (l+_)wx+_xVy-_x_ywy } (11) -- 4-4- -+- -- 4-± 4---:;: Consider the sign of the individual terms on the right-hand side for an unswept rectangular wing with a profile sym- metrical about the midchord and midspan lines. (The up- per/lower sign corresponds to ahead of/behind the midchord line.) Then, because we are solving an elliptic boundary-value problem where the terms have the indicated signs, all of the terms in Eq. (11) except _xVy give u_ <0. The smaller the aspect ratio the larger the local values of _xVy. A similar conclusion holds for v, with a _j,vy term providing a change in v_ that increases v and Vy. With a spanwise instability present in the numerics we can anticipate that as it grows in amplitude vy will grow and affect the magnitude, and eventually the sign, of u,. Figure 8 depicts the surface values u(x,y,O) computed for the supersonic region on rectangular wings of aspect ratios 3 and 2. For an aspect ratio 2 wing numerical instabilities obviously override the generally smooth nature of the flow. Inspection of the other velocity components strongly suggests that the instability has a wavelength four times that of the spanwise grid spacing and amplifies v more rapidly than u or w. The initial data for the two cases are similar, except for the larger y gradients when/R = 2. Wing Design The art of aircraft wing design involves many variables and requires knowledge and expertise beyond that of the authors. We believe, however, that by using fictitious equations with suitable baseline configurations various design goals can be met and the wings will be shock free at reasonable flight Mach numbers and lift coefficients. To aid the aircraft designer in understanding this technology we briefly describe the shock- free design process for a simple wing. The approach outlined in the previous sections results in wings and airfoils with upper surface curvatures that are less than those of the baseline configurations. Additionally, the more acute the intersection of the sonic surface with the body surface, the less the likelihood of a limit surface intervening between the sonic line and the body. For this reason, baseline configurations should have reasonable upper surface cur- vatures and more thickness than required by the final designs. Designs that are close to the limit of what can be achieved, in terms of Mach number and lift coefficient, will have limit surfaces that nearly penetrate the wing surface. This may occur near the leading edge of the wing or near the aft end of the supersonic region, or in both locations simultaneously. We will illustrate some of these points with a simple tutorial example. We take a well known airfoil, the 64A4xx, and use it for the wing sections. The planform is chosen to have straight U Y -1 cp u Y Fig. 8 Effect of numerical instability on the streamwise velocity component for: a) A_--3, MQ. =0.88; b) AP,= 2. M_ =0.89. There is Fig. 9 Intersection of the sonic surface with the wing and pressure coefficients for: the wing designed to be shock free --; the baseline 1158 FUNG, SOBIECZKY, AND SEEBASS AIAA JOURNAL Mml _ • Fig. 10 Sonic surface on the supercritical wing of the AFTI 111 using fictitious equation in the supersonic region. leading and trailing edges with sweep angles of 30 and 15 deg, respectively. We take the aspect ratio based on wing area to be 8, and the thickness distribution to be elliptical and 10°70 thick at the wing root. The twist is varied from 4 deg at the root section to 0 deg at midspan, and the angle of attack is 0.4 deg. With a freestream Mach number of 0.80 these conditions will lead [at least for the fictitious equations with P= 2 in Eq. (9)] to sonic surface data for the system Eq. (10) that are con- sistent with shock-free flow; that is, no limit surface in- tervenes before the wing surface is found. Figure 9 shows the intersection of the sonic surface found with the wing, and the center sections of both the baseline configuration and the shock-free design. The design wing is 0.7070 of the chord thinner than the baseline wing at the center section. Also shown is a comparison of the pressure coefficient at selected span stations. Both wings have a lift coefficient of 0.50. The small modifications to the baseline wing, over the portion of the upper surface wetted by supersonic flow in the solution of the fictitious equations, results in a wing, that when analyzed numerically, has the shock-free pressure distribution shown. The pressure coefficient on the baseline configuration is also shown for comparison. The inviscid drag evaluation for the original wing gave 96 counts; that for the design wing 86 counts, reflecting the changes in the pressure coefficient. Presumably a viscous calculation that correctly modeled the shock-wave boundary-layer interaction would reflect further improvements. The selection of a baseline configuration is important to the success of this method. Wings that employ traditional supercritical airfoils will lead to sonic surface data that result in a limit surface. Figure 10 shows the intersection of the sonic surface with the wing for the AFTI 111 wing; again we have used the fictitious equations with P= 2. The complex nature of this surface, and the occurrence of a limit surface when redesign is attempted, is due to the supercritical design of the AFTI 111 wing. Modifications to the baseline configuration, such as this one, are essential ingredients of any attempt at shock-free design. This is illustrated further in Ref. 9, where analytical functions are used to modify supercritical airfoils. Conclusion A procedure for designing wings that are shock free has been described in general terms and illustrated by using the small perturbation equations to modify a simple baseline configuration so that it is shock free. In using this procedure the designer must select the baseline configuration to be modified and the fictitious equations to be used. These determine the flowfield on the sonic surface of the ultimate design. A good choice will allow high Mach numbers and lift coefficients to be obtained. A poor choice will result in a limit surface in the supersonic domain at the design Mach number and lift coefficient. Instabilities in the numerical calculation of the supersonic flow that provides the wing design occur whenever the spanwise gradients are large. These can be suppressed by smoothing both the chordwise and spanwise data at each successive step of the calculation. Acknowledgments This research was carried out by the Computational Mechanics Laboratory of the Department of Aerospace and Mechanical Engineering under AFOSR Grant 76-2954G and ONR Grant N00014-76-C-0182. ,,g iip P tl, References mSobieczky, H., Fung, K-Y., Yu, N. J., and Seebass, A. R., "A New Method for Designing Shock-Free Transonic Configuration," AIAA Journal, Vol. 17, July 1979, pp. 722-729. 2yu, N. J., "An Efficient Transonic Shock-Free Wing Redesign Procedure Using a Fictitious Gas Method," AIAA Journal, Vol. 18, Feb. 1980, pp. 143-148. m 3Yu, N. J. and Rubbert, P. E., "Transonic Wing Redesign Using a Generalized Fictitious Gas Method," Boeing Document D-180-25309- l, May 1979. 4Sobieczky, H., "Die Berechnung lokaler r_'umlicher 0bet- m. schallfelder," ZAMM 58T, 1978. 5Sobieczky, H., "A Computational Algorithm for Embedded Supersonic Domains," University of Arizona Engineering Experiment Station Rept. TFD 78-03, 1978. 6Jameson, A. and Caughey, D. A., "Numerical Calculations of the - Transonic Flow Past a Swept Wing," NASA CR-153297, 1977. 7Caughey, D. A. and Jameson, A., "Numerical Calculation of Transonic Potential Flow About Wing-Body Combinations," AIAA Journal, Vol. 17, Feb. 1979, pp. 175-181. SMason, W. H., Mackenzie, D._ Stern, M., Ballhaus, W. F., and I" Frick, J., "An Automated Procedure for Computing the Three- ,.. Dimensional Transonic Flow Over Wing-Body Combinations, In- cluding Viscous Effects," Vols. 1 and II, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, AFFDL-TR-77, Ohio, w 1978. 9Sobieczky, H., "Related Analytical Analog and Numerical Methods in Transonic Airfoil Design," AIAA Paper 79-1556, July 1979. I°Payne, L. E., private communication, Cornell University, Sept. 1978. 3(a ,ll ADAPTIVE AIRFOILS AND WINGS FOR EFFICIENT TRANSONIC FLIGHT H. SOBIECZKY DFVLR GOETTINGEN, FRG. A.R. SEEBA SS UNIVERSITY OF ARIZONA TUCSON, USA PAPER PRE3ENTED AT THE 12. CONGRESS OF THE INTERNATIONAL COUNCIL OF THE AERONAUTICAL ,.SCIENCES HUNICH , GERHANY, OCT. 12.- 17., 1980 3¢ ( ICA $ ) ADAPTIVE AIRFOILS PND WINGS FOR EFFICIENT TRANSONIC FLIGHT by H. Sooieczky DFVLR ann A.R. SeeWass University of Arizsna SUMMARY A simple Oesign methoO for two- and three-dimensionaZ ShOCk-free configuratians is useO for systematic airfci! moOification to maintain shock-free Flow at varying oOe- rating conditions. A mechanical realization is proposed since only minor and local changes of the contour are re- Quired. INTRODUCTiGN High soeeO aircraft Oesign has Become one of the mos_ challenging fields of the aeronautical sciences. With availabil__ty of large computers new tools for design an_ analysis of aircraft components b@came available within the last Oecade, which encourageO the introOuction of new aero- oynamic concepts to increase fuel efficiency which is prc- portional to the ratio of lift over drag, multiplieO by the flight Macn number. Raoi_ly increasing fuel-costs within _ha Last years unOerlined the uroent call for tecnniaues 2tc imorove efficiency of t,me next generatian transoort air- _ossioilitv for increasing efficiencv by drag reduction is to avoid the occurrence of snook waves WhiCh reouires a complicated iterative orocess of aerodvoamic shaming arrieo out usino eno_ o __c . _n_ering exoer_nce, computational fac__ities and wind tunnels. The resulting wind shames for tne flight regime just below _he smeeo of sound have Oe- ccmo known _ "susercr_ tical winms" they a_ desiooeo to _e completely or nearly free of recomoression shocks at certain ooerating conditions, wnile wings with conventional sections nave strono shocks and, therefore, additional drag. _eore_icaliy isolated within flow fielns containing shocks 1 "f the coeratino conditions are slightly changed , Such _-,ock-fre=_ flow_ ha,re been consid_red_ o r not much o __a_zlc=l_" vm _u__ for some vear_,_ but oioneering" exmer.m_nts_= 2,3 also _imuleteO the Oeveiooment of commutetional methods to oO- -ci,m orec_icallv interesting shock-free airfoil shames 4, 5 . ; esa design methods are restricted to two-dimensional flow, --n_v work im the hooonr_oh olane and _re, therefore, rele- -ivel V ccmolicete_. A similar methoo 6 allowed an extension _f the aocroecn into ohvsicsl soace 7 . Tne ability to solve _ransonic design oroblems was then couoleo to the ¢evelop- man_ sf re_iacle, flow __n_lvs___= alocritnms" b V _hi_ aporoach. i_ ie_ to efficient design methods which became known as "Elliptic Continuation" or "Fictitious Gas" methods. They form computational tools for the aerodynamic concept of sea,rive aircraft geometry for adjusting contours to obtain _im =_ eff_encv even at variable operating conditions DESIGN PROCEDURE FOR SHOCH-FREE FLOWS The purpose of this paper is to il!ustrste some recent results obtained with a systematic computational procedure for suoercritical airfoils and wings which are shock-free at orescribed operating conOitions. Since the design method may be developed by extension of any reliable analysis aloo- rithm, we give a short description of the concept with a physical interpretation in order to allow for an implemen- tation of the idea into new and more sophisticated analysis methods becoming operational now and in future. A local suoersonic domain embedded into a subsonic flow fiela is enclosed in general _v a surface consisting o_ the sonic isotach and a recompression shock. If the flow is shock-free the sonic surface forms a smooth convex bubble situated on the body surface. In this latter case the struc- ture of She flow is cualitatlvely similar to a subsonic flow: isotachs of velocity higher than velocity a_ infinitv form also bubbles with smooth transition of the flow pro- perties. This relationshio of suOsonic _nd shock-free transonic flaws gave rise to the following idea to calcu- late examples of shock-free flow (see Fig. 1): In a first step we salve a partly fictitious problem by altering the governing isentropic density - velocity, _is(q) relation in the domain of supersonic velocities. An artificial compressibility relation j_f(o)> a*/q where q>a * (a _ the soeed of sound) defines a fictitious super- sonic flow with subsonic flow quality. The basic different!eL equation of the complete flow is now of elliptic type, locally describing physically realistic subsonic flows end fictitious supersonic flow. Such a flow will have no re- compression shock, the sonic line will qualitatively re- semble one of a physically realistic shock-free transonic flow. Examples of such flows may Oe obtained with use of numerical elliotic solver routines, we observe that only the local supersonic domain is physically not real, the 4surrounOing flow field is a locally correct sclution. We zsk now for e possibility to use the subsonic part tc construct a complete real shock-free flow. -km ec _ ,.,_ s ond s_ep of the _roceours consists of an _ntegration of the real supersonic differential equations, with restored oensi_y _i_(q)._ initial conditions of this hyperbolic type _roolem are prescribed aloog the given sonic surface with velocity directions resulting from the previous solution of _he fictitious problem. This ensures e smooth connection metween the two physically reel parts of the solution. r_umerical marching procedures based on the method of charac- teristics allow an integration of the potential equation, starting at the sonic surface and proceeding toward the ._y surface. The lat+er was part of the first step ellip- tic Ooundary value oroOlem but the resulting _ody stream surface from the hyperDolic initial value proWlem (initial values at the sonic surface) will be different from the civen Oody where we_ted by suoersonic flow. The body will ce flattened proviOing more space for the real flow than for the fictitious flow to pass because of _is _ _f. The analytical oackground 7's as well as the numerical emp=cts_ _Io. of this method are described elsewhere, this paper is intended to present some illustrative results in the lign_ of an aop!ication to advanced technology compu- .__onal aircraft Oesign tools. SELECTION OF A_JALYSIS ALGORITHMS, FICTITIOUS GAS MODELS ;ND B_SEL!NE CONFIGURATIONS Our design procedure resuires in its first step a reliable analysis algorithm for elliptic _artial differential equa- tions to solve the saPsonic dart of the flow and provide flow properties along the sonic surface. Many comouza_icnal codes are operational for inviscid flow past airfoils. We prefer solvers for the basic eouations in conservation Form. A finite difference relaxation coda was extended to be a design tool II . A boundar 7 laver method and - for_ the_ analysis version - a method to treat shock - boundary layer 12 13 interaction was added . Another computer code based on the same analysis algorithm Crests viscous interaction be- tween Ooundary layer and wake 14 Results o_t=ined with _ =. . h_se comuuter orogrems will be illustrated in the following. Wing design codes based on the outlined method have been developed, tad, but an implementation of 3D viscous effects still needs to be done. Both non-conservative finite diffe- rence and fully conservative finite volume codes have been extended to be shock-free wing design programs 15,16 . With rapid progress in numerical methods more efficient codes will become operational, examples given her are intended to stimulate the engineer to introduce the idea into new computer programs for transonic flow problems. Given an analysis algorithm for transonic flow we have to introduce the design option by providing an alternate formula for the isentropic flow density #_/@* = ((2' " 1)/2 - (2' - 1)/2 • (q/a*) 2) t/f,_-lJ ensuring elliptic partial differential equations. The _m 6formula gf/C_*= c P..(q/a * ;, c - 1) -p _ZZows a 2-parametric variation of fictitious gas proper- _ies and elliptic equations if p _ 1, c _ p. A continuous ___ope at sonic conditions O = a _ , where _is is switched _o _f, is o_tained if c = p, but useful results with smooth _coy surface modifications may also be obtained for c _ p. The value c = 1 gives r/ * ) -P(_ : ( q/a" J results of this gas model have been studied extensively. cas oropertles are defined by p and a result is illustrat- ed in Fig. 2 for Oifferent values of p to demonstrate the 'nf!uence cf this parameter on the resulting new surface shape. A conventional NACA 0012 airfoil is flattened by the design procedure, we observe that a long flat sonic owD_le on the airfoil is obtained by low values of p, here p = O, which describes an incompressible fictitious []aS. _O _'= .T_Ce changes between 2 and 47 percent chord are _eouireO, the maximum deviation of the new contour is 0.0054 percent chord. For nigher values of O the surface deformations are smaller and more local, but surface curvature changes become sub- stantial if p _ 1. This examole illustrates the fact that shock-free modification of a given (initial-) confi- duration for prescribed lift coefficient and flight Mach number does not result in a unique new shape. A variety of shaoe changes within certain limits is possible and the criterion of choice of the fictitious gas model is y? 7the desired resulting pressure distribution cn the airfcil. All types Between "oesk,,"_ and "roof-too" c - distributions_ P are possible and selection is at the designer's disposal. OFf-design properties of an airfoil or wing are dominated b V the occurrence of shock-waves and ccmp!icated by viscous interaction, but the design pressure distribution is crucial for prediction of these effects. This leads us to the selec- tion of baseline configurations° Extensive experimental work was performed to arrive at the widely used and well documented NAGA airfoils. One of the first results of this method was a sari_- of shock- free modification__ _ _ of a NAGA 64A4"!0 airfoil. The results are illustrated in Fig. 3, in a Hach-CL-diagram. We see the amount of thickness reduc- tion and the limits for shock-free redesign of this airfoil and chosen gas parameter (p = 0). A thickness reduction usually tends to shift the occurrence of shock-waves and drag rise toward higher Nacn numbers. So a shock-free modification requiring thickness reduction seems not very surprising. A shcck-free modification without reduction of the maximum thickness seems imoortant for prac- tical design requirements. Fig. 4 illustrates another re- sul_, the verification of a Known shock-free inviscid flow (MORN airfoil 75-06-12) with our method. A local surface thickness _ump had to be added to the upper surface, a care- ful variation of its shape and the gas parameter p finally resulted in equal thickness addition and subsequent design thickness _eduction so that the original MORN ai_foil_ and its pressure distribution was verified. These inviscid test results illustrated above lead us to the conclusion that we have computational tools to - modify conventional configurations to be shock-free at transonic operating conditions, - specify the type of shock-free flow by a selection of fictitious gas model and initial configumation geometry changes, yy o_ain e whale series of neiohboring shock-free flow solutions for variable ooerstino conditions. it is this third capacity of the methoO we will investigate in _.._ followlnc. SH2CM-FREEAIRFO!L SERIES: _SF_CEPT OF ADAPTIVE CONFIGURATION Aercovnamic efficiency of a wing is defined by the ratio _if. over Crag, multiplie_ by the _l,_ight Mach number. With _rao_ r_____n_ sharmlv.... if the Mach num_ approaches unity, efficiency drops and it is therefore e principal goal of high subsonic speed aircraft desion to delay drag- rise to higher Macn numbers for prescribed lift. Th_s is usually achieved by oelaying the occurrence of shock waves _c nioner Mach numbers through a careful variation of wing sn=__., man,,_ analysis_ _ computations and very costly wind tunnel experiments. Our design methad seems to be a useful tool to obtain be_ter airfoils end wings for transonic flight. Moreover, t:_s co_p_ _-+ ..... _o_ionsl oeT_n_tion or surface modifications for ve_,'_no.:_ free str_am_ conditions gives an idea about _ossibl= mecnsnical adjustments of the configuration in order to m_in_ain efficient operation even at different flight conOitions. We choose e d_=ign examole for illustration of the required surface modifications at varying flight Mach number at constsnt lift coefficient, Fig. 5. A given airfoil A is -_igned to be shock-free at Mach= 0.73, o L = 0.55. We ask for its performance at Macn= 0.75 and c L = 0.6. Analysis incluoing viscous interaction gives a result with a recom- yS 9oressicn shock. A bump, aOded to the upper surface gave an initial configuration 0.3 percent thicker than airfoil A, original thickness was obtained from the subsequen_ design computation. The new airfoil B is investigated bv the ana- lysis version of the code to confirm the desion result. Fig. 6 shows off design analysis results, we see that an increase of 0.01 of the Orag rise Msch number has been achieved. Geometry modifications whicn leo from airfoil A to B are Oeoicted in Fig. 7. Addition of a bumo (a) which extends from 0 to 85 percent chord and subtraction of a design _ 2 tobump (b) within the supersonic region f__m 68 oercent chord leaves two small bumas to be aOded tc the original airfoil. These bumps are only 0.0021 and 0.0013 percent chord high. At this point we might think about a tecnnical realization of such a bump addition in order to have both airfoils evailaCle for operation. Experiments with a possible use of elastic or pneumatic devices should be carried out. Another concept is a controlled distribution of suction and blowing as already investigated for laminar flow con- trol anO similar efforts to influence flow quality. Boun- oaly laver displacement sF t;he Flow past _i_foil 5 at design conditions is drawn in Fig. 7 (curve c) for comparison with - rt__e modifications.the requi_ed su " Another example to investigate sensivity of the calculated shock-free design is shown in Fig. 8. NACA 0012 airfoil was modified to be shock-free using incompressible ficti- tious gas (p = 0). We are interested now in an approximate representation af t_e calculated surface modifications by a smooth analytical curve. For simplicity we choose a spline function with few supports which is the mathematical model of an elastic beam deformed by single loads. Analysis results are comoared with design pressure Oistribution and to our pleasant surprise we find that this airfoil with an elastic section is practically shocK-free, too, I0 even thougn the oressure Oistribution and sonic line are :ifferent from the original design. The reason for this is ooviowsly related to the muLtiolicity of possible shock- f-ee desions with different fictitious gas p_rameters, Fi:. 2. _-_aving proved that desiraOle flow Quality could he achiev- eosy shape chano__, generated by mechanical devices we go 17 one steo further and propose a system for automatically controlling the flow Quality, Fig. 9. In the system shown, _! flow quaiitLy sensor F determines the operating condi- tions and surface pressure et selected stations and is interrogated bV a microcomouter M that determines the :..roper changes of the effective contour necessary for snack- wa,.,e reduction. Our experience with the presented design metnod enables us to set uo the orogramming of the micro- como,_,ter which energizes a servo system 5 which appropriate- !V alters the effective shape. This is accomplished b V servo motors for mechanically adjusting sections of the wing surface, and o_her mechanical devices on the structure fcr the opening or closing of apertures on the wing surface to bleed (or add) various amounts of air from (to) the upoer surface of the wing. Any combination of the above may also be used. As seen from the illustrated examples, the surface ares mns _. ne_d___ to be changed is limited end the amount of change required is small. :DAPTIV. SUPERCR_T!CAL W Ir_GS l_e have outlined a conceot of transonic design and illus- trated some cases of air_oil flow. At this stage experiments need to be performad to prove both new design results and some realization of aSsc_ive airfoil technology in the wind tunnel. 11 Parallel to exoerimental verifica__on _n imolemen_ati_n of the idea into new and relisOle 3D wind ano wing -bocv configuration analysis codes is necessary. Our exoerience with wing design is limited to date, also because of a lack of 3D DounOarv layer and viscous interaction methods. Design studies of inviscid shock-fre_ wings are presently carried out to refine the 3D marching procedure and deter- mine the structure of 3D local supersonic shock-free flow fields. Fig. 10 shows a result obtained by extension of a finite difference analysis code 16 to s design tool. A simple wing based on _]ACA 6qA_IO section is modified to me shock-free. Viscous effects may be accounted for by adding estimated displacement thickness to the initial configuration. Results Is of a finite volume analysis code design extension for a shock-free supercritical wing with added displacement thick- ness is shown in Fig. 11. Extent of the suoersonic damain on this "flying wing" without _odv defines the area of possible aOaptive surface changes. A thick span loader flying wing seems to be a suitable t_st bed for experiments with 3D adaptive devices. CCr_CLUSI Oi._ We have apalied the elliotic continuatian shock-free flow design method to some illustrative test examples to form a theoretical base for the concept cf adaptive wing geo- metry at veriaOle operating conditions. A system for auto- matic shape variations of wings based on experience with systematic comoutational design is proposed. Both special designs and the adaptive shape control system need to be tested experimentally, possibly in combination with new aerodynamic concepts for higher efficiency of transonic aircraft like variable geometry and boundary layer control investigations. 12 R£FERENCES Morawe%z, C.S.,"On the Non-Existence of Continuous T-ansonic Flows Past Profiles, I, II ann III", Communi- caticns cn Pure end Aoolied Mathematics, (1956, 1957 ann 195S), p. 45-68, 107-131 ann 139-14_. Whitcomb, R.T., "Review of NASa SuDercritical Airfoils" Ninth International Congress on Aeronautical Scienc==,_- Haifa, Israel, (197&). Pearcy, H.H., "The Aerodynamic Design of Section ShaDes for Sweot Wings," AOvances in Aeronautical Sciences, Vol. 3, Pergamon Press, (1962), o° 277-322. __n, ......... ._ sign _..n_ K_rn, _ , Numerical De Transonic Airfoils," Numerical Solution to Partial Diffe- rential E_ua_ions, Vol. !I, Acaoemic Press, New York, ci°78), o. 25_-?_ 8serstoel, _._.,_" "De_on__ ann Analysis of a Hodooraon Me,boO fcr _he Calculation of Suoercriticel Shockfree Airfoils," NLR Rapt. TR 770_6U, Amsterdam, The Nether- lanes, (1977). Sobieczky, H., "Entwurf Oberkritiscner Profile mi: Hills _er rheoelektrischen Ana!ogie," DFVLR Rapt. DLR-F8 75-&3, G_c_ingen, West Germany, (1975). Sobieczky, H., "Dim Berechnung loka!er r_umlicher OOer- scnailf=lde-" ZAMM 58T (1978), o 21_ 916 Sozieczky, H., "RheograDh Transformation and Continuation Methods," Von Karman Institute for Fluid Dynamics, Lecture Series 1980 - 4, "Mathematical MethoOs in Fluid Mechanic=" (198C) SoOieczky, H., N.J. Yu, M.-Y. Fung, A.R. Seebass, "New Me_hod for Des_gn_no_ _ Shock-f._=_ T-ansonic Configurations" AiAA _ _ournal, Vol. 17 No. 7 (,979). 10 Funo, H _v H Sobieczky, A R Seeb=ss,r • -, • • . - "Numercial As- _e_s or the Design of Shock-free Wings and Wing-Body _mbinations." AIAA paper 79-!557 (1979) 13 11 Jameson, A., "iterative solution of transonic flows over airfoils and wings, including flows at Mach I." Comm. Pure Aool. Math. 27 ('1974). !2 NanSanen, M., E. StanewsKy, G.R. !nger, "A cemoutaticnai procedure Far transonic airfoil flow including a soeciai solution, for shock-bouncary layer interaction"_ , AiAA paper 80-1389 (1S80). 13 MeBnik,_ R. E. , Chcw, R. end Mead, H.,.° , "Theory.. of V_sccus_ Transonic Flow Over Airfoils at High Reynolds N:jmOer", AIAA pacer 77-680 (1977). "I_ NeOeck, H.E., A.R. Seebass, H. Sobieczky, "Inviscid- viscous intersc_ions in the nearly-direct design of snook-free sueercritical airfoils", AGARD FDP Symoasium on computation of viscous/inviscid interaction (1580 . 15 Yu, N.J., "Efficient Transonic Shock-Free Wing Redesign Procedure, Using a Fictitious Gas Method." _IAA Journal, Vol. 18 No. 2 ('L980), p. I_3-748. 16 Jameson, A. and Cauohev, D.A., "Numerical Calculatian of the Transonic Flow Past a Sweot Wing", New York University ERDA Report C00-3077-IL_0, (1577). 17 Sobieczky, H., A.R. Seebass, "Shock-Free Suoarcri_ical Aerouy,_=miu Stiuc_ure and Method _or Jasigniag Same" Invention Disclosure, University of Arizona, Tucson AZ., EES Report TFD 75-02, (1979). q8 EOerla, A., "Eine Methode der finiten Volumen zur 8erecn- hung oer transsonischen Potentialstr_mung um FlOgel aus Cam Druckmiminumintegrs_" MBB React* UFE 1407(0), (!578) q ,: a" /_. Q " l-nl ,a" '' q,siq) / (_ " (_t(q) _ a" "_n ,_ - ( U'. v', / °v Jx,, /\ , , I ,IN w I a q FICTITIOUS COr,!PRESSIBILITY Q_(q _ a*), EL.'RTIC EQUATIONS FOR Qf . q-I , _,.U ...._ VALUES, ELLIPTIC SOLVER ISE{_TROPIC COMPRESSIBILITY Q,s(q _ a ), HYPERBOLIC EQUATIONS FOR q _a* INITIAL VALUES, MARCHING PROCEDURE, Fi_. Elliptic continuation shock-free design. a) First step: Fictitious gas flow analysis b) Second step: Supersonic domain integration ,.,_7 Cp -I 0 -I.1 100_ZlCp. 0°° OOo -/ • _. ... .oo o.lo o.eo 0.30 .'..4o o._,o o._o o.bo ' o.eo _._o _._:o ].' ?",)_ \?,'_'. ,">:..'>..".>'... '7 ', " ' ' .' " " ' '..: ,., Fig. 7 Surface modifications: (a) addition for fictitious gas analysis (b) resulting additicn after redesign; (c) for comparison: upper surface boundary layer displacement thickness. " _f_-_spline \ P:.,O¢, _, " i / \ . ,,, :_ II _ t \,\z / /_-'\,. \ .-" ,. ','k. ', _, .-<7 /" /'_- _'" -- _ " "/ \ x.- .._"q .-" ,._.>._.," \ _._ -- ...--" ./--/ (.." '\_ ', .." Fig. 10 Shock-free redesign of a rectangular wing with NACA 64A410 section. ,!?......... k ! °, i °co e + _ I • 46_ ILI$ zz • x III I I * • • . 0 |11 I I • • • • • • • e • o i X_e** * • ""I""ii:..-. 6 # e osj • a w eeo_ 4 f m °_T i l°ooeooe I ° e • • ,..'" .. ..;__ Qe "° ** k C WING 803;73 AR 6.5 MA CH .78 oc -1.0 q. .52 col .018 c,., -.G,_ mig. 11 Shock-free desxgn of a swept wzng. Viscous diplacement model added. Z_ Ho E. Nebeck and A. Ro Seebass University of Arizona Tucson, Arizona and H. Sobieczky DFVLR GSttingen, West Germany INVISCID-VISCOUS INTERACTIONS IN THE NEARLY DIRECT DESIGN OF SHOCK-FREE SUPERCRITICAL AIRFOILS AGARD Fluid Dynamics Panel Symposium COMJ_UTATION OF VISCOUS-INVISCID INTERACTIONS INVISCID-VISCOUS INTERACTIONS IN THE NEARLY DIRECT DESIGN OF SHOCK-FREE SUPERCRITICAL AIRFOILS by H. E. Nebeck and A. R. Seebass University of Arizona Tucson, Arizona 85721 and H. Sobieczky DFVLR G_ctingen, West Germany 3-1 S U_RY An especially simple procedure for finding airfoil shapes that have desirable aero- dynamic characteristics and tha= will be shock free at Math numbers close =o the highest values possible is described. The procedure accounts for overall inviscid-viscous interactions that are weak and includes the locally strong interaction at the =railing edge as incorporated in the Grumfoii algorithm. INTRODUCTION In the las= decade substantial progress has been made in the computation of aerody- namic flow fields in which the global inviscid-viscous interaction is weak. For two- dimensional transonic flows the fundamental interactions, carefully delineated nearly a decade ago by Green (1), are now adequately modeled by computational analysis, provided chat the global interaction is weak and that the boundary layer suffers at most mild separations. This is accomplished by coupling a numerical analysis of =he inviscid flow, which assumes the main body of the flow is irrotationa!, hence derivable from a single scalar poten=ial, with a numerical solution of an integral formulation of the boundary layer equations. Much more sophisticated computational tools, such as large eddy simula- tion, are beoouzing available but they are not yet efficient enough for their application to design problems. The application of these analysis tools in design can take many forms. Perhaps the most basic is that of prescribing a pressure distribution and determining the airfoil that will generate this pressure field. This inverse problem is not well-posed and extensive computations are usually required in order to obtain useful results. A variant of =his procedure is to prescribe desired changes co an already existing pressure field chat is generated by a given airfoil. Here the inverse problem may be linearized about an existing flow and the resulting problem is more amenable to analysis. Another proce- dure is to use numerical optimization Co find which of a family of possible airfoils will provide the best airfoil performance. Here the main limitations are the family of air- foils considered and the compu=ational expense of exploring incremental changes for improvements in performance that are only marginally larger than the errors in the compu- ted performance parameters. A third, but more limited, possibility also exists, namely, deter-mining the changes required in a baseline airfoil co make che flow past it shock free at a prescribed lift coefficient and Math number. This presumes that an airfoil baseline which meets performance goals for subcritical .Math numbers is known. One then invokes the "fictitious gas" procedure of Sobieczky (2) to find a new shape for the upper surface of =he airfoil thac will produce shock-free flow at the prescribed conditions. This shape is not unique, nor is it possible To find such shapes for all flow conditions. For a prescribed lift coefficien= and airfoil thickness, there is a frees=ream Mmch number above which a shock-free shape is not possible. While the feasibility of this procedure has been amply demons=raced for inviscid flows, indeed, even for three-dimensional flows (3,A), its success for flows with inviscid-viscous interaccion has not previously been documented. The goal of this paper is to demonstrate the ability of the fictitious gas procedure to design advanced shock-free airfoils at little computational expense even when inviscid- viscous interactions are taken into account. ANALYSIS ALGORITHM_ A number of numerical algorithms have been developed to calculate the transonic flow past an airfoil in the presence of weak £nviscid-viscous interactions. One of the early successes was The algorithm of Bauer etal. (5), which we will call B_KJ. lc employs a nonconse-_vative formulation of the potential equation, coupled with the £ntegral boundary layer code of Nash and .MacDonald (6) and a constant thickness wake model. The nonconser- vacive difference scheme fails co conserve =ass and underprediccs the irrotational shock strength. As a consequence, the inviscid version of =his algorithm gave results that agreed well with experimental resul=s for the pressure coefficient when compared at the same Math number and lift coefficient. Collyer and Lock (7) modified the inviscid analysis portion of this program co include a combination of conservative and nonconser- vative differencin_ in order =o better caocure the correct shock pressure rise, and c_-_led _ with Greens lag-entrainment method (8) for computing the" turbulent boundary layer. They included the modification of the inviscid flow due to wake curvature but did not model the strong interaction =ha= occurs at the trailing edge. Nandanan, S=anewsky, and !nger (9) have used Jameson's conserva=ive version (i0) of the BGKJ algorithm toge- ther with Rotca's in=egral dissipation me=hod (Ii) and Ingot's model of shock-boundary layer interaction (12) co compute flows with weak embedded shock waves. Perhaps the most advanced algorithm of =his type in the U.S. is that due to Melnik, Chow, and Mead (13). They coupled Jameson's inviscid algorithm with Green's lag-enTrainment method for =he boundary layer. The effects of the s_rong interaction near _he trai!in_ edge are also included in the algorit_hm. This is accomplished by incorporating the results of Melnik and Chow (14) for the multi-layered Turbulent boundary layer at a cusped trailing edge. This coupled calculation provides a self-consistent resu!: for the inviscid flow, the boundary layer, and =he wake. It also removes the singularity in =he inviscid pressure associated wit5 the Trailing edge of the airfoil. Wake curvature effects are included and are found to have a significant effect on the results. This algorit,km has been called "Grumfoil" by its authors, and we will use the same appellation here. While no shock-boundary layer interaction model is included, the algorithm seems to be accurate for shock strengths for which the irrotationa! approximation is itself satisfacto_!. SHOCK-FREE DESIGN Inviscid analysis algorithms such as those discussed above have amply demonstrated the generality of Morawetz's (15) result of the mid-1950s, namely, that shock-free flows are ma_hematically isolated one from another. Despite this isolation they have played an important role in providing moderate increments in aircraft performance. Wind tunnel research byPearcey at the National Physical Laboratory (16) and Whitcomb (17) at NASA LanBley Research Center first demonstrated that such flows could be realized and would have important applications. Subsequently, Garabedian and Korn (18), Nieuwland (19), Eoerstoel (20), and Sobieczky (21) developed analytical tools for the prescription of shock-free airfoil shapes. These cools relied on the hodograph transformation, and viscous effects could only be accounted for to the exten_ _hat the boundary layer could be computed independently of the inviscid flow. The extension of this capability to =he physical plane by the introduction of a fictitious gas for a preliminary calculation of the supersonic portion of the flow field makes i= possible to design shock-free airfoils with a proper accounting of the coupled nature of the inviscid and viscous flow fields. This capability requires one crucial approximation _hat has now been Justified by numerical experiment, namely, =ha= the boundary, layer displacement thickness is not altered in any consequential way by the difference between the pressure field of the fictitious gas and that of the real gas. Given no essential differences in the boundary layer displacement thickness in the supersonic domain due to the difference bev;een the real and fictitious pressures there, nor any due =o =he minor change in airfoil thickness, =hen =he inviscid and viscous flow fields must be correct and correctly coupled. We begin the design process with the selection of abaseline airfoil. Normally, this would be an airfoil that meets subcritical design goals and that has a reasonable amount of upper surface curvature. Here we have used a supercritical section, both for conve- nience and to see if we can improve its performance somewhat. Our goal here, however, is not advanced airfoil design, but rather, to show that the fictitious gas design procedure is feasible in The presence of coupled inviscid-viscous interactions. We then modify the Grumfoil algorithm to incorporate a fictitious density-flow speed relation when the Math number exceeds one. Both the baseline airfoil and the fictitious density relation- ship are aT the investigator's disposal and they interact in a way that allows the generation of a limited family of candiaate airfoils. Because our concern is with demon- strafing =he feasibility of the proposed procedure we limit our attention to the simple relation _/=* - (a*/q) P, (1) where _ and q are the fictitious density and real flow speed and o* and q* are their real sonic values. The exponen_ P is a parameter that varies _he gas law (1). For values of P less _han one relation (I) insures tha= =he governing equation, namely, div(p S) - 0, (2) ";here c - 7_, and # is the velocity potential, remains elliptic. For such relations the fictitious mass flow in the supersonic domain is greater :han it would be for a real gas. The solution to this fictitious flow problem is used to provide values for the flow deflection on the embedded sonic line. To insure the accuracy of this data we generally insist on convergence to a maximum residual of 10-6 on a 160 by 32 grid. This data on the sonic line is used to compute the velocity potential and stream function there. A simple charac'.eristics routine is then used to march down from the sonic line To find :he streamline consistent wi_h stream function values at The sonic line airfoil juncture. (This must be done in a way that is consistent with the mass flow added to the flow by =he boundary layer displacement effect.) This defines the new body streamline if the characteristics calculation succeeds. It may not, but may rather signal =he intervention of a limit line and indicate the failure of the sonic line data to be consistent with snook-free flow. In =his event the baseline airfoil and fictitious density law can be modified in an a_tempt =o circumvent _he failure. The new airfoil surface will be :hinner than the baseline airfoil because the real densi_.y is less than the fictitious density and, hence, for the fixed mass fl0w entering =he sonic line,_he real gas requires more area. ._ne amount that the airfoil's vertical coordinate is thereby reduced can be added to the baseline airfoil and the process repeated until the baseline airfoil =hick- hess is retained in the shock-free design. Changes in the choice of =he gas law, here simply limi:ed To changes in P, also have an effect on airfoil Thickness, albeit a small one. Further progress in airfoil design requires an understandinB of the relationship bec';een the baseline airfoil and the fictitious gas law chosen. We do not explore that question further here. 3-3 The designed airfoil must now be analyzed by the original algorithm =o see how well it performs. The boundary layer calculation used in the design process was based upon the pressure gradient of the fictitious gas calculation. The designed airfoil will have a somewhat different pressure gradient. If the displacement thickness were correct, then only the pressure field in the supersonic domain is altered. However, this change in =he pressure field al:ers the boundary layer displacement thickness and, to some degree, =he extent of the supersonic domain. We find in practice, however, that the error in che displacement thickness is so small that we need not correct for it. RESULTS We selected -he VFW airfoil Va2 for our baseline as characteristic of the wing sections that will be used on future co-_ercial transport aircraft. This airfoil, which is 12.57 thick, has a design Mmch number, M,, of 0.73 and a design lift coefficient, C_ of 0.525. Our coordinates are slightly different from those for the Va2 and we have used our own designation, 47070M, for this baseline airfoil. As this airfoil is already an advanced design we only exa_Lined a modest increment in design conditions, namely, M_, I 0.75 and C_ - 0.550 at a flight Reynolds number of 35 million. Figure I depicts the pressure distribution and sonic line-shock wave shape for the baseline airfoil at =he new flight conditions. It has a drag of 67 counts. When we repeat this calculation using the fictitious density law (1) with P _ 0.9, we find results like those depicted in Figure 2. Because we are solving an elliptic equacion we obtain a smooth pressure distribution and well behaved sonic line. Although it has no physical meaning we noce the drag is now 61 counts. None of this can be wave drag because we solved an e!l!ptic equation and we must a=tribu=e it to viscous effects. The undulations in the upper surface pressure near the leading edge are due to irregularities in the airfoil shape and the boundary layer's transition to turbulen_ flow. We now use the flow speed and deflection angle on the sonic line to calculate the flow in the supersonic domain in the manner described above. This defines the new body surface. Since it i_ thinner than the original body, we choose _o repea= the process with a baseline airfoil shape that is _he original airfoil plus a mul_.iple of the differ- ence between the original and the shape calculated using the design procedure outlined above. The pressure distribution and sonic line for the fictitious flow past this air- foil are depicted in Figure 2. Performing the design process on _his new baseline results in an airfoil _.hat has essen=ially the same thickness as the original baseline airfoil. We now compute =he flow past this new airfoil (47073) to see if we have indeed found improved performance. The resul=s of this final calculation are shown in Figure 3. Shock-free flow has been achieved. The calculation was done with the lift coefficient set to 0.550. The drag coefficient, not too surprisingly, is the same as tha_ calculated using the fictitious gas. Figure 4 compares the redesigned airfoil and the original baseline air- foil with the vertical scale magnified five times so =hat the small differences bet;con the two airfoils can be observed. The improvement in lift to drag ratio over the base- line at .M, = 0.75 is i07.. The entire design process required about 500 CPU seconds on a CDC 7600. It could be repeated to recover the very small loss in airfoil thickness. The success of ".he design procedure depends on the boundary layer displacement thickness for the fictitious gas analysis being essentially the same as that for the real gas analysis of the new airfoil. Figure 5 compares the displacement thickness for the original airfoil, 47070M, that for the fictitious gas flow past a thickened airfoil 47070B, and _hat of the final design, 47073. There are very minor differences between the displacement thickness for =he 47070B and the 47073 airfoils at about 627, of the chord. This is more readily noticed in Figure 6, which compares the skin friction coefficients for _he three airfoils. This small difference in the displacement thickness manifests itself in a small difference in the subsonic portion of the pressure distribu- tions of Figures 2 and 3 just downstream of the sonic line airfoil Juncture. The differ- ence is so small that it can really only be disce---ned when the t-wo pressure distributions are overlayed. The supersonic pressure is, of course, always different. If we compare the off-design performance of the t-_o airfoils we discover tha= Ehe improvement in the drag divergence Mmch number is not even 0.01. Figures 7 and 8 comoare the pressure distributions and the sonic line shades for the two airfoils at M, - 0.7h o and 0.76. From Ehese results it is easy to see how the new design achieves the modest improvement in drag at M. - 0.75. The variation in the drag coefficient with Math number at the design lift coefficient and with the lif= coefficient at the design Math nu_.ber for the r-;o airfoils is depicted in Figures 9 and lO. If we repeat the design process at a higher Mmch number, and with a somewhat thicker baseline airfoil, we find another new airfoil, 47081, which achieves a 0.015 increment in the drag rise Math m number. The drag coefficient as a function of Math number for the 47081 is also shown in Figure 9. Figure Ii depicts the pressure distribution and sonic line shade for the 47081 airfoil an_ Figure 12 compares _his airfoil with the 47070M baseline airfoil. _qnile _he improvement in drag rise M,Ich number is very small, _here can be no doubt that, other things being equal, such small improvements are of major importance in =he design of a co_ercial transport. _.There is a limit to the M_ch number ah which shock-free designs can be found for a rLxed ll,= and alr_o, il ".hickness. With the 47070M as a base- line this would seem to be about M, - 0.77 for C_ - 0.550 and 12.57. thickness. CONCLUS iON We have described a co_putationally efficient method for finding airfoil shapes wi_.h desirable aerod'_na-_ic proper-des that are also shock-free a= supercritical :-:ach 3-4 numbers. Theprocessof making_hemshockfree can be successfully carried out even in the oresence of coupled inviscid-viscous interactions. The shock-free design process is as reliable as the analysis algorithm used to compute the flow field. A wide range of airfcil shapes can be found, dependin B on modifications to a selected baseline and on the choice of the gas law. There is a limit to the maximum drag divergence }_ch number that can be achieved for a prescribed airfoil thickness and lift coefficient. Our expe- rience with this process in inviscid flow gives us confidence that these limits can be achieved with an artful seleczion of baseline airfoils and fictitious gas laws. ACKNO'_EDGE>_NTS _n.is research was supported by Grant Number AFOSR-76-29541 from the Air Force Office of Scientific Research an_'by Grant Number NOOOIA-76-C-OI82 from =he Office of Naval _esearch, REFERENCES i. 2. 