顾为凯

Institute Of Mechanics,Chinese Academy of Sciences

1702 AIAA JOURNAL VOL. 24, NO. 10Technical Notes.TECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes cannot ex-ceed 6 manuscript pages and 3 figures; a page of text may be substituted for a figure and vice versa. After informal review by the editors, they maybe published within a few months of the date of receipt. Style requirements are the same as for regular contributions (see inside back cover).Second-Order Thickness Termsin Unsteady Wing Theory at z = 0 (12)Weikai Gu*Academia Sinica, Beijing, ChinaRECENTLY, there has been some speculation as to whichof the unsteady flow terms arising in the calculation ofsecond-order theory on a three-dimensional body should beretained. A part of the necessary conditions for the second-order theory of irrotational compressible flow past a thinthree-dimensional body have been set forth by Qian.1 An ex-tension of his analysis to flow past three-dimensional wingsof thickness ratio 6 shows that the necessary conditions forthe second-order theory in the neighborhood of the wing arethat all of the following conditions be satisfied:(M<5)2<U (1)(2)(3)(4)(5)(6)where M is the freestream Mach number and K the reducedfrequency (or, more generally, a dimensionless measure ofthe time ratio of change). In this case, we have\M- 1 1 > (MS)2= 0 (7)+ </>Qy<t>Qyy+ <t>0y<t> 0ytwhere the equation of the wing surface is assumed to bez = H(x,y,t) = 0(6), <i> is the nondimensional velocity pertur-bation potential, 00 the linear part of $, and (f>l the second-order part of 4>. Note that <5 is the expansion parameter.For high Mach numbers,we have(13)= 0 (14)at z =at z =(16)(17)In this case and at the usual reduced frequencies, the prob-lems can be reduced to a sequence of familiar steady liftingsurface problems that can be solved by existing methods.1For a transonic flow, Eq. (5) will fail and the first-orderequation (7) will include three more nonlinear terms in theright side,M2 [ (7+ l)0to«ox*+ (7- l)0o/«ox* + 2^00*]where 7 is the ratio of specific heats. The second-order equa-tion (8) will include seven more terms in the right sideM2 [2(j)0x ((t>0y<t>0xy + 00+ (N-where 7V= (7+ l)/2. In addition, the pressure coefficient willinclude four more termsa t z = 0 (9)0U = <t>0xHx + 4>QyHy - <t>QzzH at z = 0 (10)a t z = 0 (11)Received Jan. 7, 1986; revision received Feb. 21, 1986. Copyright ©American Institute of Aeronautics and Astronautics, Inc., 1986. Allrights reserved.* Research Associate, Aerodynamics Division, Institute ofMechanics.When Eq. (1) fails and(18)the first-order equation will still be linear, but the second-order equation will include five more terms in the right side,2M2OCTOBER 1986 TECHNICAL NOTES 1703In this case, the boundary conditions [Eqs. (9) and (10)] willalso fail and they must be calculated on the surface of thewing. The expansion of the pressure coefficient will bedivergent and Eqs. (11) and (12) will fail; hence, its originalexpression must be used.When we deal with a rolling wing,2 we can choose a set ofaxes fixed to and rolling with the wing. The nondimensionalvariable x axis coincides with the axis of roll. If theconditionsKa/fjL = 0(l) or 0(6) (19)M2<a2 (20)are satisfied, the additional term due to the Coriolis accelera-tion in the second-order equation is onlywhere the dimensionless parameter a is a measure of thelateral extent of the boundaries and /x is a measure of thevertical extent of this neighborhood. The pressure coeffi-cients include two more linear termsforEq. (11)-2(Kz^y-Ky<t>lz) forEq. (12)If the condition(21)is satisfied, besides Eqs. (1-6), the second term on the leftside of Eq. (7) can be neglected. This flow is a local two-dimensional one.In addition to the above, there are many simpler cases, forexample, when AT<I0(1), which are explained in detail inRef. 3. In general, the second-order approximation will notbe uniformly valid in the neighborhood of the wing edgesand the Mach cone emanating from the vortex of the wing.But this difficulty can be circumvented by the PLK method.1The validity of the second-order theory beyond the variablerange described here is very interesting for engineeringcalculations. Hence, this theory should be used with cautionoutside of this range until sufficient test data are available toindicate what range of parameters mentioned above shouldbe used for any particular body.References*Qian, F. X. and Gu, W. K., "Accuracy and Application of aSecond-Order Theory for Three-Dimensional Supersonic and LowHypersonic Unsteady Flow Around a Thin Wing," ActaAeronautica et Astronautica Sinica, Vol. 2, No. 1, 1981, pp. 1-9.2Gu, W. K., "The Second-Order Roll Damping of Rolling Wingsat Supersonic Speeds," to be published Journal of Spacecraft andRockets.3Gu, W. K., "The Analysis of Second-Order Theory," Instituteof Mechanics, Academia Sinica, Beijing, China, Res. Rept. D.V. 15,1981.Wake Periodicityin Subsonic Bluff-Body FlowsJ.L.F. PorteiroUniversity of South Florida, Tampa, FloridaIntroductionLSIGNIFICANT part of the total drag experienced bybluff bodies is generated by the low pressures existing inReceived Oct. 30, 1984; revision received Feb. 4, 1986. Copyright© American Institute of Aeronautics and Astronautics, Inc., 1986.All rights reserved.