Two methods are described for calculating unsteady flows over rapidly pitching airfoils. The first method is based on an interactive scheme in which the inviscid flow is obtained by a panel method. The boundary layer flow is computed by an interactive method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier-Stokes equations. The solution of these equations is obtained with an approximately factorized numerical algorithm, and with single block or multiple grids which enable grid embedding to enhance the resolution at isolated flow regions. In addition, the attached flow region can be computed by the numerical solution of compressible boundary layer equations. Unsteady pressure distributions obtained with both methods are compared with available experimental data.Approved for public release; distribution is unlimited

Ekaterinaris, J.A.

Platzer, M.F.

English

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveDSpace RepositoryFaculty and Researchers Faculty and Researchers' Publications1992-03Computation of unsteady flows over airfoilsEkaterinaris, J.A.; Platzer, M.F.NASA. Ames Research Center, Physics of Forced Unsteady Separation; p. p 177-185http://hdl.handle.net/10945/60111This publication is a work of the U.S. Government as defined in Title 17, UnitedStates Code, Section 101. Copyright protection is not available for this work in theUnited States.Downloaded from NPS Archive: CalhounN94- 34977Paper Proposed for Presentation to theWorkshop on Physics of Forced Unsteady Separationto be held atNASA Ames Research Center April 17-19, 1990COMPUTATION OF UNSTEADY FLOWS OVER AIRFOILSJ. A. EkaterinarisNavy-NASA Joint Inst. of AeronauticsDept. of Aeronautics and AstronauticsNaval Postgraduate SchoolNASA-Ames Research CenterMail Stop 258-1Moffett Field, CA 94035M. F. PlatzerNavy-NASA Joint Inst. of AeronauticsDept. of Aeronautics and AstronauticsNaval Postgraduate SchoolMonterey, CA 93943SUMMARYTwo methods are described for cMculating unsteady flows over rapidly pitching air-foils. The first method is based on an interactive scheme in which the inviscid flow is ob-tained by a panel method. The boundary layer flow is computed by an interactive methodthat makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flowequations. The second method is based on the solution of the compressible Navier-Stokesequations. The solution of these equations is obtained with an approximately factorizednumerical algorithm, and with single block or multiple grids which enable grid embeddingto enhance the resolution at isolated flow regions. In addition, the attached flow regioncan be computed by the numerical solution of compressible boundary layer equations.Unsteady pressure distributions obtained with both methods are compared with availableexperimentM data.ABSTRACTThe present paper addresses the prediction of unsteady airfoil boundary layer flowsby two methods. These two methods are briefly described in the following section. Thefirst is based on the extension of the steady interactive boundary-layer method of [1] andthe second on the Navier-Stokes method of [2].PNfiGIg)ffqG FAGE BLANK !_IOT FILMED 177https://ntrs.nasa.gov/search.jsp?R=19940030471 2018-09-26T20:41:36+00:00Z1.1 Viscous-Inviscid Interaction Method: The interactive method for steady highReynolds number incompressible flows is described in [1] and [3], respectively, and makesuse of an inverse boundary-layer method coupled to a panel method with an interactiveformula suggested by VeldmarL [4]. The extension of this method to unsteady incompress-ible flows, again makes use of a panel method [5], which is similar to that of Hess andSmith [6]. This method utilizes the procedure of Basu and Hancock [7] to model the wake.The wake is represented by a series of free vortices shed from the trailing edge in responseto incidence changes so that the total vortlcity in the field is conserved. The airfoil'slift response then is obtained by subdividing the incidence history into sufficiently smalltime steps and computing the source and vorticity distributions for each time step. Theunsteady interactive method is described in full detail in Refs. 5 and 8.The unsteady boundary-layer equations are expressed in terms of an eddy viscosity,era, so that continuity and momentum equationsOu Ov+ _ = 0 (1)0-_ uyOu Ou Ou OUra_ + u_ + v-gy_ atu OU_ 0 Ou+ 0_ + _[(v + _m)_] (2)are solved subject to the boundary conditionsy = 0, u -- v = 0; y _ _, u --_ v_(_,_) (3)on the airfoil and with y = 0 denoting the dividing streamline that separates the upperand lower parts of the invlscid flow in the wake, subject to the following conditionsy _ ±c¢, u _ Ue(x,t); y = 0, v = 0 (4)with U_(z, t) given by U_ = U°_ + SUe(x, t). The eddy viscosity formulation of Cebeci andSmith [9] is used with special emphasis on the transitional region.1.2 Navier-Stokes Methods : The Navier -Stokes method is briefly described