3. Green, J. E. "Some Aspects of Viscous-lnviscid Interactions at Transonic Speed, and Their Dependence on Reynolds N_._ber," AGARD CP-83-71, 1971, pp. 2-i, 2-12. Sobieczky, H. "Die Berechnung iokaler Riumlicher L"_erschallfelder," Z_, 58T, 1978, pp. 215-2!6, Sobieczky, H., Yu, N. J., Fun B, K-Y., and Seebass, A. R. "New Method for Designin_ Shcck-free Transonic Configurations," AIAA J., Vol. 17, No. 7, July 1979, pp. 722- 729. ! Fung, K-Y., Sobieczky. H., and Seebass, R. "Shock-Free Wing Design," AIAA J., (=o appear). Also AIAA Paper 79-1557. 5. Bauer, F., Garabedian, P. R., and Korn, D. G. "Supercritical Wing Sections," Lec'.ure .No=es in Economics and Mathematical Systems, No. 66, Springer-Ver!ag, 1972. 6. Nash, J. F. and MacDonald, A. G. J. "The Calculation of Momentum Thickness in a Turbulent Boundary Layer a= }_ch Numbers up to Unity," ARC CP-93, London, 1967. 7 Coiiyer, .'.!.R. and Lock, R. C. "Prediction of Viscous Effects in Steady Transonic Flow Past an Airfoil." Aero. Quart., August 1979, pp. 485-505. 3. Green, J. E., Weeks, D, J., and Broomen, J. W. F. "Predic'.ion of Turbulent Bounds1 Layers and Wakes in Compressible Flow by a Lag Entrainment Method," RAE Tech. Report, 72231, 1973. 9. Nandanan, M., Stanewsky, E., and Inger, G. R. "A Computational Procedure for Transonic Airfoil Flow Including a Special Soluzion for Shock Boundary Layer In=er- ection," AIA._ Paper 80-1389, 1980. iO. Jameson, A. "Numerical Computation of Transonic Flow with Shock Waves," Symposium Transsonicum II, Springer-Verlag, New York, 1975. ll. Rot=a, J. C. "Turbulent Boundary Layer Ca!cula=ion with the Integral Dissipation '.!e_.hod,'' Computation of Turbulent Boundary Layers, 1968 AFOSR-IFP Stanford Confer- ence, 'Col. i. p_. 177-181. 12. inBer, G. R. and Canzrel!, J. C. "Application of Shock-Turbulent Boundary Layer Interact!on Theorlf _o Transonic Aerodynamics," 4oh U. S.-F. R. G. Meeting on Viscous and Interacting Flow Field Effects, B:._IG-FB_'r 1979, pp. 100-!lb. !3. Yelnik, R. E., Chow, R., and Mead, H. R. "Theory of Viscous Transonic Flow Over Airfoils at High Re_.rnolds Number," A!AA Paper 77-680, 1977. l&. :.[elnik. R. E. and Chow, R. "As v-m_ptotic Theory of ?co-dimensional TrailinB EdBe F!o:;s," N'.SA Conference on Aerod:.-n.a-ic Analysis Requiring Advanced Computers, ";ASA SP-3&7, 1975, pp. !77-249. !5. "-'orawe_.z, C. S. "On -.he Non-Existence of Continuous Transonic Flows Past Profiles, I, Ii, and iIi," Cor-_---unica=ions on Pure and Applied :.'_=hematics, 1956, 1957, and 1958, ?. 45-68, 107-131, and 139-!4A. 16. Pearcey, H. H. "The Aerodyna=ic Design of Section Shapes for Swept Wings," Aivances in Aeronau'.ica! Sciences, Vol. 3, 1962, pp. 277-322. 17. :_nitco_-.b, R. T. and Clark, L. P., "An Airfoil Shape for Efficient Flight a _. Super- cri-ica! Hath :;umbers," XASA ._.'! X-ll09, July !965. 13. Gar-=bedian, P. R. and Korn, D. G. "Numerical Design of Transonic Airfoils," l:umerica! Solutions to Par-ial Differenzlal Equations, Voi. il, Academic Press. ";ew York, !978, pp. 253-271. 19. :;ieuw_.and, G. Y. "Transonic Potential FIo_" Around a Family of Quasi-Elliptical .-'it- foil Eectior.s," :;LR Repot-. TR-T !72, !967. 20. Boers=oe!, J. W. "DesiEn and Analysis of a Hodo_raph Method for the Calculation of Supercri=ical Shock-free Airfoiis," NLR Report TR-77046 U, 1977. 21. Sobieczky, H. "Enc_rf uberkritischer Profile mi= Hilfe der rheoelek=rischen Analogie," DFVLR Report DLR-FB 75-43, 1975. 3-5 ,j..-- ,-:. -_- ;-,. I--. ,-:.). "0_ l;=- O ,"- "' i,..,° v i.-r, _ i'_ '-_ p:.. r- ?" C _ r'- (.r. e--. O':- r.- "_0_ m O 0 -C !: o Q g o o G / / / / '1 i / o @ I 0 ,_ o° °° / i/ i I I" ,o 0 -i_ -C. _, -C. i," -0 .I_? #-'! 0 o iI o oo o@ °_ 'I! -; .:: .: ,_- ._ i z • o e %, ; F .c % -I .2: -_ .6C • i % Z• _={E 09.. _ :J _ :J i ;-_._ °_ I 3-7 ! o yo 1 i Y! / , !j' i • o • x. II • io o= :: i " To I- _ Ta _J T° _4 i i -,-I ,_ O "-J 0 • I., m _ °,,,I % ,- .;, "% t "" " \ T I ° _. -o i° ;; i _L r_ ; ! m o. f 0_ • i E OT L u • :3.?4 ¢L ° 0._5 _e " )S" 1:35 17071 EP 9' ?, 8 ?. ? a_ oT ! q • _.7S C t • O.SG ;re • 35" tO G o ! o °:-. 47373u Fi--ur-== ",. Pressure dis_-ribucions and sonic line - shock :;ave sha_es, for :he baseline and shock-free airfoils a= _-_ = 0.74. _r =- ==. .° 3 EP ae • 35" 105 ?, -0. S_"_ ° ?. / / o i / % • 0.;_ C t o 0.';$ _e • )S-10 6 _00_0_ 0 o ° @ o / Pressure dis_rlbu:icns and sonic line - shock wa':e shapes for :he baseline and shock-free airfoils a: :_ = _.76. 3-9 Ca el s0.55 . _ .i" _ T47081 "*3 "_ 't't Dra_ coefficient as a func:ion of Mach number for :he 47070_:. 47773, and L7751 airfoils a: C_ = 0.55, Re = 35.!3 o. Ct i, 'I _or- I l M_=¢.7 c. Re, ]SxfC I _11 ,5 C.:x i{.'z oRIGINAL PAGE I_ OF p0_R OUA!,I._._ Figure 13. LEft coefficient as a funczion cf dra_ coefficient of :he 47073:-!, _7073, -J_8! airfoils a: :._== 9.75, Re = 35.106 ql i/ / \, / Figure 12. // / Comparison of :he 47070}[ and 47081 airfoils. I/n.everEical scale is magnified five times. i ,-n ",i I I -.._:-q.; "t 9," L i i ! I I I I i \.\ / \ k . \ \ I '\ /,' ., ", '\ .:if':;0_ ?,- i' Figure il. Pressure dis:ribuzicn and sonic line shape for :he &7_8! airfoil a: :_ -9.76, C[ _ 0.55, Ke - 35'!0 _. 526 AIAA JOURNAL VOL. 18, NO. 5 ARTICLE NO. 80-4035 Analysis of Two-Dimensional Incompressible Flows by a Subsurface Panel Method Jack Moran,* Kevin Cole,t and David Wahl_ University of Minnesota, Minneapolis, Minn. A new approach In panel methods is explored for two-dimensional steady incompressible flows. The meth;_d uses linear distributions of sources and vortices on straight-line panels, hut satisfies boundary conditions on the actual body surface, at nodes thai are also end points of the panels. The result is continuity in body-surface velocity distribution, without recourse Io numerical quadrature for the veh_city influence e;_effieienls. The method is unusually sensitis'¢ to the distribution _f the nodes. For example, it almost always fails to gi,,e ac- ceptable results when line nodes are distributed randomly. However, Ihe c;mtinuity of the velocity distrihulion makes possible a unique node redistribution scheme, _hich may be iterated to give accurate results reliably. Background ANEL methods are now widely used for calculating linear potential flows past aerodynamic bodies. The steps involved in setting up a panel method are as follows: I) Represent the perturbation potential by a distribution of sources, doublets, and/or vortices of unknown strength over the body surface and its wake. 2) Approximate the body and wake surfaces by the union of panels of relatively simple geometry. 3) Parameterize the singularity strength on the panels; e.g., represent it by a polynomial of degree two or less. 4) For each unknown parameter in the representation of the singularity strength, demand that the potential and/or velocity field satisfy an appropriate boundary condition at some control point. 5) Solve the resulting system of linear algebraic equations for the parameters underlying the singularity strength. Once these steps are completed, the velocity and potential may be evaluated anywhere in the flow by summing con- tributions from the individual panels. Almost invariably, approximations made in the for- mulation of panel methods lead to singularities at the panel boundaries, and so restrict the usable output of the methods to points near the panel center. This is certainly the case if the panels are plane or piecewise plane. However, even when curved panels are used, spurious singularities sometimes result. 1-_ The integrals which give the potential and velocity fields due to the singularity distributions on the panels cannot be evaluated in closed form unless they are approximated through series expansion by integrals over plane or piecewise- plane surfaces. In effect, the source, doublet, or vortex distribution over a curved surface is replaced by a series of multipole distributions over a plane surface. At the panel edges, each term of the expansion is even more singular than the one preceding. Received May 7, 1979; revision received Aug. 27, 1979. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1979. All rights reserved. Reprints of this article may be ordered from AIAA Special Publications, 1290 Avenue of the Americas, New York, N.Y. 10019. Order by Article No. at top of page. Member price $2.00 each, nonmember, $3.00 each. Remittance must accompany order. Index categories: Computational Methods; Aerodynamics. *Associate Professor, Dept. of Aerospace Engineering and Mechanics. _Graduate Student, Dept. of Aerospace Engineering and Mechan- ics. {Na_ional Science Foundation Summer Intern, Dept. of Aerospace Engineering and Mechanics. 73 A related problem of existing panel methods, at least in three-dimensional situations, is the rather complicated way in which they approximate the body surface. Since panel edges are not available as control points, there is generally just one control point per panel, and thus (approximately) one unknown per panel as well. Quadrilateral panels are therefore preferred to triangles, since the latter would double the number of unknowns for a given number of points at which data on the body shape are specified. Howcver, to avoid numerical quadrature, the surfaces on which the singularilies are distributed should be plane, and a curved three- dimensional surface cannot be approximated by a continuous system of plane quadrilaterals. Some methods simply allow gaps between neighboring panel edges; others use a piece_ ise- planar quadrilateral (four triangles surrounding a planar parallelogram whose corners are the midpoints of the sides of the quadrilateral). The use of (roughly) one control point per panel also complicates the paranteterization of the singularity strength. Recent methods are based on quadratic doublet distributions. In order to specify the ten coefficients of the quadratics without creating discontinuities at panel or subpanel boun- daries, one method uses a singularity spline based on a least- squares fit of the quadratic in one panel to the doubter strength in twenty surrounding panels. A more local support for the singularity splines is desirable for a number of reasons. First, the more local the spline, the easier it is to match the singularity strength at boundaries between distinct networks of singularity distributions (e.g., at wing-body junctions). Also, a wide support suggests that the effective mesh size is much larger than the distance between nodes. Finally, a more local spline would probably simplify and, hence, expedite the analysis. The objective of the present research, therefore, is to develop a panel method with Ihe following characteristics: I) the velocity distribution on the body surface should be continuous, even at panel edges; 2) integrals giving thc velocity field should be evaluated in closed form; and 3) both the body surface and the singularity strength should bc specified by splines of local support. Thus far we have suc- ceeded in implementing a two-dimensional version of a method which promises to meet all three objectives. This paper reports our progress. Analysis We want to determine the aerodynamics of an airfoil of specified geometry immersed in a uniform steady in- compressible inviscid flow. A general representation of the perturbation velocity potential may be constructed by MAY1980 INCOMPRESSIBLEF OWSBYASUBSURFACEPANELMETHOD 527 distributing sources and vortices over a surface S: 'I4)p= -_ s (°Qf"roP-'Y_)OQ_)ds (I) Here P is a typical field point, Q a point on S, and rc)e,O,., polar coordinates of P relative to Q. The quantities (JQ and 3%? are the strengths per unit length of the source and vortex distributions, respectively, at Q. For any given source and vortex distributions, Eq. (I) satisfies the conditions of continuity and irrotationality and also the boundary condition at infinity. To make Eq. (1) the solution of a particular flow problem, it is only necessary that it also meet the flow-tangency condition on the airfoil surface and the circulation condition. Before ,xe can proceed further, we must specify the surface S--the "panels" of the method--on which the sources and vortices are distributed. Equation (1) is essentially equivalent to Green's third identity, according to which the potential outside S i.'; representable by a source distribution of slrcnglh ig¢/On and a doublet distribt, tion of strength O on S. lhc restrictions on the validity of Eq. (I) are therefore the salne as those on Green's third identity; namely, that ¢ is a continuous single-valued solution of Laplace's equation outside the surface S on which the sources and vortices are distributed. A safe choicEr for the panel surface, therefore, is the surface of the body under study. However, it is difficult to evaluate the requisite integrals if the panel surface is curved. In the subsurface panel method, therefore, we distribute the sources and vortices on straight- line panels, whose endpoints are nodes on the body surface, as shown in F'ig. I, but continue to satisfy the flow.langcney condition on the actual body surface. This distinction between the panel and body surfaces is consistent with the limits on the validity of Green's identity, provided the flow has an analytic continuation across the body surface to the panels if, as is usually the case, the panels lie within the body. Since the panel surface can be made to approximate the body surface as closely as desired simply by increasing the panel density, this condition is not expected to be overly restrictive. However, it does necessitate the special treatment of (if not exclude from consideration) flows that are truly singular; in particular, flows past bodies with convex corners or sharp edges (aside from edges at which a Kutta condition removes the singularity). Even with the panel surface Sof Eq. (1) specified, the flow tangency and circulation conditions do not determine the source and vortex strengths uniquely. It is possible to specify one of them almost arbitrarily and then to determine the other so that Eq. (I) meets all the conditions it should. A convenient way to supply a closure condition is to specify a fictitious velocity field inside the surface S. In our work, this fictitious field has zero velocity. Then o and 3' are, respectively, the normal and tangential components of the total (not per- turbation) velocity on the outside of the panel surface. In order to discretize the problem, we approximate the source anti vortex distributions as linear over each panel. Such distributions may be parameterized in terms of the source and vortex strengths at the vertices of the panels, which in turn equal the velocity components normal and tangential to the panels at their vertices. Since the vertices are also nodes on the K_ NODES FiR.I Suhsurfaeep-nneling. body surface, these are components of the total fluid velocity on the body surface. Specifically, let thejth panel be the straight line between the jth and (j+ I)th nodes, tj its length, and _ the distance from thejth node, as shown in Fig. 2. Then we take o(E)=Oo+(Or-Oo)(U_,), v(_l='yo+(-r_-q,o)(t,/(,) (2) in which, because the singularity strengths at _ =0 and _'l arc components of the total fluid velocity at thejth and (j+ l_th nodes, respectively, ( °° )=[R(Oj-[3J)]( V'J'Yo_ ) ( °¢ ) =[R(Oi+t-[3))]( I""/+1 ) (3_ "rt " I 9+/ Here V,,/ and V,j are the velocity components normal and tangential to the body surface at the jth node, 0j the in- clination of the body surface at the same point, ;3i lhe in- clination of thcjth panel, and [R] a rotation matrix: cosO sin0 ][R(0)] = -sin0 c(,s (4/ This parameterization allows us to achieve our first major objective, continuity of the body-surface velocity. The velocity field due to any one panel does blow up at the ends of that panel. However, with our parameterization, the singularities due to neighboring panels cancel exactly, as is shown in the Appendix. Note, on the other hand, that the source and vorlcx strengths themselves are not continuous from one panel to the next. The singularity strengths are, as previously pointed out, velocity components normal or ta_}gential to the panel. At a node, the source and vortex strengths on the panels which meet there are componenls of the same velocity, the local body-surface velocity. But, because panels generally meet at an angle, the components of that velocity normal and tangential to the panels differ, and so, then, do the local source and vortex strengths. Since V,,j is known at each node from the flow-tangency condition, our unknowns are the nodal tangential velocity components. These we determine by satisfying an integral equation for V t at each node, / V,= _ .Is [aoL'Xr/"ro'- 3'C'/"" Xl'OoAds + V® ./'v (51 which is derived by differentiating Eq. (1). Contributions to Eq. (5) from the panels adjacent to the ith node must be subjected to a limiting process, in which the field point ap- proaches the ith node from the outside S; see the Appendix for some of the details. Vii j + I Fig. 2 Nomenelalure used in formulas for source and vortex strengl hs. "7 MORAN, COLE, AND WAHL AIAA JOURN,%. J An alternative approach is to require the velocity tangential to the body surface to vanish on the inside of each node. Because of the jumps in velocity across the panels, the resultant formula is exactly equivalent to Eq. (51, and we shall refer to Eq. (5) both as an "integral equation for the external tangential velocity" and a "requirement of vanishing internal tangential velocity." Circulation Condilion For sharp-tailed airfoils, tile circulation is fixed by tile Kutta condition, that the velocity be finite at tile trailing edge. Unless the trailing edge is cusped, this implies that thc trailing edge is a stagnation point. Although we can show that, in principle, our solution yields finite velocitics at sharp-but-not- cuspcd trailing cdges if, and only if, the flow stagnates there, simply replacing Eq. (5) at the trailing-edge nodes by the stagnation condition i/,i = 0 at trailing-edge stagnation points (6) yielded obviously incorrcct results, inch,ding pressures higher on the leeward side of the trailing edge than on the windward. Similarly poor results were obtained when we attempted a variety of alternative circulation conditions, including equation of the tangential velocities at the trailing-edge nodes and requiring the trailing-edge bisector I(_ bc a streamline, with such conditions used in place of Eq. (5) at otle Or both trailing-edge nodes. What such formulations seem to ignore is that the circulation condition is a requirement that the solution must satisfy over and above the integral equation approximated by Eq. (5). In any case, we obtain good results only by solving an overdetermincd system comprised of an equation like Eq. (5) for each uode and a circulation con- dition. For bodics with trailing-edge stagnalion points, v,e use Eq. (6) in place of Eq. (5) at tile trailing-edge nodes. l'or bodies with sharp trailing edges, the circulation con- dillon used is that the velociti,' component normal to the trailing edge bisector vanishes at a point very close to the trailing edge; specifically, at a distance from the trailing edge of about 10 -5 times the avcrage length of the two panels adjacent to the trailing edge. For tile present case of steady two-dimensional flow, it makes little difference whether this point is inside or outside the airfoil. The rationale for this form of the circulation condition was the suppression of the lrailing-edge singularity which would follow from failure to satisfy the Kutta condition. However, the singularity it only logarithmic, and, as we discoverd by accident, the results change very little if we simply delete the logarithmically near-singular terms in calculating the velocity at the control point near the trailing edge. On the other hand, no logical alternative circulation condition suggested itself. In particular, we cannot simply require tangential velocities at the two trailing-edge nodes to be equal and opposite. This is already aecomplsihed, in effect, by setting Eq. (5) or (6) at the two nodes. Real airfoils--certainly if viscous displacement effects are taken into account--do not have sharp trailing edges. A cutoff trailing edge is usually modeled by hypothesizing a constant-pressure wake to emanate from the edge. Thus we require that the tangential velocities at the nodes on either side of the trailing edge be equal and opposite. The trailing edge is closed by a single panel, on which the velocity is determined solely by requiring the velocity to be continuous at its two nodes, where it is tangential to the main airfoil surfaces. Thus, the trailing-edge panel supports a fairly strong source distribution. Equation (5) is imposed at the trailing-edge nodes, which implies that the internal velocity tangential to the main airfoil surface vanishes at those nodes (as well as all the others). The various forms of circulation condition for sharp, cusped, and cutoff trailing edges are summarized in Fig. 3, in i) / ht / "_..l_t _ _\\ t') EQUAL VELOCfTPE _ I:ig. 3 SiIililll',lr._ _| |_oundur) conditions impo,,ed 11 Ir:iiling et|ge., ot" ,,:,riottr, types: a) cusped, b) sharp hul nol eu'_l)ed, zn(| ¢1 ¢,l,_fl'. which tire X'd arrows indicate velocity components tha_ are set to zero. As noted above, adding the circulation condition to the system which governs the nodal tangential velocities over- determines those unknowns. To solve the resuhant system, we generally follow Bristow r in introducing an extra unknowT1 for each extra equation, namely, a constant error term in all equations like Eq. (51, the constant varying from one element to another in multiclement problems. We also provide, m our program, _he option of _tsing a least-squares technique. Generally, the two methods give quite comparable resuhs. Bristow's is much cheaper, and so is preferred, but the Icas_- squares method is occasionally more reliable in thc node redistribution process to be described below. Input Requirements It should be noted that, to obtain a solution for the source and vortex strengths, the only data which must be known about the body shape are its coordinates and slope at the nodes. No assumption or approximation is made concerning the body shape between nodes. Thus, the only approximation made in the analysis is that the source and vortex strengths are assumed to vary linearly with distance along the panels. By invoking the momentum and moment-of-momentum theorems, we reduce force and moment calculations to in- tegrals over the panels rather than over the body surface, so that that part of the calculations, too, is indcpenden{ of tile form of tile body surface between nodes. Further, the assumed linearity of the velocity distribution on the panels makes it possible to evaluate in closed form the integrals over the panels required for the force and moment. "7..¢' MAY 1980 INCOMPRt::SSIBI.E I.[.OWS BY A SUIJSURFACE F'ANL_L MEFJ It)O _ LI ° I'3Z• 4 History of root-mean.squared I _ f error d.rio_ node redislribuliou prnce._ I _ _ -/ r.r ellip_ +.f lhicl. ,_'+mtalk, 0.125. Nodes I + _ EI.UI_I initialt.s distributed by cosine formula Eq. ] i / _:'+ (IlL Citrics. 16 panel,:; triangles, 32 lie_ at . / panels;squirts, 6.+ panels. I _ / . ItS% '!meKNeIS I | J ++AUTvm. COS.+' E 01STlUSUTION uamuuu P_meLmarion Error Estimation in which e is chosen so as to fix the locations of the starling A unique feature of our method, made possible by the and ending points of the group of nodes being redistributed.§ continuity of its results for the velocity distribution, is a I_'esuhs obtained with this node redistribution algorithm for capacity for ch .'cking its own accuracy a posteriori. As noted symmetric now past an ellipse of thickness ralio I/, are shown above, the bob' approximation made in the analysis is the in Fig. 4. For three different numbers of panels (16, 32, and assumption that the velocity distribution is linear on the 64) the nodes were initially distributed according to a cosine i panels. Once t_'e nodal tangential velocities V,+ are calculated, law Eq. (5) can b: _dapted to ealculale the velocity induced by [..he panels and the onset ['low at the middle of the ilh panel, Vs. _r(i- I) Then the non,inearity of the velocity distribution on the/th x+=cos_ for i---I ..... N (! I) The root-mean-squared errors in Cp at the nod_; observed in successive redistributions of the nodes arc shown connected by solid lines• In every case shown in Fig. 4, the redistribution algorithm reduced the maximum panel error; i.e., the history or the algorithm goes from right to left in Fig. 4• Usually it also reduced the rms error in the nodal C/,, although, given the order-of-magnitude basis of the algorithm, it would be too much to expect such a reduction every time. Indeed, Ibe initial result in the 16-panel case was somewhat better than the final results, at least according to the measure of rms error in Cp at the nodes• However, with higher numbers of nodes, the algorithm improved the nodal C,_ by factors of ]-5. (AI least part of the worsening of the rms error with redistribution of 16 nodes is due to the fact that the initial distribution did not concentrate so many.points near the stagnation points, where Cp is more rapidly varying.) Because the algorithm cannot be guaranteed to optimize the node distribution, its efficient utilization required some experimentation• Since the objective is to reduce the maximum panel error by smoothing out variations in the panel error from one panel to another, the redistribution process is generally terminated when the ratio of the maximum panel error to the minimum is less than some fixed number on every network. Our experience shows little im- provement in the solutions when this max/rain ratio is reduced below about 4•0. This is illustrated in Fig. 4, in which Ihe itcralions were pt_rsuedin the 32-panel case until the max/min ratio was under 1.2. Points for which the ratio was under 4.0 are flagged. Note that the iterations do appear to be converging (though no! necessarily to an "optimum" solution so far as the rms error in nodal Cp is concerned). This has been the case in all our calculatinns, at least when we use, as a §Such a group of nodes, with fixed starting and ending points, is nailed a ne/u,'or_ in what follows. For example, separaec networks are used for a wing's upper and lower surfaces, with their leading and trailing edges being fixed in the node redistribution process. panel is measured by a v+- P,- _ ( v,+ v,+,) (_) in which V, V/++ are the (now known) velocities at the end points of the ;th panel• Now the contribution of any panel to any quantity of in- terest (the potential or velocity, for example) at any field point is calculated :)y integrating over the panel the products of the source and vortex strengths with appropriate kernel func- tions, or, what is the same thing, the dot product of the velocity distribution on the panel with some vector kernel K• The error incurred in this calculation due to the nonlinearity of the vclocity distribution on the panel is, therefore, I,h,.. av+-Kds-t,av+ (8) whele t+ is the length of the/th panel. This product, f,_V,, is called the panel error function. Again, note that it can be calculated for each panel a posteriori, once the V,+ have been determined, whether or not the exact solution is known• From calculations of flows past ellipses, Joukowski ain'- t foils, and Karman-Trefftz airfoils, using various numbers andi distributions of nodes, we found the root-mean-squared errdr in the pressure coefficients at the nodes to correlate fairly well with the maximum value of the panel error functions• Th_s • i suggested a node redistribution algorithm, in which a soluuon is ohlaincd with a given set of nodes, which are then relocate so as to even out variations of the panel error function and so to reduce the maximum panel error. Since the nonlinearity of ._ the velocity distribution on the ithpanel is of order t_, • f,a V, = O(f;) (9) Thus we stretch the panels so that their new lengths t; are given by t;=et,/+_ ¢'6 (10) svtu_,,_l_l, _ULE, AND WAHl+ AIAA JOURNAl. closure condition, stagnation conditions within the panels. An alternative approach, in which the closure condition was that the velocity inside the panels equaled that of the onset flow (so that the source and vortex strengths were components of the perturbation velocity normal and tangential to the panels), yielded divergent results for the case that is the subject of Fig. 4. To avoid oscillations which otherwise were found in some cases, the node movement is under-relaxed by a constant factor when and if necessary to avoid an increase in the average node movement from one trial to the next. Even so, as will be seen later, we occasionally find that the maximum panel error increases with node redistribution. This usually indicates that further redistribution would not improve the solution significantly. Therefore, if the maximum panel error increases, we terminate the redistribution process and reject the distribution which yields the higher panel error. Calculations similar to those reported in Fig. 4 were per- formed in which the initial node distribution was random. The histories of the rms error were quite similar to those shown in Fig. 4, except that the initial errors were much higher. In three trials with different initial distributions of 32 nodes, the rms error in the nodal Cp decreased during the node redistribution process from 0.3-0.4 to about 0,04. While the final results obtained with the node redistribution algorithm are usually quite acceptable, the very magnitude of the improvement it brings is evidence that the subsurface panel method is extremely sensitive to the node distribution. This is brought home still more graphically by Fig. 5, which shows the history of the pressure distributions ca|culated at various stages of the node redistribution process. In that case, the ellipse initially exhibited a lift coefficient of about 1.5! Studies of flows past a variety of other airfoils--including NACA four-digit airfoils and other formula-generated sections as well as Karman-Trefftz airfoils of varying thickness, camber, and trailing-edge angle--confirm the _i X4 Xr ,_ 0 s Za + 3 _ G _ EXACT SOLUTION 1.00 0.50 0 -O.GO i .t.O0 -1.50 I_ 2 (10 { k_ l ---.-2 J .'L00 -0.G0 -0.20 0.20 0S0 'L00 Fig. 5 Pressure distributions in successive stages of node redistribution process for ellipse of thickness ratio 0.125, starting with random distribution, 24 panels. Numbers indicate successive solutio.s. _'7 implications of Figs. 4 and 5, both as to the sensitivity of the subsurface panel method to the node distribution and the effectiveness of the node redistribution algorithm. The troubles of the basic method are not restricted to initially random distributions. The results of Figs. 6 and 7 were ob- tained starting with the "reasonable" distribution of Eq. (I I). The implausible crossing over of the pressure distribution near the trailing edge shown in Fig. 6 is seen in Fig. 7 to lead to a 25% deficiency in the calculated lift coefficient, even when 64 panels are used. A possibly related anomaly was encountered near the sharp-but-not-cusped trailing edge of the highly cambered Karman-Trefftz airfoil to which Fig. 8 refers. In both cases, the anomaly was eliminated after a single application of the node redistribution algorithm. Our final example, a multi-element airfoil, is displayed in Fig. 9 to demonstrate the prowess of the program used Io implement the method. Discussion and Conclusions The effect of node distribution on results is not often discussed in the literature. Hess _ gives a few resuhs which indicate the effect can sometimes be substantial, but not nearly to the extent observed in the present study. From private, communication with several of the authors of Ihe present paper's references, and some limited experiments of our own with the classical constant-singularity-strength flat- panel method, 4 the subsurface panel method seems to be unusually sensitive to node distribution. The cause of our oversensitivity to node distribution is not known. It is not the case, as has been suggested, that wc are not satisfying the Kutta condition; the velocities we calculate are properly directed near the trailing edge. Nor does it seem to have to do with the O (sl.s} behavior of the velocity near the nodes. Similar results were obtained when we used cubic distributions of sources and vortices to reduce the singularity ==:zm-.------- Fig. 6 Pressure distributions on simple uncambered airfoil wilh thickness proporliona! to ',_(l-x) _, angle of allack=5 deg. Triangles oblained wilh cosine node distribution, circles after 4 node redislribplions, 32 pands. MAY 1980 INCOMPRESSIBLE FLOWS BY A SUBSURFACE PANEL MEIHOD 531 1.4 1.2-- _ _o% THICKNESS _= $" _ O4 MAXIMUM PANEL ERRO_ r 1 I J L t0.4 10-3 tO 2 Fig. 7 Ilish)ry of lift coeffieienl during node rediseribulion process for airfoil .f Fig. 6. N_des initiall) dislrihuted by cosine formula Eq. (i I). Triangle_,, 32 panels; squares, 64 panels. x 15 14 13 1.2 1.1 1.0 I I 100tO'3 10"2 10-1 MAXIMUM PANEL ERROR Fig. 8 Hislory nf lift coefficienl during nnde redistribution prncess for highly cambered Karman-Trefflz airfoil (22070 camber, 10% Ihickness) at zero angle of attack. Nodes initially distribnled by equally spacing angular variable in circle plane. Circles, 16 panels: triangles, 32 panels; squares, 64 panels. to O(sae_s) and to make dV,,/ds vanish at the nodes (along with V. itself). On the other hand, if we compute, a posteriori, the velocity induced by the panels at points along the body surface, we find that our worst solutions are characterized by a large amount of leakage. In order to eliminate this problem, we tried to control the normal velocity component in a way that would account for the velocity induced by all the panels rather than just the local flow-tangency condition. Simply replacing Eq. (5) by one which would set the panel-induced normal velocity com- poneht to zero leads, as is fairly well known, to an ill- conditioned matrix and to oscillations in the velocity from node to node. Similar results were obtained when we sought to minimize, with respect to the tangential velocity components V,_, the integral along the inside of the panels of either the square of the velocity component normal to the panels or the square of the magnitude of the velocity. An attempt to make the potential constant inside the panels also led to an ill- conditioned matrix. As would be expected our parameterizing the source and vortex strengths with derivatives of the potential diminished the diagonal dominance of the matrix, enough to give oscillatory answers. Less well understood are the failures of the results to improve when the equation set was amplified by the addition of requirements that the net 1.60 T 0,8 _ _ _, (_ ! ... ", _5:. r-, ,4.3" T _4d_ 160 * 240_ _', 1 2 -:120 _ ] I ,ooI ....... _....... __.......... ,__ , • 0 0.20 0 40 0 60 0.8_ 1 O0 1.20 140 Fig. 9 Pressure distrih.lions .. tv,,_-clcment airfoil c.n_isling of NACA 0012 sections: mP, e of flap I% of wing chord I)clo_ _ing Irailing edge; flap chord 30"0 of v, ing el)(Jrd; flap deflection 10 (h'g. angle of allat'k 3 (leg. Ni,ml)ers indicate s.c¢cssi_,e stagc_ (_f rcdislrihulion process. source strength vanish, or that the integral of a4_/#n along the inside of the body surface vanish (which is required Io make the solution of the internal problem unique when its bound:_ry condition is that the tangential velocity vanish on the inside of the body surface). Late in the course of the research reported herein, it _sas recognized that the parameterization of the source and vortc',: strengths in terms of the nodal velocity components as described by Eqs. (2-4) was not unique in so far as obtaining a continuous velocity field is concerned. As noted in the Ap- pendix, the V,, and 1"t in Eq. (3) can be replaced by arbitrary ./ J . , functions of ttie node undtces wnthout altering our conclusions so far as continuity is concerned. In particular, we can devise a continuous version of the Douglas-Neumann method a by replacing the I/, by say, a constant F and the V._ by variables .j. - S/, and determmmg those unknowns so as to make the panel- induced normal velocity component cancel the normal component of the onset flow at every node, and to satisfy ,a Kutta condition (such as equality of tangential velocity components at the trailing edge). The results we obtained with this method were even worse than those reported herein possibly because of reduced diagonal dominance in the coefficient matrix. As shown in the Appendix, the diagonal coefficient of the matrix associated with Eq. (5) (the coef- ficient of I/,i) is -l+l/2x(_i_t +v i) in which (_,__ +v i) is the angle subtended at the ith node by the (i- l)th and (i+ l)th nodes, an angle that is generally less than _r, much less so at the leading- and trailing-edge nodes. In our subsurface version of the Douglas method, the corresponding coefficient is simply (v,_/ +vi) which can be relatively small. Kemp -_ has devised a panel method very similar to the present one, in that it yields a continuous body-surface velocity distribution. His "loading vector" is essentially the nodal tangential velocity I/,_ of the present method, and he sets the nodal normal velocity component 1/,,_ in accordance with the local flow tangency condition, as do we. A major I41, difference is that Kemp determines the V,, so that the velocity induced by the panels is normal to the panels at their nrid- points. Following publication of his report, wc performed some limited experiments with Kemp's melhod. For sym- metric flows past ellipses, we found it considerably more accurate than the present method. However, il seems to share our sensitivity to node distribt, tion, yielding spurious results when the nodes were distributed randomly on the ellipse, and for an airfoil with a cuspcd trailing edge. Our node redistribution algorithm did not help in such cases. Of course, it could be that our implementation of Kemp's rnelhod ,.,,'as in error. In any case, satisfaction of boundary conditions at panel midpoints, while no problem in two dimensions, complicates the trealment of three-dimensional problems. As wilt be discussed in the cpiloguc which tk_llows, use of the nodes as the sole control points leads to a very attractive singularity spline. It is hoped that the evidence prescnted ot_ the utility of the node redistribution scheme is convincing. In every case we have tried, it has enabled us Io overcome the sensitivity of the method to the starting node distribution. Except when we play with random distributions, five or fewer iterations are suf- ficient to reach our criteria for terminating the redistribution process. Of course, this means that the method may require as much as six times the computation time of other methods for the same number of nodes, without any improvement in accuracy relative to higher-order methods like those of Hess, Johnson and Rubbert," and Rristow. _ Therefore, the method is of practical interest only as a prototype of a method for three-dimensional flows. Epilog Whatever their worth for two-dimensional flows, panel methods are ntuch more interesting in three-dimensional situations, where their ability to treat complicated geometries far exceeds that of currently available field methods. A three- dimensional version of the subsurface panel method described herein would enhance these advantagc_ considerably. Wc have such a version in the debugging stage and enumerate here some of its features, since it may be of interest to know how much of the existing two-dimensional analysis which is the subject of this papcr can bc carried over to the three- dimensional case. The method is based on distributions of sources and vor- tices (see Ward 6 for the basic equation) on plane triangular panels whose vertices are nodes on the body surface. Their strengths are components of the total fluid velocity on the outside of the panels and arc assumed Io vary linearly over the panels. The parameters of the source and vortex strengths are then the three componcnts of the body-surface velocity distribution at the nodes which are vertices of the panel. Since the velocity component normal to the body surface is known, we have two unknown singularity-strenglh parameters per node. For a given set of nodes, this gives us about twice as many unknowns as would be dealt with by other panel methods. This disadvantage may be overcome by the extremely local nature of our singularity splines; on any panel, our source and vortex strengths depend only on parameters associated with the vertices of that panel. Thus, as noted earlier, our effective mesh size may be considerably smaller than that of another panel method which t, ses the same node distribution. The use of triangular panels greatly simplifies the geometry problem. All we must know about the body are its coordinates artd the direction of its surface normal at each node. No approximation need be made for the body shape between nodes, even in calculating forces and moments. To be sure, we do require continuity of the body surface velocity, which excludes from current consideration flows past wings of zero thickness and other idealized problems. The method yields a continuous body-surface velocity distribution. This will allow an estimate of the deviation from tat I :it/at J', )U K _',_,_ linearity of the veloci'y distribution on the panels, just as . the two-dimensional case...M this point, v,e do not know wh, would be the optima) way tc, use such informalicm Io imprm the node diszriln,tion, but thai remains v.cll wilhin Ih¢ reah of possibility. For example, we could use (lac pre_c_ algorithm directly to redistribute the nodes ira a given nv and/or column. If the number of iterations required to cir cttmvent the sensitivity of this method to node distributi-v which musl be assumed to be as bad in the t_o-dimcn',inm case, can be kept to the five or fe,.,,,cr required in 1he Iv,, dimensional case, the Hmptification of the singtdariiy splint. and elimi,lation of lh,: geometry problem should mgtke rh subsurface approach quite comf)etilive in lhrce-dimem;iom. situations. Appendix Let Vj(x,y;O) bc the contribution of the jih panel Io tiv component of velocity in the direction 0 at P(x.v). r; evaluate t',. it is helpful to introduce local c(w, rdinme- oriented with thcjih panel: .v" = (.v-x,)cosB., + (y-I'+)sini3, y" = (v -y, )cost_ - (x-x,)singT, (\1_ Then, if u* and t '° are the velocity components at I _along the (x', y" ) axes, V, = u°cosu: - Bs ) + i," sin U_- B, ) IA 21 Dift'ercntiating Eq. (1) and laking the source and ,.c,rle'_ strengths to follow the linear laws of Eqs. (2"1, we find I,,'(x',y')" 1 _,,o t,o ,,, ,', 1= 1 'v t,'(x',y') L Vo-Un rs-ul ,I ") ri O (A3) where and "'3r "" .o (x',v')=. 5-,7- _ ---t', s' ---i' /' ] X* v* ,,, - r,.r,+-g,s, (") '7t' o (x',y °) = 1 - 7 f, - -/',"11 x" v* ?r, (A4; fF = ( l127r)(.(rs+ tlr s) (A51 ,I_ = (112r) P, (A6) Here i".,.is the distance from thejth node to P(x,y) and v, is the angle subtended at P by thejth panel. When P approaches either thejth or (j + I)ih node, J;. blows up and f, depends on the direction of approach. Hov,ever, if we consider the jth and (i- I)th panels i_ combination, then, near lhejth node, we find ,-/q 2_-( 15__+ Vj ) = g.,_,Io5 cos(0 - 8j ) - 0 5 _ cos(8 -/3 i_ s ) - yosin(O - 13_) + "r,.. t sin(8 - B,_ # ) I MAY1980 INCOMPRESSIBLEF OWSBYA SUBSURFACE PANEL METHOD 533 + _, 13,;.,cos(0 - _j ) + Oojsin(0 - _; ) I +J,, ;1"_,,). cos(0-_j_;) +o% sin(0-__ I)l + terms regular near thejth node (A7) Itcr¢ subscripts j and j- 1 have been put on the source and vortex strengths to indicate the panel with which they are associated; recall that the singularity strengths are not con- linuous from one panel to the next. On substitutiag for the o's and 3,'s in Eq. (A7) from Eq. (3), wc find that the logarithmically singular terms cancel one another, while the remaining terms reduce to 1," _ / + I,_ = 1/27r( _j + _,j_ t ) [ V,,j si n (0- 0, ) + 1"9co s(O -0j )] + terms regular near thejth node (AS) While _,, and v ; are indeterminate as P approaches the jth node, their sum approaches the angle subtended at the jth node by the (j- Ijth and (j+ I)th nodes. Thus the velocity due to the linear singularity distributions Eqs. (2), with parameters givcn by Eq. (3), is continuous on and outside the pancl_,. Neat Ihc nodes, it has an sfi,s-typc bchavior. It is easy to show that these same results would obtain if V,,, and V,, in Eq. (3) were replaced (consistently) by arbitrary functions of the node indicator i. lo apply these results to Eq. (5), simply let 0 in Eq. (A8) become O. Then the integral in Eq. (5) becomes 1 ,'_z':,r( _, + v, _ t ) I.',, + terms regular near the ith node lhcrcfore, in forming the coefficient matrix which governs the nodal tangential velocity components, we simply set t,tr: and vj equal to zero whenever the), lhreaten to become singular and indeterminate, respectively, and compensate by adding (v, +J',-i )/2r to the coefficient of V,,. Acknowledgmeni This research was supported in part by NASA Ames Research Center, tinder Grant NSF-2316. A large fraction of the work was completed while the first author was a guest of A. R. Scebass, first at the University ol Arizona and then at the University of Washington. It was a pleasure to acknowledge many simulating conversations with Seebass and his colleagues, and also the additional support given the research during that phase of it by the Office of Naval Research, under Contract N00014-76-0182. References I Itess, J. L.. "Higher ()rdcr Numerical 5olutions of the Integral Equation for the "l'wo-Dimensional Neumann I'roblem," Com- putaticmul Methods in Applied Mechanics and Engineering, Vol. 2, 1973, pp. 1-15. -'Johnson, F. T. and Rubbert, P. E., "'Advanced Panel-type In- fhlcnce Coefficient Methods Applied to Subsonic Flows," AIAA Paper 7%50. 1975. _Bristow, It. R., "Recent Improvements in Surface Singularity Methods for tile FlowficM Analysis about T_xo.Dimensional Air- foils, °' AIAA Paper 77-641, 1977. 'lltc'_s, J. L., "The I'roblem of Thr,:e-Dimensior|al Lifting I'otcntial Flow and its Solution by Means of Surface Singularity Distribution," COml_Uttttional Method_ in . tpplied Mechanics arid k)tgitu'ering, Vol. 4. 1974, pp. 2,_3-319. • _Kemp, W. B. Jr., "A Vcctor-('onlinuot:s l.oading Concept for Aerodynamic Panel Method,,," NASA TM 80104, 1979. 