*Associate Professor, Chemical and Mechanical EngineeringDepartment.the near wake. This drag component is known as base dragand is strongly dependent on the fore- and afterbodygeometry. This note reports an experimental study of the in-fluence of the boundary-layer thickness and base masstransfer on the time-dependent, near-wake pressure andvelocity fields of an axisymmetric blunt-based cylinder insubsonic flow. Both of these parameters are known to have asignificant influence on the base pressure.1"3Experimental Apparatus and TechniqueThis investigation was conducted in the Rutgers axisym-metric near-wake tunnel (RANT II). This is the same facilityused by Porteiro et al.4 in their studies of the influence ofmass transfer on the wake of a blunt-based body. The pres-ent work is a continuation of that study. The tunnel is anopen-jet facility designed and constructed for interference-free studies of turbulent, subsonic, axisymmetric near-wakesbehind a 1.9 cm diam cylindrical model. The model supportsting is hollow, allowing the transfer of mass to and fromthe boundary layer and base region. Boundary-layer blowingand suction is carried out through a porous metal sleeve of1.9 cm o.d. extending from the model support sting to adistance 3 diam upstream of the model base. Base masstransfer takes place through a porous metal plate 1.9 cm indiameter and 0.159 cm thick. Pressure regulators were usedto stabilize the bleed air pressure and flow meters were usedfor metering the airflow.Boundary-layer velocity measurements were made at alocation 3 diam upstream of the base with a miniature totalpressure probe and a Statham ±3.45 kPa-g transducer. Theprobe location was zeroed by electrical continuity and itsposition could be determined within 0.025 mm in 152.4 mmof total travel. Total pressure measurements were taken at 27radial locations chosen to provide detailed information onthe velocity profile. Reference static pressure measurementswere also taken at an axial location 3 diam upstream of thebase. Base pressure was measured with a static pressure taplocated at the center of the base of the model. An alcoholmicromanometer providing readings to 0.05 mm of waterwas used for these measurements.The fluctuating component of the base pressure wasmeasured with a Kulite XCS-093-5 pressure transducer witha natural frequency of 70 kHz. The transducer dc outputwas analyzed with a Hewlett-Packard 3490A wave analyzerup to a frequency of 62 kHz.Near-wake velocity studies were made with a constant-temperature hot-wire anemometer. An analysis of the fre-quency spectrum of the hot-wire signal was carried out at thestagnation point, in the far wake, and in the shear layer atpoints located in the vicinity of the base and at a halfwaypoint in the near wake.This investigation was carried out with a nominal Machnumber of 0.11. The corresponding Reynolds number was2.57 x 106/m. The nominal stagnation pressure was 102 kPaabs and the value of the stagnation temperature 280 ±10 K.The approaching boundary layers were turbulent.Experimental Results and DiscussionThe fluctuating component of the near-wake velocity wasstudied by analyzing the frequency spectrum of the hot-wiresignal up to a frequency of 62 kHz.When the hot wire was placed in the shear layer very closeto the base, analysis of the frequency showed no peaks in thesignal amplitude for any combination of boundary-layerthickness and base mass transfer, except those involvingmoderate suction rates. Moderate suction rates produced aslight but noticeable increase in the signal amplitude at a fre-quency corresponding to a Strouhal number of 0.19. Alter-ing the boundary-layer thickness brought about changes inthe frequency of the amplitude maximum.Analyses were also carried out at the location in the shearlayer having the maximum level of turbulence (X=Q.5D) atthe stagnation point and on the far-wake centerline at an ax-

Second-order thickness terms in unsteady wing theory

Abstract

Similar works

Full text

Available Versions

Institute Of Mechanics,Chinese Academy of Sciences