6Ward, (5. N., Linearized Theor.v of ttigh-Speed I"lo w, Cambridge University Press, 1955, pp. 40-42. From the AIAA Progress in Astronautics and A eronautics Series... TURBULENT COMBUSTION v. 58 l:'dited by Lawrence A. Kennedy, State Unit,ersity oJ.New York at BuJ]ulo Practical con_buslion systems arc almost all based on turbulent combustior|, as distinct from the more elementary proce'_',e,, (more academically appealingl of larninar or even stationary combuslion. A practical combustor, whether emploved in a pox_ cr generating plant, in an automobile engine, in an aircraft jet engine, or whatever, requires a large and l asl u|a_,s fb3v, or throughput in order to rueet useful specifications. The impetus for the study of turbulent combustion is therefore >,t r (._tl_. In ,,pile _.,I this. our understanding of turbulent cornbustiort processes, thai is, more specifically the interplay of fast oxid,uh.c chertileal reactions, strong lransporl fluxes of heal and rnass, and intense fhtid-mechaniczl ttlrbulence, is still incomplelc. In Ihe last few years, two strong forces have emerged that now compel research scientists to attack the subject of turbulent combustion anew. One is the development of novel instrumental techniques thai permit rather precise nonimrttsive measurcrnent of rcaclant concenlrations, turbulent velocily fluctttalior|s, temperatures, etc., generally by optical means using laser beams. The other is the cornpelling demarKI Io soh'e hitherto bypassed problems such as iden- tifyin_ the nlechanisms responsible for the production of the ntinor compounds labeled pollutants and discovering ways to reduce such emissions. Fhis nev. climate of research in turbulent combustion and the availability of new results led to the Symposium from which this book is derived. Anyone interested in the modern science of combustion will find this book a rewarding source o f it11't_rmat ion. 485 pp., 6 x 9, illus. $20. O0 Mere. $35. O0 List [0 ORDER WRITE: Publications Dept., AIAA, 1290 Avenue of the Americas, New York, N. Y. 10019 4nn. Rev. Fluid Mech. 1980. 12 : 181-... Copyri#ht _ 1980 by Annual Ret:iews Inc. All ri_lht5 reserved TRANSONIC FLOW PAST OSCILLATING AIRFOILS l x8157 H. Tijdeman Nationaal Lucht-en Ruimtevaartlaboratorium, Amsterdam, The Netherlands R. Seebass Departments of Aerospace and Mechanical Engineering, and Mathematics, University of Arizona, Tucson, Arizona 85721 INTRODUCTION Under certain conditions structures like airplane wings and tail surfaces may experience vibrations of an unstable nature. This phenomenon, called "flutter," is an aeroelastic phenomenon governed by the interaction of the elastic and inertial forces of the structure with the unsteady aero- dynamic forces generated by the oscillatory motion of the structure itself. In general, two or more vibration modes are involved, e.g. bending and torsional vibrations of a wing, which, under the influence of the unsteady aerodynamic forces, interact with each other in such a way that _he vibrating structure extracts energy from the passing airstream. This leads to a progressive increase in the amplitude of vibration and may end with the disintegration of the structure. For a given wing structure the aerodynamic forces increase rapidly with flight speed, while the elastic and inertial forces remain essentially unchanged. Normally there is a critical flight speed, the "flutter speed," above which flutter occurs. Because of the potentially disastrous character of this phenomenon, flutter speeds of aircraft must be well outside their flight envelope. In many cases this requirement is the determining factor in the construction of wings and tail surfaces. Because the vibration characteristics of the structure at zero airspeed can be determined ac- curately by current numerical methods or by ground vibration tests, the i The authors thank the NLR. the AFOSR, and the ONR for their suppo_'t of this review and related studies. 181 0066-4189/80/0115-0181 $01.00 182 TIJDEMAN & SEEBASS accuracy of the flutter prediction depends mainly on the knowledge of the unsteady aerodynamic forces. In subsonic and supersonic flight unsteady aerodynamic forces can be predicted reasonably well by theoretical and numerical means. For tran- sonic flight, with its mixed subsonic-supersonic flow patterns, prediction methods are less advanced. The current practice for wings of general planform is still rather arbitrary, with interpolations and extrapolations being made on the basis of calculated airloads for pure subsonic and supersonic flow. And, in many cases, one must resort to very expensive wind tunnel experiments. Currently there is a renewed interest in transonic flight for both military and civil aircraft. For military aircraft this stems from the need for a new generation of air combat aircraft, like the F-16 and F-17, which require an optimal maneuverability at transonic speeds. In civil aviation there is a great need for more efficient aircraft, made possible by techno- logical advances that include the so-called supercritical wing. Such wings make it possible to cruise at transonic speeds without the usual drag penalty associated with the presence of shock waves. This is achieved by shaping the wing geometry in such a way that the transition from local flow regions with supersonic flow to the adjacent subsonic regions does not take place with strong shock waves as it does on the conventional- type wings, but with only very weak shock waves or even without them. In the present review we describe the nature of transonic flows past oscillating airfoils and discuss recent developments in unsteady transonic flow calculations. We place emphasis on plane flows because most of the published studies deal with this type of flow. The first section starts with a general description of the flow past airfoils. Experimental results are then reviewed and used to illustrate the interaction between the steady and unsteady flow fields, the periodic motion of shock waves, and the effects of frequency and amplitude of oscillation. In the subsequent section we discuss the inviscid equations forming the basis of the various theoretical methods and review techniques for their solution, all essentially numerical. Viscous effects and calculation methods are then described. Finally, in the last section we assess the present status of the field and the future develop- ments expected. The reader should be aware of four related reviews. Landahl (1976) reviews the unsteady aspects of transonic flow, while a recent review of unsteady fluid dynamics by McCroskey (1977) also has informative sections on unsteady transonic flow and unsteady boundary layers. Computational aspects of steady and unsteady transonic flows are re- viewed by Ballhaus (1978) and the dissertation of Tijdeman (1977) contains a more complete discussion of many of the topics addressed here. TRANSONIC FLOW PAST OSCILLATING AIRFOILS 183 FLOW PAST AIRFOILS Steady Flow A brief survey of the behavior of steady transonic flows past airfoils provides an introduction to the discussion of the transonic flow past oscillating airfoils. When the free-stream Mach number of a purely sub- sonic flow past a symmetrical airfoil is increased, the flow pattern usually develops in the manner sketched in Figure 1. The so-called critical Math number M* is reached when the maximum local Mach number in the flow becomes unity. Beyond the critical Mach number a supersonic region appears on the airfoil, which, in general, is terminated by a nearly normal shock wave through which the flow speed is reduced from supersonic to subsonic. With a further increase of the free-stream Mach number, the shock wave moves aft and the size of the supersonic region and the shock strength both increase. After the pressure jump through the shock wave has become sufficiently large, so-called shock-induced separation M.=I Mx ": M* C_ c SUBCRITICAL FLOW M= 1 SONIC LINE I I M*< M,,: < lM'cIIM_I M_ 1 I:_ (z_ -- SUPERCRITICAL FLOW © / M"cl / / \ \ \ M=I f i M>I M* .,.) c" c-..,I <21 INCIDENCE X/c= .04 .19 .34 .40 .46 .64 .80 1 -"- _- ) ) ) _ _"" ( ¢__,__ c ) .....-_ " ,_ D _' ¢ _..._ •,-...._( :_ ( _...+...- ., ( _ (...,__. ¢ UNSTEADY PRESSURES LIFT MOMENT TIME Figure 5 Unsteady pressure signals and overall loads on an oscillating airfoil: (a) subsonic flow ; (b) transonic flow. TRANSONIC FLOW PAST OSCILLATING AIRFOILS 189 similarity of the flow past an oscillating full-scale airfoil and its wind- tunnel model representation it is required that the reduced frequencies be the same. Figure 5a shows the time histories of the local pressures and the resulting lift and moment on an airfoil performing oscillations in pitch in a subsonic flow. Both the pressures and the overall loads show almost sinusoidal variations around their mean values. In this case the pressures and loads may be described by the first harmonic of a Fourier series, viz., P = Ps + Ap' cos tot + Ap" sin _ot = Ps + [Ap[ cos (cot - qg) where p and ps denote the local and mean pressures, while Ap' and Aft' are the components of the fundamental. The coefficient Ap' can be interpreted as the actual pressure perturbation at the instant the oscillating airfoil reaches its maximum deflection, while Aft' represents the pressure per- turbation at the instant the airfoil passes its midposition. This way of describing unsteady pressures or loads is valid only if the aerodynamic quantities vary sinusoidally in time, or, in other words, as long as a linear relationship exists between the displacement of the airfoil and the unsteady pressures. This is usually the case for moderately sub- sonic and supersonic flow, at least as long as the flow remains attached. For transonic flow, however, this is no longer true, particularly in the region of a shock wave, as illustrated in Figure 5b. In such cases one has to give the complete time history of the signals or to add higher harmonics to the Fourier series. Nonlinear Character of Unsteady Transonic Flow The combined influence of airfoil thickness, incidence, and amplitude of oscillation is different for moderately subsonic and supersonic flow than for transonic flow. For subsonic and supersonic flow both the equations and the corresponding boundary conditions can usually be linearized. This implies that the problem of an oscillating airfoil can be decomposed into a steady problem (thickness + incidence) and the unsteady problem of an infinitely thin plate oscillating in a uniform flow. The unsteady-flow problem can be treated independently of the steady-flow problem. The main parameters for the unsteady flow then are the reduced frequency k, the free-stream Mach number M_o, and the mode of vibration. For transonic flows at low to moderate reduced frequencies the equa- tions governing the motion cannot be linearized. This implies that the unsteady flow field can no longer be treated independently of the steady flow field. For the aeroelastician this means a considerable complication. In addition to the aforementioned parameters for subsonic and supersonic flow, he has to consider also the mean steady flow field around the airfoil, J_ 190 TIJDEMAN & SEEBASS which is determined by the geometry of the airfoil and its mean incidence with respect to the oncoming flow. For a normal flutter investigation in subsonic and supersonic flow, unsteady airloads already have to be computed for 50 to 100 combinations of reduced frequency, free-stream Mach number, and vibration mode. For the transonic flow regime this number increases considerably because then the computations have to be performed for different values of the incidence. The complication becomes even worse if it is not possible to linearize the unsteady transonic-flow problem by assuming the unsteady flow to be a small perturbation super- imposed upon a given mean steady flow field. Then the unsteady airloads are no longer linear functions of the amplitude of motion, which implies that in aeroelastic calculations, where the unsteady aerodynamic forces have to be combined with inertial, stiffness, and damping forces of the aircraft structure, linear systems of equations no longer apply. So it is quite evident that from the practical point of view there is a strong demand for some sort of linearization. Of course, in theory, this linearization can always be enforced by making the amplitude of oscillation small enough, but the question arises whether the amplitudes that occur in practice will be that small. OBSERVATIONS FROM EXPERIMENTS Results Available The first transonic-flutter accidents occurred during World War II with aircraft of advanced design at that time that were able to penetrate the transonic regime during a diving flight. These accidents gave the transonic regime its veil of mysticism and contributed to the many myths about the difficulties associated with crossing the "sound barrier." At that time it was impossible to get aerodynamic data for the transonic speed range because there were no transonic wind tunnels and there was little or no support from theory. During the first fifteen years after the war, the knowledge of transonic flows improved considerably through experience gained with research aircraft, like the Bell X-l, and the development of transonic wind tunnels with slotted and porous walls. The latter greatly enlarged the possibilities for obtaining aerodynamic data under controlled conditions. From that period stem a number of unsteady aerodynamic load measurements on oscillating wind tunnel models. The majority of these experiments had an ad hoc character and were directly related to problems encountered in flight. The first measurements of local unsteady pressures on an oscillating wind tunnel model in transonic flow were made by Erickson & Robinson (1948). Their method, in which electrical pressure cells installed flush TRANSONIC FLOW PAST OSCILLATING AIRFOILS 191 with the model surface are used, was applied successfully by Wyss & Sorenson (1951) and their colleagues at the then NACA. Although they actually measured the pressures on oscillating control surfaces and on airfoils and wings oscillating in pitch, only overall aerodynamic coeffi- cients were published, along with some typical oscillograph records of local pressure fluctuations. The first detailed unsteady pressure distribu- tions in the transonic regime were reported by Lessing, Troutman & Meness (1960) and by Leadbetter, Clevenson & Igoe (1960). A series of exploratory wind tunnel investigations on some characteristic airfoil sections was initiated by the NLR in the late sixties. With the aid of a special technique in which scanning valves and pressure tubes were used (Bergh 1965), steady and unsteady pressures were measured on the conventional NACA 64A006 airfoil with oscillating trailing-edge flap, on the shock-free NLR 7301 airfoil oscillating in pitch (Tijdeman & Bergh 1967, Tijdeman & Schippers 1973, Tijdeman 1977), and on the same airfoil with an oscillating control surface (Schippers 1978). Parallel with this basic research program, unsteady pressure distributions were measured on a variety of three-dimensional wings, under contract with aircraft industries (see, for example, Bergh, Tijdeman & Zwaan 1970, Tijdeman 1976). About the same time Triebstein (1969, 1972) performed experiments on a rectangular wing with and without control surface. In his tests he also applied the measuring technique with tubes and scanning valves. In France, successful tests on a supercritical airfoil section with control surface were reported by Grenon & Thers (1977), who used a large number of miniature pressure sensors placed inside the model. Recently, a similar technique was applied by Davis & Malcolm (1979), who performed exploratory tests on the conventional NACA 64A010 airfoil and the NLR 7301 airfoil of Figure 3. In the NASA experiments a sophisticated test rig that makes it possible to drive the models into pitch as well as plunge motions was used; moreover, the tests could be performed at large Reynolds numbers. Finally, a preliminary series of measurements on an NACA 64A010 airfoil oscillating in pitch has been reported (Davis 1979). Interaction Between the Steady and Unsteady Flow Fields Some of the results for an airfoil with flap (Tijdeman 1977) will be recalled here to demonstrate the mechanism of the interaction between the steady and unsteady flow fields. This example is chosen because it lends itself well to physical interpretation. Low-speed steady and unsteady pressure distributions on the sym- metrical NACA 64A006 airfoil with flap are shown in Figure 6-1. Figure 6-1a shows the steady pressure distributions along the upper surface of the airfoil for three flap angles, viz., - 1.5, 0, and 1.5 degrees, respectively. q/ 192 TIJDEMAN & SEEBASS From these steady distributions, the "quasi-steady" pressure differences, AC_ = Cp(6o - A_)- Cp(6o + A_) 2A& are determined, and the resulting chordwise distribution is shown in Figure 6-1b. This quasi-steady pressure distribution can be interpreted as the "unsteady" pressure distribution when the oscillations are infinitely slow. Figure 6-1c depicts the first harmonic of the unsteady pressure distribution for a frequency of 120 Hz and an amplitude of about one degree. The unsteady pressure distribution very much resembles the quasi- steady distribution; both show the characteristic peaks at the leading edge of the airfoil and at the hinge axis at 75% chord. For the unsteady example the results of "thin-airfoil theory," assuming an infinitely thin - 0.50 Cp -0.25 . _ j/_ 0 o .oX/c NACA 64 A 006 M® = 0.S0 _Cp 4 2 .0 x/c a: STEADY b: QUASI-STEADY ; k- 0 Figure 6-1 MAGNITUDE IACpl 6 PHASE ANGLE 1_' 0 /_:_l.0X/c _" THIN-AIRFOIL \--THEORY -ioo_ C: UNSTEADY ; f =120 Hz ; k = 0.392 Steady, quasi-steady, and unsteady pressure distributions in low subsonic flow. q_ &TRANSONIC FLOW PAST OSCILLATING AIRFOILS 193 wing in a uniform main flow, are also given. At this low speed, the agreement with the experimental amplitude and phase distributions is satisfactory, which indicates that the unsteady-flow problem can be treated independently of the steady flow pattern around the airfoil. Results for the same configuration in high subsonic flow are shown in Figure 6-2. Around the 50_o chord point, where the flow is almost critical, a bulge occurs in the magnitude of the measured distribution of both the quasi-steady and unsteady pressures. This bulge and the phase variation are not predicted by the thin-airfoil theory. Another characteristic feature is that the phase lag on the front part of the airfoil is consistently larger than that given by the theory. Cp -0.50 -0.25 NACA 64A006 Mao = 0.825 Cp ¢/ Ak_ .,,_,o 2 1.0 c 0 8: STEADY /%Cp 1.0 b:QUASI - STEADY ; k = 0 Figure 6-2 MAGNIIUDE JA CpJj_j_ 1.0 PHASE ANGLE 1000 olT _9."_ HIN - AIRFOIL _ "_'_ I THEORY - 100°7 _ 200o C: UNSTEADY; f=120 Hz; k=0.248 Steady, quasi-steady, and unsteady pressure distributions in high subsonic flow. "7",_' 194 TIJDEMAN & SEEBASS Results that are typical for a transonic flow with a nearly normal shock wave are given in Figure 6-3. It is clear from this pressure distri- bution that a change in flap angle is followed by a shift in shock position, and this leads to a peak in the magnitude of the quasi-steady and unsteady pressures in the vicinity of the shock• This peak, which is a significant contribution to the overall unsteady lift and moment, cannot, of course, be predicted by a fully linear theory. Note that the pressure perturbations in front of the shock wave are smaller than those predicted by theory, Cp -0.50 - 0.25 NACA 64A006 Moo =0.875 I 4_6=1.5 ° o° ._____Llll Cp" 2 1.0 _Cp L0 b:QUASI-STEADY; k =0 MAGNITUDE J_cpJ 4 THIN - AIRFOIL _t-_ ..4" _ ° _0o''_ 1.0 PHASE ANGLE 1000 0 THIN-AIRFOIL _ 1.0 __oo°- /// / // - 200°L_ C: UNSTEADY; f = 120 Hz; k = 0.234 Figure 6-3 Steady, quasi-steady, and unsteady pressure distributions in slightly supercritical flow. fTRANSONIC FLOW PAST OSCILLATING AIRFOILS 105 and that the measured phase variation shows a sharp change in gradient in the region of the shock wave. To illustrate that the observed effects in the preceding examples for high subsonic and transonic flow are caused by the interaction of the steady and unsteady flow field, a graphical experiment has been performed. A pulsating pressure disturbance is assumed to be located at the airfoil's hinge axis. Acoustic waves propagate from this disturbance into the sur- rounding nonuniform flow. The acoustic wave patterns, as obtained with the well-known construction of Huygens for the airfoil under considera- tion, are shown in Figure 7. This figure displays the position of the wave- fronts after equal time intervals At for two different Mach numbers. The part of the figure above the airfoil depicts the time histories of the wave- fronts in the actual flow field. Below the airfoil the same wavefronts are shown, but now for a steady uniform flow field in which the local Mach number everywhere is equal to the free-stream Mach number. The corre- sponding travel times (time lags) are given in the diagrams at the bottom of Figure 7. At M = 0.8 the flow is subcritical and the upstream-moving wavefronts encounter more "head wind" in the actual flow than in the NON-UNIFORM FLOW FIELD .... UNIFORM FLOW FIELD Moo = 0.80 Mm = 0.875 4 ! !/__SO 10 8 2 t ESTIMATED. \ At SONIC LINE \\,_\_III / / URCE 112_/_1J//_ SOURCE ' I i 10.75 1.0 l It l, \ 1 _ li0111 i i I I Ii I I I I _ \_. I I ' I I,\ -A, I I 0.75 \\ _ t 2 0 0.75 "LY 3E 10-3 TIME LAG (sec) a 4 1.0 0 0.75 1.0 i TIME LAG (sec) Figure 7 Upstream propagation of wavefronts generated by a source at the hinge axis: (a) wave propagation;(b) time lag derived from wave pattern. 196 TIJDEMAN & SEEBASS X sh FREQUENCY 120Hz =0 ° FLAP POSITION 1o o F"----X ' / J \ J /-'--"_ up/\. / .. 1o_ _ TIME lt,,,i) SHOCK POSITION Xshf c .8 I Moo = 0.90o,,,,.o" "°" "-o.. ,.o. o,. .o. ,o" "°" "o- .,o...o. .6 "°_ _"o" "P" "°'"°"(7"° TYPE A Ma0 = 0.875 0-.,0.. ,0_ 0.- ,-0-..0, _ "0,,0 " 'b" _C:K"x::k /SHOCK "O,, ,.O. aOI DISAPPEARS "(3.-.13... O FORMATION OF A NEW SHOCK _/_b _x AIRFOIL AND _\ PROPAGATES \ UPSTREAM Figure 8 Observed types of periodic shock-wave motion. 9_ TYPE B TYPE C TRANSONIC FLOW PAST OSCILLATING AIRFOILS 197 uniform flow, as can be seen from the closer spacing of the fronts and from the time-lag curves. Moreover, the velocity gradients normal to the chord in the actual flow cause a forward inclination of the wavefronts. When it is recognized that the spacing of the wavefronts is a measure of the intensity of the local pressure perturbation gradient, while the time lag is a measure for the phase shift, it becomes clear that the high subsonic effects observed in Figure 6-2 can be attributed mainly to the influence of the nonuniform steady flow field. At M = 0.875, when a supersonic region is present and terminated by a shock wave, the inclination of the wavefronts is essential to enable the waves to penetrate the region of supersonic flow. Some portions of the upstream-moving wavefronts close to the airfoil surface merge into the shock while other portions bend around the top of the shock and penetrate the supersonic region. This is reflected in the time-lag curve. Since the energy content of the. wavefronts penetrating the supersonic region has decreased, due to geometric dilata- tion, only small pressure changes occur in front of the shock wave. These findings correlate very well with the effects observed in the wind tunnel results presented in Figures 6-2 and 6-3. Note that the main contribution to the peak at the shock position is due to the oscillatory displacement of the shock waves, which is not included in this simple graphical result. Periodic Motion of Shock Waves From the preceding discussions it should be clear that the periodic motion of shock waves makes an important contribution to the overall unsteady airloads. Optical flow studies on an airfoil with flap (Tijdeman 1976) have shown that in the oscillating case three different types of periodic shock- wave motions can be distinguished. They have the following main characteristics, which are depicted in Figure 8. SINUSOIDAL SHOCK-WAVE MOTION (TYPE A) This type resembles, more or less, the shock-wave motions discussed by Lambourne (1958) and Nakamura (1968). The shock moves almost sinusoidally and remains present during the complete cycle of oscillation, although its strength varies. Due to the dynamic effect, phase shifts exist between the model motion and shock position and between shock strength and shock position. The maximum shock strength is not reached during the maxi- mum downstream position of the shock, as in quasi-steady flow, but during its upstream motion. INTERRUPTED SHOCK-WAVE MOTION (TYPE B) This motion is similar to Type A, but now the magnitude of the periodic change in shock strength becomes larger than the mean steady shock strength and, as a consequence, the shock wave disappears during a part of its backward motion. 198 TIJDEMAN & SEEBASS Figure 9 Periodic shock-wave motion for various frequencies. TRANSONIC FLOW PAST OSCILLATING AIRFOILS 199 UPSTREAM-PROPAGATED SHOCK WAVES (TYPE C) At slightly supercritical Mach number a third type of periodic shock-wave motion is observed, which differs completely from the preceding types. Periodically a shock wave is formed on the upper surface of the airfoil. This shock moves upstream while increasing its strength. The shock wave weakens again, but continues its upstream motion, leaves the airfoil from the leading edge, and propagates upstream into the oncoming flow as a (weak) free shock wave. This phenomenon is repeated periodically and alternates between upper and lower surface. The type of shock-wave motion that occurs in a given situation depends on the Mach number, its chordwise distribution ahead of the shock wave, and on the amplitude and frequency of the motion of the airfoil. The types of shock-wave motions mentioned above are observed not only on oscillating airfoils, but also on steady airfoils with severe flow separation downstream of the shock waves (see, for example, Finke 1975, McDevitt 1979). Under these conditions the shocks do not always remain normal, but cyclically become lambda-type shocks. Optical flow studies on an airfoil oscillating in pitch (Tijdeman 1977) reveal that the amplitude of the shock-wave motion decreases with fre- quency, as shown in Figure 9. This is consistent with the observation made in the next section dealing with the theoretical developments. As a consequence, the large pressure peaks generated by the periodic shock motion become smaller with increasing frequency, or, in other words, the contribution of the shock waves to the overall unsteady airloads, which forms one of the dominant effects in transonic flow, will be largest at low to moderate frequencies. Another interesting feature, observed in the mentioned experiments and in those of Grenon & Thers (1977), is that an almost linear relationship is found between the frequency of oscillation and the phase shift between the motion of the airfoil and the motion of the shock wave for low to moderate reduced frequencies. This means that there is a constant time lag between the motion of the airfoil and the shock-wave motion. This corresponds with the findings of Erickson & Stephenson (1947), who observed a fixed relation between the phase lag of the shock motion and the time required by a pressure impulse to travel from the trailing edge to the shock wave. In a flow pattern with a well-developed shock wave, the shock motion takes place nearly sinusoidally, with an amplitude of the shock motion that is proportional to the amplitude of the motion of the airfoil. Further, in spite of the presence of the oscillating shocks, the overall lift also varies almost sinusoidally while the moment sometimes shows irregularities (Figure 5b). Locally, underneath the foot of the moving shock wave, the _q 200 TIJDEMAN & SEEBASS pressure signal is highly nonlinear. However, as illustrated in the data of Figure 10 from Schippers (1978), the distributions of the higher harmonics of the pressure signals along the chord show the characteristic that they do not contribute noticeably to the overall lift and only slightly to the moment. This behavior can be explained very well by considering the pressure changes at such a point to be entirely due to a sinusoidal oscillation of a shock that spends part of its time on each side of the point (Tijdeman 1977). Neglecting the local pressure gradient, then, gives a field whose higher harmonics do not contribute to the net lift and only to second order in shock amplitude to the net moment. Effect of Frequency and Amplitude of Oscillation As mentioned earlier, it is important for the aeroelastician to know to what extent a linear relationship holds between unsteady airloads and NLR 7301 WITH FLAP 10 ,_c_ Mao == 0.756 k = 0.07 Figure IO \ HINGE Unsteady pressure distribution showing the first three harmonics. / t'-( TRANSONIC FLOW PAST OSCILLATING AIRFOILS 201 the amplitude of motion of the airfoil. For his purpose, linearization of the unsteady airloads makes sense only as long as small, but still practical changes in incidence or flap angle (say of the order of 0.5 and 1 degree, respectively) give rise to linear changes in the aerodynamic loads. Also, he must have a procedure for selecting a minimum number of suitable mean steady flow conditions for which the unsteady airloads should be determined. At this moment the experimental and theoretical evidence is not yet conclusive in this respect, but there are a number of observations that certainly throw some light on the problem. Theoretical analysis (see the next section) shows that the unsteady transonic flow problem becomes a linear one for sufficiently high fre- quencies. From this, coupled with the observation that the amplitudes of periodic shock-wave motions are largest at low frequency, we can expect that the nonlinear characteristics of unsteady transonic flows will manifest themselves mainly at low to moderate frequencies and in quasi-steady flow. Therefore, we can expect that the behavior of the slopes of the steady lift and moment curves versus incidence or flap angle may serve as an effective guide to detect possible nonlinear regions and to select the mean incidences or flap angles around which linearization is possible. As noted above, we may also expect that at sufficiently high frequency the measured results will approach the results of thin-airfoil theory. Un- fortunately, the experiments show that this does not happen within the frequency range of interest for flutter investigations (k < 0.5). Figure 11 compares some of the results of Davis & Malcolm (1979), whose studies cover a considerable range of frequencies, with a thin-airfoil theory. It is clear that the unsteady airloads cannot be calculated with a linear theory that does not account for the mean steady flow field. However, if the NACA 64A010 AIRFOIL Morn0. 8 ORe . 12.6 x 106 PITCH AXIS .25C ARe . 3.3 x 106 10 "_ 7.5 ..J U 5 2.5 THIN AIRFOIL THEORY E I I I .I .2 .3 .I .2 k .3 O 8 8 0 Figure I 1 Effect of frequency on unsteady airloads. /G/ 202 TIJDEMAN & SEEBASS amplitude of oscillation is small, we can expect that in most cases linearization for a practical range of amplitudes of oscillation should be possible as long as the flow remains attached. This expectation is con- firmed very well by the tests Davis and Malcolm performed on the NACA 64A010 airfoil in a flow condition with a well-developed supersonic region terminated by a relatively strong shock wave. They found a linear relation- ship between the real and imaginary parts of the lift coefficient with amplitude for 0.25, 0.50, and 1.0 degrees, and further demonstrated that the unsteady airloads for a motion that was a linear combination of two other modes were a linear combination of the airloads for those modes. THEORETICAL AND NUMERICAL METHODS In this section we discuss the theoretical and numerical methods developed in recent years to calculate the inviscid transonic flow past oscillating airfoils, stressing the conclusions we may deduce from them. The dis- cussion proceeds from the so-called Euler equations, which can be con- sidered as the most complete set of equations describing inviscid flow problems, through increasingly less complete models to the small- perturbation equation. Euler Equations Because the flows of interest here occur at high Reynolds numbers, viscous effects are, for the most part, confined to the boundary layers and wakes. For flows in which the boundary layer remains attached over most of the airfoil, the inviscid flow is the correct first approximation for most of the flow field. For inviscid flows, the conservations of mass, vector momentum, and energy give a system of four first-order partial differential equations in five scalar unknowns. These equations are usually referred to as the Euler equations. An additional equation relating the state variables is required. The system of equations is hyperbolic and quasilinear. Weak solutions to a hyperbolic system, viz., solutions with discontinuous behavior, may be found numerically if the difference equations are deduced from the conservative form of the equations, that is, from the equations in the form of a space-time divergence of a vector unknown. The physics governing the structure of shock waves is one of a balance between viscous and inertial terms involving viscous dissipation in the wave. Thus, the dif- ference schemes are usually constructed so that the truncation error is predominately dissipative rather than dispersive. In such calculations the mesh size must be small in order that the viscosity implied by the TRANSONIC FLOW PAST OSCILLATING AIRFOILS 203 truncation error be small compared with the length scale typical of flow in the vicinity of "captured" shock waves. Numerical solutions of the Euler equations have been carried out by a number of investigators. Warming, Kutler & Lomax (1973) describe an effective third-order and consequently dissipative, finite-difference scheme with minimal dispersion. Beam & Warming (1976) report com- parable results for a dispersive-dominant algorithm, with a switch from central to one-sided differencing where appropriate. The desire to compute steady flow fields originally motivated such computations. Some of the first methods used were explicit in time, time-accurate solutions being computed until a steady flow was achieved. A series of publications by Magnus & Yoshihara (1975, 1976, 1977)and Magnus (1977a, 1977b, 1978) provide many useful results for oscillating airfoils and include studies of the influence of various approximations on the accuracy of the solution. Other interesting results were reported by Beam & Warming (1974), who made a computation for small unsteady perturbations to an already established steady flow, by Laval (1975), and by Lerat & Sid6s (1977). Magnus & Yoshihara (1975) gave detailed unsteady pressure distri- butions for an airfoil oscillating in pitch; subsequently (1976) they gave analogous results for an airfoil with an oscillating flap, including here an ad hoc procedure to account for the shock-boundary-layer interaction. More recent studies were for the NLR 7301 supercritical airfoil (Magnus 1978). The results, among other things, reproduce the three types of shock motion observed experimentally. To illustrate this, Figure 12 depicts the instantaneous surface pressure distributions for pitch oscillations of an NACA 64A410 airfoil displaying a Type A shock motion. An interesting feature that appears in all numerical results is that the lift and moment vary nearly sinusoidally, despite the presence of strong shock waves. Figure 13 illustrates this behavior for an NACA 64A006 airfoil with an oscillating flap. The paper of Magnus (1977a), which summarizes his previous studies using the Euler equations and comments on errors introduced by the boundary conditions used, is of considerable interest. For the low to moderate reduced frequencies, the finite computational domain often leads to the reflection of disturbances from the far-field boundary and their interference with the flow field near the airfoil before a harmonic motion is established. In Magnus' studies the far-field data were determined from vortex and doublet singularities located somewhere near the airfoil. Another approximation invoked in these studies was that the unsteady boundary conditions were applied at the steady-state location of the oscillating airfoil. Substantial differences are found when the correct /c:.: 204 TIJDEMAN & SEEBASS boundary conditions are imposed, with the incorrect boundary conditions causing larger disturbances in the flow. Other studies, e.g. those of Lerat & Sides (1977) and Steger (1978), avoid this approximation at the computational expense of time-dependent mappings. It is clear from these studies that it is essential to impose boundary conditions on the airfoil unless the amplitude of the unsteady motion is small, and to use a computational region that applies boundary conditions that either allow --1,4 Cp NACA 64 A 410 k =0.2 M = 0.72 g 0. 0.6 1.0 0.8 Figure 12 Instantaneous pressure distributions for an NACA 64A410 airfoil pitching around midchord calculated using the Euler equations. TRANSONIC FLOW PAST OSCILLATING AIRFOILS 205 waves to pass through it, or that is sufficiently large that incorrect boundary conditions on its extremities do not cause unacceptable errors. Explicit calculations such as those discussed above require substantial central processing (CPU) time. Three or four cycles of a harmonic motion may require a comparable number of hours on a CDC 7600. Their main inefficiency occurs at the low to moderate reduced frequencies of praclical interest, where improved computational times can be achieved using implicit methods. Beam & Ballhaus (1975) report that the numerical effort per time step for the implicit solution of the Euler equations is four times that required for their explicit solution. For reduced frequencies below about 0.2, however, the larger time step allowed by an implicit calculation results in reduced computational times. The Potential Approximation The flows of primary interest here are, for the most part, irrotational, with vorticity introduced by viscous effects in the boundary layers and VARIATION IN LIFT .04 AC L .0_ 0 -.02 -,01 NACA 64A006 AIRFOIL WITH FLAP M®= 0.875 dt o . 0° THEORY ------ EXPERIMENT (1 ° HARMONIC ONLY) .0_ _CM ,01 0 -.01 -.0; VARIATION IN MOMENT /i _,_/ "_ \ I , / "_ 90° 180_V 270° 360° Figure I3 Time histories of the unsteady airloads calculated using the Euler equations. /L-s 206 TIJDEMAN & SEEBASS shock waves. In the inviscid approximation shock-free flows that are originally irrotational will remain so ; flows with shock waves will have a rotational component downstream of the shock wave. Crocco's theorem implies that in steady flow the vorticity behind the shock wave will be proportional to the cube of the change in the pressure coefficient across the shock wave multiplied by the speed of sound and divided by the vertical extent of the shock wave. For many practical situations this means that the flow behind the shock wave, and hence the overall inviscid flow, will be affected to a nonnegligible extent by the vorticity introduced by the shock wave. However, the assumption of irrotational flow makes it possible to simplify the problem to one with only a single unknown, the velocity potential. This leads to a considerable reduction in CPU time and storage requirements. Under the assumptions mentioned above, the following equation for the velocity potential, _, can be derived : 0,, + (V_)_ +½V_. V(VCl,)_ - a2720 = O, (1) where a = a_ - (_- 1)[_, + ½(w) 2- ½u 2], and the symbols have their usual dimensional meaning. This is subject to the boundary condition that the flow remains tangential to the airfoil surface: F,+(V_)- VF = 0 on F = y-fY(x)-3_'(x,t)= O. Here 6 and 3 are amplitude parameters for the steady and unsteady motions. Numerical algorithms for the steady potential equations, due principally to Jameson (1974, 1978), are now highly developed and provide reliable results for shock-free flows. Isogai (1977) has solved (1) using a non- conservative time-marching algorithm with unsteady boundary con- ditions applied at the airfoil's mean surface, a mean steady-state far-field, and an approximate vortex-wake condition. Results for a supercritical airfoil at its design condition oscillating in pitch (lsogai 1978) are shown in Figure 14. The more nearly linear behavior of the unsteady pressure distribution with an increase in frequency is clearly evident. More recently, Chipman & Jameson (1979) have developed a conserva- tive alternating-direction algorithm that uses a time-varying coordinate system to satisfy the exact boundary conditions. A conservative calculation /'d'_' ,,e TRANSONIC FLOW PAST OSCILLATING AIRFOILS _907 will capture "shock" waves that conserve mass but add momentum to the flow to balance the wave drag on the body. These will be stronger dis- continuities than Rankine-Hugoniot shock waves for a given Mach number ahead of the "shock" wave (van der Vooren & Slooff 1973). More definite comparisons of numerical solutions to (1) with those for the Euler equations are needed to determine quantitatively the adequacy of the potential approximation when shock waves appear in the flow. Isogai's computations indicate, as expected, that the results are unreliable if the cube of the pressure jump across the shock wave is not small. The calculation of one cycle of harmonic motion requires the equivalent of a minute or so of CDC 7600 CPU time; the number of cycles required to achieve harmonic results will depend on the reduced frequency, with even the lower frequencies requiring at least three cycles. The Small-Disturbance Approximation The most basic approximation in inviscid aerodynamics is that of a small disturbance. With the assumption of small disturbances, the vorticity ,_C'p 6 -1.0 I I I I X -.$ 0 .5 1.0 70-10-13 AIRFOIL c_= 0°÷ OSIN. KT M:.70 k:.l IS I0 $ M:.70 k=.3 i"--...5 -S 15 I 15 I X -LO -. 0 . 1.0 _ol °# r I FULL POTENTIAL EQUATION"-- 0 =2 ° .... PLAT PLATE THEORY (DOUBLET LATTIC METHOO) 10 i °/ ..../_ ° __C p ,,-" -S / . J 10 a I X -10 -1.0 -.S 0 .5 1.0 -1,0 i t i I l I PX -.5 0 .5 1.0 Figure 14 Unsteady pressure distributions on a 70-10-13 supercritical airfoil oscillating in pitch around the design condition calculated using the potential equations. /C' '," 208 TIJDEMAN & SEEBASS introduced by a shock wave will be small and the flow may be assumed irrotational. The least trivial balance of terms then leads to the equation - 4,,,,,= O, (21 where the perturbation velocity potential _b is defined by • = Uc[x +,_o¢d, where • is the velocity potential. Here the time has been nondimensional- ized by the circular frequency and the coordinates by the airfoil chord, and the vertical coordinate scaled by d_o.Application of the mean-surface boundary conditions requires that 6_o¢),,(t,x, O) = - 6 Y'(x) _ 6 I_ + L(x, t) J (3)-7- Z ' and hence that 3o_ = max (6,_;,k_). For k = O(1) a linear theory applies. This theory is fully developed in the monograph by Landahl (1961) and will not be discussed further here as the cases of most practical concern are those of small reduced frequency. The theoretical limit of interest here is that with k = 0(60). In practice, however, k is often large enough that contributions of this size, viz., O(k), are important, and retaining them provides a bridge to the linear theory. Neglecting the first term of(2) makes the equation parabolic and is equivalent to disregarding one of the characteristics and requiring disturbances to be propagated downstream with infinite speed. There are numerical advantages to doing this. Couston & Angelini (1978) and Houwink & van der Vooren (1979) have shown that marked improve- ments are obtained at larger values ofk when terms of order k are retained in the transport of shed vorticity, the boundary condition (3), and the pressure coefficient. In the strict limit k = 0(30), that is, for low reduced frequencies, a fully nonlinear theory applies and the conservation of mass takes the form -(k/bo)MLdA,,+{(1-M2)/bo-(7+l)M2dpx}C_xx+dp.v,. = 0, (4) a result first given by Lin, Reissner & Tsien (1948). The corresponding "shock" jump relations are /c',# TRANSONIC FLOW PAST OSCILLATING AIRFOILS t60 s- _o (),+ I)M2__ _(j_x]]2-1"-_t_y[_ 2 = O, d(_)=s -_x_/E_,] 209 (S) FLAP POSITION 0t_ _, ,byu_,L_/ c_/ -.6 Cp -.4 SHOCK POSITION XS/c .l .5 .4 NO SHOCK NO SHOCK _ Cpt -.2 _ ,xlc .02 .2 .4 .6 • _.0 --.6 Cp tot. 90° .2 .4 .6 .g",_ 1.0\ _6 NACA 64A0_6 AIRFOIL WITH FLAP tdoo-0.854 !- 90 Hz, k • 0.179 BALLHAUS & GCK)RJIAN MAGNUS & YOSHIHARA -.6 Cp -,4 -.2 0 .2 °n'lS0 o .2 .4 .6 _0 o_t.330 o NO SHOCK _t = 270o .2 .4 .6 .8_.0 r_._t- 210 ° __ . . %° b n 1 I" 1_ t _x/c .2 .4.6 _.0 Figure 15 Instantaneous pressure distributions on the upper surface, showing type-B shock motion, calculated using the Euler and the small-disturbance equations. 2.10 TIJDEMAN & SEEBASS with _-x the average value of _x, and _[_bx]]the jump in (hx, across the discontinuity. Explicit procedures for the solution of (5) are unstable unless the time step is much smaller than that required to resolve low-frequency motions. A fully implicit alternating-direction algorithm with a time-step limitation consistent with that required for accuracy has been developed by Ballhaus & Steger (1975). The far-field boundary conditions used are mean steady- state values on all but the downstream boundary where (Px = 0. This NACA 64A006 AIRFOIL ± _c,' _r 4.ol 3.0 _.,._... M oo= 0.8 t_to= 2° CL = 0.49 ..... LINEAR THEORY ------ INFINITELY THIN I THICKNESS+INCIDENCE J LTRAN 2/NLR .l..r i.lL _ .m..- ...... I I I i • 0.2 0.4 0.6 0.8 1.0 k ioI 0.2 / t Z_.--" .... 0 ,/_.f_i;" I , , S...7 0.4 0.6 0.8 l.O -1.0 -2.0 Figure 16 frequency. Unsteady lift coefficients for a pitching airfoil showing the effect of reduced //d TRANSONIC FLOW PAST OSCILLATING AIRFOILS 21 1 algorithm resolves shock waves and their motion well, provided the flow changes from supersonic to subsonic across the shock. Using this al- gorithm Ballhaus & Goorjian (1977) were able to reproduce observed features ofexperimental studies. To illustrate this, Figure 15 gives the time history of pressure distributions, which reveal a Type B shock-wave motion. Another interesting feature, shown in Figure 15, is that good agreement with results obtained with the Euler equations can be achieved. This is accomplished by introducing arbitrary powers of the Mach number into the equations and boundary conditions to "tune" the results to provide this agreement. As another example of small-disturbance theory, Figure 16 shows the two components of the lift coefficient for a pitching airfoil as a function of reduced frequency. These computations, by Houwink & van der Vooren (1979), include wake vorticity transport in the Ballhaus-Steger algorithm. They agree reasonably well with the linear theory that applies for k = O(1), but which also retains the _, term. Figure 16 clearly shows that results for the airfoil with thickness and incidence approach the results for the infinitely thin plate as the frequency increases. Calculations of this type may be useful to determine, for a given airfoil, the frequency range within which a transonic computation method should be applied. T ime-Linearization The small-disturbance equation (2) can be linearized by assuming the unsteady flow field to be a small perturbation superimposed upon a given mean steady flow field, or, in other words, _p(x, t) = CPo(X,y) + (3/,_)_(x, y, t) + o(3/,_) where 3/,_= o(1). The linearized version of(5) must be retained to account for shock motion. The steady flow field may be defined either by experimental or numerical means, providing an accurate description of the shock wave's geometry and strength. Because of the ease of the practical implementation in aeroelastic computations, time-linearization is attractive for the flutter specialists. Their main interests are the magnitude and phase of the lift and moment perturbations for the relevant modes of motion. Since the un- steady loads are supposed to be linear, they can be solved in two ways, namely, in the frequency domain or in the time domain. In the frequency domain a steady equation with the frequency as parameter has to be solved for each frequency of interest. In the time domain the time history of the aerodynamic response to a step input (indicial response) has to be III 212 TIJDEMAN & SEEBASS calculated. The results for a harmonic motion of the required frequencies then can be obtained by linear superposition, using Duhamel's integral. Solutions in the time domain have been obtained by Beam & Warming (1974), who considered small unsteady disturbances to a basic steady state defined by the Euler equations. The results compare favorably with the linear results obtainable from the theory of Heaslet, Lomax & Spreiter (1948). Noting the advantage of the indiciai approach, Beam & Warming further observe that a ramp change (i.e. linear growth for an appropriate time) can be used in place of a step change, avoiding some difficulties in the numerical computations. Ballhaus & Goorjian (1978) have used the nonlinear algorithm of Ballhaus & Steger (LTRAN2) to calculate indicial responses. They must use an amplitude that is small enough to produce linear results and yet large enough for the response to be computed correctly, a minor difficulty that is avoided by a strict time-linearization. Fung, Yu & Seebass (1978) have given a time-linearized version of the LTRAN2 algorithm that includes explicitly the effects of shock-wave motion through the time- iinearized analog of (5). Steady-state shock jumps are obtained from another modification of Ballhaus' algorithm that uses shock fitting (Yu, Seebass & Ballhaus 1978). A detailed comparison of nonlinear and time- linearized calculations (Seebass, Yu & Fung 1978) verifies that the latter is accurate when _/6 is < 10-_. If shock-wave motion is not included, the lift and moment variations are incorrect. It also appears that most indicial responses are approximately exponential for the frequency range of most interest and behave as exp (- t/r), where z is usually large, say 15. This implies that the amplitude of any response is the asymptotic change times [l+(2kr)2] -_, while the phase angle is 2kr[l+(2kr)2] -½. This behavior is consistent with the experimental observation noted earlier, viz., that the amplitude of shock-wave excursions is proportional to k- for moderate reduced frequencies, and the phase shift is proportional to k for low reduced frequencies (K.-Y. Fung, private communication). Time-linearized algorithms for the frequency domain have been de- veloped by Ehlers (1974), with subsequent studies by Weatherill, Sebastian & Ehlers (1977, 1978), by Traci, Farr & Albano (1975), and by Fritz (1978). They solve for _ using a relaxation procedure. Computations in the frequency domain have. an inherent limitation on kM2/(I-M_) that depends on the mesh size. This is a consequence of the generation of standing-wave solutions, which Weatherill, Sebastian & Ehlers (1978) have unsuccessfully tried to eliminate by various means. This restriction is a serious one. More importantly, none of these studies allows for shock- wave motion, although in principle the procedure of Fung, Yu & Seebass (1978) can be carried over to the frequency domain. i/:_ 4TRANSONIC FLOW PAST OSCILLATING AIRFOILS 213 To conclude this section on theoretical methods, a few words have to be said on the integral-equation method, since this type of approach has been highly successful for unsteady flow problems governed by linear equations. Such methods were probably the first to provide results for transonic flows with shock waves. The integral method is more suited for the time-linearized equations and, as described by Nixon (1978) for the low-frequency small-disturbance equation, shock-wave motions can be included. A disadvantage is that time-tinearized calculations require an effective definition of the steady-state solution. Because finite-difference type time-accurate methods are frequently competitive with other methods of finding the required steady-state solution, one most probably will select a time-accurate procedure to determine the steady-state solution. How- ever, in that case it is more convenient to continue the study of the unsteady response with the same method, instead of switching over to the integral-method approach. Before the recent advances in computational methods, local lineariza- tion proved to be a useful but limited tool (for a review see Spreiter & Stahara 1975; for some recent results see Dowell 1977). Better tools, namely finite-difference algorithms, are now available. Remarks on the Kutta Condition With the approximation of inviscid flow, we must impose a Kutta con- dition at the trailing edge in order to obtain a unique solution. The imposition of the Kutta condition in the form of continuity of the pressure at the trailing edge requires that neither the average velocity nor the jump in velocity be zero if the circulation is to change with time. Consequently, in the strict inviscid limit the flow must follow either the upper or the lower surface of the airfoil at the trailing edge. Which surface it follows depends upon the past history of the motion. More specifically, the rate of change of circulation, F (measured counterclockwise), is given by --_ d (qu + qt). = - _ (qu- q,), (6) where the right-hand side is to be evaluated at the trailing edge and the velocities are those sketched in Figure 17a. Though this view is satisfactory from a computational standpoint, it is rather too narrow. The Kutta condition is an idealization of the behavior for infinite Reynolds numbers. The correct picture is given by Sears (1976), who notes that for viscous flow, Equation (6) applies provided u and I are 1i,7 214 TIJDEMAN & SEEBASS a b Figure 17 Flow patterns at the trailing edge for inviscid and viscous unsteady flow. the instantaneous locations of the upper and lower points of boundary- layer separation adjusted for their motion, as shown in Figure 17b. Un- steady computations using the Euler equations normally impose some requirement equivalent to the (steady) Kutta condition, e.g. that the flow leaving the trailing edge bisect the trailing-edge angle. While this is in- consistent with results given above (Basu & Hancock 1978), the error involved in doing so is usually inconsequential, nor are time lags of consequence at the frequencies of interest here (see, for example, McCroskey 1977). Viscous Flows Viscosity determines, through the Kutta condition and the presence of a boundary layer and a wake, the basic structure of the flow past airfoils. The important parameter here is the Reynolds number, which has a significant influence on the thickness ofthe boundary layer, on the location of transition and separation points, and on the way in which the boundary layer interacts with a shock wave. A relatively simple method to determine the viscous flow past airfoils is the use of a combination of an algorithm to compute the inviscid flow field with an algorithm to compute the boundary layer. For steady, attached flows such methods are available: first, the inviscid flow field is determined; next the boundary layer is computed and the displacement thickness of the boundary layer is added to the airfoil contour. For this new airfoil the inviscid flow field is calcu- lated again, followed by a new calculation of the boundary layer and so on. An illustration of a result obtained in this way with the method of Bauer et al (1975) is given in Figure 18. This figure reveals that both the steady and the associated quasi-steady pressure distributions are signifi- cantly altered by the presence of the boundary layer and that a consider- ably improved agreement with experiment is obtained. For quasi-steady flow (Figure 18b) the effect of the boundary layer is even of the same order of magnitude as the effect of wing thickness. This indicates that reliable predictions of the unsteady airloads on actual airfoils can be obtained ;/¢ .w' 4 ud ,i g TRANSONIC FLOWPAST OSCILLATING AIRFOILS I 0 \o o o 215 216 TIJDEMAN & SEEBASS only if the boundary layer is included. Provided the flow remains un- separated, computations as shown here for quasi-steady flow are satis- factory also for unsteady flow, as was demonstrated recently by Grenon, Desoper & Sid6s (1979). Computations for steady flow that model viscous effects at the trailing edge and the shock-boundary-layer interaction have been reported by Melnik, Chow & Mead (1977); they achieve good agree- ment with experimental measurements. To take the shock-wave- boundaryqayer interaction into account the unsteady methods are still limited to the simple ad hoc procedure devised by Magnus & Yoshihara (1976). They incorporate, in a quasi-steady manner, a wedge-nosed dis- placement ramp at the foot of the shock wave, providing a qualitative improvement in inviscid results. There is much yet to be learned about both the steady and unsteady coupling of the boundary layer with the inviscid flow. The main problem areas are the modeling of the interaction between shock waves and boundary layers (Melnik, Chow & Mead 1977 have made considerable progress in this respect) and the accurate treatment of the flow past the trailing edge. Further, a better physical understanding of turbulence is essential if the more complex models are to provide acceptable results for modeling separated flows. The equations that govern the complete viscous flow are the Navier- Stokes equations for a compressible medium. Questions regarding the existence and uniqueness of the solutions to these equations, even when the medium is incompressible, are generally unanswered. From an en- gineering point of view, flows of practical interest have turbulent boundary layers, and the Reynolds-averaged form of the Navier-Stokes equations, in conjunction with suitable turbulence models, is an appropriate and necessary basic approximation. The main difficulties that arise are in determining the adequacy of turbulence models, delineating their ability to produce reliable results over a range of conditions, and providing the central processing time needed, and storage required, to effect a numerical solution of the equations. Comprehensive reviews of such calculations may be found in Peyret & Viviand (1975) and MacCormack & Lomax (1979). Algorithmic advances for the Reynolds-averaged Navier-Stokes equa- tions have recently improved computational efficiency by more than an order of magnitude. One marked improvement was the time-split method due to MacCormack (1976) that separates the equations into a hyperbolic part, which is treated explicitly with a local characteristic method, and a parabolic part, which is solved by an implicit method. The efficiency of this algorithm is sufficient to allow complex three-dimensional flows to be calculated (Hung & MacCormack 1978). Beam & Warming (1978) TRANSONIC FLOW PAST OSCILLATING AIRFOILS 217 have extended an earlier algorithm for the Euler equations to the Reynolds-averaged equations, and Steger (1978) has implemented this algorithm with unsteady grid generation and the "'thin-layer" version of these equations. The "thin-layer" approximation is essentially the boundary-layer approximation, except that the normal momentum equa- tion is retained, obviating difficulties in matching viscous and inviscid calculations (see Baldwin & Lomax 1978). In the framework of the Reynolds-averaged Navier-Stokes equations, Seegmiller, Marvin & Levy (1978) studied the flow past an 18_o thick circular-arc airfoil in a channel at Mach numbers of 0.76 and 0.79 for a Reynolds number based on a chord of 107. This study is part of a con- tinuing investigation to determine the adequacy of numerical algorithms. They use the time-split algorithm of MacCormack. Separate turbulence models are used for the boundary layer ahead of the shock, the separation bubble following the shock, the wake of the separation bubble, and the outer boundary layer and wake. Each is modeled with a simple scalar ? / 0.0 2.6 \ \ ) ! 5.2 . \\ ' . \'_) ! ,/ t' s \ \ 7.8 CHORDS TRAVELED Figure /9 Computed Mach contours for flow past an 180o thick circular-arc airfoil at Mac = 0.754, Re = |O _, from Levy (1978) //7 218 TIJDEMAN & SEEBASS eddy diffusivity. This numerical simulation yielded an unsteady solution when the Mach number was 0.76, but a steady solution was found again when the Mach number was increased to 0.79. This unsteady motion, an alternate fore-aft motion of the shock wave with shock-induced separation on each side of the airfoil, was also observed experimentally over a narrow range of Mach numbers for the Reynolds number used in the calculations. The numerical results reproduce well (within 20_o) the frequency of this oscillation at the lower Mach number. Figure 19, from Levy (1978), depicts the Mach contours exhibiting this behavior for a free-stream Mach number of 0.754. Velocity profiles at various chordwise stations were found to be in qualitative agreement with those measured. The main difficulties encountered in this numerical simulation were the inadequacy of the turbulence modeling near the trailing edge and a tendency of the numerical algorithm to capture strong (supersonic to subsonic) shocks where weak (supersonic to supersonic) shocks were observed. The time to carry out a cycle of the unsteady computation on a CDC 7600 was 1.8 hours. In an exploratory study, using the "thin-layer" algorithm mentioned above, Steger & Bailey (1979) simulated the aileron buzz observed in flight tests of the P-80 and subsequently documented in the Ames 16-foot wind tunnel. These coupled aeroelastic computations were able to re- produce the Mach number of the observed onset of buzz at two angles of attack. This result, and that discussed above, demonstrate that con- temporary algorithms and computer hardware are able to simulate complex flow phenomena. CONCLUDING REMARKS During the past five years sufficient experimental observations and measurements have been made to provide a good understanding of the transonic flow past oscillating airfoils. Furthermore, recent studies have provided results essential for the design of transonic aircraft. The main limitations of these experiments are their failure, for the most part, to duplicate full-scale Reynolds numbers and an inability to duplicate free- flight conditions due to wind tunnel wall interference. Experimental studies, both in progress and planned for the future, will be more nearly at full-scale Reynolds number, and eventually these Reynolds numbers will be obtained with minimum wall interference in new facilities now under development. Paralleling this progress has been a rapid development of reliable, and in the small-perturbation approximation, efficient numerical algorithms for the computation of inviscid flows. Numerical results from these 'O TRANSONIC FLOW PAST OSCILLATING AIRFOILS 219 methods are in qualitative agreement with the experimental observations, with the main discrepancies in quantitative prediction as a consequence of the inviscid approximation. For steady flows coupled inviscid-boundary- layer calculations of unseparated flows obtain quantitative agreement with experimental measurements. We can expect this to be true for unsteady flows in the near future. The numerical simulation of unsteady separated flows is demonstrably possible, but the two orders of magnitude improvement in computer speed that is projected for a special-purpose aerodynamic computer will be essential for this simulation to have practical consequences. It is the authors' opinion that the satisfactory prediction of unsteady airloads for aeroelastic applications is within reach. This can be accom- plished by "tuning" inviscid boundary conditions to model an experi- mentally determined steady flow and then computing its unsteady response using an inviscid small-perturbation algorithm. Thus, the time is ripe to start with the incorporation of the new methods in aeroelastic practice as recently demonstrated by Ashley (1979). Of course, the use of two-dimensional methods is justified only for large aspect-ratio wings. To treat the low aspect-ratio configuration the next, and not difficult, step has to be made, namely, the development of prediction methods for three-dimensional flows. Literature Cited Ashley, H. 1979. On the role of shocks in the "sub-transonic" flutter phenomenon. AIAA Pap. 79-0765 Baldwin, B. S., Lomax, H. 1978. Thin layer approximation and algebraic model for separated turbulent flows. AIAA Pap. 78-257 Ballhaus, W. F. 1978. Some recent progress in transonic flow computations. Numerical Methods in Fluid Dynamics, ed. H. J. Wirz, J. J. Smolderen, pp. 155-235. New York : McGraw-Hill Ballhaus, W. F., Goorjian, P. M. 1977. Implicit finite-difference computations of unsteady transonic flows about airfoils. AIAAJ. 15:1728-35 Ballhaus, W. F., Goorjian, P. M. 1978. Com- putation of unsteady transonic flows by the indicial method. AIAA J. 16:117-24 Ballhaus, W. F., Steger, J. L. 1975. Implicit approximate-factorization schemes for the low-frequency transonic equation. NASA Tech. Memo X-73,082 Basu, B. C., Hancock. G. J. 1978. The un- steady motion of a two-dimensional aero- foil in incompressible inviscid flog'. J. Fluid Mech. 87:159-78 Bauer, F., Garabedian, P., Korn, D., Jameson, A. 1975. Supercritical wing sections I1. Lect. Not. Econ. Math. Syst. No. 108. Berlin : Springer Beam, R. M., Ballhaus, W.F. 1975. Numerical integration of the small-disturbance potential and Euler equations for unsteady transonic flow. NASA SP-347, Part II, pp. 789-809 Beam, R. M., Warming, R. F. 1974. Numerical calculations of two-dimensional, unsteady transonic flows with circulation. N 4SA Tech. Note D-7605 Beam, R. M., Warming, R. F. 1976. An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. J. C omput. Phys. 22:87-110 Beam, R. M., Warming, R. F. 1978. An implicit factored scheme for the compres- sible Navier-Stokes equations. AIAA J. 16:3o3-402 Bergh, H. 1965. A new method for measuring the pressure distribution on harmonically oscillating wings. Proc. 4th ICAS- Conyress, Paris, 1964, ed. R. Dexter. London: MacMillan. See also NLR MP.224, 1964 Bergh, H., Tijdeman, H., Zwaan, R. J. 1970. High subsonic and transonic effect_ on 220 TIJDEMAN & SEEBASS pressure distributions measured for a swept wing with oscillating control sur- face. Z. Flu.qwiss. 18(9/10): 339-46 Chipman, R., Jameson, A. 1979. Fully con- servative numerical solutions for unsteady irrotational transonic flow about airfoils. AIAA Pap. 79-1555 Couston, M., Ang61ini. J. J. 1978. Solution of nonsteady two-dimensional transonic small disturbances potential flow equa- tion. In Nonsteady Fluid Dynamics: Proc. ASME Winter Annual Mtg., Dec. 10-15, 1978, San Francisco, pp. 23.'_-44. See also ONERA TP-No. 1978-69 Davis, J. A. 1979. Unsteady pressures on an NACA 64A410 airfoil: experimental and theoretical results. AIAA Pap. 79-0330 Davis, S., Malcolm, G. 1979. Experiments in unsteady transonic flow. AIAA Pap. 79-0769 Dowell, E. H. 1977. A simplified theory of oscillating airfoils in transonic flow: Review and extension. Proc. AIAA Dynamics Specialist Conf., San Diego, 1977, pp. 209 24 Ehlers, F. E. 1974. A finite difference method for the solution of the transonic flow around harmonically oscillating wings. NASA Contract Rep. 2257 Erickson, A. L., Robinson, R. C. 1948. Some preliminary results in the determination of aerodynamic derivatives of control sur- faces in the transonic speed range by means of a flush type electrical pressure cell. NACA RM A8H03 Erickson. A. L., Stephenson, J. D. 1947. A suggested method of analyzing for tran- sonic flutter of control surfaces based on available experimental evidence. NACA RM A7F30 Finke, K. 1975. Shock oscillations in tran- sonic flows and their prevention. Syrup. Transsomcum 11, ed. K. Oswatitsch, D. Rues, pp. 57-65. Berlin: Springer Fritz, W. 1978. Transsonische Str6mung um harmonisch schwingende Profile. Dornier GMBH Rep. 78/16 B Fung, K-Y., Yu, N. J., Seebass, R. 1978. Small unsteady perturbations in transonic flows. AIAA J. 16:815-22 Grenon, R., Desoper, A., Sid6s, J. 1979. Effets instationnaires d'une gouverne en ecoulement bidimensionnel subsonique et transsonique. AGARD CP-262 Grenon, R., Thers, J. 1977. I_tude d'un profil supercritique avec gouverne oscil- lante en 6coulement suhsonique et transsonique. AGARD CP-227 Heaslet, M. A., Lomax, H., Spreiter, J. R. 1948. Linearized compressible-flow theory for sonic flight speeds. NACA Rep. 956 Houwink, R., van der Vooren, J. 1979. Results of an improved version of LTRAN-2 for computing unsteady air- loads on airfoils oscillating in transonic flow. AIAA Pap. 79-1553 Hung, C. M., MacCormack, R. W. 1978. Numerical solution of three-dimensional shock wave and turbulent boundary-layer interaction. AI AA J. 16:1090--96 lsogai, K. 1977. Calculation of unsteady transonic flow over oscillating airfoils using the full potential equation. Proc. AIAA Dynamics Specialist Co_f, San Diego, 1977 Isogai, K. 1978. Numerical study of tran- sonic flow over oscillating airfoils using the full potential equation. NASA Tech. Pap. 1120 Jameson, A. 1974. Iterative solution of tran- sonic flows over airfoils and wings, in- cluding flows at Mach 1. Cotton. Pure Appl. Math. 27:283-309 Jameson, A. 1978. Transonic flow calcula- tions. In Numerical Methods in Fluid Dynamics, ed. H. J. Wirz, J. J. Smolderen, pp. 1-87. New York: McGraw-Hill Lambourne, N. C. 1958. Some instabilities arising from the interaction between shock waves and boundary layers. AGARD Rep. 182 Landahl, M. T. 1961. Unsteady Transonic Flow. New York/London: Pergamon Landahl. M. T. 1976. Some developments in unsteady transonic flow research. Syrup. Transsonicum II, ed. K. Oswatitsch, D Rues. Berlin: Springer Laval, P 1975. Calcul de 1'6coulement instationnaire transsonique autour d'un profil oscillant par une methode _i pas fractionnaires. ONERA TP No. 1975-115 Leadbetter, S. A., Clevenson, S. A., lgoe, W. B. 1960. Experimental investigation of oscillatory aerodynamic forces, moments and pressures acting on a tapered wing oscillating in pitch at Mach numbers from 0.40 to 1.07. NASA TN-DI236 Lerat, A., Sides, J. 1977. Calcul numerique d'ecoulements transsoniques instation- naires. AGARD Spec. Mtg. on Unsteady Airloads in Separated and Transonic Flow, Lisbon, April 1977. See also ONERA TP No. 1977-19E Lessing, H. C., Troutman, J. L., Meness, G. P. 1960. Experimental determination of the pressure distribution on a rect- angular wing oscillating in the first bend- ing mode for Mach numbers from 0.24 to 1.30. NASA TN-D344 Levy, L. L., Jr. 1978. Experimental and computational steady and unsteady tran- sonic flows about a thick airfoil. AIAA J. 16: 564--72 Lin, C. C., Reissner, E., Tsien, H. S. 1948. /..o£' G ¸ TRANSONIC FLOW PAST OSCILLATING AIRFOILS 221 On two-dimensional non-steady motion of a slender body in a compressible fluid. J. Math. Phys. 3:220--31 MacCormaek, R. W. 1976. An efficient numerical method for solving the time- dependent compressible Navier-Stokes equations at high Reynolds number. NASA Tech. Memo X-73, 129 MacCormack, R. W., Lomax, H 1979. Numerical solution of compressible viscous flows. Ann. Rev. Fluid Mech. 11: 289-316 McCroskey, W. J. 1977. Some current research in unsteady fluid dynamics. Trans. ASME, March 1977, J. Fluids Eng., pp. 8-39 McDevitt, J. B. 1979. Supercritical flow about a thick circular-arc airfoil. NASA Tech. Memo 78549 Magnus, R. J. 1977a. Computational re- search on inviscid, unsteady, transonic flow over airfoils. ONR CASD/LVP 77-010.68 pp. Magnus, R. J. 1977b. Calculations of some unsteady transonic flows about the NACA 64A006 and 64A010 airfoils. AFFDL Tech. Rep. 77-46 Magnus, R. J. 1978. Some numerical solu- tions ofinviscid, unsteady, transonic flows over the NLR 7301 airfoil. ONR CASD/ LVP. 37 pp. Magnus, R. J., Yoshihara, H. 1975. Un- steady transonic flows over an airfoil. AIAA J. 13:1622-28 Magnus, R. J., Yoshihara, H. I976. Calcula- tion of the transonic oscillating flap with "viscous" displacement effects. A 1 A A Pap. 76-327 Magnus, R. J., Yoshihara, H. 1977. The transonic oscillating flap. AGARD-CP-226 Melnik, R. E., Chow, R., Mead, H. R. 1977. Theory of viscous transonic flow over air- foils at high Reynolds number. AIAA Pap. 77-680 Nakamura, Y. 1968. Some contributions on a control surface buzz at high subsonic speeds. J. Aircraft 5 : 118-25 Nixon, D. 1978. Calculation of unsteady transonic flows using the integral equation method. AIAA J. 16: 976-83 Peyret, R., Viviand, H. 1975. Computation of viscous compressible flows based on the Navier-Stokes equations. AGARDograph No, 212 Schippers, P. 1978. Results of unsteady pressure measurements on the NLR 7301 airfoil with oscillating control surface. NLR TR 78124 C Sears, W. R. 1976. Unsteady motion of air- foils with boundary layer separation. AIAA J. 14:216-20 Seebass, A. R, Yu, N. J., Fung, K-Y. 1978. Unsteady transonic flow computations. AGARD-CP-227 Seegmiller, H L., Marvin, J. G., Levy, L. L. Jr. 1978. Steady and unsteady transonic flow. AIAA J. 16:1262-70 Spreiter, J. R., Stahara, S. S. 1975. Unsteady transonic aerodynamics--An aeronautics challenge. Proc. Unsteady Aerodynamics Syrup., Univ. of Arizona, Vol. II, pp. 553- 82 Steger, J. L. 1978. Implicit finite-difference simulation of flow about arbitrary two- dimensional geometries. AIAA J. 16: 679- 86 Steger, J. L., Bailey, H. E. 1979. Calculation of transonic aileron buzz. AIAA Pap. 79- 0134 Tijdeman, H. 1976. High subsonic and transonic effects in unsteady aero- dynamics. Part of AGARD Rep. 636 : see also On the motion of shock waves on an airfoil with oscillating flap. Syrup. Trunssonicum I1. ed. K. Oswatitsch, D. Rues, pp. 4_56. Berlin: Springer Tijdeman, H. 1977. Investigations c_f the Transbnic Flow Around Oscillatin 9 Air- fi)ils. Doctoral thesis. Technische Hoge- school Delft, The Netherlands. 148 pp. Tijdeman, H., Bergh, H. 1967. Analysis of pressure distributions measured on a wing with oscillating control surface in two- dimensional high subsonic and transonic flow. NLR-TR F.253 Tijdeman, H., Schippers, P. 1973. Results of pressure measurements on an airfoil with oscillating flap in two-dimensional high subsonic and transonic flow Izero incidence and zero mean flap position). NLR TR 73078 U Traci, R. M., Farr, J. L., Albano, E. 1975. Perturbation method for transonic flows about oscillating airfoils. AIAA Pap. 75- 877 Triebstein, H. 1969. Instalion/ire Druck- verteilungsmessungen an einem schwing- enden Tragfliigel im subsonischen und transsonischen Geschwindigkeitsber eich. DLR FB 69-49 Triebstein, H. 1972. Instation/ire Druck- verteilungsmessungen an einem har- monisch schwingenden Fliigelmodell in drei-dimensionaler, kompressibler Str6m- ung. Z. Fluywiss. 21(11):400-12; see also DLR FB 72-55 van der Vooren, J,, Slooff, J. W. 1973. On inviscid isentropic flow models used for finite difference calculations of two- dimensional transonic flows with em- bedded shocks about airfoils. NLR MP 73024 U Warming, R. F., Kutler, P., Lomax, H. 1973. Second- and third-order noncentered 222 TIJDEMAN & SEEBASS difference schemes for nonlinear hyper- bolic equations. AIAA J. 2:189-96 Weatherill, W. H., Sebastian, J. D., Ehlers, F. E. 1977. Application of a finite dif- ference method to the analysis of transonic flow over oscillating airfoils and wings. AGARD-CP-226 Weatherill, W. H., Sebastian, J. D., Ehlers, F. E. 1978. The practical application of a finite difference method for analyzing transonic flow over oscillating airfoils and wings. NASA Contract Rep. 2933 Wyss, J. A., Sorenson, R. M. 1951. An in- vestigation of the control surface flutter derivatives of an NACA 65-213 airfoil in the Ames 16-foot high speed wind funnel. NACA RM A51JIO Yu, N J., Seebass, A. R., Ballhaus, W. F. 1978. Implicit shock-fitting scheme for un- steady transomc flow computations. AIAA J. 16:673--78 Zierep, J. 1966. Theorie der Schallnahen und der Hyperschallstr6mungen. Karls- ruhe: G. Braun r __ J. Fluid Mech. (1979), vol. 95, part 2, pp. 223-240 Printed in Great Britain 223 Lifting three-dimensional wings in transonic flow By M. S. CRAMER Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (Received 5 September 1978) The far field of a lifting three-dimensional wing in transonic flow is analysed. The boundary-value problem governing the flow far from the wing is derived by the method of matched asymptotic expansions. The main result is to show that corrections which are second order in the near field make a first-order contribution to the far field. The present study corrects and simplifies the work of Cheng & Hafez (1975) and Barnwell (1975). 1. Introduction This paper is concerned with the transonic flow over thin lifting wings. In particular, the flow far from the wing is discussed. A number of authors have studied the transonic flow far from wings and bodies. One of the most important contributions to our understanding of these flows is the transonic area rule. This area rule (see, e.g. Oswatitsch 1952) states that the transonic flow far from a wing-body combination is the same as that produced by an equivalent body of revolution having the same axial distribution of cross-sectional area. This rule has been established for slender bodies by Oswatitsch (1952) and Cole & Messiter (1957). Spreiter & Stahara (1971) extended this to non-slender wings, i.e. wings having an aspect ratio of order one. Ashley & Landahl (1965) also extended the theory to include wings at an angle of attack comparable to their thickness. Generally, the area rule is deduced by deriving the boundary-value problem governing the flow far from the wing or body; this boundary-value problem is seen to be identical to that for a slender body of revolution, provided that the streamwise rate of change of cross-sectional area is the same for both. Hayes (1954) has pointed out that the transonic area rule fails when the volume of the wing is sufficiently small. Cheng & Hafez (1972, 1973, 1975) and Barnwell (1973, 1975) have studied the effect of lift on the transonic area rule. The study presented here treats the case of a lifting wing with no thickness. With this simple case it is easy to illustrate the basic theory and the main effect of lift on the far field. The boundary-value problem governing the flow far from the wing is obtained through a straightforward application of the method of matched asymptotic expansions. Our main interest here is in the basic theory of lifting wings in transonic flow; for a more complete discussion of the flow from aphysical point of view, and of extensions to the basic theory, we refer the reader to the references cited above. In §6 we review and discuss the previous investigations of Cheng & Hafez (1975) and Barnwell (1975). In most respects our work agrees with the above authors. How- ever, there are important differences in our expressions for the boundary condition 0022-1120/79/4336--7770 $02.00 © 1979 Cambridge University Press /3_ 224 M. S. Cramer / U q) for the outer flow; the disagreement with the results of Cheng & Hafez is due to fundamental differences in the matching. 2. Mathematical formulation The co-ordinate system used is sketched in figure 1 ; the origin is taken at the nose of the wing, the x axis is aligned with the undisturbed uniform flow, and the z axis is taken as approximately perpendicular to the wing surface. A typical wing has been sketched; it has a length 1and a span of 2b. The equations defining the wing are for al(x/1 ) _ y/l _ a2(x/l); the functions a 1 and a2 give the leading edges of the wing as well as the outer edges of the trailing vortex sheet. The function Z is taken to be some sufficiently smooth function of x and y. Because it defines a single surface in space, the wing has no thickness. The aspect ratio is taken to be of order one; i.e. b/l will be assumed to be of order one. To eliminate unnecessary writing, the independent variables x, y and z will be scaled by l; otherwise the quantity b/l = 0(I) will appear throughout the calculations. For the sake of simplicity Cheng & Hafez (1975) assume that Z is such that there are no singularities at the leading edges. In fact, they assume that the velocity perturbations are zero at the leading edges and the outer edges of the trailing vortex sheet; here we assume this as well. By making this assumption we avoid the difficulties associated with leading edge singularities and separation addressed by Barnwell (1975). The small parameters of the problem are _, which gives a measure of the angle of attack of the wing, and M02- 1, which indicates that the flow is transonic; here M o -- U/ao, where U is the speed of the undisturbed uniform flow and ao is the ambient sound speed. Lifting three-dimensional wings in transonic flow 225 The flow is assumed to be irrotational; a velocity potential ¢* therefore exists and the inviscid equations of motion of a perfect gas may be written where a_=_a_+Z_-_[U 2 V¢ 2V¢*.], a is the local speed of sound in the gas and the _ is the ratio of specific heats. The last equation is just the Bernoulli equation for steady isentropic flow. The velocity potential _b* contains a part due to the uniform stream and a part due to the per- turbation of the wing. It will be convenient to work with the equation for the perturba- tion potential, ¢, defined by ¢* _=Vx+¢. In terms of ¢, the equation of motion is v cx + v 2 ]v¢1 2 ] _x IV¢[-t-V¢.V(_) -- a V ¢, (1) a2 = a_- ('- l ) [ UOz + _V_¢2]2] ' I exactly. The boundary condition on the wing is ¢. = (v+¢x)_x+ ¢_ (2) on z = _* (x, y; l; b; _). As x _+ y2 + z2_> oo, it is further required that IV¢] -_ 0. In the following sections, solutions to (1) and (2) are sought which are valid for small and M_ - 1. In § 3 the solution valid near the wing is derived; this will be called the inner solution. Because the inner solution neglects certain nonlinear terms in (t), it fails to give a valid description of the flow at large distances from the wing. An approximation to (1) which is valid at large distances from the wing is derived in § 4. There it is shown that, to lowest order, the flow is governed by the small disturb- ance transonic equation; the region in which this is valid is called the oute_ regior_. The boundary condition satisfied by the first term of the outer expansion is obtained by a matching with the inner solution; this is done in § 5. There it is seen that every term in the inner solution contributes to the boundary condition for the outer problem; the resultant boundary condition for the outer problem will therefore be an infinite sum of terms. The outer expansion will be written ¢ = Ulfo ¢Po(_,_, to) + o(fo) , where _ and _ are just scaled values ofx and r (see figure lb); the function _o will he shown to satisfy 1 1 M_- 1 ¢_+_0_+_0_= _f_ O0_+(_+l)(I)_Oa_ , (3) ¢0(_,_, oJ) ~ +b(_)[ln2_+cos_o]+c(_)ln_+d(_)+ .... as _ -_ O, and ¢-1¢o_, ¢o_, ¢o_ -+ O, as _-_ oo. Here the dots indicate terms which are of 226 M. S. Cramer order _ In 3 _ and higher in _. An analogous result has been obtained by Cole & Messiter (1957) for the case of a slender axisymmetric body in transonic flow, see, e.g. their equation 5.9. Note also that the terms shown in (3) are singular as _--> 0, whereas the unwritten terms vanish in this limit. In §5, it is further assumed that ¢o may be uniquely determined by a specification of the singularities at the axis; we may there- fore truncate the infinite series and write the boundary condition as a (P0 ~ _ + b[ln2 _ + cos_ w] + c ln_ + d as _-_0. 3. Inner solution In the inner region the velocity potential ¢ and the independent variables x, y and z may be sealed as follows: _ = Ul_, x = l_, y = I_, z = l_, provided b/1 = O( 1); where _, _ and 5 are of order one in the inner region. Equations (1) and (2) may now be written as [ _'_0IV_Ig+V_? v(IV_I-----_)] = aS_02 ' (4) a° = 1- (7- with on 5 = (5) where all derivatives are now with respect to £, _, 5. Equations (4) and (5) will now be perturbed for small a and M_-1; the inner expansion is written _t = go_o+ g_ _ + 0(92), (6) where the 9i's are the as yet undetermined inner gauge functions. When (6) is sub- stituted into (4) and (5) and when the coefficients of like powers of a and Me_- 1 are equated, there results a set of boundary-value problems, each of which is of the form V_ = _(_,_) t (7) _k_(_,0±) = f+(_), a 1 _0; hence . F sinto {[S' (F'2) ' ]fo_Po($,_,to) .., o'_-_n-T--+a2 _--_-(y+l)_ln_ ln_ +.)_, + (7 + 1) (F'2) ' }2 16_. _ (ln2_+cos2to) . (18) Fraenkel (1969) has pointed out that terms containing logarithms, viz. the term having a s as a coefficient in (18), should be matched as a single term. With this in mind, we see that the appropriate choice for f0 = 82 is a211n31, which further implies that $ = $(a) is given implicitly by S/lln l" Thus, the matching requires that, as _-_ O, 1 F sinto IS' 1 (F'_) '1lln 1½2 r e e. nlln3l+( +l) 1-i'g n Jlne _* 1 7+ 1 (F'S) ' (lnS_+cos_to). (20) According to the analysis presented so far, this is the boundary condition for the 13_- ,J Lifting three-dimensional wings in transonic flow 233 outer problem. When the third-, fourth-, and higher-order terms of the inner expansion are calculated, they also make a contribution to this boundary condition; in fact, the actual boundary condition is an infinite sum of terms. This is to be expected as we seek the asymptotic expansion of (I)0 as __ 0 rather than its value at _ = 0. As an example, we could continue the inner expansion (6) to include third- and fourth-order terms ?t = go ?o + 91 71 + 92 ?2 + ga % + o(ga) where an inspection of equations (12) and (13) shows that 92 = (M_- 1)a and g3 = aa. The procedure of this paper could be applied to these higher-order terms to determine their contribution to the boundary condition (20). It may be shown that when _z and ?a are included in the inner expansion we must add the following quantity to (20): _e F". _oW_} 1 {_nl, sin to[_(F'B' )'Iln [i {¥_ _s'ntoln_+ +__ lnS_ + (F'(A' - B' - 2 In $B'))' ln2_ + (F'(_B' - A' + 29/"*' +B' In _))' ln_ - (F'B')' cos2 _o] + $9_*}. Here ,g=_ M_- 1/_ 2, A = S'/27r, B = T+ l(F")'/32zr_ and 9¢* andg/f_ are harmonic functions proportional to _/3. In like manner we could also determinethe contributions of higher-order terms; these contribute terms of even higher order in _. In order to simplify the boundary condition for the outer problem, we now make the assumption that the outer problem is well-posed provided that the singularities in q)o at # = 0 are specified. Because the higher-order terms, i.e. the terms g_ _;, i 1> 2, in the inner expan- sion contribute terms which vanish as _-+ 0, we may truncate the boundary condition to include only those shown in equation (20). Thus, the outer problem may be written i°_-¢Po_ + 1 _o_ + ¢I)o_ = _2 ¢Po.*_+ (Y + 1 ) _o._ _o:_._, 1 F sinto IS' 1 )(F'2) ']I1n81_2. _ _ _tln$lq'(y+l 16fro] ln_ _* 1 (y) (ln,_+cos2o_), (21)2 and, as_-+oo, _-aq)o_ , (I)0_, (I)0_-_ 0. Here we recognize the first term as a doublet and the second term as a soureehaving strength (S'/2n) (1/lln $1) + (9' + 1) (F")'/IOn'. The first part of the source is due to the nature of the second-order velocity perturbations on the wing and the part depending nonlinearly on the lift is due to the fact that the flow in the neighbour- hood of the wing, i.e. _ = 0(1), appears as a source flow when viewed from the far field. We note also that in theories of transonic flow not involving lift, the solution to the outer problem only depends on M_)- 1/(y + 1)Sz, i.e. the similarity parameter of the problem. Here the solution also depends on (y + 1) and In _; hence, in lift dominated flows, no simple similarity rule holds. Furthermore, it is clear that no conventional area or equivalence rule applies for the wings treated here. We refer the reader to Cheng & Hafez (1975) and Cheng (1977) for a further discussion of equi- valence rules applied to lifting wings. . / 5"-v 234 M. S. Cramer Throughout this paper we have confined ourselves to wings having zero thickness. The effect of a wing's thickness is easily incorporated and we will now give a brief discussion of it. The equation of a wing having thickness can be written = ccZ _+rZ v where the subscript v will always denote functions associated with thickness effects. The inner expansion corresponding to (6) would be q_i = ce%+7_v+a2Tl+ .... If we now proceed as we did in § 3, we should find that 1 a: % = - [ Zv_ In r 1dy 1+ :,vr_, 71"Jal and, as _-->oo, where , o(,) asSv = 2 Z_dyl. When this is cast in terms of the outer variables we have (_)° ~ _ Svln' + 9if* + O(_), where _* - _. - (S,./2tr) In 3 = 0(1). Thus, the thickness would contribute 7[(S_/21r) lnO + 9_*] to equation (18). Inspection shows that the thickness and lift have an equal effect on the outer problem provided • = O(fo) = O($2), where _ is related to a by (19). An examination of higher-order terms shows that this is the only additional singularity generated by the introduction of the thickness; thus, provided r = 0(32), the thickness contributes r/32[Sr/2rt lnO + _g/*] to the boundary condition in (21). Generally, when r 4 0(32(a)), where 3(a) is given by (19), we may neglect either the lift or thickness when calculating the outer flow. For example, when 7 = O(a), the matching requires that f0 = r, 3 = r½ and that the boundary condition is S_ ¢0 ~ _-_ln_+ 9ff_ as _--> O. When r = 0(_3), the matching yields the same results as in the zero thickness case; in this case the thickness effects may be considered negligible for the purposes of calculating the far field. 6. Discussion of previous investigations In this section we discuss the investigations of Barnwell (1975) and Cheng & Hafez (1975), comparing their results and procedures to ours. Both papers give derivations of the boundary-value problem governing the flow far from a lifting wing; their procedures are seen to differ considerably in both appearance and content from each other and the present study. _7"/ _"/ aL Lifting three-dimensional wings in transonic flow 235 We first discuss the work of Barnwell. It should first be mentioned that Barnwell uses a body oriented co-ordinate system in contrast to the wind oriented system used here; hence care should be exercised in comparing Barnwell's work to either ours or Cheng & Hafez's. Barnwell also provides a discussion of the effect of leading edge separation; this complication will not be discussed here. Barnwell first presents an inner expansion which contains gauge functions which are logarithmic in the ratio of the inner and outer length scales. Although our inner expansion proceeded in integral powers of a and M02- 1, we allowed 3/_ and _H(/_,_) in (9) to depend on a; thus, the resultant inner expansion is seen to be equivalent to that of Barnwell. A further examination of Barnwell's inner expansion shows that Barnwell has omitted the following term, 82¢1 _ 82¢1 _, _4 852 g= h4_'_Y from his boundary condition (10). This produces an error in the strength of the equi- valent source given by H(_) in his equation (68). Barnwell also presents a very careful study of the outer solution. As we mentioned in §4 he finds the equations governing three terms of the outer expansion; in his notation these terms are el (Pl + e2 (I)2+ es Os, where Q, % and % are the outer gauge functions and (I)1, (P2 and (I)a are independent of an), small parameters. The lowest-order term satisfies the small disturbance tran- sonic equation and 02 and (Pa satisfy linear equations which have coefficients dependent on the lower-order (I);'s and their derivatives. To match the inner and outer expansions he needs a small _ expansion of the ¢;'s. To obtain this he uses the iterative technique of Cole & Messiter (1957) to solve the differential equations governing (I)1, (D_.and (Pz for small values of ?; this assures us that the inner expansion of the outer solution satisfies the outer equations. For the sake of simplicity, we have presented a more intuitive approach to this than that presented by Barnwell. Essentially, we have tacitly assumed that a small _ expansion of our outer solution will contain all the terms necessary to match. It is easy to show that when such an iterative procedure is applied to our outer solution, a boundary condition results which is identical to the one pre- sented here. Once Barnwell obtains his expansion of the outer solution he matches this to the large _ expansion of the inner solution. Except for the error in the source strength mentioned above his results are in agreement with those given here. As a final remark we note that Barnwell states that an intermediate expansion is necessary in order to match the inner and outer expansions. He bases this on an examination of the large f expansion of the inner solution (his equation 67) and the small _ expansion of the outer solution (his equation 68). Because the leading term in (67) is a dipole and the leading term in (68) is a source, he concludes that an inter- mediate expansion is necessary. In § 5 we used a rule concerning the matching of logarithms; if this is applied to Barnwell's expansions (67) and (68) it is clear that they may be matched without recourse to an intermediate expansion. We now discuss the results of Cheng & Hafez. Of the two previous investigations the procedure of Cheng & Hafez has the closest resemblance to ours. Their inner expansion can beshownto be the same as ours and they use a one-term outer expansion similar to that given here. Throughout their paper, Cheng & Hafez use an elaborate aD 236 M. S. Cramer parameterization scheme. They also correctly state that their results are valid for = b/l = 0(1); this appears to be inconsistent with the parameterization scheme. Specifically, in equation (2.10) they introduce a parameter r, _--8/(y+ 1)_211nel, where e is the ratio of inner and outer length scales analogous to our 8; they further require that F, be non-vanishing as e--> 0. This would seem to imply that A must vary as I1nel-½which violates their ;( = 0(1) assumption. However, this inconsistency in the parameterization does not affect the final results. In § 4.3 the outer equation is introduced and a small y _ er expansion of the outer solution is given. In §4.5 the matching is carried out for the case corresponding to the one discussed here. The boundary condition for the outer problem is given by their equation (4.12); this is seen to disagree with our boundary condition (21). Specifically, the terms _ [Dx(x ) (y,)-i cos _o+ Iln e'l-½ ma2(_?') -2 20]sin appear in their equation (4.12), but are absent in ours. It may be shown that these terms correspond to the O($2/_ 2) term found in the outer expansion of _0 and the O($_-1 In _/$, 3/_) term found in the outer expansion of _1; these higher-order terms must be truncated in the matching. In a later publication, Cheng (1977) discusses the application of this theory to particular wing configurations; the boundary condition used in this study is equivalent to the one derived here. With the exception of the errors mentioned above, the results of Barnwell (1975), Cheng & Hafez (1975) and the present study are in agreement. The study presented here approaches the problem from a more fundamental point of view and is therefore believed to be more accessible to the reader. 7. Conclusion We have presented a theory of thin three-dimensional wings without thickness in transonic flow. The boundary-value problem governing the flow far from the wing has been derived. The calculations presented here are intended to be simpler than those of the previously published studies; they also correct errors found in these earlier studies. Both the previous investigations and the present study show that there are effects which are of second order in the near-field which produce first-order effects in the far-field. This research was carried out while the author was a Research Associate in the Department of Aerospace and Mechanical Engineering and the Program in Applied Mathematics at the University of Arizona, Tucson, Arizona. It was sponsored by the N.A.S.A. through Grant NSG-2112 and A.F.O.S.R. through Grant 76-2954. The author is indebted to Drs R. C. Ackerberg, R. E. O'Malley, W. R. Sears and A. R. Seebass for valuable comments and criticisms. Lifting three-dimensional wings in transonic flow 237 Appendix A. Solution for _01 Because of the linearity of the Laplacian it is permissible to break _1 up into three parts: _i = _'+ _"+½(y+ I)_", where _' satisfies v_+'= o _=(%_Z)_ on _=0±, a 1_<_ oo. This condition, combined with the fact that _f_ is symmetric in _ and satisfies V_f_=0 for all _,_,5, and _= 0 on _ = 0±, a I _<_ < a_, implies that $(t_ is a function of£ alone. If we absorb this function of£ in 3/"(_) we may now write _1 = _1 _;[%_] Z_-yl_-'_dyl+l_(_0)e-_f:f[_0]zeelnrldyl T + l {w" -l f_'[_t_"] ln ra dyl} + _(_), (A9) ÷ --_ d al and as _--_¢_, S'(2) ' Y+ ! (F'_) ' (ln_+eos_o)+ \--_-, (A 10) _pl ~ --_- In _+ 9U(_) _- 3---_-_ REFERENCES As_u.:v, H. & LA_tDAHL, M. 1965 Aerodynamics of Wings and Bodies. Addison Wesley. B_WE_.L, R. _V. 1973 A.I.A.A.J. 11, 764. BA_NWE_.t., R. W. 1975 N.A.S.A. Tech. Rep. TR R-440. CH_N_, H.K. 1977 A.I.A.A.J. 15, 366. C_o, I'[. K. & HA_EZ, M. 1972 A,I.A.A.J. 10, 1115. CH_ro, H. K. & HAF_z, M. 1973 USCEngngRep. no. 124. 240 M. S. Cramer C_E_G, H. K. & HAF_Z, M. 1975 J. FluidMech. 72, 161. Cozy, J. D. & MESSXTER, /k. F. 1957 Z. angew Math. Phys. 8, 1. F_a_KEL, L.E. 1969 Proc. Camb. Phil. Soc. 65, 209. HxYES, W. D. 1954 J. Aero. Sci. 21,721. OSWATITSCH, K. 1952 Proc. 8th Inst. Cong. Theor. Appl. Mech. Istanbul. SPR_.ITER, J. I_. & STAHARA, S.S. 1971 A.I.A.A.J. 9, 1784. VA_tr DYKE, M. 1964 Perturbation Methods in Fluid Mechanics. Academic. OAIAA 78-1114 New Method for Des!gningShock-Free TransonicConfigurations H. Sobieczky, N. J. Yu, K-Y. Fung, A. R. Seebass Reprintedfrom AIAAJournalVolume 17, Number 7, July 1979, Page722Copyright AmericanInstitute ofAeronautics and Astronautics, Inc., 1978. All rightsreserved °E OF AERONAUTICS AND ASTRONAUTICS o1290 AVENUE OF THE AMERICAS °NEW YORK, NEW YORK, N.Y. 10019 722 AIAA JOURNAL VOL. 17, NO. 7 ARTICLE NO. 78-1 114 . New Method for Designing Shock-Free Transonic Configurations 4_ H. Sobieczky, ° N. J. Yu,l" K-Y. FungAt and A. R. Seebass§ University of Arizona, Tucson, Ariz. A new method for the design of shock-free supercritical airfoils, wings, and three-dimensional configurations is described. Results illustrating this procedure in two and three dimensions are given. They include modifications to part of the upper surface of an NACA 64A410 airfoil that will maintain shock-free flow over a range of Mach numbers for a fixed lift coefficient, and the modifications required on part of the upper surface of a swept wing with an NACA 64A410 root section to achieve shock-free flow. While the results are given for inviscid flow, the same procedures can be employed iteratively with a boundary-layer calculation in order to achieve shock-free viscous designs. With a shock-free pressure field, the boundary-layer calculation will be reliable and not complicated by the difficulties of shock wave, boundary-layer interaction. Introduction ELL-KNOWN requirements for increased efficiency and. in the case of commercial aircraft, productivity have forced the operating conditions of compressors, tur- bines, propellers, wing sections, and aircraft into the tran- sonic regime. Unfortunately, once local regions of supersonic flow occur, shock waves are likely with the attendant wave drag and boundary-layer separation losses. In the mid 1950s, Morawetz_ proved that shock-free, two-dimensional, irrotational, near-sonic flows are mathematically isolated. In other words, any arbitrary¶ changes in the flow or boundary conditions that provide a shock-free flow will lead to the formation of a shock wave. Thus, Morawetz's theorem stated that the shock-free inviscid flow solutions, if and when they existed, were isolated by neighboring solutions that contain shock waves. Recently, this result has been extended to three dimensions by Cook. 2 Fortunately, it was recognized that such flow would have practical significance if, as seemed likely, the shock waves that occurred in neighboring flows were very weak. Wind tunnel research by Whitcomb 3 at the NASA Langley Research Center and Pearcey 4 at the National Physical Laboratory, United Kingdom, led to the develop- ment of practical "shock-free" airfoil sections. Subsequent analytical studies by Garabedian and Korn, s Nieuwland, 6 Boerstoel, 7 and Sobieczky a established theoretical design procedures for two-dimensional inviscid flows. More recently, the development of sophisticated numerical codes for the analysis of transonic flowfields has led to the design of both airfoils and wings by numerical optimization. 9'_° The practical success of the preceding efforts, as documented by the recent NASA Conference on Advanced Technology Research, _t has been substantial. Further progress, as reported here, seems likely. The senior author recognized that Presented as Paper 78-1114 at the AIAA llth Fluid and Plasmadynamics Conference, Seattle, Wash., July 10-12, 1978; submitted Aug. 18, 1978; revision received Jan. 26, 1979. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1978. All rights reserved. Index categories: Transonic Flow; Configuration Design; Com- putational Methods. *Adjunct Professor; Research Scientist, DFVLR, G6ttingen, Germany. Member AIAA. tSenior Research Associate; presently, Research Scientist, Boeing Commercial Airplane Company, Seattle, Wash. Member AIAA. _scnior Research Associate. Member A1AA. §Professor, Aerospace and Mechanical Engineering and Mathematics. Associate Fellow AIAA. ¶One of the consequences of our research is that, in two dimen- sions, for any small change in the flow Mach number, there are an infinity of small changes in the airfoil shape that will insure that the the procedure he was using in the hodograph plane implied an analogous procedure in the physical plane, and furthermore, that this procedure did not seem to be restricted to two- dimensional flows. _z._3This paper reports the success to date in using this idea to provide shock-free designs in two and three dimensions. The design procedure invoked here is, in principle, a simple one. While there is no guarantee that a shock-free flow will necessarily result from the procedure, our experience in two dimensions has been that if the hodograph method will work for specified flow and airfoil parameters, then the procedure outlined here will work too. Also, it provides neighboring shock-free airfoil shapes for fixed lift coefficient with varying Mach numbers and varying lift coefficient for fixed Mach numbers, as well as providing a multiplicity of closely related shapes that are shock free at fixed lift coefficient and Mach number. This wealth of shock-free, two-dimensional designs is no great surprise; therefore, it is not surprising that they are found with minimal computational effort. Two- dimensional inviscid flow potential airfoil designs require less than a minute of CYBER 175 CPU time and only a few seconds of CDC 7600 CPU time. For three-dimensional flows our results are less extensive. Also, while it is clear that the procedure we use rests on a sound mathematical foundation in two dimensions, this is not the case in three dimensions. Indeed, for three-dimensional (that is, nonplanar and nonaxisymmetric) flows we solve an ill-posed boundary value problem.* ° That this problem can be solved successfully is a result of the practical requirement for specifying chordwise modifications more densely than spanwise ones. We have demonstrated the ability to modify three- dimensional wings so that, within the context of the numerical algorithm used, shock-free flows are obtained. We have not yet demonstrated an analogous wealth of shock-free flows in the three-dimensional case, but see no reason to believe that this situation is different there. The practical consequences of this wealth should prove to be of interest to the aircraft in- dustry. _4 Its success will depend on the designer being able to choose baseline configurations that will result in good off- design performance and not compromise nonaerodynamic requirements. Design Procedure The procedure we use to find shock-free designs assumes that a reliable numerical code is available for computing the flow past a given configuration, such as that sketched in Fig. **The authors are indebted to A. Jameson of the Courant Institute ....... ° • ,.r_ ,. JULY 1979 DESIGNING SHOCK-FREE TRANSONIC CONFIGURATIONS 723 Z M< Fig. ! Sketch of shock-free flow past a lifting wing depicting the sonic surface obtained by introducing fictitious behavior inside this surface thai results in elliptic equations. I. Such codes are available for two- and three-dimensional inviscid flows. When they are coupled with a reliable boundary-layer code, the design procedure outlined here can be used to calculate shock-free viscous flow designs. While this would require some modest iteration, it is certainly possible both in practice and in principle. With the existence of a reliable analysis algorithm presumed we modify this algorithm so that once the flow become hyperbolic we alter the basic equations so that they revert to elliptic behavior. This may be done in a number of ways, but it should be done in a way that it conserves new, but fictitious, "mass" and "momentum" fluxes to a satisfactory degree of accuracy. We may, for example, change the density's dependence from the usual one to one that returns the equations to elliptic form. We might suppose, for the purpose of illustration, that once the equations become parabolic (i.e., sonic) on some surface, then at higher velocities the density will be maintained at its sonic value, giving elliptic equations. We use a numerical algorithm to compute this fictitious flow past a configuration of interest, chosen perhaps on the basis of previous design experience. Because the equations are elliptic, this will result in a discretized, pseudoanalytic description of the velocity, density, and pressure fields on the embedded parabolic surfaces, and this description will be consistent with the correct governing equations. These initial data on the parabolic surfaces are then used to calculate the correct flowfieid inside such surfaces. This new flowfield may or may not contain shock waves. This depends on the choice of the fictitious equations, or perhaps better, fictitious gas, used inside the parabolic surfaces. This new flow will define a stream surface that is tangent to, and has the same curvature as, the stream surface at the intersection of the sonic surface and the original body. Inside this surface a new body shape is defined by the stream surface of the new, but now real, flow. Here, of course, we must also address the question of well- posedness. In two dimensions there is no difficulty because either of the spatial coordinates may be designated as the time- like variable. This is not the case in three dimensions where only the spatial coordinate aligned with the flow is time-like. Because shock-free flows are reversible, the domains of dependence and influence may be interchanged. But neither the normal (nor the binormal) to the stream direction can be considered time-like in the three-dimensional initial-value problem. Thus, data are given on surfaces that are not in the usual domain of dependence and the problem is ill-posed. It is this fact that has made us stress that a reliable analysis algorithm should be the basis for the design computations. An variations in the spanwise direction that are on scale, that is, small compared to the nominal axial (flow direction) distance, will amplify; thus, the success of the numerical algorithm depends upon not introducing such disturbances. This is not the first time ill-posed problems have been solved to obtain results of engineering interest (see, for example, Ref. 15, pp. 448-472). Fictitious Gas As just mentioned, modifications are made to the basic equations to retain their elliptic behavior once the flow has accelerated to sonic speed and a parabolic surface, with the needed initial data, has been generated. The possible modifications are manifold. Our discussion is limited to those used to obtain the results reported here. For two-dimensional flows we have used Jameson's _6._7 circle-plane algorithm for the full potential equation. Thus, in the analysis mode, we are solving with [P_xlx+ [PO: 1:=0 (la) 3,-1 -I//_-tp/p,= l+---ff--M_ (l-ep_-cb2:)] (lb) where _ is the velocity potential and p the density. If we limit our consideration to fictitious gases for which the density is a function of the square of the velocity, viz., p =p (q2), where q2 = U2[_ + ¢_1, then gas laws of the form p/p, =(a./q) P Pa, (lc) will insure elliptic behavior; P= 1 gives parabolic behavior and the fictitious and real gases have the same value of (dp/dq),. An alternative choice, and the one used most extensively here, is P=0; in this case, Eq. (la) becomes Laplace's equation. When the flow would normally be hyperbolic, Eq. (la) is now solved with the density-velocity relationship of Eq. (lc). A fictitious mass flow, which matches the real mass flux at the sonic surface, is thereby conserved and the velocity field remains irrotational. For three-dimensional flows we have used the Ballhaus- Bailey-Frick algorithm,Z_ as implemented by Mason et al. _9 This is a small perturbation calculation, and the classical conservative formulation is adopted here. Thus, in an equivalent form, we solve the system -½(3,+ l)[u2l_+Vy +w: =0 (2a) v:-wy=O (2b) uz-wx=O (2c) wherethe velocity vectoris q=a.[(l + u)i+vj+ wk]. A simple modification, Eq. (2), is to replace l u 2 Ix by - sgn(u) lu z Ix for all u. This system is elliptic, except on the sonic surface where u = 0. We may think of the first of Eqs. (2) as being the consequence of the small perturbation ex- pansion for the density, viz., -1= - u- 3,--- u _1 (3)& p, 2 whereas the fictitious equation, with u replaced by - lul for u > 0, results from 3,+3 . o 1= -u+--u _ (4) ,o. 2 This fictitious gas has the same value for (dp/du), as the real 724 SOBIECZKY,YU FUNG, AND SEEBASS AIAA JOURNAL solve Eqs. (2) with I u 2 Ix replaced by -sgn(u) [ u 2 ]_; this corresponds to using the densities given by Eqs. (3) and (4) for u < 0 and u > 0, respectively. Calculalion of Ihe Hyperbolic Flowfield As previously described, we calculate the flow past a body using the correct equations when the flow is subsonic and a modified, incorrect, set of equations when the flow is supersonic. This calculation serves to define sonic surfaces on which the flowfield calculation is switched from the correct equations to the modified ones. Outside this surface, presuming the trailing edge of the wing is subsonic, the solution satisfies the correct equations, and the potential at infinity has the correct value for the circulation. If infinity in the physical plane is not mapped to a finite part of the computational plane, then there is, in principle, a need to correct the doublet and nonlinear contributions; in practice, these contributions are small and changes in them negligible. Thus, the flow in the elliptic, subsonic domains is fixed and known, as are the initial data we need on the parabolic sur- face. For two-dimensional llows, the calculation of the correct hyperbolic behavior is carried out using the method of characteristics. This is done in a hodograph-like working plane in which the characteristics are orthogonal straight lines. If we take /j=0+v and _=O-v, where 0 is the flow deflection angle and i, the Prandtl-Meyer turning angle, then the velocity potential and stream function satisfy (5) or, equivalently, d_ _._=const = ::t:K-I where the ± signs refer to _,n = const, respectively. Here K(J,) =Kl_,(q)] = I IM2(q) -111 '/'p(O)/p(q) Values for the velocity potential on the parabolic line, z=z* (x), and the shape of this line are used along with the usual relations between the spatial coordinates and + and _bto find _, on the sonic line. These initial data are then integrated using Eqs. (5) to determine the locus _b(x,z) = 0 which passes through the intersection of the sonic line with the body sur- ...... , P :_(.,+. ,._, z ....... i Z't' X J Fig. 2 Sketch of two neighboring isolach surfaces used in the tl+ face. The values of z for which _b(x,z) =0 determine the new body shape. This shape will have the same slope and, at least theoretically, the same curvature, as the original body at the sonic points. This follows from the observation that flow quantities are not changed at the sonic line; thus, the streamwise momentum and normal pressure gradient are unchanged. Consequently, the local flow curvature must be the same. For three-dimensional flows, the calculation of the hyperbolic flowfield is carried out by a procedure that marches inward from the sonic surface by successive surfaces of constant density (isopycnics) for the full potential equation, or constant axial flow speed u for the small per- turbation equation. We limit our discussion to the small perturbation equations, as all the results reported here are derived from them. Preliminary results using the full potential equations have been obtained by N. J. Yu. We may either write Eqs. (2) in the appropriately scaled form or work with them directly as we will do here. We are given an isotach surface z ° (x,y), as shown in Fig. 2, on which we know u=u"=const, w=w*(x,y), and v=v* (x,y). We use the data on this surface, and the surface shape, to calculate • ° * v_. (6)z;, z,',, w,, W,*' v,, Because these data satisfy Eqs. (2), we can verify that v7= z_.w7- z _w7 which can be used, if needed, to check the consistency of the initial data. The values given in Eq. (6) can now be used to calculate the z derivatives of u, w, v on z*(x,y), where u(x,y,z*) =const, by using u: = [ztv,*- z_'vt - w'J/./ w. = [(7 + 1 )u'z._w.*_ - z.*w._.+ v*l/ J v: = [(_,+ ! )u*z_v_ - w,*.- z,rv,*l/.I where J, the Jacobian, 0 (u,v,w)/0 (x,y,z), is: J= ('y + l_u*z .2 - z *2 - 1 (7a) (7b) (7c) When the Jacobian, which is initially negative, vanishes, we can no longer compute the z derivatives; this corresponds to the subsequent formation of multivalued solutions, i.e., limit surfaces. If J=0 occurs before the calculations produce a suitable stream surface defined by w(x,y,O), v(x,y,O), then they must be rejected. With Eq. (7a) inverted to give (dz/du) ,, we take a set in- crement in u, Au, to form a new isotach surface z* (x,y)+Az*(x,y). This new shape, along with the mean value of u between the two surfaces and the second and third of Eqs. (7), provides the new values, w*(x,y)+Aw*(x,y), v* (x,y) +Av* (x,y), of w* and v* on the next isotach. These values and the shape of the subsequent isotach are then converted to continuous functions by one-dimensional cubic splines in the x and y coordinates. This "onion-peel"-iike process is then continued until z =0, unless a limit surface intervenes. In the latter event, the solution must be rejected. A more detailed discussion of this procedure is given in Ref. 20. Two-Dimensional Results We have explored rather extensively some of the modifications that can be made to an existing airfoil, namely, an NACA 64A410 airfoil, to obtain shock-free flow. We will call this the baseline airfoil; the airfoil shapes we generate are ide,ntieal with thi_ airfoil aver lhat nortion wetted bv subsonic JULY1979 DESIGNINGSHOCK-FREETRANSONICCONFIGURATIONS 725 CP A T" ?- ?. B. O (9 @® (9_ (9 ÷÷+_1_" ÷+ ÷ ÷ + + + + ÷ + + ÷ ÷ ÷,t,.÷ ÷ ee + ÷ e(9 ÷÷ O + (9 ÷++ (9 (9 I (9 0 ; ÷_ ÷÷÷ . " ÷÷;(9 ÷_ CP T 8. O. e:l c;' ÷ ÷ ÷ ÷+ -N÷ +++ + ÷ ÷ ÷ ++ _÷++÷+÷++÷÷÷+÷+÷ + ÷ + ÷ ÷++ ÷.¢_ ÷** -÷ ÷ +++ +_ 'l[. +÷+ + +÷++++_ t ÷ (9 O ÷ MFICH = ."/20 flLPHR = 0.40 DESIGN ( ÷ I OR[GINflL ( O ) CL 0 .?029 0.7799 CO 0.0000 0.0064 CI1 -.1397 - .t60t Fig. 3 Comparison of the pressure coefficients and sonic lines for the baseline NACA 64A410 and the shock-free airfoil obtained from it by the direct design procedure. M_ICH = .720 IqLPHR = 0.40 DESIGN ( @ ) ANALYSIS ( ÷ ) CL 0.701Z 0 -7029 CO O.OOOt 0.ooo0 Crl -.1395 -.t397 Fig. 5 Comparison of the pressure coefficient and the sonic line obtained by the design calculation that modifies the airfoil shape with those obtained by compuling the flow past the modified airfoil. • SHO(;K-FREE 0[$15N LIMIT (_¢ SHOCK FREE-O[SIGNS CLj io- 09O OBO 07O ¢ ¢ ¢ 066 068 070 .05// \ / \ ¢ #- T2 0 ?4 0 76 I_ Mao 0 78 Fig. 4 Parameter space explored for the shock-free airfoils thai can be obtained when the baseline cunfiguration is an NACA 64A410 airfoil. flow; we need only modify the airfoil over a limited portion of its upper surface to obtain shock-free flows. Further, this modification is not unique for fixed flight conditions; rather, if one such shape exists, there usually will be a family of modifications of the baseline airfoil that will produce shock- free flow corresponding to different gas laws. Modifications to NACA 0012 and 64A410 airfoils thai resulted in shock-free flows were reported by Eberle in Ref. 21. With a baseline airfoil selected here mainly for illustrative purposes, we then pick a set of flight conditions for which we wish to find a modification of the airfoil shape that will result in shock-free flow. We choose M® =0.72 and the angle of attack _=0.4 deg. At these conditions inviscid flow I'/,a of 0.78 and a C O of 0.0064. The design procedure discussed earlier results in an airfoil that is 9.3% thick with a lift coefficient of 0.703. The original and design pressure coef- ficient, sonic lines and body shapes are compared in Fig. 3a; these results and all other "analysis" results were computed using the numerical algorithm of Ref. 16. Figure 3b compares the pressure coefficients and sonic lines determined by the design procedure with those computed for the design airfoil shape. With this shock-free design established at M=0.72 and with C L =0.70, we now wish to determine the families of shapes that provide shock-free flow for a fixed lift coefficient as the Mach number varies, and a fixed Mach number as the lift coefficient varies. This has been done with P= 0; that is, with a constant density fictitious gas (at the critical value), - requiring an iterative procedure for the case at fixed lift coefficient. Other shapes were then explored that will produce the same lift coefficient, 0.70, at a fixed Mach number by taking P to be -0.5, 0.5, and 1.0. Also, for P=0 we have. """ determined the maximum Mach number for which the desil_n " procedure will produce a shock-free airfoil as a function, of lift coefficient. This Mach number is nearly a linear function of lift coefficient at larger lift coefficients. The slope e,f this variation is consistent with that given by Boerstoel.-': Preliminary studies also indicate that for a fixed lift coef- ficient of 0.6-0.7, an 0.1°70 increase in the maximu,n Mach number requires about an 0.2°7o reduction in the thickness for shock-free flow when the nominal thickness is about 10%. This result is less optimistic than the envelc)pe of the hodograph designs given by Boerstoel, z2 who found that only an 0.1% reduction was required, in our study the generic 726 SOBIECZKY,YU FUNG, AND SEEBASS AIAA JOURNAL it" ORIGINAL AIRFOIL M¢O = 066,_ = 0.89 ° ///_ .68 70 ,70 54 ///,_742;o _ ORIGINAL AIRFOIL P : 1.0, a=0.26" 0.5 36 ° O _40" Y SCRLE : 5m[X sCIqLE) Fig. 6 Shock-free airfoil shapes for fixed lift coefficient CI. =0.70 and varying Mach number. The fictitious gas has a constan! density in the supersonic domain (P=0). The baseline airfoil is an NACA 64A410. Y SCALE = 5.IX SCRLE] Fig. 8 Shock-free airfoils for fixed Math number M= =0.72 and lift coefficient C t =0.70, varying the exponent P of Eq. (Ic) and thus changing the densily's dependence on flow speed. The baseline airfoil is an NACA 64A410. -- ORIGINAL AtRFOIL --CL=0701, a = 04" .776 8 ° 849 I 2 ° 922 116° Y SCQLE = S-IX SCQLE) Fig. 7 Shock-free airfoil shapes for fixed Mach number M= =0.72 and varying tilt coefficient. The fictitious gas has a constant density in the supersonic domain (P=0). The baseline airfoil is an NACA 64A410. modifications required when the baseline airfoil is near the envelope of hodograph designs. Positive values of P provide less airfoil thickness reduction, since the fictitious and real- gas densities are more nearly the same. The range of our airfoil studies is depicted in Fig. 4, with shock-free airfoils being determined for the points indicated. Also shown in Fig. 4 is the maximum Mach number for which a design was found as a function of lift coefficient for P = 0. The accuracy of the design procedure was studied at a number of design points by comparing the design's pressure distribution and sonic line shape with those obtained using the unmodified numerical algorithm to analyze the design airfoil shape. Typical results are shown in Fig. 5. The sonic line shape and initial data on the sonic line are determined in the circle-plane; then they are mapped back to the physical plane. The method of characteristics in the hodograph variables is used to compute the design pressure coefficient corresponding to the calculated airfoil surface shape. The agreement, as shown, is excellent. For designs that approach the Mach number at which a limit line first penetrates the surface, obtain a converged solution. These designs have very rapid expansions immediately following the sonic line. Indeed, as Boerstoe122 has noted, the analysis code used with an op- timization scheme will not produce designs of this character. The shock-free airfoil shapes that are obtained for fixed C L and P, fixed M= and P, and fixed M® and C L at various Ps, are shown in Figs. 6-8. One can overlay the results for fixed C L and find quite similar airfoil shapes that are shock-free over a range of Mach numbers. Because modifications to the baseline airfoil are required only over a limited portion of the upper surface, and a family of specified changes in the airfoil curvature is known for each set of flight conditions, a closely related family of shock-free airfoil shapes can be generated. Thus, the minor modifications to a limited portion of a wing surface needed to produce shock-free flow over a practical range of flight conditions can be easily determined. Three-Dimensional Results Our first design results using the method just described were for two-dimensional, small perturbation flow past a parabolic arc airfoil. Consequently, we initiated our three- dimensional studies with a rectangular, unswept wing having an aspect ratio of six and a parabolic arc airfoil. We utilized the small perturbation approximation, Eqs. (2), and a parabolic thickness distribution; the airfoil was taken to be 6% thick at the centerplane. The flow was calculated using the algorithm of Ref. 19, modified to return the equations to elliptic behavior as described earlier. The initial data on the embedded sonic surface were then used to compute the correct flow in the supersonic domain using the "onion-peel" algorithm of Ref. 20. This defines new wing surface slopes. The flow past this shock-free design was then analyzed using the modified numerical algorithm. Figure 9 compares the pressure distributions on the original and design wing at various lateral positions for M= = 0.87. Also shown are the cross sections of the sonic surface at the same lateral stations. The only essential differences in the pressure occur in the supersonic domain, which is consistent with the design process. The modifications made to the wing s!ope, shown in Fig. 10 for several lateral stations, have eliminated the shock wave. A subsequent, more realistic, calculation was made for the planform sketched in Fig. I 1. The wing section chosen was an NACA 64A410 profile at the center section and an elliptic thickness distribution. The leading-edge sweep was taken to i,_ ,_¢_ J__ .t ...... _t_ .... J. _ It J . __J .l ......... L--J JULY 1979 DESIGNING SHOCK-FREE TRANSONIC CONFIGURATIONS 727 Z*(x,y) y/C / ANALYSIS OF DESIGNED WING ..... ORIGINAL WING Fig. 9 Sonic surface for the shock-free rectangular wing obtained by modifying a wing with a parabolic arc airfoil section and the pressure coefficients on the oril_inal and modified wing, as calculated by the numerical algorithm of Ref. 19. The thickness distribution of the baseline wing is parabolic in y/c. x/c I dz -3 4.0 O- -40- y_ = 0.11 "-"" O: 3 _.5 07 ,'J L X/C Fig. 10 Changes required in the surface slope at various lateral stations to provide shock-free flow over the rectangular wing of Fig. 9. ratio was 5. The sonic surface is also depicted in Fig. II. Figure 12 compares the pressure coefficients on the upper surface of the original wing and the wing designed to be shock free. While the reduction in drag for this wing is small compared to the induced drag, it is clear that the wing modifications have essentially eliminated the shock waves, and, consequently, the wave drag. More importantly, shock wave induced boundary-layer separation is avoided. Im- pressive results for the ONERA M6 planform have been obtained by Yu 23 using the full potential equation. At this point we stress that the preceding comparison is obtained by computing the flow past the original wing and the (_P(lion tlt/;no ii¢:;no lh_ = ¢¢im_ nnlm_rit-Ql _lnnrilhm Th. nr_o_e¢ that leads to the new wing shape also provides the pressure on the wing. Conclusion A novel and simple procedure for determining modifications that will make a baseline configuration shock- free for supercritical flight conditions has been delineated. For two-dimensional, inviscid flows, shock-free designs are obtained in seconds on a CYBER 175. Families of airfoils that are shock free at fixed, as well as varying, flight conditions are found. The same procedure has been applied to three- dimensional wings, resulting in wing modifications that make the wings shock-free when the flow is analyzed with the numerical algorithm that was modified to become a design tool. It can also be applied to the design of shock-free cascades. A unique feature of the procedure is that any code that is effective in computing the flowfield may be modified in various ways to be a design algorithm, if it is coupled with a method for calculating the solution in the supersonic domains for given data on the sonic surfaces. A straightforward marching technique for such computations is described for three-dimensional flows; in two dimensions, either the marching procedure or the method of characteristics may be used for the supersonic domain. The algorithm for the supersonic domain serves to define the modifications needed in the configuration to achieve shock-free flow; these modifications will be limited to that portion of the design 1" I f t 728 SOBIECZKY, YU, FUNG, AND SEEBASS AIAA JOURNAL j, z'(x,y) I0" o o 25 20 Fig. I I Sonic surface on the shock-free swepl wing designed from a wing with an NACA 64A410 center section profile and an elliptic thickness distribution. The leading ed2e sweep is 30 deg; the trailing edge sweep 15 deg. _R=5 0 0 y/c 25 20 / I0 ANALYSIS OF DESIGNED #lING ..... ORIGINAL WING x/C _0 Fig. 12 Comparison of the computed pressure coeffcien! on the wing of Fig. It with an NACA 64A410 center section profile and an elliptic Ihickness dislribution, with the pressure coefficien! obtained bv compuling the flow pasl the modified wing using the same numerical algorithm. Acknowledgment This research was carried out by the Computational Mechanics Laboratory of the Department of Aerospace and Mechanical Engineering. The work was supported by AFOSR Grant 76-2954E, NASA Grant NSG-2112, and ONR Contract N00014-76-C-0182. The authors are indebted to the technical monitors for constructive comments and encouragement during the course of this investigation. They also wish to *k_b L] T_,T_k_r_[_ I_rlr r.,_lrrv, llno G'llll tho P_lPlll_til')ll_ for the airfoil studies and P. DeShazo for the development of sup- porting algorithms. References t Morawetz, C. S., "On the Non-Existence of Continuous Transonic Flows Past Profiles, I, I1, and III," Communications on Pure and Applied Mathematics, Vols. 9, 10, and 11, 1956, 1957, and 1958. PP. 45-68. 107-131, and 129-144. JULY 1979 DESIGNING SHOCK-FREE TRANSONIC CONFIGURATIONS 729 2Cook, L. P., "A Uniqueness Proof for a Transonic Flow Problem," Indiana Uniuersity Mathematical Journal, Vol. 27, Jan.- Feb. 1977, pp. 51-72. 3Whitcomb, R. T. and Clark, L. R., "An Airfoil Shape for Ef- ficient Flight at Supercritical Mach Numbers," NASA TM X-1109 (confidential report), July 1965. 4 Pearcey, H. H., "The Aerodynamic Design of Section Shapes for Swept Wings," Aduances in Aeronautical Sciences, Vol. 3, 1962, pp. 277-322. 5Garabedian, P. R. and Korn, D. G., "Numerical Design of Transonic Airfoils," Numerical Solution to Partial Differential Equations, Vol. II, Academic Press, New York, 1978, pp. 253-271. 6Nieuwland, G. Y., "Transonic Potential Flow Around a Family of Quasi-Elliptical Airfoil Sections," Nationaal Lucht-en Ruim- tevaart Laboratorium Rept. TR-T 172, Amsterdam, The Netherlands, 1967. 7 Boerstoel, J. W., "Design and Analysis of a Hodograph Method for the Calculation of Supereritical Shock-Free Airfoils," Nationaal Lucht-en Ruimtevaart Laboratorium Rept. TR 77046 U, Amsterdam, The Netherlands, 1977; also Boerstoel, J. W, and Huizing, G. H., "Transonic Shock-Free Aerofoil Design by an Analytic Hodograph Method," AIAA Paper 74-539, Palo Alto, Calif., June 1974. SSobieczky, H., "Entwurf fiberkritischer Profile mit Hilfe der rheoelektrischen Analogie," Deutsche Forschungs-und Ver- suchsanstalt f_ir Luft-und Raumfahrt Rept. DLR-FB 75-43, Gdt- tingen, West Germany, 1975. '_Hicks, R. M. and Vanderplaats, G. N., "Application of Numerical Optimization to the Design of Supercritical Airfoils Without Drag-Creep," SAE Paper 770440, Business Aircraft Meeting, 1977. )°Hicks, R. and Henne, P. A., "Wing Design by Numerical Op- timization," AIAA Paper 77-12,17, Seattle, Wash., Aug. 1977. 11Ach'an('ed Technolog), Atrfoil Research, Proceedings ATAR Conference, NASA Langley Research Center, March 1978. 12Sobieczky, H., "Die Berechnung Iokaler Rfiumlicher Uber- sehallfelder," lecture at Gesellschaft for angewandte Mathematik und Mechanik, Copenhagen, May-June 1977, ZAMM 58T, 1978, pp. 215- 216. JSSobieczky, H., "Transformation Methods for Three- Dimensional Compressible Flow," Deutsche Forschungs-und Ver- suchsanstalt ftir Luft-und Raumfahrt Rept., Gdttingen, West Ger- many, in preparation. 14Sobieczky, H. and Seebass, A. R., "Adaptive Airfoils and Wings for Shock-Free Supercritical Flight," Invention Disclosure, University of Arizona, Tucson, Ariz., May 1978. ISHayes, W. D. and Probstein, R. F., Hypersonic Flow Theory, Vol. I, lnuiscidFIows, Academic Press, New York, 1966. 16jameson, A., "Iterative Solution of Transonic Flows Over Airfoils and Wings," Communications on Pure and Applied Mathematics, Vol. 27, 1974, pp. 283-309. 17Bauer, F., Garabedian, P., Korn, D., and Jameson, A., "Supercritical Wing Sections," Lecture Notes in Economics and Mathematical Systems, M. Beckmann and H. P. Kunzi (Eds.), Vol. 108, Springer-Verlag, Berlin, Heidelberg, New York, 1975. 18Ballhaus, W. F., Bailey, F. R., and Frick, J., "Improved Computational Treatement of Transonic Flow about Swept Wings," Aduances in Engineering Sciences, NASA CP-2001, 1976. JgMason, W. H., Mackenzie, D., Stern, M., Ballhaus, W. F., and Frick, J., "An Automated Procedure for Computing the Three- Dimensional Transonic Flow over Wing-Body Combinations, In- cluding Viscous Effects," Vols. I and II, Air Force Flight Dynamic Laboratory Rept. AFFDL-TR-77, Wright-Patterson AFB, Ohio, Feb. 1978. 2°Sobieczky, H., "A Computational Algorithm for Embedded Supersonic Flow Domains," University of Arizona Engineering Experiment Station Rept., Tucson, Ariz., July 1978. 2tEberle, A., "Profiloptimierung for transsonische Strdmung mittels der Methoden der finiten EJememe und Charakteristiken," MBB Report UFE 1362 (0"), 1977. 'JBoerstoel, J. W., "Review of the Application of Hodograph Theory to Transonic Airfoil Design and Theoretical and Experimental Analysis of Shock-Free Airfoils," Symposium Transs'onicum II, K. Oswatitsch and D. Rues (Eds.), Springer-Vertag, Berlin, Heidelberg, New York, 1976, pp. 109-133. -'-_Yu, N. J., "An Efficient Transonic Shock-Free Wing Redesign Procedure Using a Fictitious Gas Method," AIAA Paper 79-0075, New Orleans, La., Jan. 1979. Iqq RelatedAnalytical,Analogand Numerical Methods _nTransonic Airfoil gssign HELMUTSOBIECZKY DFVLRGOTTINGEN,WEST GERMAMY ORIGINAL PAGE ]_ 0_'I_DOR (,tUABI'rY AIAA 12thFLUIDABJDPLAS_]A DYB]AI_I]-CSCOHFEF:JE_JCE July 23-25, 1979/ Williamsburg,Virginia. For _erm:s;ion to ,:o_yor raoubllsh contact the American !'_sl]tute ol Aeronautics anOAstronautics.lZgOAvenue o!the ,_mericas. New Y,irk..=;.Y. I C0i9 /.:_C RELATED ANALYTICAL, ANALOG AND _NUMERICAL METHODS i_:4TRA.XSO:NIC AIRFOIL DESIGN Helmut Sobiec zky*, DFVLR G_;ttingen. West Germany, Abstracl Design methods for _ransonic airfoils are pres- enzed together with a review of the preceding anatytica! and analog flow szudies both in hodo- graph and physica_ space. Transonic analytical results lor cusped airfoils in sonic f_ow provide detai_s of solution structures in hodograpll planes to formu!ate a new transonic boundary value problem. This is used subsequently for extension of the c_assical rheoelectric analog- for sub- sonic flow modelling, into the transonic regime. Airfoil design with lhe resulting hybrid tech- nique is described. Rep'_acement of the analog}, by numerical so_ver routines in H_e hodograph p_ane finally leads _o an application in physical space and the development of effective computer codes suitable for design of shock-free airfoil families including adaptive wing sections. 1. IX TRODUC TIO.N The deve'_opment of airfoils in the past has ahvays been an importam fn-st step in design aerodynamics. _Iore recentt,,, requirements for increased efficiency nave forced the operating conditions of aircraft and turbomachinery imo the _ransonic regime. Broad emphasis is s'dU. 'mid, therefore, on the developmen_ of two- dimensiona _. design componems like airfoils and cascades in _he high speed regimel . Vari- ous computational algorithms have been devel- oped for analysis of ;ransonic f_ow past given a_.rfoi[s, b'a_ only few metl_ods are avaitabteto tl_e design engineer for airfoil shape definiti0n wi_h specified properties. Adjunct Professor, University of Arizona; Member, AIAA iKi ! Research reported here was carried ou: within tile past six years at the DYVLR in German} and since 1977 at the University of Arizona. It is \he purpose of this paper to demonstrate coherence belween the first basic ana'.ydcal n_c,dets and recent effective computer codes resulting fron] our research, including a method which belongs actuaU.y into pre-computer ,.ime but proved to have a high educational va_ue: the Rheoe_ectric Analog'. This method was used two decades ago for investigation of cornp'_icated two- and three-dimensional po_en_.lai distributions and had useful appl.ications in aerodynamics. tlowever, principal difficulties arised in the higt: speed subsonic flow regime which prohibited an application for transonic f_ow research when s'A:_ no practlcalty useful calculation me:hods for :ius field of growing importance were avai_abW. These difficulties, however, were overcome by H_e author in the case of design problems with the aid of the aforerr, emioned analytical findings and an experimental ana;og procedure was deve'_- oped to obtain airfoi_ designs of sausfactor 5" accuracy. In the meantime, within the past five years, we witnessed the rapid development and widely dis- tributed avaitabitizy of numerical soh'er routines forpartial differential equations. It was, "_here- fore, necessary to _ransfer gained experience and techniques from the analog f_ow expert.meres to more economical digital computer programming. Application of numerical subroutines and finally the conversion of whole well-established analysis codes into effective design too'.s are our currem practical results resting on previous basic experimems and solutions. 2. B_LTRAMI D_F_ "FR_..NT._L. EQUATION ;_N PLANE POTENTIAL THEO}IY fi ! Potential f_ow in 2D physical space V,e consider s!eady, two-din_.ensiona, lsenlropic and irrotaziona' flow o:'a po_.y'.ropic, inviscid gas The basic equations ol too-ion are ',l_en de_ez'mined by dl\" {_ 7) = 0 , {l a_ cur', (_,) : 0 , (1 b) :he continuity equation and irrozationati_y, re- .4 spectivety, with _ the density and v the ve[oc- i',y vector in pt_ysical space. Isentropic gas properties determine \eloci:y q , sonic velocity a and density _ as funcuons of the Math num- ber A1, for given stagnauon condizions, denmed here v,'izhsubscript 0: q : _' = aA1 2 2 _, -1 2 (2) a = a0---- 4- q - 1 -_ 2) =Q:' : (1 - ,-'7-- hi ",-1 F (AI) ('0 \\i'll (la, lb_ we n.ay define a \e'mcib potential q) and a s_rearr, funcnon _', with _helr gradients in me two directions x, v of 2D physical space equal to the velocity components u, \' in these d: rections: (% lt- _>._: _ ," Ro (3! % : u = q cos t}, =v =qsin@, where @ is the f:ow angle. The system (3) is a generalisation of the Cauchy- Riemann equations, socatted Behrami equations. E[iminaUonof l( or (_) yields Poisson equations for (_) or If, respectively: c @v It'xx " _,-v.. q _'x " -Q- _:\'. (4 b) wi:h _ a function (2t of _I, and therefore = = {j- _ , (5) the systen: ;3a and _he equations _4: are non'dnear Fu:'tnerrnore, the systen, is o:" el'-ip'ic '_ype when ..%!e _., and of hyperbolic type.. \vhen _'k] > i , with a parabo'ic :ype dividing Line %[ : 1, ti,e son:c '.ine. 2. '2 Po_en:ia'. flow in the Hodog:apl_ plane The aforementioned non[ineari: 3 of the basic system (3} may be avoided if a new palr of inde- penden', variables is introduced _o replace phy-z:- ca', coordinates x, 5. These variables are suit- able functions of the velocity components, ;hey are ca[[ed hodograph variables. A speela', pair of such variables is consisting of :h_ flow angle and a func'-ion of the Mach number, known also as the Prandtl-Meyer _urning angle " q __{, O v = :M--I q wuh a" defining the critical ve:oeiD. The coefficient Qo _ , o K = K(M(u) :-- M-- i _7} e wilt also be used in lhe foFtowing system. The new variables u, @ may eitl_er be used directly to define a hodograph plane wherein ti,e basic Be[:rarni system becomes _.inear: > > qb = : K(v)I_" 8, (v < o, .M < l i f};} _ ) or _ and @ more generally are functions of a computational working plane obtained frorr, _l_e basic v, @ hodograph by conformat (for hi < 1) or characteristic (for M _ 1) mapping. For sub- sonic flow including sonic conditions, (El < I ,. v < 0!, conformat mapping defines a worl:in_ plane _, with the mapping function E. The basic system in _ becomes then v.'ilh (10) -1 Iv _s, t! : Re (E ([)_ 8 is, "" : I.r: {E -1 ([)). (II' Equations (I0) fornt, as (8), a Linear Beltrami system, white i3) is nonlinear. Elimination of h" or _) yields linear Poisson equations for (_ or _', respectively: K K + _xt s + K K K t 'fss "__::t : - -_ 'is " -ff _ft (12a) (12b] As we will see tater, boundary value problems for" practicaLLy interesting solutions of the basic system (8_ may be significantly simpler to solve in a working plane _ with (i0) raxher than it. the ori_na[ _(} where (8) is valid. The same is ".rue, in principle, for tDe super- sonic part of =lle flow. Here we introduce characleris:ic variables with a suitable mapping !unction H, (13) yielding lhe sys_ern vahd in the _, _I plane (_¢ : K(v(¢,_)) h_ (_n --'n(v({,_)) _'n (14) or equivalently, dI_" : K-1 d--_ : HS) _, _ = consT which is the basic re¿aLton for the me:hod of characteristics to integrate !he flow equations (8) for 51 > 1. 2. 3 .Near sonic flow in tl_e hodograph plane A g:ven solution of (8) a_Lows the integration of physical coordinates x, y with the formul_ dz : dx + ldy : e C+)- i-- dIS_!q ,,, For f'mws with on:)" sn-,a"L perturbations :0 a sonic parallel flow, (hl - l) _l, (17_ •b < r/2 we may e_iminate _) and l( so tha: a bamc system for the physica_ plane coordinates x. y is obtained. Furthermore, imroducuon of a similarity parameter a aKows the use o:" re- duced variables forptace (x, y) and state Iq, _ which comain the we'l-known Transonic simi- Larity _aws _ 3-I I I y_ _ ,_ ,oS = : '' _" (-}-'1) '" 1--= '""- - k a_ ; -I 7 =g • x_ X = (_)/'a* 1 .1 y = oi/3. 31/312-1 (.).l)t;.-I 3 . _/'a- xvi:!: posi:i,,,e S for q__ a* and nega:i\e 5 for q __ a*, thus S = 0 equ_\,atent to sonic f[ox_ con- di:ions. The basic system (8) n_en yields a corresponding Bettrami system for the reduced ph>'sxcal plane Variab_,es X, Y in the reduced \_ariab_es of sta:e working plane $, T: XS = : SI/31 YT' XT = S1 / 3, YS Lineari_y again, and the simple s_ructure of the coefficient gave rise to extensive s:udies of this svstern and :he struc:ure of its solutions. I, is equivalent _.o the well known Tricomi equation for near sonic flow _. ALso, i: is a special case 4 of GeneraLized Axisymrnetric Potential Theory Numerous particular solutions were described $ and_.se¢for betterunderstandingol experin_en- tallyobsem,edtransonicflowphenomenaat a ume,whencomputersandnumericalmetl]o_s werestill notavailable.Ananatytica'_example fo:"_ransonicalrfoi! fLov:wigi'Austra_ethe possibilitiesofthisapproach 2 4 Electric Po_entia'.. in a Plane Conductor Let us consider the distribution of electric cur- ten: _n a three-dirn.ensionaL conductor. Let E be rile electmcMpoten_iat and k (x, y, zl be the conductiviLv The intensity of current, di, which crosses a surface element, dS is given byOhm's law: dE di : - }, --r-- dS (20) on where n is the surface normai to dS. In the case o5 a :wo-dtmensional (x, yl conductor, variable conduc:ix'iLv can be sin,ulated by constant conduc- ',:\icy bu_ variable thickness distribution, h (x, y), o: ".he conductor The current imensiLv, di, •":ossm_ The surface e_ement, dS, described b\ 0, 0). A transonic flow example wilh occurrence of mixed subsonic - supersonic fk,w, say, a local supersonic region embedded in subsonic flow, with smooth :ransition of the f_ow properties across the sonic line wit[, there- fore, map into contacting regions E and H in Rheograph ¢0' see Fig. ! a, b. \\e wish to de- scribe quantitatixety a solution of system (8) representing such a flow and ask for a method. J J First, we omit the change of sign in the !l"s', of equations (8). \Ve take toe negative sign for bolh > hatf-p[anes u < 0, thus !aa\ing an elliptic system for' the subsonic and lhe supersonic Rheograpl. _0" \Ve now define a boundary ,'a_ue problem for In._- linear, elliptic system, as sketched in Fig 2 a. I: is well posed and we assume to ha', e a method to obtain a sotutLon. Tiffs solution wiK. locally, be one of the correct mixed type system (8) in region E 1 where V < 0, bu_ it is a f!c':- tious one in E o for v > 0, because real con,- pressible flow requires solution of the hyper- bolic mart of (8} witl_ the possitive sign for _ > 0. The solution in E o has l_ere the purpose to pro- ride a reasonable solution in E 1 with sonic line data (271 This can be achieved a_so with some modificauor. of the fictitious e_%iptic system in E_: The coefficient Kiv) can be changed in some prescribed way, as tong as it stays rea'_ and positive in E o. One po-_sibi'_itv is taking sin',ply K (v • 0) : K r = const (28) :...-, Fig. 1 E, ii t:,,",!i ;'//y a a a 2 3 b b b • ), u. v _ •;6V ::_" a 4 b _- tl, "V E, Figure I - 4. EUiptic Contznuation PrincipLe \vh:¢hwould resuh in a Cauchy-Riemann system in ]22• In this case, zhe part in E,) of the etD.ptic soLu',1on can then easily be dvscribed ana;.ytzcatly if '.he resuhmg data _)_, t(*(O] at v = 0 are ex- panded In terms of a t_armonic analysis . Tl_ere is also a physical in'_erprelation for this artificia_ solution in the ftoxv p_ane x, y if V in E:_ is reimerprezed q]E : __ ¢9q]V (E2'b = YE _n a_ _-_ and K E takes the value _o/Q*: The solution of the eLhptic systerr: in E 2 repre- sents an example of "supersonic incompressible "_Low"with critical constant densi_yQ*, en_bedded into tl_e subsonic compressible solution obtained in E l , see Fig 2b. Sirean_tines, and most importan:, the s]ream_ine _' = _'l = ¥[B = const defining our flow boundary for this hctitious _Low, are integrated by use of (16) with the re- "_ociLv variables qE' _' and Q = Q* ]'lie whole solution in E 1 "- Eo results tn aftov. with densi:y obeying isentrop_c f_ow relations (2) only up to sonic ve_ociLv, beyond i_ density is ::-ozen zo ",he critical value. This interpretation Led to a design 11 method which is no_ restricted "_otwo-dimen- sional flow, results wiZt be presemed !a'er \Ve re_urn now to our problem in tlae plane _0" We stilt have to solve the equa'.ions for the rea _. supersonic part of thefLow, represented by tl]e hyperbolic system (8) with positive sign and \%_id in the half plane _0 (v > 0}. We choose _he characteristic form of this system as ou_hned in (1 3) - (1 5). \Vi'_h the given data _)" , li'* along tl_e _%-axis in the given imer\at A B we car, solve this initial value problem a _, the sonic },ine with the me_hod of characteristics..A'Ahough welt known and used for man}" practical prob'.ems we would like _o stress the fact, tha_ we solve /o_L, the sys',em in tl_e charac:erxsUc triangle ABC (Fig. 3a) by ca_.cu!at:ngdowns_ream aLc, ng characteristics _ = const, and u_strean, along characteristics N = consL with 6, D defined in (13). Starting at AB v:e proceed toward C, the method therefore being a marching procedure normal to the flow direcuon, from _he sonic hne to a suriace s:reamtine ye_ to be determined. Tt:is concep: _s, in principle, also used in a procedure to calculate three-dimensionaL flow fields 12'13. A line It: -- I_ = lfB = const is found in triangle ABC (it is different from the pre- scribed boundary in E 2 t) and if it does not inter- sect one of the characteristics _ = const and -- cons'., more than once, then i_s integration (i6l in the physical plane, see Fig. 3b, will give a new streamline arc AS and, along iL a veioc- it)' and pressure distribution. V,e use on'_y the part between this s_ream_ine and the some line for oar fIow example and call this flow field tI 1. \Ve go back now to our a'.l-e_hp:ic solution E 1 ,- E_, Fig. 2b, and replace the par: E o and also _he surface, streanAine arc AB by the solution ti 1 and i_s new arc A B of Fig. 3h This gives us a rrnxed subsonic - supersonic flow field which is a solution of the linear mixed system ,8, in "he hodograph pl.ane, Fig. 4a, bul also one of Ire nonhnear rr:ixed system (3}, or equations ;4_, :n the pnysica', plane, Fig. 4b. It can be snown, lha: the new arc A E of H 1 fits smooti_ty into the E 1 subsonic flow boundary, streamline curvature across an?' poin I on lhe sonic line is conlinuous. We l_,ave oultined a method to obtain solutions for transonic flow, to be applied mainly to sub- sonic f_ows with embedded local supersonic regions. Applications to flows with predommantty supersonic flow and embedded subsonic regions lnvoLx'e [!-_e treatment of bow and Iai', shock waves, resut:s have been obtained for alrfOi _. flow with sonic or s_ight[ N supersonic frees:ream conditions on".y in special cases where anatyucai solutions of the near sonic equations 119) were applicable. An example will be iLLustrated ta_er, to shoe: _rans,-_lo.'-. fron, the prob',err, of supcreri:_ca'. airfoil flow with subsonic freestrean conditions. to sonic and slightly supersonic frees'-rean_ con- diiions. However, supercrltlca[ f'_ov:is our' matin concern here, and more prec'isel.v, :he use of "rileidea ouQined for design of such f_.ows wD.-ch are sl]ock- free. _. 2 The Rheograph structure of supercm::ca- airfoil flow The structure of sul, ercriticaL airfoi __floe' is wet'. known and needs no explanation here However, some details are treated here shor'&y because they are of consequence for the p:.-ac:_ca! ,.ndlrec: desiyn method which will be outlined -a_er. IVe know from incompressible flow past iif:_.ng airfoils, that the iso_achs in the flow field near the pressure (_ower) surface exhibit a saddie- point. This is the resuh of locally ccntractin@ s_reanAines due to 'he far fie/d-effec_ive .:'ir- cuLation and the near field-effec:ive body :n:ck-. ness. For compressible flow incLud!n_ sul,er- critical conditions with or without a recompres- sion shock, this is equally true, Lines of cons=anl local .Math number forn: a saddlepoin- N be:.ov.- the Lifting airfoil, see Fig. 5 a. Tins poim is of mterest for the mapping of a, say, g:,en rests/; of airfoil flow into oar t{heo_raph !_,Lane _0' • _)c, _z'_M • const. o ........ l_ i .r_._ ,._-c-- : \ I b f:---------_- c Figure 5. Sadd_epoint, lift coefficient and Rneographs _'0' _2 because we want to know the principal structure of tl]e boundary condltions for sucl: f:ows in order to design new examples. The airioiLimage in 40 for shock-free flow shows two complications in view of formulating a closed eLLiptic boundary \'aLue prob'Lem according to tl_e firs: step of our design procedure: First, the stagnation point of the airfoil is mapped into w (!%1: O) : - co. Second, a part of the flow-field obviously covers the plane _0 twice, as indicated by tile loop in the airfoil image. The structure of tile field image has to be completed now wi_h the mapping of the afore- mentioned saddtepoint i'<, defined by v (MN), __N A second Riemann sheet provides the sec- ond deck of _0' it is connected with the basic deck along a cut from the airfoil mapping inter- section to the point N, forming a branchpoint in the Rheograph _0" A detailed description of the mathematical structure of these flow prop- erties has been given in 14. In order to arrive at a single-sheeted boundary value problem of c_osed, finile structure we perform now two n,appings (9): firs_ the stagnation point S is moved into a finite domain with the mapping C0 CI : e (30) An(_ther mapping unwraps the loop of the airfoil image and we obtain a single sheeted domain by 11/2 (31) : eV(M N) + i with c an arbitrary sca_ing constant. The airfoil image maps in _o into a closed curve including the stagnation point S as illustrated in Fig. 5c. The aforementioned saddtepoin_ maps into the origin, the sonic hne into a Cassini curve or outer temniscate with half axes a, b. The ratio b/a is a function of the teen[ Mach number M N in the saddiepoint: b/a = ,/1 - e v(MN) I + e v(AIN) (32) "I'iae value M N is related to the Mach number at i infinity gl in a similar way as tile velocities do in tl:e sadd[epoint and at infiniD" for an intern_ pressib_e f_ow example past a Joukov.'sky airfei! or a circular cylinder with circulation. These latter examples are known anatytica'Ay and fron- these we arrive a_ the ratio MN/'Mco as a func- tion of the Lift coefficient c L : A o _N/Mco ~ I -A_L (331 wittt a constant A. The circular cylinder exa.,.ote gives at least an idea about the magnitude of A: A _ I/(2 =21. (34) These relations invite to be checked on airfoil flow examples. We have a possibility to do this with existing results for hodograph supercritica! airfoil design examples by Nieuwtand Is, Boerstoe'_ 14, or by Garabedian and Kern 16. Some of these authors' designs are evaluated in Fig. 6, we see that the given relations [33t, (341 are fulfilled satisfactory for not too ta,"ge c L. \Ve conclude that for given (b/a) in (321 and for given hl co obsioustv, a certain band of c L is possible, We stress this fact because we wilt later use an electric analog flow too. which -l, work with devices designed for fixed b/a where the given relations and the diagram Fig. 6 pro- , b _a). vides possible lift coefficients c L (hi [ lit! ' i Ii , Nieuwlond, Boer S toe! x ( vorious designs] Gorabediom, gorn + ( various designs ) 0 (designs and redesigns} J M. 2 r_" 03 O9 l 1 \I \ * I i * 08 0 10 c, __ Figure 6. Mach number ratio vs. lift coefficient ":"3 Free strean, s!n_u[am,;es zn :he Rneo_ra!.,!_ nlane \Ve further tnves_!{.ate tile structure of supe:- cril:ca[ flow in our working plane _o \Vith the mapping of _he airfoi', into a closed cur',e as sketched in Kig 5c. the domain enclosed is ! to sonic free stream _nvo'tving tn_ C12 far fle[d of a delached bow wave, The !after , . 18 prob%em is solved anaqwtzcalLy with use of _he [:'ansonic shock po_.ar mapped into the near sonic Rl_eograph Fig. S c \V, giv_ some detailed i'_ustrations for the afore- mentioned ana_.ytica / resuhs of cusped a::';¢,i_ f'_ow in F_gu:-c-s 9- 11 ahhougl_ theiF va',ue for practical fmws _s [irr.i_ed. On the other ha_d, these resu'As represenl educatlona'_ _xan,p'.es -o:- lI'ansonlc flow phenon_ena, whe,'e ihe ;.,robLem is sc,_.\ed for _he subsonic part firs:, with the supersonic part either given analytical:) to_ether Wilh the subsonic res_:ts, or being ca[cu'.ated s:arting at suitab'_e initia" conditions proxuded by tile st,'bsonle sol.uiiorL % : l -_ ! l I b "k_4"_ 11 ,/ Figure 8. Rheograph _o for freestrean-, conditions M __ i (3O -._ 5 l---r - _./J i 5- |)" _.C [ IT'C . [y::T _ (7- C} 22_r -'2"23"_'_5s/c( )"" I.': terms of alrlof, geometry, this extent :s illustrated in ]:'lg. !I b for a Gude,*te) c___p Stand - off distance of a detached bow v."..,, e _s obtained, e.g. for a I0_ thick airfoil the boy. wa_e attaches at I',! = i. IT. The exter; t e: the O0 Locai. subsonic field norn-,a _.to H_e fLo_ d!:e.ztion i_ Large, a_ the far fte_d solution Fig !] a mall- cares. This is important for _'md tunne', te_:s with detached bow waves, wllere the tunnel wa'A should no t be reached by the subsonic field. A 10{ thick Guderley - airfoi', placed in a wind tunnel at hl = I. 15 for instance, requires a CO distance from airfoil to wall, of about /our times the chord Length, wl_iLe the bow wave stand off distance from the cusp is then on[)" a f;hh ¢,f the chord Length. Figure 9. Cusped Lifting airfoil in sonic freest re a _-, in Fl£ 9 the cusped airfoi', and its geometry !o:'n._'._ _ d:awr, ]'tle _ont( fYee-strearr, ) = ittas a co.--tart, an_.ie o:" attack, a . whlcl_ Le.',dF to sn:ooth entry eond::ions, with G but also tt, e ioca: pressure on the airfoil, [if: and d:'a_ "'_nction_ Of tt_e camber./ttnckness ra;io _z, I", see ]rig lOa, b. -l'lus anaLytica: resuh :_: a zenera[isation of Gude:'!c:,"s cusp, whe_-e 17 .; r : (: a detailed cescmpt]on :s _iven in There are results also for supersonlc Mach nun.bets, Fig. 11 a shows "_configuration of a btm w;.',(, and. the _.oca': £u't_£omc :a: fleid In a stfr.i_arltv _i , ..o_ p_ane, wLich iLLustrates the extent of a Local subsonic reg,on for hl -* 1. (.IO r [ " / ,/ "_"i': 5 J: C_ 2- C! ; • f _ ;_ '2 _ Ig Yi_,,:'_ iO. Cusped _.ifting airfo[U Smoo:r_ entry I / t, 2"J_7,, ','z /, / / • / conditions. Lift, drag ,/ /' // : o' \ _'//!,,_ , / I "\ ¢ -. I \ '[ : _.4 ..... \, [-:t.f.:"-,U i: I:] .2,'2 Figure 11. Detached bow wave: a_ Similarity so'.ution for .M -' : • (t.* b) Cusped air!oil stand-off d:stance 4. RHEOELECTRIC ANALOGY 4. I. Rheoe[ectric tank for transonic flow analog) The structure of tt_e particular solutions u;ec for tt:e ;Kustrated examples g:x'es informa:_o:-. also abou: the detai_s of ttae n.appinZ near t;,_- sonic hne. b' = O. It ;s easy to verify, tha: a SOlUtiOn for _), _" Of system fd_, describing an e;en:enz of curved P,ow in transition fron. sub- somu to supersomc flow, or reverse, i_ de- scribed by the "oca'_%y valid expansirm in _0' IL • .') q,_v.+_=<.r*v_'-c'- z,.4/3. V_v_ -O,v-,e! (3(;a' t';Cv,_?. _-_(O!-c ''" _- - O_.'i-_'_-C(v',O_ (:_(;b,) This ts a very weakly .¢.inguLarbehavior into the 1_'- direction, "_ is a consequence of _eneral.ized axis.vrr, n-e:ric polen'`ia! _heory n-_entloned ear;de:" and k.low_edge of this sl:-u,c'ure enables us lo a.oid certain d:fficul'.les occurrin_ when a so'_u- ,'.ion (_) or (10, in _ has to be evaluated The :o_towing descriplion of transonic rheoeieclric analoG, n:ainly concemrates on outlining a tech- nical solution for problems stemrnin._ fron_ _his s _n gu _a :'i'..v. F tort; definilions (25), (2(;_ for the two types of a:-a'Lo_y we see "ha" local thickness h of a p'Lane conductor, rr, ut',ip[ied by i_s conductivity )t, is ana;.o_ovs _o _he coefficien5 Ii in analog'5." B, or -1 . _,:, :he reciproke K in analogy A. As we see direc:'..y frolr. ',he nea" sonic equations (19), the coefflc_en_ has a cubic root zero at the sonic !._ne ,I'3 Ii (u -' 01 : K* _ I 1_ {37_ which to.ads '.o _he above mentioned exponents 4. 3 and 2/3 occurring in expansion (3a/. The r'..ecessary requirement of a s_owly \'ariing thick- ness I_ of a p':ane conductor for \'alidity of the t,'_ant- t-,_'Ara.'_.i _ys_er:- {24 _ for _he e[eclrica_ ' ariab[es is therefore no', fulfilled near lht2 sonic "_!ne: ¢onduclor thickness wou_d t:ave io go to :nfini_?." or to _'ero wilh steep gradienl. A'n?os, c_ass:ca[ apphcauons of the ana_o_y to simulate aerodynamic problems some decades a_o include tile con_.pressib'_e flow hodograph for subsonic f!.ow _'I9, but _ess actuality of _rans- onic f'_ow a_ this time in genera_ and the above n_emioned iin:itations s_emming fron-, lhp zero in :he coefficiem (37} prohibiled an efficient exten- s:on of tne analogy into :he ,Math number uni:\- reg:rne The frst aE_hor's research on tile aforemeniioned ana_x'tica- slructure of Iransonic f'_ow in ',he modiI'_.ed hodograph plane, especially re[allan '31) h:'d it, a pr'ac,_ca', design of a new _ect:-o_.)'ti( tank - with wa:er used as cond:mto:- - which a'_Lows an electric Conlirluatiori of ti_e ana'_o_ f_ow beyond the sonic 'dne in a Rheograph _ Yh_- bas'w _dea is the use of an inchned wa',, bcu:nc- ary for tl_t_ _ank sinluta:ion of _ne sonic _ine. as shown in Fig. i2. The _dea is "me use of a _¢,ca,:.v '`hree-din-.'ensiona _. e'_ecric poten_ia: _(, br e-,a'_- ua-_ed on "d_ surface For Analog- A . Fig. i 2 a. sonic line electrodes are inchned forming a 6T 5 degree wedge of the water body in'd-_e {_,, O, _'_- space. (.,n the surface N : 0, where electric potemla', is evaluated, exponen_ 4/3 for repre- senta:_on of (36 a', is observed and understood eas:'.y as a result of tt:e [oca _,poten'-ia_ dis',r_- butio,: )n tt_e (1_, _)-p_ane. h: Fig. 12b "he idea is i[iustrated for .-',nakc,_-v B: here we have an undercut sonic lint. with a 135 ° water body Io represen_ the exponent 23 _n I36b). .. ,'//' H'_ _ -_-.'_. "4 a_ Analog}- for vc!ocily po!en:_ai ; -\ "\ / b) Analogy for stream func'Aon _N Figure 12. Electrolytic _ank for transonic' fmw analogies I) "I) I(_ I The _dea is, :ike some of :h_ shah-tiroL p:opc_- ties of transonic f_ow in our Rheog:'aph pLan,_.. a cenerahza'don of a::isymn-,e/rlc po:en:ia I d:s- _--ibu'don: the "Inclined ELectrolytic Tank" c_'" a nyperbohc shape hot:ore wi',h a 45 degree wedge wat(r body is fanci',lar to researchers hay!n9 used the ana'_ogy to represent incon-:- p;'_ssib[e axisy.,T, rnet:uc flows I'l° 4. 2. Apphcauon of the anaPesT for des:fin of ._upercr:tical airfoiis Fhe outhned idea of the inehned tank walls a\ oI<]_ the _ecl:nica', difhcu[ties with inhnite o:r zero depth of the e[ect:-o!ytic tank, but also a.Lows the oleo:rio continuation of the potentla'.. 0is:r_bution beyond tl]e sonic Line In order to t-stabhsll a certain distribution on it, see Fig. !2 White the tank with water as conductor is a ,.erv accura'e way to mode', the analogy', there ape other possibihties avoiding this "wet" lecll- nlque and s:tL'_ o: acceptable accuracy One is :hi _,se (,: _ond:,c'tm¢ g:'apnite {,aper It is, o5 (rl,l:'_t. (/. COnSta.%i I.l]lCl,_nt'ss and has theFefore :_, C,t-ir, llonlc)ge:2.1zed in order 1o simulate \'art- ab-e tank depth Per:oration of the graphite paper 21 was dsed in some experln:ents with transonic :.ow ana',og representation. It is _he first s'.ep _.nto _he discl-etisat_.on of an ¢'.ectric network. ] ':'s :,: an expens!' _ too'. ::" tl.t' g:'ld ;_ flne _--noug'.L it reqdlres au_on,aled eva[uatlon, being par: therefore of a nybr:d con.pu:ationa: sys[en: :nc._d:ng a dig_ta: compute:- A ne',wor!: for SO- :.I:_IOF. C,f t:'ansonlc hodograph [)tobLL, n:s. was used 12 I;; F:'ane_ , where the ana'.og5 l_as a tong tra- d:tion. The present author used the [es_ ex[,ensi\e pos- sibilities given by the [:'at,hi:e conciucting paper At 3970 digital con:puta:iona[ codes for trans'onlc ;,nlen:;:,.. fio_v analysis wert- j'as: beginning to appear, desist, n,etl_ods were not axai',.abte A p:->iec', a" .'.lqe DFVLR in Gern,any was, u_e:'eio:-e, • he deve[opmen; of con:putationaL methods tot ::'ansonic. f:ow wl'.h the aid of the ana'ogy 2a -Fh:s 'tee to expe,untents wit..h sin.p_ set-ups and data /¢._, 12 eva'_ation on the digits" con,pater. --",. si::.pLe sin:u.ation of the rneoe.ec::-ic "ank vlth var,ab:e de.r;:h was acRieved with acceptab'_e accuracy tl_:-ouzn the use o: compressed streets of graph:'._. paper, the shee:s shaped para.Le: to iso'au!:s :r. the anaL.ogy ptane _. :n order' lo SllT_d'_a:t: }: il/;. Fig I 3 shows a cut view H_rou_h a rectangular "dry tam_" outfiued with a grid of probes B,.,- :ween top and bottom plate, and an elastic c-,shton Layer, tl:e grapl_ite paper sl_eets are placed. Thebasie sheet extends into the regime v > (, c< the Rheograph plane, where the Qow rr, ay be influenced b) source distributions. The basic sheet also is provided with electrodes for s:n_eu- Larily represents:ion, in tile case of a s_'bsan:,, Lifting al:'foi[ with a quadl't:po'.e in order to rat.- resent a dipole with arbitrary orientatior, A sev-up for Analog).' B lias been established ant. for air'foiL de.=Agr, application, Rheogra-)h _. > _ the working plane with lile gaseiu _.relations (21 : - ('_4! fo" h-'tink" airfoils. In F_g. 14a tile _:'I('. o: probes is drawn, with exte:'na, f[ov: and s_ng:,- la,'i_y feeding eiec_rodes. The _a:te:- are [/.ac'ed _nto a nearly para_!e',, electric fiow and crea:e a Line of constant E (_ 11 ) with sadd_epo:ms and _ _HEGG_ADH FL;.I,_ --. _- _-,, <:-, _ S_n_2U_ity sources _ Outer how SourCe _- _°*'_"°"---- I ! i I ] ,' '_ i _9os,¢ comdurtor . ._- _'_ / Additional ,< ' ¢O_taglG'3 4: _ "N " "'x _tt _o_... ... • _c -_ " :-7 Bottom rote. -- "f- --_ CornDreS_,o:_ Sc,ew Figure 13 Analog flow table using compressed graphite paper f f , --e .---- e--- _-._t p --÷÷-t .... -_ -----.-4] i=_A_ L-__-._---. I ' • ! ! ,'----" i , c ;:'-" i ': • --o ..... k2__° • ,,,: /-_ h 1 F:_ure 14a Analog flow table for l_heograph _o Evaiualion grid, equipotentia_ line interpolation formm_' a c',.osed domain, Fig. 14b. One of the s:_dd'Lepoints is shifled _nto the mapping of the su-:cnat'.on point, S, by \-aming the potentimeters , lt]3 The subsonic f_.ow domain is shown bvR A d:-av,'_.ng of _he sonic fine [en-_nisca_e If the (>n- cb:_sed domam ex_ends ou_ide of it, then the dcn,ain may be evaLualed as an elliptic continu- a:ion analog airfoil f_ow representation. The hne I( = const, is toca_ed _n a limited number o _ inlerva%s between probe grid points, Fi[. 14a, and lhe potent_a'_ va!.ues are :ransferred to ti_e dif_,.taI computer, atong with :l_e In:anuaD.yl selected position of these probes. InterpoLation. poten_ia _. gradiem evaLua',ion and da_a spline f,_t:ing along 1t" = 0 and along the sonic line, as weK as integra:ion of _he supersomc fief.d, bound- ary _ayer computation and viscous dispLacemen_ subtrac'ion, is carried out in the digital compme:- The analog part of trns hybrid techmque is shown :n Fig 1 5 /k_ Figure 15. Anal.og f_.ow evatua!ion with Ana!.og f!.ow tab_.e (t), Solution orientation (2_, Bridge circuit (3), FLow adjustment Potentiome.ers (4), Digital Vot_n-:eter (5t, Da_a transfer unit and Scanner f6), Termina_ and tape punch (71 13 i ne :n_thod ouT:n_-d wa_ the.ended ".o give ,p:_,,- n:a"on ab:,u: pe_s'_b'e s:n-.p.:::cations o" l,n&,- _r'aD:, _ec'Ly.lque_ -n the :rans(,r.lc :'egln,e. A',a,U ability of ,:un_erica I ana:<.,_=:s O) o. 8,2 difflcutties in the e',a'..uatior: occur as a resuh of the slender "wais:" of :he son:c- "ine :emnisc.a:e tl_e conf:_uration r_i_ T b :'es.::t_ fro,n- a p_'ojected e Lof 0. 2 and ?,[ = 0. _;.3. The • Of_ resuttiI',g airfoil is 5 4_ _hict< i: is drawn in Fig. 18 with _.l_e designed su.:_erson:.c region. Analysis eatcutations were carried oul :or stightly different ?dach numbers and ang'es of attack, with the resul" din: useiu'_ degi_r._ ob'.i- ous b' can be obtained btlt design Mach >.:n,be:- and -_-ngLe of _ttack differ somewhat fron. ti_e vatues defining singu',.arily toc_tion :r: :he am:'oF flow. The airfoil was tested in the Braunschwv:ig [ransonic Wind "I,anne_. =_ some of the e:.:roe:'i- mental polars are sl_own in Fig. 19. ! ,(iii/ / 0.._,L i!i , \ \ 0 OOg_ Og.'8 002_ _o 9 1Yigure l_. DFVLY{ 49_0_ Airfoil Polar=: Experimem Re = 4-106 14 ',. N['51ERICAL ItODOGRAP}I ._IETHO[-}S -, i _"" : MetP, od_ . rnF;_'. is aultab'..e fop flus torn_u.atic.n. In }']g 2(: the c'oPAput[_liona', grid for" a ?hosen arblt:'ar: but;nd- a: T is d:'awr. [ I,:. :.Pect, d::la d_.scrlption of a_":o! / design re- sd'.ttng :tom Rh_oe;.ec:tric armLc}_) expePzmen_s ","as lntencea to give a back[Pound of educati,_nat '.a.d_ IO." furl.heN, i-Tlort, econort;ica2 -rvatn:e.'_- oi transonic de,;i_n aerodynanncs \v:th ncn, erica2 methods. Tne hodograph methods dex, e_.ooed by i' (I <_':: ,. TayLor, G. I., and C. Y. Sl;arma::. A n_ect]anica: n_ethod fo:" sob.ing pr.,i en _ of flow _r. c'o::_'_l'_.ss_b:_- f.,ads. A!:* ,." a:;_ [;. la?'_o:, G. i. , and C. F. Sl_arn,an. i_:,:!, - Lcn.s o:" f_ow it: (ompressib;.e {Lu_ci__. I':'uc l_o.x. Soc of k:mdon. Serie A. : 121 ( !9 2;-;'. Ll=:'.a',ard, L. C , "[!_.e use of rheoe.ec':r':ca ana_og_es in aerocivnarnics, AG'.l',iJ.:.g:'a! h 9. Alorawelz, C. S., _"(or_exlstence o: ::.ans- onic flow past a profil, e, Comm. P::,,'e and AppL Mail:., voL. !-. (19(,4)p. _.'%7 - :J?- I0. 11. ]:errari, C. , and F. G. Tricc, ml, Transonic Aerodynan,ics Acaden,ic Press, New YorI,: _ 1 ondon (I{)(bq). Sobieczky, H., K-Y. Funs, A. R..qeebas._, N. J. Yu, A New Method for Desi_ning Si_ock- free T:-ansonic Configurazions, AIAA paper * 78-1114 l197 _",1. 12. gobiec:zky, It. , A Con.putationa: A[¢oritlm, for Embedded Supersonic FLow Don_,a_,ns, Thv l:niversi:,v of Amzona. Tucson, AZ, L'ngmeering Experiment Station, P, epor_ -lFlT) 78-03 (1978). o Funs, K.-Y. , 11. Sobieczky, A. R. :4eebass, .N:,.n_erica _. aspects of the design of shock-free wings and wing-body cornb:,nations, AIAA paper 79-1557 (1979), /Is 20 :4. B._erstoe. J. \Y. , De_l_c and anaLys_ o: a l,(.dograph n:eq_od :or thv ca[cul.ation tC _upercr'hica,. shock-free airofoi[s, .NL!] ]'H 77(}46 ,U (!977L : -. X:euw:.and G. Y. , Transomc bo;entia I "',_ou around a ,a.n.,, of quas:-e"iptica '. aere:oi_ sections, ,X'LF:-Tf', 172 (19(;7_. Bauer', F. P. Garabed:an, D. Korn. (A. Jameson,, Supercrltica; wm_ sectton_a !, I!, III L.ee_ure No '! UNSTEADY TRANSONIC FLOW COMPUTATIONS by A. R. Seebass, N. J. Yu, and K-Y. Fung Department of Aerospace and Mechanlcal Engineering University of Arizona Tucson, Arizona 85721 SUMMARY We investigate the effects of unsteady modes of motion on two-dimensional transonic flows; we do so in the context of the invlscid small perturbation approximation. The study is a numerical one and draws upon the alternating-dlrectlon implicit procedure developed for such calculations by Ballhaus and his co- workers at the NASA Ames Research Center. Our numerical algorithm treats shock waves as moving discontin- uities. Results of nonlinear and time-linearlzed calculations of the transonic flow past an NACA 64A006 airfoil experiencing harmonic motions in several of its modes are presented and discussed. i. INTRODUCTION In unsteady transonic flows, relatively small periodic changes in the boundary conditions can lead to substantial changes in the loads and moments with marked phase lags. These are of major concern in the aerodynamic design of aircraft that operate in the transonic regime. A short, but timely, review of various aspects of unsteady transonic flow may be found in Reference I. Of particular concern are aero- elastic behavior, and flutter and buffet boundaries. Here the unsteady perturbations may sometimes be small enough that linearlzatlon about a nonlinear steady flow, as suggested by Landah] (2) long ago, is possible. In such flows the behavior of the boundary layer, especially as it is affected by the pressure rise caused by any shock waves in the flow, is clearly of major importance. Additionally, in the neighborhood of the leading edge the flow perturbations are large; consequently, highly accurate _nvlscld results require the use of the full potential equation. It seems likely that eventually the computational algor- ithms used in routine studies of unsteady transonic flows will use the Reynolds averaged Navler-Stokes equations now used in research studies. However, such algorithms (3) currently require substantial computer time and are too inefficient for exploratory studies such as this one. The ability of these algorithms to model complex unsteady transonic flow phenomena, such as buffet, has recently been demon- strated (4). An important consideration in constructing an algorlthm for unsteady transonic flows is the treatment of moving shock waves. The experimental observations of Tijdeman (5-7) indicate that even for simple airfoil motions shock wave motions can be complicated, and that they can strongly affect aerodynamic force and moment variations. Tlme-llnearlzed methods, i.e., methods that assume the unsteady perturbations are small compared to the basic steady disturbance have not usually considered shock motions (8, 9), although they can be modified to do so for small shock excursions (I0). Time-integration methods (11-18) treat shock waves by "capturing" them, a procedure that can present a number of difficulties. Unsteady experiments (5-7), analysis (I0) and numerical studies (i0) all indicate that the amplitude of the shock wave motions increases inversely with reduced frequency. Thus some of the most important effects occur with low-frequency motions. This is not surprising; nonlinear behavior Is suppressed at higher frequencies, with the small perturbation equation becoming linear for frequencies higher than the two-thirds power of the airfoil's thlckness-to-chord ratio. Explicit finlte-difference schemes are not efficient when applied to low-frequency cases because the stability restriction on the time step is sub- stantially more severe than that required for accuracy. As a result, efficient semi-lmpllclt methods (13) and even more efficient fully implicit methods (ii, 12, 17, 18) have been developed. Caradonna and Isom (17) use an iteratlve implicit procedure, i.e., the nonlinear implicit finite-dlfference equations must be solved iteratively at a given time level. In an earlier, unpublished, study we also used such a pro- cedure. Ballhaus and Steger (II) and Beam and Warming (18) constructed more efficient algorithms that solve the nonlinear equations directly by the solution of simple matrix equations generated by an alternating-direction implicit (ADI) procedure. This method has proven to he so efficient that it is now used as an alternative to successive line over-relaxatlon (SLOR) for steady flow calculations (Reference 19 and Yu and 6eehass, unpublished). As mentioned above, these implicit schemes "capture" shock waves, i.e., shock waves evolve automati- cally as part of the numerical solution. Shock capturing produces shock profiles that are distorted in a manner that depends on the truncation errors in the finlte-dlfference scheme. The use of mixed-dlfference schemes (ii, 18) can improve the situation for cases in which the flow changes from supersonic to subsonic across the shock. However, when this condition is not satisfied the differencing cannot be switched across the shock and shock resolution is poor. In any case, shock capturing requires spatial grid spacings, in regions where shock waves are anticipated, that are sufficiently small to resolve the shock waves. The grid spacing required to do this is usually much smaller than that required to resolve flow variable gradients in most of the rest of the flow field. Shock fitting removes the ]arge gradients from the finite difference solution and permits equivalent flow field resolution with fewer grld points, both in space and time (20, 21). If shock waves are not treated as discontinuities, but are to be captured correctly, the difference equations must be solved in conservation form. This imposes an additional con- straint on the construction of ffnlte-dlfference schemes that can be difficult to satisfy. A need for shock fitting also arises in computing tlme-linearlzed solutions for very small unsteady perturbations. Tlme-llnearlzed solutions for indlclal motions can be used to determine force and moment coefficient variations at various reduced frequencies, obvlatlng the need for a numerical solution at each reduced frequency (see, e,g., Reference 22). /_3 li-_ Traci, et al., (8, 23, 24) have developed relaxation methods for solving the resulting time-linearlzed equations of motion for harmonic disturbances. Less complete, but comparable, studies have been made by Weatherill et el. (25); these derive from an earlier study by Ehlers (26). In both of these studies shock motions, which contribute substantially to the time-varying loads and moments, are neglected. Difficul- ties also arise in the convergence of the iteratlve numerical scheme. Unsteady small amplitude motions have shock wave excursions that are the order of the amplitude of the motion divided by the reduced fre- quency of the motion. Consequently, these shock motions dominate other low frequency contributions to the lift and moment coefficients. Such time-linearized shock motions can be computed in a rational way, but the accuracy of the results depends critically on an accurate resolution of the steady flow field in the vicinity of the shock wave (i0); this is best accomplished by shock fitting. This paper briefly reviews the numerical procedures we have developed for computing nonlinear and time-linearized small perturbation unsteady transonic flows. We use an ADI scheme and treat shock waves as discontinuities in the flow. Calculations of the transonic flow past an NACA 64A006 airfoil experi- encing harmonic or indicial pitching and flap oscillations are discussed. 2. FORMULATION We write the unsteady small disturbance equation for low frequency transonic flows in the commonly used form -2KM2_xt_ + (I - M 2_ - (y + l)M_x}#xx + Cyy = 0. (i) The spatial coordinates, the time, and the velocity potential in (i) have been non-dimensionalized by the chord, the reciprocal of the angular frequency, and the free stream velocity times the chord, respectively. Other, perhaps more useful and suitable, forms are given in References 21 and 27. This equation results from a systematic expansion of the velocity potential in the thickness ratio r and applies for reduced frequencies K = 0(32/3 ) where K = mc/U, i.e., the angular frequency multiplied by the time it takes the flow to traverse the airfoil chord. Lin, Relsner and Tsien (28) showed that, with restriction to small perturbations throughout the flow, this is the only nonlinear equation that arises. For moderate frequencies the equation 2 - 2K_xt + {i M 2 I)M2[#x + _ K%t]}_x x + = 0 -K #tt - _ - (Y + y + 1 _yy is frequently used, with or without the _t term, and may provide results that apply at higher frequen- cies than those obtained from (i). The boundary condition on the body takes the simple form ,y(X.O,t) - tY$ +! + KY )I, 1 < 1-__x!_, (2) where Y(x,t), the instantaneous body shape, has been decomposed into a steady part, Y°, and an unsteady 2/3 part, yU. Here 6 is the amplitude of the unsteady oscillation. Because K = O(x ), the last term u _ " ' v 'in (2) is dropped unless Y = 0 or is small. For thls reason, the tlme-linearlzed perturbation eloclty x (Y: 0) is times that for the analogous pitching motion, wherepotential for plunging motions = just K yU(x,t) = (x- Xo)Sin t. Numerical studies conducted by Magnus (15) show that erroneous boundary data on a finite domain can lead to significant errors. The low frequency approximation implies that any changes in the circulation are communicated instantly downstream to infinity. Consequently, the slmplest boundary conditions are ¢. = 0 on the downstream boundary and _ = 0 on the other boundaries. Ballhaus and Goorjian (12) used t_ese boundary conditions in their study and obtained satisfactory results. The validity of such far- field boundary conditions can only be justified by numerical experiments; i.e., near the boundary the disturbance quantities #x' and # , must be much smaller than the values at the alrfoil surfaces. For the lifting case, # depends on th_ instantaneous circulation, F. This dependence can be derived theo_etlcally by assuming that in the far field all the perturbations are small compared to the basic steady state (see, e.g., Reference 27). Here we use a stretched coordinate system that maps the doubly infinite domain into I_I < I I_I < i, and set _ - 0 on the downstream boundary _ - i and _ = 0P -- x elsewhere on the boundary of this domain. As a numerical test for this procedure we have computed the steady state circulation about an NACA 64A006 airfoil for various flap deflection angles, using the ADI method with appropriate far-field values of ¢, corrected for the usual steady state circulation con- tribution. These results have been compared with the results obtained by the ADI calculations with the boundary conditions employed here for an unsteady flap deflection to the correct angle. These results are identical within the accuracy with which we have computed the solutions. Any shock wave that exists in the flow field must satisfy the Jump relation derived from the conser- vative form of the governing equation (i), namely dx -o(_--_) - {1- M2®- (_' + + OCy_ 2 s together with the condition derived from the assumption of irrotatlonality, " -"x'/°*y'/I'/ (4) 11-3 Here _x refers to the mean value of _x evaluated, on each side of the discontinuity, and _x B cares the jump in _x across the discontinulty; the subscript "s" denotes the quantity evaluated at the shock surface. indi- The pressure coefficient, defined so that it vanishes at sonic conditions, takes the form C = -2 + p l)H2o_ _x " In the small disturbance approximation, the Kutta condition is imposed by requiring that C uous at y = 0 for x • 1/2. P 2.1 Time-Linearized Equations We now assume that the unsteady disturbances, characterized by (5) be contin- 8, are small enough that we may write _(x,y,t) = _°(x,y) + 6_(x,y,t) + 0(6) (6) and neglect higher-order terms in 8. The restriction imposed on 6 for this to be true will depend on the other parameters of the problem, viz., K E (i - H_)/[(y + l)M_r] 2j3 and K. This gives - - : x + );y" o, -..-,(x), l 1{1 - M2 (¥ + 1) _ )# x - _ <- x ! [ ; (7) -' i)xt + (tl - - (. + + ,. - o. ) .u+x _ x (g> The solution to (7) must satisfy the steady version of the shock relations (4) and (5). The shock rela- tions for (8) are discussed in Section 2.2. We avoid writing _(x,y,t) ffi Re{_(x,y)e imt } (9) as this restricts the study to harmonic motions. Because indicial motioDs can be superimposed to obtain the results f_r any frequency, they seem more important. Equation (9) results in an equation for a com- plex-valued _ which may be solved by line relaxation. Our experience with unsteady ADI techniques has been that they are at least as effective as line relaxation for problems of this type, and hence there is no advantage to the decomposition (9). The numerical algorithm developed in Reference 21 and described briefly in Section 3 can be used to solve the basic equation (I) subject to the boundary conditions (2), the shock conditions, (3) and (4), the far-field boundary condltions.and the Kutta condition. Steady state solutions, #*(x,y), may be obtained rapidly by subjecting a basic steady state, such as undisturbed flow, to rapidly changing boundary data until a new steady configuration is prescribed. This, then, determines the steady state result: for (7) needed to solve (8). 2.2 Shock Fitting The basic algorithm for shock fitting in mixed flows was developed in a previous study of steady transonic flows (20). A different approach to shock fitting has also been used by Hafez and Cheng (29) in their study of steady transonic flow problems. Their procedure essentially replaced the shock-point operator of Murman (30) by an analogous difference statement derived from the shock jump conditions. Sub- sequently, the velocity potential on each side of the shock wave is extrapolated to locate the shock wave. To understand the shock-fitting procedure for unsteady transonic flow calculations it is necessary to recall how shock waves form in an unsteady field. Shock waves are generated when the local flow becomes supersonic and compressive. While the initial shock formation may not be predicted exactly by the numer- ical solution when shock fitting is used in the early stages of shock wave formation, it eliminates spurious oscillations in the numerical solution and does provide the correct development of the shock wave in later stages of the calculations (31). The criteria that we set for the initial shock formation is that the local flow become sonic (relative to the airfoil) and compressive. In the body-flxed coordinate system, a shock wave can exist both in the usual supersonic-supersonic and supersonic-subsonlc transitions, but also in a purely subsonic flow field, sometimes referred to as a "subsonlc-subsonlc" shock. In any case, the flow ahead of the shock relative to a coordinate system fixed on the shock must be always super- sonic. Consequently, the correct judgment for the existence of a shock wave in the unsteady field is to evaluate the local flow velocity ahead of a prospective shock with respect to the coordinate system fixed on it; i.e., if the local flow is supersonic a shock may exist, if the local flow becomes sonic the shock strength diminishes, and if it is subsonic a shock cannot exist. Any shock wave that exists in the flow field must satisfy the jump relations (3) and (4). In two- dimensional small perturbation transonic flows the shock waves that usually occur are nearly normal to the flow direction. While it is not necessary to do so, in the numerical calculations reported here we have assumed that if the basic steady flow has a shock wave, then this shock may be approximated by a shock wave normal to the free stream flow. To be consistent with this approximation we must also assume that the motion of any shock wave that arises from unsteady changes in the flow, as well as the motion of existing shock waves, is also calculated by this normal shock approximation. For this simplified model, (3) and (4) reduce to /7_ I II-4 M 2 - 1 g¢0 = o on _ _ dx :_{ ® +ix), s (d-'{)s 2K (7 + 1)Mi which gives the speed of the normal shock in the flow field. For steady flows Cx is a function of x alone; this, of course, still permits U_xD to vary with y. For unsteady flows, while x is a function of t alone, the strength of the shock will still vary with y. s For time-llnearized flows the steady state result for _° with normal shock fitting will give a steady state shock position x °, 0 < ]Yl < Y* • We now determine the shock wave's motion by writing the • S o-- -- + perturbed sh_ck .posltion as x s = x s + 6X(t) -and using the time-llnearlzed version of (i0); we note that an expression of the form x ° + 6X(t)/K would probably be more appropriate. From (10), we conclude that the shock motion is governedSby (i0) dx = ,x(X,0,t) with [_qb0 = |¢°8 + 6_$0 : 0 (11)dt 2K on the shock. Linearizing the expression in (Ii) for the velocity potential about the steady shock posi- tion we find ¢(Xs,Y,t) = ¢(x_,y,t) + Cx(X_,y,t)6dx =-,°%,y)+ + **%,y,t)+ o(62). Because we have treated the shock as a normal one, y appears here simply as a parameter. _¢(Xs,Y,t)_ and _¢°(x_,y)_ are both zero; consequently we have t (y + 1) [" 0,t)dt _#(x_,y,t)U 2K=- -- $ O which must be integrated in time in conjunction with the solution to (8). Now (12) 3. NUMERICAL PROCEDURES In a preliminary study of the unsteady transonic flows a normal shock-fltting procedure was imple- mented in an implicit-iterative scheme. Satisfactory results were obtained, but the procedure was time- consuming because of the iterative process required at each time step. The recent studies of Ballhaus and Steger (ii) and Ballhaus and Goorjlan (22) show that an ADI scheme is more efficient that the Implicit-lterative scheme in treating the low frequency transonic flows. The shock-fltting algorithm was modified and implemented with an ADI scheme. In this section the ADI procedure and the method used for unsteady shock fitting are briefly reviewed. 3.1 Coordinate Stretchin_ To minimize the far-field boundary effects on the numerical results a relatively large computational region is usually required. For some of the cases studied in this paper the shock excursions are large and the unsteady disturbances carried several chord lengths away from the airfoil; thus, the use of a relatively large computational domain seems desirable. A simple and straightforward way of computing the solution in a large computational domain is to use nonuniform mesh distributions with most of the mesh points concentrated in the region of interest. An alternative is to introduce analytical coordinate stretchlngs. In the present study, we use the following coordinate stretchings: O_IG]_NA L PAGi = i {I - exp(;alx)} for x _ 0 and n = ± {I - exp(+a2Y)} for y _ O, _)_, p(_.);{ Q[[#,[, where a I and a9 are constants that control the mesh distributions. The infinite physical domain is transformed into Ehe finite computational domain bounded by I_] ! l, and lql _ i. The transformation provides a concentrated mesh distribution near the airfoil which is suitable for the present study. While this scaling is not consistent with the known algebraic decay of the perturbations, calculations made with an algebraic scaling, viz., _ = x/(Ix I + a ) etc., gave essentially identical results. The exponential variation used here seems more desirable n_ar the airfoil. The governing equation (i), written in the stretched coordinate system, is h ) _{ .2_1 I- [ _ + al(1 _ [_])¢_]2) + { al(1 _ _[) Cn } = O. (13) { 2a_(l - In)) (Y + I)M 2 _ na_(l- Inl) t Because (13) is in divergence-free form, a Conservative difference approximation can be constructed if the shock wave is to be "captured" rather than "fitted." The normal shock Jump relation follows directly from (13); this relation and the boundary condition on the airfoil surface are now /_6 ! 1-5 N ,i(1 - S_I)(_ + 1) H_- 1 (_)o " 2_ ¢ (T+ l)H2" *-1Cl -Icl)._ i I &rid al._ (1- I¢I)ay(_,t)+___Y_c,t)-1+ exp(,i/2)<_ • 1-_n(o,t) exp(-al/2) o i _ a 2 _t ' - - lquatlons analogous to (13) and (14) for the tLme-llnearized results are given in Reference 10, I 3.2 AlternaClna-Direction Implicit _ADI) Hethod The lov frequency equation in the stretched coordinate system is solved by the alternating-dlrection laplielt scheme developed by Ballhaus and Steger (11). To s/mpllfy this discussion, equation (13) Is rewritten in the form I Y_C + ¥_ + Gnu O, (15) I vhere the function T, F and G su, y be determined by comparing equations (13) and (15). The solutlonis advanced from tlme level "n" to level "n+l" by the following lye-step procedure: l÷n÷nl_l:ld(c_l I _-_ (V_ - T_) + D_F + 6nO - O; _ ( - V ) +_ - Gn) - O. (16) Here "+" refers to an intermediate value of T, D[ is the type-dependent difference operator for _-derivectves end 6 the central-difference approxlsstion for n-derivative. The backward difference approxization for ¥_ can be either a first-order or a second-order difference approximation, with the n latter _iv£ng Improvld results. The nonllnear terlm F is evaluated, using a llneerlzetlou somewhat differett frnl the two-tile level averaging procedure of Ballhaus and Stager. The dlfference approxima- tions described above provide flrsc- or second-order accuracy for T_ , second-order accuracy for F, and Gn In subsonic regions, nnd first-order accuracy for F in s_ersonic regions. A local analysis shows t'_ac the procedure is unconditionally stable. In the first step a quadradlagonal system Is generated end can be easlly solved by direct ellmlnatlon. For llftlng calculations two grld llnes are u_e_ _o _epresent the lo_er and upper surfaces of the alrfoil. The cir:ulatlon, F, is calculated by F - _TE - @ITE _hrough each sweep. Here "ITE" denotes theupper a_d lower values at the first grid point behind the trailing edge. This elrculatlon is Incorporated Into th_ construction of the _-derlvaclves behind the airfoil for _ = O. In the second step a trldlagonal system Is generated by the body. Ahead of the leadlng edge end n behind the trailing edge the duubie grid notation for _ = 0 destroys the trldiagonal system. However,ahead of the eading edge, _U , _L and behind (he--trailing edge, 4U , sL + F; thus the difference equations can he reordered to give a tridiagonal systm. On _he airfoil surface, the matrix equa_ions above ,nd below the airfoil are decoupled; they can either be solve_ separacel_ or simultaneously by paok- l lag tb_ matrix equat£on_m_ether. ALeln. analogous but somevhat silpler equations and procedures are used for the tlme-llnearlzed ca] :u- 1arlene. In these calculatlons the type dependent operator, D[, changes at the steady state sonic llne and shock rave. The coefficient, f(_,n), tha_ nppesrs in (8) In the form (_(_._)_r)_ depends on the n steady state results _'(_,n) and lUSt be stored. On the other hand, the matrices u_e_ do no_ depend onthe sol,]floe # and. consequently, need only be Inver(ed once. In i_s present form our algorlthmdoes not tak._ advantage of this feature. 3.3 S}.ock Ftttln_ _._ s_ar_ the unsteady nonlinear flow eeicalations by using an AI)I scheme. When the local flow be- . comes _ontc end compressive, we introduce the shock-fitting algorithm described in detail in Referent( 21. Sonic,,compresslve points are treated as shock points _here _l_fereutlaLion In t and _ across dls- contin, ities is avoided. Initially, the shock has zero strength and Is statior_ry. The flow propert. _e ahead ef and behind the shock can be easily extrapolated [rum neighboring points. The shock wave can eitho_ increase or decrease In strength during the unsteady process. Thls results in three poeslblllt_es for sn_ek mo_ion that have to he considered separately in the flttlng procedure: The shock moves upstream and crosses grid points; the shock remains stationary or moves vithin a grid spacing; the shock movesdomlet-eam and crosses grid points. At each new time level the s ock position is determined by applying (10). The formulations of the difference approx/_atlone for each case are quite slm£1er. F_r ttme-llnearlzed calculations the solution le advanced In tile using the ttle-llnearized analogues I of (16) coupled with (12) In the form • +,.,"; + ,.,+ I Here and d.ate, el.d,stetspositionf ,hock.reThls procedure corrects the ) values for shock lotions as the solutlon progresses. The shock motion is (14) " ! f L..,.... k........ I1_ easily determined simultaneously by using (11) and (12) in the fot_ XB+I(O,t) . -It(s_,o,t)In+llltx(X;,O)l. Further details are 81ven in Reference 10. 4. RESULTS A/_ DISCUSSION kch the nonlinear and tlme-llnearized algorithms have been used to compute the flow past an HA_ 6ttO06 airfoil subjected Co indicisl, I,e.o step. changes and harmonic motions in pitch end flap oscilla- tion. The latter calculations have included a range of Math numbers, amplitudes for the nonlinear algorithm, and reduced frequencies for the harmonic changes. The nonlinear algorithm has also been used to compute the flow past a pulsating parabolic arc airfoil. In this latter flow, at V_. - 0.85, as the airfoil thickens a shock _ave forms and moves downstream until shortly after mid-cycle. As the airfoil thins, the shock nave moves upstream with increasing speed, eventually leaving the alrfoll. A comparison of the results, with and without shock fitting (21), indicates that shock fitting predicts the far,arian of the shock wave more accurately. It also p_operly defines the shock wave when it becomes "subsonic- subsonic" in the fixed grid system. The shock wave decays slowly as it propagates into the free stream after passing the location of the leading edge'when the airfoil's thickness has Just become zero. t.l NACA 64A006 Airfoll r Nonlinear Calculatlons Steady state solutions were computed as discussed Ir Secclon 2.1 for an NACA 64A006 airfoil for various values of the freestrsamHach number by using th_ ADI scheme with shock fitting outlined In Section 3. The free stream Kach number was varied between 0.8 and 0.9. The mesh system had 101 by 82 grid points in the x- and y-directions respectively, kbout 250 to 450 ttJme steps were required for the solution to converge IA_Jmax _ 10 -_. These steady state solutions are used as initial data for the _onlinear and t/Jae-linearized unsteady flow calculations Result were computed for the airfoil with quarter-C,ord flap for various values of the reduced frequency, the free stream Hach number, and the oscillat::on amplitude, in order Co simulate the shock motions observed by Tijdeman (5, 6). These motions were classified by him as: type A - small shock oscillation; type B - the shock becomes very weak or disappears during part of a cycle; type C - the shock leaves the airfoil. Results for type A motions ar_ not given, as they are easy to treat compuca- tionally. For all cases studied it took three to six cycles for the flow field co become periodic. Stability seems to require that the t_e step be small enough Chat At(in degreea)/K < 10. Figure 1 illustrates the pressure coefficients on the airfoil surface at various ttJnes for H - 0.854, g - 0.358 and 6 = 1". For these conditions Ballhaus and GoorJian (12) were able to simulate type B motion where the shock disappears during some pert of th_ cycle. Here the shock does not disappear during the cycle; rather, it becomes quite weak during a small portion of the cycle. This difference Is probably due In part to the aasuaq)tion of a nornal shock, which eaults in • stronger shock than would norwally occur, and to the use of shock fitting, which £s able c_ resolve very weak shock waves. Fi;ure 2 depicts the pressure coefficient on the el"fell surfaces for N.= 0.822, K = 0.496 and 6 " 2", e/mulating type C shock motion. Because we ha'e used less spatial resolution and have not sca_ed the equation and boundary conditions with various power; of the Hath number, a slightly larger deflection angle seems co be needed in order to generate the type C shock motion; that is, ve need s 2" deflection engle rach _r than the 1.5" of _eference 12 to obtain a_alogous behavior. In this case the flow field is subcrtclcal during most of the cycle, where the shock va, e is barely "captured" in the non-shock-fitting procedure. During the unsteady process the shock moves reward the leading edge. However, the strong singular behavior in pressure at the leading edge prevents the shock from propagating off the airfoil. The perturbation velocity becomes large and is negativelthus, the flo_ used to calculate the relative velocity ahead of the shock can no longer support a shoc_ wave. Normal shock-fitting calculations deter- mine the _hock speed from the pressure Jump across the s_ock at the airfoil surface. This eliminates the possibll ty chat a portion of the shock may propagate o'f the leading edge in the computations, gut this does not haply it cannot occur; rather this is a limital:ion of the normal shock fitting. HagnuJ and Yoshihara (15) have solved the Euler equations using an explicit procedure for the condi- tions cf I'igure 1. Their results are compared with our :alculation In Figure 3 for t_o angular times chosen to represent the least and the largest dIserepanc:es. These discrepancies are thought to be mainly due to the inaccuracy of the small perturbation aolutio" near the leading edge, Small errors there change the stze e_ shape of the sonic line and influence the clock's position. For the conditions considered, the shock is neerly normal and the normal shock approximation should be a good one. _ther good agreement is obtaineC. Kddittonal nonlinear calculations have been carried out for H - 0.BB0 and K " 0.48. Both pitching and flap _otions have been calculated for indictal and _atwontc changes. For these conditions very small unsteady changes lead to _ery small shoek_otione and t_e shock wave regains between grid points. Because of the extrapolation procedure used in the shock-fltti_'_, the _ mesh distribution used here can intro- duce errors, elbelc small ones, In the shock's poeitlon _hen a grldltne is crossed, _e wished to elLea_- nate these errors In order to usa the nonlinear cslcula;ions Co Judge cite accuracy of time-linearised calculations. These results indicate that for pitching _bour mid-chord, nonlinear, amplitude dependent, behavior occurs for _/_ _ 0.1 for K - 0.48. Because the amplitude of the shock cations increases with decreasing K, nonlinear effects occur at mailer value: of _1_ at loner reduced frequencies. Indiclsl serious require about eight hundred time s'_pe of varying size to resolve the response. For hanmnlc motions, lnitl_tnd f_om rest. three to ten cycles are required for the solution to become har- monic, with large values of K and _ requiring lore _ycles. The pitching mode requires more cycles then the flap mode, The amplitude of the poelclve and nsgative phases of the mutlon could be varied from m., I I I I I i I i I i I I I I I i i I i g I I i g ! I I i I I I I i I I 11-7 cycle to cycle to reduce the number of cycles required. Each cycle requlres 60 to 180 time steps to com- pute, with more steps required for smaller values of X. Each time step takes about 5 seconds of CPU time on a CDC 6400, or about 0.25 seconds of CPU time on a CDC 7600. 4.2 NACA 64A006 Airfoil_ Time-Linearlzed Calculations T_me-linesrized results have been computed for an NACA 64A006 airfoil experiencing harmonic and Indlcial pitching and flap motions. AS noted earlier, in the low frequency approximation made here pitching and plunging smtlons lead to the same result except that the time-llnearlzed perturbations are proportional Co the maximum pitch angle for the former, end K times .the maximum amplitude _or the lat- ter. Harmonic motions Inltlat_d from a steady state become nearly periodic in three to ten cycles, with the changes induced by flap oscillations becoming periodic more rapidly than those resulting from pitch- Ing oecllletlons. Hore cycles were required for larger reduced frequencies and, to a lesser degree, higher Hath numbers. I lh order to confirm the valldlty of the tlJe-llnearlzed calculations, both the tlme-11nearlzed and ttonllnear a_gorlthas were used "to compute the response to a s_ep change in angle of attack and the har- monic response to pitching motions. Figure 4 eomparesthe nonlinear and t/me-llnearlzed results for the normalized circulation and shock position for harmonic pitching motions at N - 0.88 and K = 0.48. Res4_Ite are given for the fifth cycle; note that the nonlinear results are not yet periodic. Figure 5 compares the nonlinear and t£me-llnear£zed pressure deviation from steady sta_e at six angular times for the same conditions. Good agreement between the results is obtained for 6/x less than 0.1. Time-llnearlzed pressure distributions at slxangular positions for an oscillating quarter-chord flap vlth K - 0.06 and H - 0.875 are shoes in Figure b. The flap deflection is downward during the flrst Half of the cycle. The results for the second half of the period, for the symmetrlcal problem mhob'n here, are Just the results eho_rn vlth _he lOWer and upper surface pressures interchanged. Thus the results for 0" are nor given as they are Just those for 180" with the lower and upper surface pres-- suree reversed. Because the flap hinge occurs very close to the steady state shock location, the pressure singularity due to the change in flow direction at the hinge is missed. The circulation and shock excursion obey the folloulng reletlons: r(t)l_ = 9.z6 sin (t - 59"), x(t) - 12 sin (t - 51_). Note the substantial phase lag in the circulation and the sh0ck_s position. T/me-llnearized pressure distributions at s_tx angular positions for an oscina_lng airfoil with K - 0.12 and M - 0.875 are depicted in Figure 7. If these results are multiplied by K, then they _epresent the pressure perturbarlons for a plunging airfoil. As in the previous case of an oeclllati_g flap, changes in forces and moments of 0(6/K) occur due to shock wave motion. In this case r(t)/d - 5.4g sin (t - 70°), x(t) = 5.62 sin (t - 67"). Analogous computations have been carried ouc for : - 0.12, 0.24, 0.36, and 0.48. Figure 8 _eplcts the shock wave's excursion and maximum circulation as a function of K "I. The nearly linear variatlo_ of the sh)ck excursion substantiates an observation made In a one-dlmenslonal model where the shock yore excurrl_n is directly proportional to IlK (see Reference 10). I_ these calculations the circulation gtves an lmediate evaluation of the lift coefficient as , functlcn of time; the moment coefficient must be eval_ated by integrating the moment of the pressure coefficient. This is done by Integrating the smment ,,f pressure perturbations with the shock wave i. its ete&_y-stece position and then correcting these results for the moment due to the shock wave morion, assuming that the shock's strength is defined by the s'eady-state _ressure field. This makes an error in the shock strength of 0(_), but the effect on the mr_ent is 0(6 /K); because we have neglected o :her hi,her-order terms it is consistent to neglect this chmge in the strength of _he shock wave. _lgure 9 depicts the absolute value and phase angle of the normalized lift and moment coeff_cie'_ts, as a function of the inverse reduced frequency K -l, for harmonic flap and piCchlng motions ac H " 0.875. The tlme-/Inearlzed algorlrhm used here is s derivative of that used for the nonlinear calculations. Consequently, computational t/mes are not greatly reduced from those required for the nonlinear celu._Is- tlons. The llnearlty of these computations may make it poselble to greatly reduce the coaputatlona] effort required. A local stability analysls shows the_ the computations should be unconditionally stable, but numarlcal experience has shown some dlfflcultlas for At(In degrees)/K _ 50. Each time step r_u_res about two seconds of CPU time on a CDC 6400, or about 0.1 seconds on • CI)C 7600. The number of time steps required for a given computation is somevtmt less than those required for the nonlinear computatlont at small values of K, and comparable at larger values of K. . CONC '_SION Efficient and accurate methods for computing low frequency, unsteady behavior in transonic flows have been developed. They utlllze the ADI procedure developed st NASA Ames for the small perturbation e, _a- lion. but _reat shock waves as discontinuities. The time-llnearized calculations allow shock wave m_ttons, which are shove to be 0(_/K) and often dominate changes in the force and moment coef_Iclents. Com:arlson /Yq •_.I_ k... ! L.. t. g ! 1-8 of the tlme-11nesrlzed results wlth fully nonlinear calculations dellne_,tes their range of appllcabllity. The unsteady behavior due to harmonic pitching and flap oscillatlons of _n NACA 64A006 slrfol] is discussed. ACKNOWLEDCMENT This research was sponsored by the AFOSR through Cran¢ 76-2954S, the NASA through Grant 2112, snd the ONR through Contract N0014-76-C-0182. REFERENCES I. Ballhaus, W. F.: Some Recent Progress in Transonic Flow Computations. VKI Lecture Series on Compu- tatlonal Fluid Dynamics, van )Carman Inetltute for Fluld Dynamics, Rhode-St-Cenese, Belslum , 1976. L 2. Landahl, M. T.: Unsteady Transonic Flow. Pergamon Press, New York, 1961, 111-113. 3. MacCormack, R. W.: An Efficient Numerical Method for Solving the Time-Dependent Navier-Stokes Equa- tions ac High Reynolds Number. NASA TH X-73129, 1976. 4. Levy, L. L., Jr.: An Experimental and Computational Investigation of the Steady and Unsteady Transonic Flow Field About an Airfoil in a Solid-Wall Test Channel. AIAA Paper No. 77-678, 1977. 5. TtJdeJean, H.: On the Hotton of Shock Waves on an Airfoil with Oscillation Flap in Tvo-Dimensionsl Transonic Flow. [_LRTR 75038U, 1975. 6. TiJdeman, H.: On the Motion of Shock Waves on an Airfoil with Oscillating Flap. Ed. K. Oswatits:h and D. Rues, IUTAM Symposium Transsonicum II, G_ttingen, 1975, 49-56. 7. TiJdeman, H., Schippers, P., and Feraoon, A. J.: Unsteady Air Loads on an Oscillatlng Supercrltl.:al Airfoil. NLR _ 77008 U, 1977. 8. Traci_ R. M., Albano, E., and Farr, J. L_: Perturbation Method for Transonic Flows about Osctlla:tng Airfoils. AL%A J., Vol. 14, No. 9, 1976, 1258-1265. 9. Weatherlll, W. H., Ehlers, R. E., and Sebastian, J. D.: Computation of the Transonic Perturbatto t Flow Fields Around Two- and Three-Dimensional Oscillatln_ _inKs. NASA CH-2599, 1975. 10. FunK, K-Y., Yu, N. J., and Seebaas, A. R.: Small Unsteady Perturbatlons in Transonic Flows. Infor- mal report, submitted for publication 1977. II. Ballhaus, W. F. and Stager, J. L.: Implicit Approxlmate-Factorizatlon Schemes for the Low-Frequtncy Transonic Equation. NASA TM X-73082, 1975. 12. Ballhaus, W. F. and GoorJlan, F. M.: Implicit Finite Difference Computations of Unsteady Transonic Flows about Airfoils, Including the TreaUnent of Irregula: Shock Wave Motions. AIAA Paper 77-20 ;, 1977. 13. Ballhaus, W. F. and Lomax, H.: The Numerical Simulation of Low Frequency Unsteady Transonic Flo_ • Fields. Lecture Notes in Physics, Springer-Verlag, Vol. 35, 1975, 57-63. 14. Magnus, R. and Yoshlhara, H.: Unsteady Transonic FLows Over an Airfoil. AIAA J., Vol. 13, Ha. 12, 1975, 1622-1628. 15. Magnus, R. and Yosh£hara, H.: The Transonic Oaclllltin g Flap. AIAA Paper No, 76-327, 1976. 16. H_Enus, R. J.: Computational Research on Invlscid, Unsteady, Transonic Flow over Airfoi]s. ONR Re)art CASD/LVP 77-010, 1977. 17. Caradonna, F. X. and Isom, M. P.: Numerical Calculation of Unsteady Transonic Potential Flow O_er Helicopter Rotor Blades. AIAA J., Vol. 14, No. 4. 1976. 482-488. 18. Beam, R. M. and Warmlns. R. F.: An Implicit Flnit_-Dlfference Algorithm for Hyperbolic Systems :n Conservation-Law Form. Journal of Computational £1ystcs, Vol. 22, No. 1, 1976, 87-110. 19. Ballhaus, W. F., Jameson, A., and Albert, J.: Implicit Approximate Yactorlzatlon Schemes for th_ Efficient Solution of Steady Transonic Flow Probleme. AIAA 3rd Computational Fluld Dynamics Confer- ence, June, 1977, 27-34. 20. Yu, N. J. and Seebass, A. R.: Invlscld Transonic Flow Computations wi_h Shock Plttlna. Ed. K. Oewatttech and D. Rues, IUTAH Symposium Transsonlc_a II, G_tttngen, 1975, 449-456. 21. Yu, N. J., Seebass, A. R.. and Sallhsus, W. F.: An Implicit Shock-Fitting Scheme for Unsteady Transonic Flow Computations. AIAA ]rd Computational Fluid Dynamics Conference, June, 1977, 13-_6. 22. Ballhaua, W, F. and GoorJlao, P. M.: Computation of Unsteady Transonic Flows by the Indlcial _lethoJ. AIAA Paper No. 77-477, 1977. s 23. Tracl, R. M., Alberto, E. D., and Farr, J. L., Jr.: Perturbsclon Method for Transonic Flow about Osctllatln8 Airfoils. AIAA Paper No. 75-877. 1975. /_ I I I i I i I ! ! ! ! / 1 I-9 24. Traci, R. M., Albano, E. D., Farr, J. L., Jr., and Cbeng, H. K.: Small Disturbance Transonic Flows about Oscillatlng-Airfoils. Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, AFFDL-TR-75-100, 1974. 25. Weatherill, W. A., Ehlers, F. E., and Sebastian, J. D.: Computation of the Transonic Perturbation Flow Field Around Two- and Three-Dimensional Oscillating Wings. NASA CR-2599, 1975. 26. Ehlers, F. E.: A Finite Difference Method for the Solution of the Transonic Flow About Harmonically Oscillating Wings. NASA CR-2257, 1974. 27. Krupp, J. A. and Cole, J. D.: Studies in Transonic Flow IV, Unsteady Transonic Flow. UC_% Engineering Report 76104, 1976. 28. Lin, C. C., Reissner, E., and Tsien, H. S.: On Two-Dimensional Non-Steady Motion of a Slender Body _n a Compressible Fluid. J. Math. and Physics, 27, 1948, 220-231. 29. l{afez, M. M. and Cheng, H. K.: Shock-Fittlng Applied to Relaxation Solutions of Transonic Sma11- Disturbance Equations. AIAA J. Vol. 15, No. 6, 1977, 786-793. 30. Murman, E. M.: Analysis of Embedded Shock Waves Calculated by Relaxation Methods. AIAA J., Vol. 12, No. 5, 1973, 626-633. 31. Moretti, G.: Thoughts and Afterthoughts about Shock Computations. PIBAL Report No. 72-37, 1972. I, 1 ll-lO ° I TIME: 0 ° -c 8 t t I I -O .50 "-I) • I .SO -0 .SO I 0 -C P 0 0 ! TIME: 90 ° I l..50 TIME: 30" I I 1 .sO -0.50 -C p 0 0 T 'I TIME: t20 ° £ u . . I .50 0 0 -0.s0 % 0 0 ,-m o TIME: 60 • • 1.50 -0.50 0 0 00 $ TIME: ISO ° • 1.50 FIGURE I. Cp ON AN NACA 64A006 AIRFOIL AT M= = 0.854, K = 0.358 WITH _ = i ° sin t (NONLINEAR CALCULATIONS). oTIME: 0° -C c_ -2 P P (3 0 C_ 0 -o.so 1.oo _ _'.so 2;0 % ° ?- TIME: 80" !so TIME: 30' -C -C P P 0 I .... _ I. I I -o.so __ t oo ,, l.so -o.so =6 o TIME: 120 ° 0 u_ TIME: 600 o -C p 0 0 I -I _'_'I I I I I °lTIME: 150 °-Cp 8 I 2 | --AJ I I ? FIGURE 2. Cp ALONG AN NACA 64A006 AIRFOIL AT M= = 0.822, K = 0.496 WITH 6 = 2 sin t (NONLINEAR CALCULATIONS). o CD tm i i MRONU$ RNO YOSHIHRRR -C ,,,: UPPER SURFRCE P o • : LOWER$URFRCE m c_ 0Q U] c) I MRONU5 RND YOSHIHRRR =: UPPER SURFRCE A: LOWER SURFRCE C3 C_ ,-4. TIME: 60' TIME: 180 = FIGURE 3. COMPARISONS OF THE RESULTS FOR C_ ON AN NACA 64A006 AIRFOIL, AT M_ = 0.854, K = 0.358 WITH _ _ 1 ° sin t, WITH THOSE OF I_AGNUS AND YOSHIHARA (NONLINEAR CALCULATIONS). r(t)/6 -i -2 • t x(t) 1 0 -1 "--L FIGURE 4. NONLINEAR (***) AND TIME-LINEARIZED(--) CIRCULATION AND SHOCK POSITION FOR THE PITCHING MOTION OF AN NACA 64A006 AIRFOIL. RESULTS SHOWN ARE FOR THE FIFTH CYCLE. THE NONLINEAR RESULTS ARE FOR 6 = 0.i ° AND ARE NOT YET PERIODIC. M® = 0.880, K = 0.48. ..- -(c P l - co)/6 P TIME: O' l 81 _}_o 0.2o O.SO _t .OC -(cp - c_)/a T.,,E. !20" M _ /\ _t , - _ _ .-.-_ I _T I _2 -(c c°)l_ p P :i I 1 o _to N 11-13 -(cp - c_)l_ TIME: 80" t_o II X -(c = c°)16 P P T[ME: 180° ¢_ ==. I • X -(c P - co)/_ p TItlE: 300" " l Q _ 4 B¢_ oo 0.20 Q-6Q i,,, X FIGURE 5. NORMALIZED NONLINEAR (****) AND TIME-LINEARIZED (--) PRESSURE PERTURBATIONS ON THE UPPER SURFACE OF AN NACA 64A006 AT SIX TIMES. PITCI|ING MOTION WITH M = O.B80, K = 0.48. FOR THE NONLINEAR CALCULATIONS 6 = 0.i °. 11-14 °o 0_'0 _J I o m- I 0_'0- 8 ,j. _o i # ! OZ:'O 00"0 ° 07;0- 0_"0- / °. L_J 3r- ! OIP'O I OZ,'O 8 ,j. 0 1:3 f • 4- ---d_" I OPO- It II 8 ! I ! 0_" 0 8 • • # 00"1 OZ'O- I OZ" O r,. 0 F", [--i .< ,.-a ,.-.1 P...I 0 .-1 < m < ,.,,, N Z Ul ..n m g 0 M IM ,-1 [-.. M 0 &_F- I-- U m LD .o _Z I.- p- V % .° LLJ p-- _._ I I 0_" 0 B __: =., O_'O O0 OZ'O- 4 I 01,' 0 OZ" 0 00" I C_'O 8 ? .,_ • I [_ 0_'0- ¢-j O" -- ° ? rj I ,, O0.1_'_ IOZ'O 0_'0- / I 0_" O- / I 0_." O- I 0_"0- % O0 I-- U I "o ,° I.lJ .,#- I.-- I r._ I ,., 4 -.--z_ ! Ot'O :1 ..... % L_ o° LLJ I 01,'0 /#'I I I_,'0 I =;_ o_'o- I Ol,' 0 OZ'Io 00' 8 - H 8 o v (-} r_ 0 _ 0_'0- ,_1 t_ 0 ,-.1 < L_ rj < 0 ,-.1 0_'0- r.,-% 0 m÷ I 0_'0 I°. oo'o_ oz'o- / t -- 01,'0- U N < ,..-1 I Xmmx i0 0 0 0 0 J f J j f f J 7 J I I I f f 5 I0 15 I / / / / / / ;r / / / / / / / I J I 2_ / 5 i0 15 DK-I K-I FICURE 8. NORMALIZED MAXIMITM SHOCK EXCURSION AND CIRCULATION AS A FUNCTION OF INVERSE REDUCED FREQUENCY FOR AN NACA 64A006 AIRFOIL WITH OSCILLATING QUARTER-CHORD FLAP. F_ = 0.875. If o_ • _ Z_ i _0 00_ _0 O_ Z _ m_O o_ O° i o -41) 0 o I / / / / / o 0. ! .,.. /. A / \ / J \ \ I / / / / 0o o o 0 bo o I-, bo 0 o o \ \ \ \ \ \ \ \ I \ I i I o i c_ ,% m _D 41+ 0 Z_t -_ r_C_H _N Z _t_ 0_,_ _'_0 "11 0_ H>O _t'_ -],._ c3 C_ t.-_ • "_C_ Co _.-+i...-4 --_ Z tn _nC_Z i o t_ o o o o i° o /\ / /2 I_ I I I I I I ; I P t o om o- -t _;I \ \ \, \ \ o bo o ' | 0 _ o co (:3 t_ I I I' / / Y I/ / \ / \/ I \ I \ I o Reprinted [rom AIAA JOURNAL, Vol. 16, No. 8, August 1978, pp. 815-822 Copyright, 1978, by the American Institute of Aeronautics and Astronautics, and reprinted by permission of the copyright owner. Small Unsteady Perturbations in Transonic Flows K-Y. Fung, ° N. J. Yu,* and R. Seebass? University of A rizona, Tucson, Ariz. The effects of vet3 small, low-frequency perturbations on steady transonic flows, in the context of two- dimensional flows described by the small perturbation equation, are investigated. Previous lime-linearized studies failed to account for the shock wave motions that are known to occur. A method is provided that allows one to correctly account for shock wave motions due to arbitrary but small unsteady changes in the boundary conditions. Consequently, both harmonic and indicial responses may be determined. Time-linearized results for the transonic flow past an NACA 64A006 airfoil experiencing harmonic motions in one of several modes are presented. Selected results are compared with those obtained from nonlinear calculations using a shock-fitting algorithm. L Introduction N unsteady transonic flow, relatively small periodic changes in the boundary conditions can lead to substantial changes in the magnitude and phase lag of loads and moments. These are of major concern in the aerodynamic design of aircraft that operate in the transonic regime. Reference 1 contains a short but timely review of various aspects of unsteady transonic flow. Of particular concern are aeroelastic behavior and flutter boundaries. Here the un- steady perturbations may be considered small, and linearization about a nonlinear steady flow, as suggested by Landahl z long ago, would seem to be appropriate. Indeed, it has been suggested 3 more recently that the steady flow be determined experimentally. Difficulties arise, however, which detract from this procedure. Although the equation is linear, its coefficients are variable and must be determined by numerical solution of a nonlinear problem that, in the cases of prime interest, has a discontinuous solution; that is, there are embedded shock waves. Also, although a change of variables in the linear equation provides a scaling of parameters which is indicative of the tradeoffs between, e.g., Mach number and reduced frequency, the only similitude is the one basic to the nonlinear formulation. Tract et al. 4 have developed relaxation methods for solving the resulting time-linearized equations of motion. Less complete but comparable studies have been made by Weatherill et al.5; these derive from an earlier study by Ehlers. 6 In both of these studies shock motions, which contribute substantially to the time-varying loads and moments, 7.s are neglected. Also, difficulties arise in the convergence of the iterative numerical scheme. Here we pursue a different numerical course. Yu et al. 9 have developed a numerical procedure for computing solutions to the unsteady small perturbation equation for transonic flows which treats embedded shock waves as discontinuities. This procedure can be used to calculate the basic steady flow that we wish to subject to small unsteady perturbations. A simplified version of this algorithm then can be used to calculate the linearized unsteady perturbations to the flow. These calculations can be carried out in conjunction with an algorithm that determines the shock wave motion. Presented as Paper 77-675 at the AIAA 10th Fluid and Plasmadynamics Conference, Albuquerque, N. Mex., June 27-29. 1977; submitted Nov. 14, 1977; revision received March 15, 1978. Copyright © American Institute of Aeronautics and Astronautics, Inc.. 1977. All rights reserved. Index categories: Transonic Flow; Nonsteady Aerodynamics; Computational Methods. *Senior Research Associate. Member AIAA. tProfessor. Associate Fellow AIAA. The procedure that we have used to calculate the shock wave motion is a rather obvious one; it is not surprising, then, thai it, too, was given in the monograph by Landahl z (See. 10.2). An alternative procedure, related in some ways to that used here, is implied by Nixon's to study of perturbations to steady discontinuous transonic flows. Formulation We write the unsteady small-disturbance equation for low- frequency transonic flows in the commonly used form -2KM2rb,.,+ll-M2-(,y+l)M2rbxlC_._,+4a,:,.=O (1) The spatial coordinates, the time, and the velocity potential in Eq. (1) have been nondimensionalized by the chord, the reciprocal of the angular frequency, and the freestream velocity times the chord, respectively. Other, perhaps more suitable, forms are given in Ref. 9. This equation results from a systematic expansion of the velocity potential in the thickness ratio r and applies for reduced frequencies K=O(re/:), where K=toc/U, i.e., the angular frequency multiplied by the time it takes the flow to traverse the airfoil chord. Lin et al. I_ showed that, with restriction to small perturbations throughout the flov,, Eq. (1) is the only nonlinear equation that arises. For moderate frequencies, the equation ( 2 2 -2KMe(D,., + [I-ME (_-K M®dp, x [q_, + _+ _ K¢ ] ltd.,:, + ¢,,. = 0 +IJM 2 frequently is used, with or without the _, term, and may provide results that apply at higher frequencies than those obtained from Eq. (1) or the linear form of the preceding equation. Because K= O(r"/_), the boundary condition on the body takes the simple form ¢,, (x,O,t) = r[0 Y(x,t) /Ox + KO Y(x,t) /at] = r[ y* + (6/r) ( Y:,!+ KYI') 1, - ½ <_x<- ½ (2) where Y(x,t), the instantaneous body shape, has been decomposed into a steady part Y* and an unsteady part Y". The last term, KY_, is dropped except when Y:" is small or zero because K=O(r'/3). Here t5 is the amplitude of the unsteady oscillation. Far from the body we require that the derivatives of 4_ vanish. In this approximation the pressure 816 FUNG, YU AND SEEBASS AIAA JOURNAL coefficient, defined so that it vanishes at sonic conditions, takes the form Cn =_21 M_-I +4a_l (3) In the small-disturbance approximation, the Kutta condition is imposed by requiring that Cp be continuous at y=0 for x>½. Any shock wave that exists in the flowfield must satisfy the jump relation derived from the conservative form of the governing equation, Eq. (1), namely, (x) -2KM_ _0_ 2 _ -[I-M_-(y+I)M_Sx] [TO,.] 2 + [_0_.[12 =0 (4) together with the condition derived from the assumption of irrotationality, (5) Here 0, refers to the mean value of 0.,. evaluated on each side of the discontinuity, and 1_4),.]] indicates the jump in ¢,_. across the discontinuity; the subscript s denotes the quantity evaluated at the shock surface. Time-Linearized Equations We now assume that the unsteady disturbances, charac- terized by 8, are small enough so that we may write O(x,y,t) =0" (x,y) +8¢(x,y,t) +0(8) (6) and neglect higher-order terms in b. The restriction imposed on b for this to be true will depend on the other parameters of the problem, viz., _--(1 -M_ )/[(y + I)M-_r] "/3 and K. This gives l I - M_ - (3' + I ) M_O,* )4).,", + O>*,,= 0 (7a) _.,*(x.O)=rY*'(x), -½,=o (8a) ¢,. (x,O,t) = Y!((x,t), - ½ <-x<_ ½ (8b) The solution to Eqs. (7) must satisfy the steady version of the shock relations, Eqs. (4) and (5). The shock relations for Eqs. (8) are discussed later. As mentioned previously, a shock-fitting scheme that approximates the shock waves as discontinuities normal to the freestream has been developed 9 with an alternating-direction implicit scheme (i.e., AD1) to compute the solution to Eq. (7). Comparison of these results with the results obtained v2 using an exact shock-fitting algorithm and line relaxation indicates that they should suffice for most studies. At the very least, they should prove adequate for the time-linearized studies of interest here, as only small shock excursions can be allowed. We assume, then, that we have the numerical values for 0_ required in Eqs. (8). These data will be discontinuous across some vertical line, the shock wave, .x -: x*, 0_< lyl _x,, I_ ( 2(1-2) )d.f-h(t)(12)O(x,t)=3(x-x,.)+8 ,. p t+ /-_ r_.t_ lqi - -- AUGUST 1978 SMALL UNSTEADY PERTURBATIONS IN TRANSONIC FLOWS 817 m ak where the choice h(t) =x, (t) insures that 1] ¢]] =0 at x=x_ because ¢ = ¢ ° for x < x_. The shock motion must be determined by the direct in- tegration o f Eq. ( I 1): 2dx =l_&= 6 dt - _p[x_ (t),t] (13) Now, for comparison, we determine the results that are obtained by time linearization; i.e., we write ¢(x,t;b) =¢* (x) +_¢(x,t) +0(6) (14) and solve the linear equation for ¢ which results by dropping higher-order terms in 6. That is, we solve 2G, + (1-C)G_- ¢.,.[,G=0 subject to ¢_(l,t) =p(t). We now linearize the first of Eqs. (1 I) as follows: IT¢(x,t) j} =0= [[ ¢(x,*,t) +¢, (X_,t)lx_(t) -x3 + ...] = [ ¢*(xD +&k(x;,t) +Clx_ (t) -x3] Thus we conclude that (15) 6 [_¢(x_,t)] =- E ¢*(x_*)] (x_-x_) (16) From the second of Eqs. (11), with x_ (t) = x* + 6X (t), we find dx 1- 1 dt= - 5 ¢/_= - 4 _b_b where ( )b refers to the value behind the shock. Thus we may replace Eq. (11) by or o , o o[ ¢(x.t)] =- [ ¢,.(x_)] x ¢b (x_,t) = -4x(t) (17a) and dx I dt - 4 tk_h (17b) Example Consider now, for example, p(t) =sinwt; it is easy to show that a general solution to Eq. (15) is ¢ (x,t) = - (l/oOIcoso_(l-x-t) -h(t)l The function h(t) follows from Eqs. (17a) and ll7b) and assuming, e.g., that _b( 3A,0) = 0. Thus, t_(x,t) = - (1/o_)[cosw(l -x-t) - cos(w/4)] (18) and X(t) = (1/4w)[coso_( ¾ --t) - cos(o_/4)] (19) Had we solved Eq. (15) with _k-0 for xx_, given by Eqs. (12) and (13), is i x ( 2(1-.f) )d,f (22)¢=¢*+3-4xs(t)+6 sino_ t+ l-d&(.f,t) x$ where d G 6 ( l-x_'_--_" --sin_ t+ (23)dt xs = 4 1 - 2_q / The results, Eqs. (22) and (23), are consistent with the time- linearized results, Eqs. (18) and (19), to O(6), except for a slow secular drift in the shock position of O(Mt) that occurs in Eq. (23) but not in Eq. (19). This is an artifact of our one- dimensional model; even if it were not, it would not invalidate the use of the linear results for flutter studies where 6 is small and structural damping determines the time scale of interest. The main conclusions that we derive from this study are that it is essential to consider shock motion in computing time-linearized solutions if we are to determine the effects of small unsteady perturbations correctly to lowest order, and that shock excursions increase as the frequency is decreased. Additionally, this motion can be computed in a straight- forward manner. Two-Dimensional Time-Linearized Analysis The results from our simple model show that the time- linearized results must be corrected for shock motion if they are to be consistently correct to lowest order. This can be accomplished by calculating the shock motion in conjunction with the time-linearized solution. Here we follow an analogous procedure and calculate the change with time of the values of the perturbed potential behind the shock required by the linearized shock jump relations. Thus, we wish to solve Eq. (8a) with Eq. (8b) subject to the far-field boundary and Kutta conditions. As we noted, the steady result for ¢._ can be calculated adequately for most small-disturbance flows using normal shock fitting, as described in Ref. 9. Under the assumption that the shock wave is normal, the shock jump conditions, Eqs. (4) and (5), can be replaced by requiring dxs _7+1{ Me-1 +¢'1 (24)[_¢]] =0 on dt 2K (y+l)M" For steady flow, 2, = 0 and in Eq. (24) { • • • I = 0. We express the shock positon as Xs=X_+6x(t) and conclude that the shock motion is governed by dx 7+ 1 - /_ dt- 2K 6_ 818 FUNG,YU,ANDSEEBASS AIAAJOURNAL AsdiscussedinRef.9,¢_isevaluatedaty = 0. On the shock [_ = _*]+/5[_] (25) Linearizing Eq. (25) for the velocity potential about the steady shock position, we find ¢(x¢,y,t) =O(x_,y,t) +c_,.(x_,y,t)/sx =¢b ° (x_,y) +O_ (x_,y)/sX +/5_b(x_,y,t ) +0(/52 ) Because we have treated the shock as a normal one, y appears here simply as a parameter. Now [lO(x+,y,t)] and E ¢;0 (x_,y)_ are both zero; consequently, we have I'E¢lx,*,)',t)]] (_+/) [_,"(x,?,.v)ll - * " "- ¢,.(x_,O,t)dt (26)2K o which must be integrated in time in conjunction with the solution to Eq. (8). Equation (8) now is solved numerically in time and space in conjunction with Eq. (26), which is used to update the values of ¢ behind the shock. We start with a steady solution and initiate a body motion, such as the harmonic oscillation of a flap. The calculations proceed in time until they are judged to be periodic. Note that indicial as well as harmonic motions may be considered because we have not utilized the usual harmonic decomposition, Eq. (9). Numerical Procedure The numerical procedure used here derives from that developed for the nonlinear equation, Eq. (l). The main simplification occurs in the shock jump conditions. In order to minimize the far-field boundary effects on the results which, as Magnus _+has noted, can be significant, we again use coordinate stretching 9 in the form ,_=+[l-exp(-allxl)] for x_O r_= ±[/-exp(-a, lyl)] for y_0 solution is advanced in time from its initial sJ.eady state to subsequent time levels with the following two-step procedure. New values of g,, denoted by ¢, +, are calculated along rl = const lines using -I- n+ At _-_ +A, lf(_,_7)(l I,_1) + - ¢_1,_At +A36_1 (1- I'ql )d,b_'}=0 This is coupled with the computation of new values of ¢, + behind the shock obtained by using Eq. (26). With the shock located at _ such that _ < __ < _j_+/, we can express the values of ¢ ahead of and behind the shock in a Taylor series, finally arriving at the result where [ ¢]1 * = - c(,7)a,_' + [ ¢,]1" (28) C('q) = [(),+ l)/4K]a{ (1 - I_'1 ) 2 [[ _ (_',r/) ]] and ¢_' is evaluated, following Eq. (26), at rt =0. One-half the change in ¢ across the shock is accounted for in this step, effectively using the trapezoidal rule in the time integration, Eq. (26); hence Cis half the value implied by Eq. (26). With the values of ¢ + determined, the new values of ¢ at the subsequent time level, _b"+ _, are calculated using A_ A/¢'_'+_At-¢_ + -:-2 6'7[ (1- Ir/I)(_ '+_ -¢_') 1=0 in conjunction with the completion of the time integration, Eq. (26): [I ¢]]""' =-c('7)ar¢_ '+'+ E¢'] + (29) Again, j/_,+/is evaluated at r/=0.:_ The full procedure is, effectively, where a_ and a, are constants that determine the mesh distribution. This stretching transforms the infinite physical domain into a computational domain bounded by I_ I _<1 and It/I __O.I for K=0.48. Because the amplitude of the shock motion increases with decreasing K, nonlinear effects occur at smaller values of b/7 at lower reduced frequencies. Results are aiven for the fifth cycle; note that the nonlinear results are not /_,_ yet periodic. Figure 3 compares the nonlinear and time- linearized pressure deviation from steady state at three angular times for the same conditions. Good agreement between the results is obtained for 6/r less than 0.1. Time-linearized pressure distributions a_ six angular positions for an oscillating quarter-chord flap with K= 0.06 and M_ = 0.875 are shown in Fig. 4. The flap deflection is AUGUST1978 SMALLUNSTEADY PERTURBATIONS IN TRANSONIC FLOWS 821 m," 41 xl_llx lO i / 11.I J J / / t I _ _ " -1 0 5 I0 15 K rmax/6 i0 / / / / / / / / 5 / //* / / / / .tV 1 0 K0 5 I0 15 Fig. 5 Normalized maximum shock excursion and circulations as a function of inverse reduced frequency for an NACA 64A1_i airfoil with oscillating quarter-chord flap (M,_ = 0.875). downward during the first half of the cycle. The results for the second half of the period, for the symmetrical problem shown here, are just the results shown with the lower and upper surface pressures interchanged. Thus the results for 0 deg are not given, as they are just those for 180 deg with the lower and upper surface pressures reversed. Because the flap hinge occurs very close to the steady-state shock location, the pressure singularity due to the change in flow direction at the hinge is missed. The circulation and shock excursion obey the following relations: F(t)/,5=9.26 sin(t-59deg) X(t) = 12 sin(t--51 deg) Note the substantial phase lag in the circulation and the shock's position. Time-linearized pressure distributions for an oscillating airfoil with K = 0. i 2 and M® = 0.875 also were computed. The results, if multiplied by K, represent the pressure per- turbations for a plunging airfoil. As in the previous case of an oscillating flap, changes in forces and moments of 0(6/K) occur due to shock wave motion. In this case, F(t)/6 =5.48 sin(t- 70 deg) X(t) =5.62 sin(t-87 deg) Analogous computations have been carried out for K=0.06, 0.12, 0.24, and 0.48. Figure 5 depicts the shock wave's excursion and maximum circulation as a function of K -t. The nearly linear variation of the shock excursion substantiates an observation made in a one-dimensional model where the shock wave excursion is directly propor- tional to 1/K. In these calculations the circulation gives an immediate evaluation of the lift coefficient as a function of time; the moment coefficient must be evaluated by integrating the mnment nf tha nr_:trlrp c'npffi,'-iant Thic ic dr, no h,t, in_ tegrating the moment of pressure perturbations with the shock wave in its steady-state position and then correcting these results for the moment due to the shock wave motion, assuming that the shock's strength is defined by the steady- state pressure field. This makes an error in the shock strength of 0(6), but the effect on the moment is O(M/K); because we have neglected other higher-order terms, it is consistent to neglect this change in the strength of the shock wave. For the time-linearized results to be valid, we must really require 6/rK,_ I. Our numerical results indicate that, for 6/rK< 0.2, the unsteady perturbations are essentially linear. The time-linearized algorithm used here is a derivative of that used for the nonlinear calculations. Consequently, computational times are not greatly reduced from those required for the nonlinear calculations. The linearity of these computations may make it possible to greatly reduce the computational effort required. Numerical experience has shown some difficulties for At (deg)/K_>50. This is in agreement with the consistency requirement for thg ADI algorithm used here. Both the domain-of-dependence con- dition and a local linearized stability analysis show the procedure to be unconditionally stable. Each time step requires about 2 s of CPU time on a CDC 6400, or about 0.1 s on a CDC 7600. The number of time steps required for a given computation is somewhat less than that required for the nonlinear computations at small vahtes of K, and comparable at larger values of K. Conclusion An accurate and efficient procedure for computing time- linearized, small-perturbation, low-frequency transonic flows, including shock wave motions, has been developed. Shock motions must be included, as their amplitude is proportional to that of the motion divided by the reduced frequency. Both indicial and harmonic responses for various modes of motion may be computed in seconds on a CDC 7600. Acknowledgemen! This research was sponsored by the Air Force Office of Scientific Research through Grant 76-2954E and the Office of Naval Research through Grant N0014.-76-C-0182. References IBallhaus, W. F., "Some Recent Progress in Transonic Flow Computations," VKI Lecture Series on Computational Fluid Dynamics, yon K_.rm_n Institute for Fluid Dynamics, Rhode-St- Genese, Belgium, March 15-19, 1976. 2Landahl, M. T., Unsteady Transonic Flow, Pergamon Press, New York, 1961. 3Tijdeman, H., "Remarks on the Transonic Flow Around Oscillating Airfoils," ,4GA RD Fluid Dynamics Panel Symposium on Unsteady Aerodynamics, Paper 227-10, Ottawa, Canada, 1977. 4Traci, R. M., Albano, E. D., and Farr, J. L., Jr., "Perturbation Method for Transonic Flow about Oscillating Airfoils," AIAA Paper 75-877, 1975. 5Weatherill, W. A., Ehlers, F. E., and Sebastian, J. D., "Com- putation of the Transonic Perturbation Flow Field Around Two- and Three-Dimensional Oscillating Wings," NASA CR-2599, 1975. 6Ehlers, F. E., "A Finite Difference Method for the Solution of the Transonic Flow About Harmonically Oscillating Wings," NASA CR- 2257, 1974. 7Tijdeman, H., "On the Motion of Shock Waves on an Airfoil with Oscillation Flap in Two-Dimensional Transonic Flow," NLR TR 75038U, 1975. 8Tijdeman, H., "On the Motion of Shock Waves on an Airfoil with Oscillating Flap," IUTAM Symposium Transsonicum 11, edited by K. Oswatitsch and D. Rues, G6ttingen, 1975, pp. 49-56. 9yu, N. J., Seebass, A. R., and Ballhaus, W. F., "An Implicit Shock-Fining Scheme for Unsteady Transonic Flows Computations," AIAA Paper 77-633, 1977; also AIAA Journal, Vol. 16, July 1978, pp. 673-678. t°Nixon, D., "Perturbation of Discontinuous Transonic Flow," AI,,/ 822 FUNG, YU, AND SEEBASS AIAA JOURNAL l lLin, C. C., Reissner, E., and Tsien, H. S., "On Two- Dimensional Non-Steady Motion of a Slender Body in a Compressible Fluid," Journal of Mathematics and Physics, Vol. 27, No. 3, 1948, pp. 220-231. 12yu, N.J. and Seebass, R., "Accuracy of Transonic Flow Computations" (unpublished), 1977. J_Ballhaus, W. F., Jameson, A., and Albert J., "Implicit Ap- proximate-Factorization of Steady Transonic Flow Problems," NASA TM X-73202, 1977. 14Magnus, R. J., "Computational Research on Inviscid, Unsteady, Transonic Flow over Airfoils," Office of Naval Research, Repl. CASD/LVP 77-010, 1977. JYBallhaus, W. F. and Steger, J. L., "Implicit Approximate- Factorization Schemes for the Low-Frequency Transonic Equation," NASA TM X-73082, 1975. _6Beam, R. M. and Warming, R. F., "An Implicit Finite- Difference Algorithm for Hyperbolic Systems in Conservation-La_s Form," Journal of Computational Physics, Vol. 22, Jan. 1976, pp. 87-110. 17 Ballhaus, W. F. and Goorjian, P. M., "Implicit Finite Difference Computations of Unsteady Transonic Flows about Airfoils, Including the Treatment of Irregular Shock Wave Motions," AIAA Journal, Vol. 15, Dec. 1977, pp. 1728-1735. 18Fung, K.-Y., Yu, N. J., and Seebass, R., "Small Unsteady Perturbations in Transonic Flows," AIAA Paper 77-675, Albuquerque, N. Mex., June 1977. A JFocusing of weak shock waves at an ar6te By M. S. CRAMER AND A. R. SEEBASS J. Fluid Mech. (1978), vol. 88, part 2, pp. 209-222 209 Printed in Great Britain lit Focusing of weak shock waves at an ar te By M. S. CRAMER Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853 t AND A. R. SEEBASS Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721 (Received 28 June 1976 and in revised form 13 Fobruary 1978) The focusing of very weak and slightly concave symmetrical shock waves is e.xamined. The equation that describes this focusing is derived and the resulting similitude discussed. The initial conditions come from a formal matching of this nonlinear description with the linear solution. The maximum value of the pressure coefficient is shown to be proportional to the two-thirds power of both the initial strength of the wave front and a parameter characterizing its rate of convergence. 1. Introduction There are many sources of weak shock waves; they arise naturally and through the activities of man. Examples include the commonly experienced phenomena of thunder and the sonic bang generated by supersonic aircraft. In this paper we use the terms shock wave and wave front interehangeably to refer to a surface of discontinuity in the pressure and velocity fields in the fluid. It frequently happens that such wave fronts become curved. This curvature may be the result of inhomogeneities in the medium, reflexion from curved surfaces or unsteady boundary conditions. Wave fronts which are concave in the direction of propagation exhibit different kinds of behaviour depending upon the strength of the wave front and the rate of focusing. When the focusing is weak relative to the magnitude of the pressure rise across the wave front, the wave front will straighten and no focusing will occur. When the strength of the wave front is sufficiently small, the wave front will focus along a caustic surface and at a cusp in this surface, called an ar_te, if it occurs. A perfect focus occurs when a finite portion of the wave front converges to a single point. The focusing process is characterized by large pressure amplification and a nonlinear interaction between the shock and the flow behind it. Despite considerable analytical, numerical and experimental work, many important questions remain unanswered. Analytical studies are hampered by the fact that available theories, such as the shock dynamic theory of Whitham (1957) and the theory of geometrical acoustics, are inapplicable at a focus. The first fails because it does not account for the interaction between the shock and the flow behind it; the second fails because it is a linear theory and predicts infinite pressures at focal points. Important theoretical studies of the _f Present a_ldress: Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721. 8 I"_'_ ,LM 88 210 M. S. Cramer and A. R. Seebass behaviour of weak shock waves at a caustic include those of Guiraud (1965) and Hayes (1968). For the case of a smooth caustic they gave the similitude that relates the amplification of the wave front to its initial strength and geometry. An important contribution Go our understanding of the behaviour of focusing wave fronts comes from the experimental investigations of Sturtevant & Kulkarny (1974, 1976). Using shadowgraph techniques and pressure measurements they studied the focusing of curved shocks for a wide range of geometries and strengths and delineated the compli- cated wave patterns and pressure histories which occur. Because the caustics associated with smooth wave fronts are generally cusped, the ar_te is one of the more frequently observed foci. In this paper the focusing of very weak shocks at an ar_te is examined. We consider nearly straight symmetrical wave fronts and use the method of matched asymptotic expansions to determine the initial- value problem and related similitude that govern _he flow in the vicinity of the ar_te. The dependence of the maximum pressure coefficient on the initial strength and shape of the wave front is discussed; the main result is that the pressure levels at the ar_te are proportional to the two-thirds power of both the initial strength cf the wave front and a parameter characterizing the rate at which the wave-front converges. 2. Physical problem In this section we give a qualitative account of the physical effects that govern the propagation of curved wave fronts. One effect is that a wave front always propagates normal to itself and therefore has a tendency to converge. Another is that the speed of propagation of the wave front increases monotonically with its strength. The latter effect will tend to straighten converging shocks. The behaviour that results depends on which effect dominates. The behaviour of shock waves which are relatively strong w_s discussed by Whitham (1957, 1959, 1974) using his theory of shock dynamics. To understand the behaviour of such shock waves, consider the propagation of a concave symmetrical shock into a homogeneous medium as sketched in fig_lre 1 (a). The shock's strength is taken to be a maximum in the plane of symmetry and, because the amplification is greatest in this plane, the maximum strength will remain there. However, the shock speed will also have its maximum in the plane of symmetry and the resultant variation in propagation speed will straighten the shock. This behaviour is expected even for shocks with pressure coefficients much less than one, provided that the focusing is sufficiently slow. When the strength of the shock is sufficiently small its speed of propagation is approximately the sound speed of the undisturbed medium. In many respects the flow will resemble that predicted by geometrical acoustics, and we first discuss the behaviour of such weak shocks from this viewpoint. In this approximation the propa- gation speed is taken to be the sound speed of the undisturbed medium and, if we take this to be homogeneous, the trajectories of points on the shock, called rays, will be straight. Adjacent rays originating on concave portions of the wave front will intersect, and the locus of these intersections will form a surface in space, called a caustic. When the wave front has a minimum radius of curvature the caustic will be cusped and the wave front will emerge from the cusp in a crossed configuration, as sketched in figure 1(b). Experimental evidence of weak shocks which focus and cross is found in the sonic-bang measurements made by Wanner et al. (1972) and in the laboratory Focusingof weak shock u_ves at an argte 211 $. y (a) X 0 y Initial ]front I (b) FmURE 1. Focusing of (a) a moderate strength shock and (b) an acoustic diseontinuity;f"lL) _= 0, R o = l/j'"(O). investigations by Beasley, Brooks & Barger (1969), Cornet (1972) and Sturlevant & Kulkarny (1974, 1976). Geometrical acoustics predicts that when the wave front reaches the caustic surface its pressure jump becomes infinite. At the cusp in the caustic this singularity is even stronger because portions from either side of the plane of symmetry focus there simultaneously. Of course, these singularities are never observed, and they merely indicate a local failure of the geometrical-acoustics approximation. Sturtevant & Kulkarny have discussed the nonlinear effects which limit the shock strength to finite values in the focal region. They showed that as the shock approaches a cusp in the caustic it is immediately followed by a sharp expansion. This is due to the amplification of the shock relative to the rest of the flow. When the pressure gradient behind the shock becomes sufficiently great the weakening effect of the expansion as it overtakes the shock becomes noticeable, even if the shock is still weak. Because of the. shock's interaction with the expansion wave, the strength of the shock is limited to finite values. 8-2 / 212 M. S. Cramcr and A. R. Seeba_s 3. Mathematical formulation We consider the wave front shown in figure 1 (b) and take the co-ordinate system to have its origin at the point of minimum radius of curvature of the initial wave front. The + x axis is in the direction of propagation and the y axis is tangential to the initial wave front at y = 0. The equation of the initial wave front is taken to be xi = f(y_), where x_ and Yi are the co-ordinates of the initial shock and f is symmetrical about y = 0. We are considering the propagation of very weak shocks in an inviscid perfect gas with no heat conduction. In the case considered here the shock strength is always small. The results of Hayes (1957) may be used to argue that for the spatial and temporal gradients expected here the flow may be assumed irrotational. If we assume that a velocity potential ¢ exists, the inviscid equations of motion reduce to Ctt+ 2¢z_zt+ 2_v_ut+ 2¢xCvCz v = (aZ-¢_x)q_z:t+(a2-¢_)¢_, (1) where a is the local speed of sound, which for a perfect gas it is given by the isentropie Bernoulli integral for unsteady flow: a2 = ao2 - (y - 1) [¢t + ½(¢_ + ¢_)], (2) where a 0 is the constant speed of sound in the undisturbed medium and 7, is the ratio of specific heats. The velocity potential must satisfy the initial conditions ¢(x, y, 0) = ¢0(x, y), Ct(x, y, 0) = ¢1(x, y), (3) where the functions ¢o and ¢1 are taken to be zero ahead of the wave front, i.e. for x > f(y); immediately following the shock, their values must be consistent with the appropriate shock jump relations. As mentioned in previous sections, we consider the shocks to be not only initially weak but also nearly straight. We take the maximum strength of the wave front to be at y = 0 and the pressure coefficient, Cp(x, y, t) = - 2¢t/a _ to lowest order, to be small for all x and y at t = 0. We define e - c_(0-, o, 0) < 1 as the small parameter associated with the shock strength. For the shock to be practically straight we require thatfbe such that the maximum slopef' is small. If we define L to be the point of maximum slope, i.e. f"(L) - O, then the requirement $ =- L/R o _ 1, where R o is the minimum radius of curvature of the shock, ensures that the slope is small everywhere. Another restriction we shall need fx) place on the shape of the wave front concerns the rate of change of the radius of curvature of the wave front at t = 0. Denoting the radius of curvature by R(yi), we have R(y_) = [1 +f,(y_)2]g/f,(y_). (4) In §4 we shall Reed to require that R_Ro 3_ = (3 -fJv(0)Roa)_ be of order one in the limit of vanishing 3. Here R 0 denotes the second derivative of R(y_) at y_ = 0 and f,v(0) is the fourth derivative of fat yi = 0. Examples of possible wave-front shapes are Z(Y; A, l) = All - exp (-y2//_)], f2(y; c_, c_.) = _clyZ- _A6c2y6. Both shapes are smooth and possess inflexion points. The parameter $ is found to be 2½All in the first case and c_c_ ¼in the second, and it is clear that the limit $ -+ 0 may Focusing of weak shock waves at an ar$te 213 always be taken. The above formulae may be used to calculate R 0R 0o_. In the first case this equals 3(1 +$_) exactly and the requirement that RoRoo _ = O(1) as b'--> 0 is satisfied; thus the results derived in this paper may be applied to the first example. The second case is given because it is a wave-front shape which does not satisfy this condition; essentially, this is because fi2v(0 ) = 0 and, consequently, RoRo 3_ --> 0 as -> 0. In the following sections we shall always assume K- -pv(O)R_,$_ BoRon2 = O(J). Our main objective is the determination of solutions to (1) and (3) which are valid for small e and $ and, for consistency, whose pressure and velocity perturbations are always small. In particular we wish to study the case where wave-front crossing occurs. In this paper we give approximations to (1) in two distinct regions. The first is just the linear wave equation and is valid for times of order L/a o. The solution, subject to the initial conditions (3), is easily found; this is the outer solution. An approximation to (1) which is valid as the wave front approaches the ar_te, i.e. the cusp in the caustic, is derived in § 5. The initial condition for this equation is obtained by the method of matched asymptotic expansions, thus establishing the initial-value problem governing the flow in the vicinity of the ar_te. In the next section the outer solution and the expression for the caustic shape near the cusp are derived. In § 5 the inner region in which the nonlinear effects predominate is discussed and the inner equation deduced. In § 6 the appropriate initial condition for this equation is obtained through the method of matched asymptotic expansions. These results provide a similitude which shows that the solution to the inner problem involves only a single parameter. O 4. Outer region To determine the outer equation it is convenient to work in a co-ordinate system that moves with the wave front. Accordingly, we write the full potential equation (1) in terms of the co-ordinates X = x-a 0 t, y and t: ttt -- 2a0 txt + 2¢x txt + (T - 1) _t txx - (7 + 1) a 0 tx txx = ao2_bvv - [2_bv try + (V -- 1) tt try] + a0[2¢v txv + (_/-- 1 ) tx tv_] q" .... (5) where (2) has been used and the terms omitted are cubic in ¢. In the outer region it is natural to take aot= O(L), y = O(L), X = O(A*), ¢ = O(aok*), where the length scales A* and k* are yet to be determined. According to the theory of geometrical acoustics the square of the pressure coefficient immediately following the shock varies inversely with the ray-tube area. For our problem all the rays are straight lines and, consequently, the pressure coefficient behind the shock obeys (see, for example, Friedlander 1958, pp. 51-56) Cp,(t; Yi) = Cp_(O; Yi) \R(yi)- %t/ " Here R(y_) is the radius of curvature of the initial wave front at the point y = Yi; y_ effectively labels the ray of interest. Thus, for times of order L/ao, we see that the l 2t:? 214 M. S. Cramer and A. R. Seebass time rate of change of the shock's strength is of order ao/R(yi). This suggests that the appropriate outer scaling for the derivative (_/_t) x is ao/R o. Accordingly, we adopt the following outer scaling for the derivatives in (5): ao_ _ 1 _ _ 1 St R o_i' Oy L_' _X A*_J[ and ¢ = ao k*¢. For t = O(L/ao), there is no amplification to lowest order in 8; we therefore assume that k*/A* and k*/L are small. When the above expressions are substituted in (5) we find that it may be written in the following form: _iT_2 ARo.. k* R o - k* Ro2 1 - k* 1 2" - k* 1 A* - - = _ ¢_ +X; _ [ ¢_¢e* + (7-1) ¢_ 4_] X; _ _ [2¢_¢_+ (_- _)ffF4,o]+ .... We now assume that k*/A* and k*/L are small; it may be shown that this implies that the omitted cubic terms are negligible compared with the largest of the first- and second-order terms. Furthermore, when k*/A* is small, the third term on the left- hand side and the second term on the right-hand side are seen to be negligible com- pared with the ___term and the ff_ term, respectively. When these terms are dropped, the terms remaining are Ro 1,* R _ 1 k* 1 i*[2_5_+(__l)_tq_]" q_ig--2_-_2_--(_+l)_**_.°_X¢_ ----_¢_ A*_ R 0 This expression is reduced further by assuming that $ is small. An immediate conse- quence of this assumption is that the q_ term is negligible compared with the _ term. Inspection of the remaining terms shows that the only choice of A* which yields a non-trivial balance of terms, i.e. one which contains derivatives with respect to )_, and t, is A*/R o = 6_, or A* = L& Here, for convenience, we have dropped the order symbols. The resulting balance of terms may be written as £ 2_F+ (9,+ 1)_q_y¢_+_ = O. The coefficient of the nonlinear term is obtained by noting that A* = _L implies that (O/_t)z "z - ao(8/_X)t in the outer region and therefore that 1 (k.)c_ = o(_)= -_ ¢, _ _o¢_ = o _;, or simply k* = A*e. Thus, provided that e/8 _ = o(1), nonlinear effects are negligible. We now assume that this is the case, so that the outer equation becomes #_+ 5_ = 0. Transforming back to the physical variables and back to the. x, y, t co-ordinates, we see that our outer equation is just the wave equation ttt = a_(¢xx + ¢_), (6) where we have used the fact that (_/_t)z _ -ao(O/OX)e in the outer region. JFocusing of weak shock waves at an argte 215 The explicit form of the outer solution is obtained by solving the linear wave equation (6) with the initial conditions (3). The solution to (6) which satisfies (3) is given by the well-known Poisson integral formula 4.¢ lz =a, t o]. The quantities Io and 11 are the integrals of ¢0 and ¢1 over a sphere of radius ao t centred at the point (x, y, 0). This sphere has the equation (xs- x)2 + (ys- y)_ + z_ = a_t 2, (7) where the subscript s refers to points on the sphere of integration. We now consider the distance s measured along a ray from the initial wave front to any point (x, y, 0); then s = [1 +f'(y_)_]½ [x-f(yi)] = [1 +f,(y_)2]½ (Y (8) Here we have defined s to be positive ahead of the initial wave front and negative behind it. In the following sections it _'ill be useful to have the functions ¢o and ¢1 in terms of s instead of x and y. Using the fact that f/L and f' are small, we may replace (8) by Y8 = yi+o(ss), s8 = x_-f(y_)+o(s_). (9) Using (9) to replace x s in the integrands of I0 and 11, we may write these integrals to lowest order as I o = ff¢o(Ss,y.)dA, 11 --= the area element dA of the sphere (7) may be written as dA = a°tdz_dY" Jabots-(y,- yF - z ]½" Using (7) and (9), we now write s, explicitly in terms of y, and zs: s, = X + aot + [a2ot_- (y,- y)2 _ z_]½- f (y,) + o(s,), where the + sign refers to points on the sphere ofin_egrationwith x_ X x, respectively. The above results have been derived in terms of the physical variables X, y, t, etc. We formally define the outer variables aot _ z .,f_= X X _- L' Y--- ' z-_½L' A--*=_ ' The outer solution may now be written as 4n_= 1. 1 c9 _ -i x--i 1o, where in which and k*a o eSLa o" _o(gs, Y.) = ¢o(ss, Ys)/eao_L, f_:(_s,Y.) -- ¢,(s.,Ys)/a_o e (lO) (11) d._ =- dA /$LL L 216 M. S. Cramer and A. R. Seebass In determining the scaling for ss, we have used the fact that the functions ¢0 and ¢1 are identically zero for s_ > 0 ; therefore it is necessary to consider only non-positive values ofssin the above integrations. As a result of this, s._is of order SL in the outer region and, although Y8 and _ are each of order one in the outer region, their difference Y8-Y will always be of order b'tL there. Because zs is also found to be of order $_L, the last two terms under the square root are of the same order. In the outer region t can vanish; (I 1) is therefore the lowest-order expression for _ which is uniformly valid in the outer region. Strictly speaking, the outer solution is valid only for times of order L/a o. The result (I0), of course, predicts infinite pressures when the wave front reaches the caustic surface. This surface is defined by the intersection of adjacent rays, i.e. of adjacent normals to the initial wave front. These considerations imply that the distance, measured along a ray, from a point on the initial wave front to the caustic is just R(y O, the local radius of curvature of the wave front. In terms of the Cartesian co-ordinates x and y the equation of the caustic surfaces is found by substituting s = R(y_) in (8). Doing so and expanding for small _, we find that the caustic is cusped and that near this cusp it has the shape to lowest order. Or, if we eliminate Yi, --N-. / where the subscript c refers to the caustic. As discussed in §3, we require that ~ ,' ., o(1), such as theK ~ _RoRo_ be of order one. Initial shock shapes which have K -- seeond example discussed in § 3, require the inclusion of higher-order terms in the above expansions. This will change the relative sizes of the x and y in the inner scaling, thereby increasing the strength of the singularity at, the foeal point. For wave fronts with K = o(1) but such that a _ may be found and made small, the procedure of this paper may be used to obtain analogous results. The singular behaviour of the solution at the caustic suggests that nonlinear effects are important there. In the next section we assume that the initial strength of the wave front is so small that these nonlinear effects are important only in the vieinity of the caustic. We then find the appropriate nonlinear equation governing the flow near the cusp in the caustic. ,) 5. Inner region We now seek an inner expansion valid in the vicinity of the ar6te. We introduce the inner variables _, E, 0, X and ff and corresponding inner scales A,/_, A and k, viz. aot-R o y X ¢ (12)x-R° t- , O- X =" _-- - AR ° ' AR o ' _, aok" We assume that the nonlinear effects are important in only a small neighbourhood of the cusp, and we shall therefore take A = o(1) and ,_/L = o(1). The amplification is greatest in the region immediately following the shock and we expect nonlinear effects J Focusing of weak shock waves at an ar_te 217 to be most important there; consequently, we use the same scaling for x and t. Further- more, we assume that, although the strength of the shock is considerably amplified in the focal region, it remains small. Thus we shall take tx/ao = O(k/A) = o(1) and ¢_/a o = O(k/A) = o(i ). When the scaling (12) is substituted into (5) the full potential equation may be written as q_iT- 2 AR° k AR o k A_R02_ A_R], k A_R_ k A_R_ A - - As in our analysis of the outer region, the assumptions that k/A and k/,_ are o(I ) imply that we may drop not only the cubic terms, but also the third term on the left-hand side and the second term on the right-hand side. The resulting equation reads k A_R_ _2 q; t- + X-X A_R_ _ k A2R_ A On physical grounds it is clear that we must require that the inner equation contains X, _ and _ derivatives and at least one nonlinear term; the only choice of A which results in such an equation satisfies A/AR o = o(1). For small A/AR 0, the nonlinear term on the right-hand side and the q_ term may be dropped. In order to balance the remaining terms we need AR 0 = A_R_ = k A_R_ >_ 1. (13) A A _ A A _ Thus, to lowest order, the inner equation is 2_+ (7+ 1) q_xq_xx+ _ = 0. (14) We remark here that (14) also describes low frequency, unsteady, transonic flows, as one might have anticipated. We now assume that in the inner region the relative size of the x and y length scales is given by the caustic surface calculated in § 4. This requires that we take A s = O(A_/L_). Dropping the order symbol and simply substituting A _ = AaL _ in (13), we may express A, A and k in terms of A and 8: A/L = A], A/L= A_$, k/i = Ab_. (15) Thus A, A and k are known in terms of the physical parameters e and $ once :Xhas been determined. The flow in the focal region is found by solving the inner equation (14) subject to an appropriate initial condition. In the next section we determine this initial condition through matching with the outer solution. 218 M. S. Cramer and A. R. Seeba, ss 6. The matching and the similitude In this section the initial data for the inner problem are obtained through the method of matched asymptotic expansions. It is further shown that, apart from a simple scaling of the independent and dependent variables, the solution to the resultant initial-value problem depends on only a single combination of the physical parameters: the similarity parameter. We first consider the outer expansion, which we shall write in the inner variables (12) and then expand for small A and $ with the inner variables fixed. We begin by expressing (! 1) in terms of the inner variables (I 2) and expanding for small A and 8. As a result of this expansion we find that the inner scaling for the integration variables y, and zs is given by 9s = 5-_._s = A-½L-lys, "zs = ¢3½A-15, = A-1L-lz, and that, to lowest order, the inner expansion of ss is ts(A½9,, A$-½_s; A2X, A|gs, (1 + At)/$) = A2[½__fl(9,)], (16) where/7(9,) is defined by ?(9,) = ?(gs; x, 9, t) - - -&K_ + ½i__,+ 9,9 - x. In the derivation of (16) we have chosen the lower sign in (11). Any other choice corresponds to portions of the integration sphere with x s > f(Ys) and contributes nothing to the integrals TOand I r The integrals r e and r t are to be evaluated over the surface of a sphere. However, when the inner expansion of the outer area element dX is taken, we see that it may be replaced by A_3-1 d$ s dgs. Thus, in the inner expansions of r o and _r1, we may transfer the integration from the sphere to the ts, 9, plane. We need the expansion of 4 near the cusp only for times less than Ro/a o, i.e. _ < 0. The area of integration will therefore be bounded by the single closed curve ss = 0, or ts = + [2/?(9s)]½, as sketched in figure 2. The ts = 0 intercepts 9_ and 9z are just the two real roots of _ = 0. When the inner expansions of the integrals _ro and J'l are taken it will he useful to have the Taylor series expansions of 40 and 41 for small _ and Ys: 4o = ½3.[1 +O(._s,_s_)], 41 = -½[1 +O($_,,77_)]. These are, of course, consistent with the weak-shock jump conditions. The above results may now be used to show that the inner expansions of 1"0 and rlare and /"_',,£ ("4P & = J,7,Jo /.#'. £(_,8# .. A__ro to lowest order. When the _s integration is performed and the result substituted in (10), we find 1 ,,S Focusing of weak shock waves at an ar$te 219 s_:=0 Fm_m_ 2. Area of integration for I o and I t. Thus, when written in inner variables, the inner expansion of the outer solution is given by k* (A_x ' A,,,1 _.h_) = 1 A__$__:-fl½ d,s. (17) This result is essentially the expansion of the linear solution (10) as the wave front approaches the cusp in the caustic. As mentioned in previous sections, the pressure distribution associated with this pressure field can be singular. The pressure coefficient based on (17) is At t = Ro/ao or, in terms of the inner variables, _ = 0, the pressure coefficient is proportional to (R0 -x)-t along the x axis and to [y]-t along the y axis. This is more singular than the analogous result for a smooth caustic, where the pressure behaves as the - _ power of the distance along the caustic and the - ¼ power of the distance normal to it (see, for example, Friedlander 1958, pp. 67-70; or Hayes 1968). Another interesting feature of (17), which turns out to be essential for the matching, is that it is self-similar in time, i.e. it may be written as = To show this, we define similarity variables q and F by _=- r - 60, where g = X/K, y --=_/K and • - t/K. Using these definitions and replacing the integra- tion variable _8 in (17) by (- 6r)½ q, we find that (17) may be rewritten as _(-_) G(_, r), (18) where G is defined by G(q, r) -- 6 f:= ]3½(q)dq, in which fl = - q4 _ 2q_ + rq + a and q= and qt are the two real roots of _ = 0. 220 M. S. Cramer and A. R. Seebass We now consider the inner solution. Although, in general, solutions of the form (18) do not satisfy (14), the inner solution must have such a form as _ -* -oo. Thus it is most convenient to do the matching in the co-ordinates _, F and r. Accordingly, we write the solution _(X, 9, i) to (14) in terms of (r, P and r and then rewrite _ in outer variables: ¢ = _(_, F,r) = q_(d, F, (_- 1)/A_), where d -= -]J[/($Y-1) 9 and F =-- 4_/6½(1 -_)_. The usual matching principle states that, in the limit e --> 0, A --> 0, _ --> 0 with J[,/7 and i fixed, ¢(_, F, ($t- 1 )/AK) must approach (l 8) asymptotically. Thus the function _(_, F, r) must satisfy 4(_, r,r) ~ -½(½K)½(-r)la(_, r) as r -* - oo for all values of _ and F. In order that the initial condition be non-zero and finite, we must take A_ _ e/$ _ = o(1). This last result determines the scaling (15) completely in terms of the physical para- meters e and 8, viz. A A} e A e_ k e_ L _2, L $_' A When written in terms of the variables g, _/and r the initial-value problem for the inner region is 2_, + (y + 1 )K -1 _ _ + q_,, = 0, where as r _ -oo ¢(_, y, r)~ -½(½K)_(-r)OG(_,y,r) for all values of_ and 7]. Here G(_, _1,r) is just. the integral G(q, F) rewritten in terms of the variables _, y and r. We have now established the initial-value problem governing the flow in the focal region. The solution to this problem is seen to possess a similitude, i.e. it depends on only a single combination of the remaining physical parameters y and K. This is readily seen when the above initial-value problem is recast in terms of the scaled velocity potential q) -- ½(½K)½_. The problem then becomes where as r -+ - oo 2¢_, + Q¢_ O_ + O_ = 0, ¢ ~ - (- r)_ G(L _, r). The similarity parameter Q is defined by Q - ½(y + 1)/(2_)_. Except for a scaling of the dependent and independent variables, any two flows with the same value of Q will be identical. The flow quantity of especial interest here is the pressure coefficient, which is given by e et3t Cv = _ ¢_(¢, y, r; Q) = _ ¢_(_,,, r; Q). ) 'O S Focusing of weak shock waves at an ar_te 221 From this we see that the pressure levels at an ar_te are of order e|b't and the amplifica- tion is of order e-½8i. The result that the pressure is proportional to the two-thirds power of the initial strength of the shock was also deduced by Pierce (1971 ). We conclude this section with some remarks regarding the requirement that e = o(_2). An important result, but one which is outside the scope of the present theory, is the prediction of the transition shock strength; that is, for a given initial shock shape, the prediction of the initial shock strength above which the straightening of the shock associated with shock dynamic theory occurs. In § 4 we have seen that when e = 0(_) = 0(5) nonlinear effects are important even for times of ordL L/a o. Because the distance to the caustic is large compared with L, we expect that t shock straightens without focusing. On the basis of this, we conjecture that the "der of magnitude of the transition shock strength is given by e = 0(o_). Of course, a • re precise estimate must be given either by laboratory or numerical experiments or by a more comprehensive analysis. 7. Conclusion The focusing of a very weak and almost straight shock at an ar_te has been examined. The method of matched asymptotic expansions has been used to establish the initial- value problem and similitude governing the flow in the focal region. The fundamental parameters in this problem are seen to be e, a measure of the initial strength of the shock, and $, which measures the rate at which the wave front converges The maxi- mum pressures at the ar_te have been shown to be proportional to (e_)t. The results of this paper are valid for wave fronts with e = o($ _) and $ = o(1). This research was supported by the office of Naval Research through contract N0014-76-C-0182 and the Air Force Office of Scientific Research through grant 76-2954B. REFERENCES BEASLE_', W. D., BRooxs, J. D. & BAR(_ER, R.L. 1969 A laboratory investigation of N-wave focusing. N.A.S.A. Tech. Note D-5306. COlt_ET, E. P. 1972 Focusing of an N-wave by a spherical mirror. Appl. Res. Lab., Univ. Texas, Austin, Rep. ARL-TR-72-40. • FRIEDLANDER, F.G. 1958 Sound Pulses. Cambridge University Press. GUIRAUD, J.P. 1965 Aeoustique g_ometrique, bruit ballistique des avions sup[.rsoniques et focalisation. J. Mdc. 4, 215-267. HAYEs, W.D. 1957 The vorticity jump across a gasdynamie discontinuity. J. Fluid Mech. 2, 595-600. HAYES, W.D. 1968 Similarity rules for nonlinear acoustic propagation through a caustic. 2nd Conf. Sonic Boom Rea. (ed. I. R. Schwartz). N.A.S.A. Special Publ. no. 180, pp. 165-171. PIERCE, A.D. 1971. Maximum overpressures of sonic booms near the cusps of caustics. In Noise and Vibration Control Enqineer/ng (ed. M. J. Crocker), pp. 478-487. Purdue University Press. STV_VANT, B. & K_KAm_'Y, V.A. 1974 Dynamics of weak shock waves at a focus. Proc. 2nd Inter-Agency Symp. Univ. Res. Transportation Noise, North Carolina State Univ., Raleigh, pp. 402-415. 222 M. S. Cramer and A. R. Seebass ST_rR_VANT, B. & KULKAR_Y, V.A. 1976 The focusing of weak shock waves. J. Fluid Mech. 73, 651-671. WA_I_'Ea, J. C., VALLEE, J., VIWER, C. & THERY, C. 1972 Theoretical and experimental studies of the focus of sonic booms. J. Acoust. Soc. Am. 52, 13-32. WKXTI_, G.B. 1957 A new approach to the problems of shock dynamics. Part I. Two- dimensional problems. J. Fluid Mech. 2, 146-171. W_ITm_, G.B. 1959 A new approach to the problems of shock dynamics. Part 2. Three- dimensional problems. J. Fluid Mech. 5, 369-386. WHrrH_, G.B. 1974 Linear and Nonlinear Waves. Interscience. ,fi/_. "O j Reprinted from AIAA JOURNAL, Vol. 16, No. 7, July 1978, pp. 673-678 Copyright, 1978, by the American Institute of Aeronautics and Astronautics, and reprinted by permission of the copyright owner If Implicit Shock-Fitting Scheme for Unsteady Transonic Flow Computations tm N. J. Yu,* A. R. Seebass,i and W. F. Ballhaus:_ University of Arizona, Tucson, Ariz. The alternating-direction implicit scheme developed by NASA Ames for unsteady transonic flows has been modified to include a shock-fitting algorithm, as well as an analytically stretched coordinate system. The shock- fitting procedure treats shock waves as discontinuities normal to the freestream. Improvements in shock position and the unsteady pressure distributions are obtained by this modification. The various types of shock motion observed experimentally by Tijdeman are well simulated in calculations using the modified computational scheme. The method for detecting shock-wave formation and the procedure for fitting a moving shock wave are illustrated. Results for an NACA 64A006 airfoil with oscillating quarter-chord flap are presented and discussed. q.. Introduction HERE is a need for the numerical simulation of unsteady transonic flows about airfoils in order to predict unsteady. aerodynamic loads and to provide an understanding of the behavior of unsteady transonic flowfields. A number of methods j-ll have been proposed for computing such flows, and there is continuing improvement in the results obtained. This paper describes a shock-fitting procedure, _2 coupled with an implicit finite-difference algorithm, 1,2 which can accurately and efficiently simulate most unsteady transonic flows about thin airfoils. A major consideration in constructing an algorithm for unsteady transonic flows is the treatment of moving shock waves. Experimental observations of Tijdeman13"_ indicate that even for simple airfoil motions shock-wave motions can be complicated, and they can affect aerodynamic force and moment variations strongly. Time-linearized methods, 9,m i.e., methods that assume that the unsteady perturbation is small compared to the basic steady disturbance, presently do not consider shock motions although they can be modified to do so for small shock excursions. _s,_6 Time-integration methods ts treat shock waves by "capturing" them, a procedure that can present a number of problems that will be discussed subsequently. Tijdeman's experiments also indicate that shock motion amplitudes increase with decreasing frequency. This is supported by a simplified analysis of the time-linearized equations. _6 Thus, some of the most interesting transonic flowfields result from low-frequency motion. Explicit finite- difference schemes are not efficient when applied to low- frequency cases because the stability restriction on the time step is substantially more severe than that required for ac- curacy. As a result, efficient semi-implicit methods 3 and even more efficient fully implicit methods_.2.7.8 have been developed. Caradonna and lsom 7 use an iterative implicit procedure; i.e., the nonlinear implicit finite-difference equations must be solved at a given time iteratively. In an earlier unpublished study we also used such a procedure. Presented as Paper 77-633 at the AIAA 3rd Computational Fluid Dynamics Conference, Albuquerque, N. Mex., June 27-29, 1977; submitted July 26, 1977; revision received April 17, 1978. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1977. All rights reserved. Index categories: Transonic Flow; Nonsteady Aerodynamics; Computational Methods. *Senior Research Associate. Member AIAA. _'Professor. Associate Fellow AIAA. SResearch Scientist, NASA Ames Research Center and Ballhaus and Steger _ and Beam and Warming s constructed more efficient algorithms that solve the nonlinear equations directly by the solution of simple matrix equations generated by an alternating-direction implicit (ADD procedure. This method has proven to be so efficient, especially for the calculations with nonuniform mesh distributions, that we now often use it as an alternative to successive line overrelaxation (SLOR) for steady flow calculations. ]?.18 All of these implicit schemes "capture" shock waves; i.e., shock waves evolve automaticall) as part of the numerical solution. This procedure has several deficiencies associated with it. They can be eliminated, at the expense of coding complexity, by "fitting" shock waves as discontinuities in the flow. Shock capturing produces shock profiles that are distorted in a manner that depends on the truncation errors in the finite-difference scheme. The use of mixed-difference schemes I.s can improve the situation for cases in which the flow changes from supersonic to subsonic across the shock. However, when this condition is not satisfied, as in the later stages of pulsating motion of a parabolic-arc airfoil, the differencing cannot be switched across the shock and shock resolution is poor. In any case, shock capturing requires spatial grid spacings, in regions where shock waves are an- ticipated, which are sufficiently small to resolve the shock waves in a reasonable distance. The grid spacing required to do this is frequently much smaller than that required to resolve flow variable gradients in most of the rest of the flowfield. This also results in an unnecessarily severe time- step restriction because time steps for implicit schemes are chosen such that shocks move less than one spatial grid point per time step. 1.8 This is necessar2, to preserve both accuracy and stability. Shock fitting removes the large gradients from the finite-difference solution and generally permits equivalent flowfield resolution with fewer grid points, both in space and time. Finally, if shock waves are not treated as discontinuities but are to be captured correctly, the difference equations must be solved in conservation form. This imposes an additional constraint on the construction of finite-difference schemes which can be difficult to satisfy. For example, no fully conservative difference scheme for the full potential equations have been developed yet that can match the convergence reliability and computational efficiency of Jameson's non- conservative "rotated" difference procedure. 19 In principle, the shock-fitting procedure discussed in this paper could be applied to the ftdl potential equation. For steady flows it may substantially reduce the time required to obtain a converged solution. Here. as a first step, it is applied to a simpler formulation that contains the essential nonlinear unsteady behavior associated with low-frequency transonic flnw_ I_llrth_.rrnore lhe zlonrithm i_ _imnlified by treatine 674 YU, SEEBASS, AND BALLHAUS AIAA JOURNAL the shocks as discontinuities normal to the freestream. The procedures could be generalized to a curved shock with ad- ditional programming complexity. Methods of detecting shock formation and of judging the existence of a shock wave in the unsteady flowfield are described. A production code, LTRAN2, has been developed for the efficient solution of low-frequency transonic flows about airfoils in motion. 2 LTRAN2 is based on the ADI method of Ballhaus and Steger,_ and is being released on request to industry and government agencies. The modification of the ADI procedure to include shock fitting, which is the main subject of this paper, will improve LTRAN2 significantly. Substantially fewer grid points will be required to achieve equivalent flowfield resolution, and the time-step restriction due to shock motion can be relaxed considerably. Numerical simulations of various types of shock motions for an NACA 64A006 airfoil with oscillating quarter-chord flap are described. Significant improvements in shock resolution and, consequently, in unsteady pressure distributions are obtained using the shock-fitting procedure. The results show the marked effect of shock-wave motion. Formulation of Governing Equations Low-Frequency Approximation The unsteady, small-disturbance, transonic equation for low frequency commonly is written as -2KM_¢x,+ll-M_-(-t+l)M_¢xlexx+¢_=O (1) which may be derived 2° from a systematic expansion of the velocity potential with respect to the thickness ratio r and the reduced frequency K, where K=oc/U, i.e., the angular frequency multiplied by the time that it takes the flow to traverse the airfoil chord. The spatial coordinates, the time, and the velocity potential in Eq. (1) have been non- dimensionalized by the chord c, the inverse of the angular frequency _ - t, and Uc, respectively. The tangency condition and the unsteady pressure coef- ficient that are consistent with the low-frequency ap- proximation are rOY(x,t) [ 6 ]= r Y_ + - ( Yux + Yu, ) (2)CY - Ox r and [ M_-I 1Cp=-2 ('V+ I)M_ +¢_ where Y(x,t), the instantaneous body shape, has been decomposed into a steady part Ys and an unsteady part Y,,.§ Here 5 is the amplitude of the unsteady oscillation. Note that Cp is defined such that, at sonic conditions, Cp = C_p = 0. Any shock wave that exists in the flowfield must satisfy the jump relation derived from the conservative form of the governing equation (1), viz., ex) -:KM_ [[ ¢x_] _ -_ s- + [¢,]1 _=o (3) together with the condition derived from the assumption of irrotationality, (___Yx)s= [[¢xB (4)[I¢,] §In the cases studied numerically, Yu = O( Yu. ) ; consequently, the last term in 17o 19_ it nf hioh_r nrder :_n_ w:_ n_c,l,_rt_,d " t Here ex refers to the mean value of ¢x evaluated on each side of the discontinuity, E ex_ indicates the jump in ¢_ across the discontinuity, and the subscript s denotes the quantity evaluated at the shock surface. In two-dimensional small-perturbation transonic flows, the shock waves that usually occur are nearly normal to the flow direction. We assume here that if the basic steady flow has a shock wave then this shock may be approximated by a shock wave normal to the freestream flow. Furthermore, we assume that the motion of any shock wave that arises from unsteady changes in the flow, as well as the motion of existing shock waves, also may be calculated by this normal shock ap- proximation. For this simplified model, Eqs. (3) and (4) reduce to the single equation (__) =7+1 M_--I(._,+ I)M_ " which gives the speed of the normal shock in the flowfield. For steady flows, ¢._ is a function of x alone; this, of course, still permits [¢x]] to vary with y. For unsteady flows, although .i s is a function of t alone, the strength of the shock will still vary with y. Coordinate Stretching To minimize the far-field boundary effects on the numerical results, a relatively large computational region is usually required. For the cases studied in this paper, the shock excursion may be large and the unsteady disturbances carried several chord lengths away from the airfoil; thus, the use of a relatively large computational domain seems inevitable. A simple and straightforward way of computing the solution in a large computational domain is to use nonuniform mesh distributions, with most of the mesh points concentrated in the region of interest. 2 An alternative is to introduce analytical coordinate stretchings. In the present study, we use the following coordinate stretchings: I_= + I I-exp(_:alx) I for x_O (6a) rt= + [ l-exp( _a2Y) ] for y,_O (6b) where a t and ae are constants that control the mesh distributions.¶ Equations (6) transform the infinite physical domain to the finite computational domain bounded by I_l_0, the local flow is supersonic and a shock may exist; if V_--0 _", the local flow becomes sonic and the shock strength diminishes. For V, _<0, a shock cannot exist. We start the unsteady flow calculations by using an ADI scheme. When the local flow becomes sonic and compressive, we introduce the shock-fitting algorithm. Sonic, compressive points are treated as shock points where cross differentiations in t and ,_ derivatives are avoided. Initially, the shock has zero strength and is stationary. The flow properties ahead of and behind the shock can be extrapolated easily from neighboring points. The shock wave can either increase or decrease in strength during the unsteady process. This results in three possibilities for shock motion that have to be considered separately in the fitting procedure: the shock moves upstream and crosses grid points; the shock remains stationary or moves within a grid spacing; and the shock moves down- stream and crosses grid points. At each new time level the shock position is determined by applying Eq. (8). The for- mulations of the difference approximations for each case are quite similar; the specific formulas used may be found in Ref. 27. Q Results and Discussion To illustrate the advantages of shock fitting over shock capturing, we compare the flow past a pulsating parabolic arc whose time history is shown in Fig. i, as computed by the two methods. Major differences occur when the shock wave propagates ahead of the airfoil as its thickness returns to zero. Figure 2 compares the pressure coefficient for these later stages when t_>25 (chord/U) in units of At=2 computed by the ADI method, with and without shock fitting. The full time history of this motion, as computed by the two methods, is compared in Ref. 27. Later studies have shown, as suggested by a reviewer, that the compression preceding the fitted shock wave is a result of the first-order approximation made here. In subsequent calculations with a second-order procedure, which is possible with shock fitting, this behavior does not occur. Results were computed for an NACA 64A006 airfoil with quarter-chord oscillating flap for various values of the • reduced frequency K, the freestream Mach number M=, and the oscillation amplitude 6, in order to simulate the shock motions observed by Tijdeman.__'_4 The steady-state solutions at the mean flap deflection angle for each M_, are ,JI, computed first, using the ADI method of shock fitting. We have found that the ADI scheme is more efficient than suc- cessive over-relaxation in computing steady flows. The results of the ADI calculations are identical with those obtained by line relaxation and converge more rapidly when performed with the stretched coordinates. For the problems studied here, i.e., the NACA 64A006 airfoil, the freestream Math number was varied between 0.8 and 0.9. The mesh system had 101 by 82 grid points in the .v and .v directiowls, respectively. About 250 to 450 time steps were rctlt_ivcd for the ,,olution Io con- _crgc. I_.XoI;,.,, _<.l(] a Th,,c ,_l-';t__I',-,,'talc ",olution'_ are used as _r't_tt,13 :l'li!_ t¢'._ rl'ii_ :ltl.r,"*, IL lJ,_ ,'ll,'lll'l!l*_r_. _,'g'_r qll ,'_*t,'_ JULY 1978 IMPLICIT SHOCK-FITTING SCHEME FOR TRANSONIC FLOW 677 e it "b Cp -0.5 Fig. 2 Pressure coefficient in the later stages of the motion of Fig. 1 computed with (--) and without (---) shock-fitting. cp _I B _ x,,¢ £J. Cp _ TIPI(I 3O' Cp & Cp _d r+o II_i 121)1 Cp _ Tli_l I[1" -_.- _ ---.q S _t "m' ,so" | • +_i _ .:c _ -+.,o ++_ °_--_,® Fig. 3 Cp on an NACA 64A006 airfoil at M_ =0.854, K--0.358, with + = I-deg sint. studied it took three to six cycles for the flowfield to become periodic. Stability seems to require that the time step At be small enough that At (deg)/K< 10. Figure 3 illustrates the pressure coefficients on the airfoil surface for M® =0.854, K=0.358, and 6Yux=l-deg sint. For these conditions, Ballhaus and Goorjian 2 were able to simulate type B motion where the shock disappears during some part of the cycle. Here the shock does not disappear during the cycle; instead, it becomes quite weak during a small portion of the cycle. This difference is due to the assumption of a normal shock, which res_;ts in a stronger shock than would normally occur, and to the use of shock fitting. Magnus and Yoshihara's explicit results for the full Euler equations are compared with these results for two time levels in Fig. 4. The discrepancy between the two results is due mainly to the small-disturbance ap- proximation and the lack of resolution at the leading edge of *1-. ...... _"¢ ch,et*.T _;,_,Ir_ ; _:hnl_c the llmp h+mlnrv ctHrin+, Ihe tl_, so" TI_ leo + Fig. 4 Comparisons of the results for Cp on an NACA 64A006 airfoil at M_. =0.854, K=0.358, with _= I-deg sint, with those of Magnus and Yoshihara. 4,s r _s 0.03 Cim 0.6 = ;_ : . : _ n.£ V -0.03 0.4 Fig. 5 History of the circulation and shock position for the first three cycles for the flow conditions of Fig. 3. first three cycles, of the circulation and the shock position. After three cycles of oscillation, the pressure field is essen- tially periodic; the circulation requires four cycles to become periodic. The circulation reaches its maximum value, and the shock wave its most downstream location, 57 and 83 deg after full flap deflection. Results for type C motion are given in Ref. 27 and hence not repeated here. The results for Mo, = 0.822, K = 0.496, and (5Y,,, = 2 deg sint simulate type C shock motion. Because we have used less spatial resolution and the unsealed equation, a deflection angle slightly larger than that of Ref. 2 is needed in order to generate the type C shock motion; that is, we need a 2-deg deflection angle rather than the 1.5 deg used in Ref. 2 to get analogous behavior. The results are given in Ref. 27. In this case, the flowfield is subcritical during most of the cycle, and the shock wave is barely "captured" in the non-shock-fitting procedure. During the unsteady process, the shock wave moves toward the leading edge. The strong singular behavior in pressure at the leading edge prevents the shock from propagating off the airfoil. The perturbation velocity becomes large and is negative; thus, the flow used to calculate the relative velocity ahead of the shock can no longer support a shock wave. Normal shock-fitting calculations determine the shock speed from the pressure jump across the shock at the airfoil surface. This eliminates the possibility that a portion of the shock may propagate off the leading edge in the computations. But this does not imply that it cannot occur; instead this limitation is a conseqoence of the normal shock fitting. The addition of the shock-fitting algorithm to the basic ADI scheme increased the computational time, for fixed grid spacing, by less than 7070 for all cases studied here. It is dif- ficult to compare the time required by the two schemes to achieve the same accuracy. Without shock fitting, when the shock is not of the supersonic-subsonic type, a very fine grid is required for a reasonable resolution at the shock. The need for such a grid is obviated by shock fitting, and improved accuracy in the shock region can be obtained simultaneously with reduced computational time. Because it is difficult to deten'nine when the two procedures give comparable ac- curacy, a definitive evaluation of the computing time saved by shock fitting has not been attempted. Typical computation times for the NACA 64A006 airfoil calculations rer_orted here are about 5 to 10 min/cycle on a 678 YU, SEEBASS, AND BALLHAUS AIAA JOURNAL CDC 6400 computer. A mesh of 101 by 82 grid lines in xandy directions, respectively, was used in these calculations, and the number of time steps per cycle ranged from 90 to 180. No effort was made to optimize the program. The additional central memory required for the shock-fitting algorithm is about 10 to 20_0 of that for the basic ADI scheme. Conclusion The unsteady behavior of a large number of inviscid low- frequency transonic flows can be studied accurately and efficiently using the present shock-fitting procedure coupled with the alternating-direction implicit method. The ADI method relaxes the stability restriction on the time step, greatly improving the computational efficiency; the shock- fitting procedure treats shock waves as discontinuities, which resolves the computational difficulties that can arise in the usual shock "capturing" procedure. Computed results using the present procedure compare favorably with the explicit time integrations carried out by Magnus and Yoshihara. They should be sufficiently accurate for engineering studies of airfoil motions for which the normal shock approxim'_tion made here is a reasonable one. They also reproduce well the types of shock motions observed experimentally by Tijdeman, t3.t4 as well as the propagation of the shock wave ahead of the pulsating parabolic arc. Acknowledgment This research was sponsored by NASA through Grant NSG-2112 and the Air Force Office of Scientific Research through Grant 76-2954. References t Ballhaus, W. F. and Steger, J. L., "Implicit Approximate- Factorization Schemes for the Low-Frequency Transonic Equation," NASA TM X-73082, 1975. 2 Ballhaus, W. F. and Goorjian, P. M., "Implicit Finite Difference Computations of Unsteady Transonic Flows about Airfoils," AIAA Journal, Vol. 15, Dec. 1977, pp. 1728-1735. 3 Ballhaus, W. F. and Lomax, H., "The Numerical Simulation of Low Frequency Unsteady Transonic Flow Fields," Lecture Notes in Physics, Vol. 35, Springer-Verlag, Berlin, 1975, pp. 57-63. 4 Magnus, R. and Yoshihara, H., "Unsteady Transonic Flows Over an Airfoil," AIAA Journal, Vol. 13, Dec. 1975, pp. 1622-1628. 5Magnus, R. and Yoshihara, H., "The Transonic Oscillating Flap," AIAA Paper 76-327, 1976. 6Magnus, R. J., "Computational Research on lnviscid, Unsteady, Transonic Flow over Airfoils," Office of Naval Research, Rept. CASD/LVP 77-010, 1977. 7Caradonna, F. X. and lsom, M. P., "Numerical Calculation of Unsteady Transonic Potential Flow Over Helicopter Rotor Blades," AIAA Journal, Vol. 14, April 1976, pp. 482-488. a Beam, R. M. and Warming, R. F., "An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation-Law Form," Journal of Computational Physics, Vol. 22, Jan. 1976, pp. 87-110. 9Traci, R. M., Albano, E., and Farr, J. L., "Perturbation Method for Transonic Flows about Oscillating Airfoils," AIAA Journal, Vol. 14, Sept. 1976, pp. 1258-1265. t°Weatherill, W. H., Ehlers, F. E., and Sebastian, J. D., "Computation of the Transonic Perturbation Flow Fields Around Two- and Three-Dimensional Oscillating Wings," NASA CR-2599, 1975. t l Ballhaus, W. F. and Goorjian, P. M., "Computation of Un- steady Transonic Flows by the Indicial Method," A IAA Journal, Vol. 16, Feb. 1978, pp. 117-124. t2Yu, N. J. and Seebass, A. R., "lnviscid Transonic Flow Com- putations with Shock Fitting," IUTAM Symposium Transsonicum II, edited by K. Oswatitsch and D. Rues, G6ttingen, Germany, 1975, pp. 449-456. t3Tijdeman, H., "On the Motion of Shock Waves on an Airfoil with Oscillation Flap in Two-Dimensional Transonic Flow," NLR TR 75038U, 1975. 14Tijdeman, H., "On the Motion of Shock Waves on an Airfoil with Oscillating Flap," IUTAM Symposium Transsonicunl II, edited by K. Oswatitsch and D. Rues, Gdllingen, Germany, 1975, pp. 49-56. tSHafez, M. M., Rizk, M. H., and Murman, E. M., "Numerical Solution of the Unsteady Transonic Small-Disturbance Equations," AGARDMeeting, Lisbon, Portugal, April 1977. t6Fung, K-Y., Yu, N. J., and Seebass, A. R., "Small Unsteady Perturbations in Transonic Flows," to be published in AIAA Journal. tTBallhaus, W. F., Jameson, A., and Albert, J., "Implicit Ap- proximate-Factorization Schemes for Steady Transonic Flow Problems," AIAA Paper 77-634, 1977; also AIAA Journal (to be published). tSYu, N. J. and Seebass, A. R., unpublished results, 1976. t9jameson, A., "lterative Solution of Transonic Flows Over Airfoils and Wings, Including Flows at Mach I ," Communication_ in Pure and Applied Mathematics, Vol. 27, 1974, pp. 283-309. 2°Lin, C. C., Reissner, E., and Tsien, H. S., "On Two- Dimensional Non-Steady Motion of a Slender Body in a Compressible Fluid," Journal of Mathematics and Physics, Vol. 27, March 1948, pp. 220-231. 21 Krupp, J. A., "The Numerical Calculations of Plane Steady Transonic Flows past Thin Lifting Airfoils," Boeing Scientific Research Lab., Rept. 0180-12958-1, 1971. :2Ballhaus, W. F., "Some Recent Progress in Transonic Flow Computations," VKI Lecture .Series on Computational Fluid Dynamics, 1976. 23Abbon, 1. H. and von Doenhoff, A. E., Theory of Wing Sec- tions, McGraw-Hill, New York, 1949, Appendix I, p. 354. 24Krupp, J. A. and Cole, J. D., "Studies in Transonic Flow IV, Unsteady Transonic Flow," Univ. of California at Los Angeles, Engineering Rept. 76104, 1976. 25Hafez, M. M. and Cheng, H. K., "Convergence Acceleration and Shock Fitting for Transonic Aerodynamics Computations," AIAA Paper 75-51, 1975. 26Moretti, G., "Thoughts and Afterthoughts about Shock Computations," Polytechnic Inst. of Brooklyn, PIBAL Rept. 72-37, 1972. 27yu, N. J., Seebass, A. R., and Ballhaus, W. F., "An Implicit Shock-Fitting Scheme for Unsteady Transonic Flow Computations," AIAA Paper 77-633, 1977, pp. 13-26.

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