in thisparagraph. The full, unsteady, two-dimensional, compressible Navier-Stokes equationswere solved. In a curvilinear coordinate system _, 7/the governing equations are:O_l OF 0(_ 10l_ OS0-_+ _ + 0--_= R--;(-_ + _) (5)!78where _1 is the conservative variable vector q = (p, pu, pv, e) T, and _-', (_ are the nonlinearinviscid terms, and 1_, S are the viscous terms.The integration is performed with the finite difference factored Beam-Warming algo-rithm [10]. The approximately factorized form of the algorithm is:where[* + (At/2)(#_Ai_,k + (Dimpz)_)] ×[* + (At/2)(#,Bj_,k + (Di._pl),)]Aqj_,_ = (RHS) '_(6)(7)Solutions with embedded grid which provide enhanced grid resolution at isolated flowregions are possible. Thus, high grid resolution can be provited at critical flow regions, suchas the leading edge region, where supersonic flow conditions and possible shock formationmay occur even at moderate free str.eam subsonic speeds (M = 0.45 - 0.50) as the angleof attack increases. The option of solving the attached flow region with the compressibleboundary-layer equations on an embedded grid is also provided. The boundary-layerequations for a generalized coordinates system [11] are:continuity0--/ _,-7-, + _( )= 0 (8)momentumOu o_ ov op op o J2-0 [T=_.+_._] (9)normal momentumenergyOH OpPOt Ot00+N{_:[-:,uOpo_ = o (lO)OH OH+ PU-o-_ + pv-ff_ =K" Oa2" +_ K" Oa2" ]"+ _ + (-_-_)1 _[T_=+ _'_ + [-5y-y))++ %yv + K( Oa2_ K'Oa2"]"(11)179Here H = _ is the enthalpy per unit volume, and the other quantities have thePsame definitions as before. Eqs. 8-11 are supplemented by the equation of state,p pTc¢up_ Tp_(12)orT (7 - 1) [H u2 + v2Too - a L _] (13)Viscous or inviscid solutions can be obtained for the global grid by marching in timefrom an initial condition. Steady solutions are obtained by marching in time from freestream initial conditions until convergence to the steady-state. Similarly, unsteady flowsare computed by marching in time from a steady flow initial condition. After the globalgrid solution is computed the boundary layer equations can be solved in the secondarygrid using as initial condition at the inflow the velocity profile obtained by the viscous flowsolution. Boundary conditions at the edge of the boundary layer domain are provided bythe pressure and velocity distribution of a viscous or inviscid global flow solution for theouter region. Grid refinement is applied for the boundary-layer calculation and the valuesof the flow parameters at the extra boundary points are obtained by simple interpolationof the flow variables obtained from the viscous solution. For unsteady calculations theboundary layer equations are solved at each time step.2.0 RESULTS AND DISCUSSIONThe unsteady flow calculations for the NACA-0012 airfoil subject to ramp-type mo-tion, as described in detail in [12], were performed by using both interactive and Navier-Stokes methods for a Reynolds number of 2.7 x 106 and for a non-dimensional pitch ratek defined by k = &c/2Uoo = 0.0127. The airfoil chord was 10.16 cm, the pitch rate 1280degrees/sec, pitching from 0 ° to 15.54 °, at a free-stream Mach number of M = 0.3. The ex-perimental data include upper and lower surface pressure distributions for incidence anglesof 2.9, 5.8, 8.9, 11.7 and 15.5 degrees.Figures 1 and 2 compare measured and calculated distributions of pressure coefficientsfor incidence angles of 2.9, 5.8, 8.9, 11.7 and 15.5 degrees, with Figure 1 showing thepredictions of the interactive method and Figure 2 those of the Navier-Stokes method. Inboth methods, the flow was assumed to be fully turbulent due to the lack of experimentaldata about the location of transition, and the ramp change in the angle of attack wasassumed to be given by2ama_ t3 + 3ama_ t2 (14)a(t) - T3 T2180where T is the nondimensional time required to complete the ramp motion from 0 ° to area,.It is useful to point out that while the interactive method is based on the assumption ofincompressible flow, the Navier-Stokes method is for a compressible flow. Calculationsperformed with the Navier-Stokes equations for a Mach number of 0.2 and 0.3, however,showed no effect of compressibility on the results.Figures 1 and 2 show that the predictions of both methods are in good agreementwith the experimental data,although the Navier-Stokes computations slightly underpredictthe suction peaks at the lower incidence. Figures 3 and 4 present a comparison betweenthe velocity profiles computed by both methods at two Chordwise locations correspondingto z/c = 0.5 and z/c -- 0.9 at several angles of incidence. While there is reasonablygood agreement at low incidences, the two profiles begin to deviate significantly at higherincidences. Figure 4e shows, however, that both procedures predict the onset of flowreversal at a = 15.5 ° for z/c = 0.9. Unfortunately, there is no experimental data availableto verify this prediction and to assess the accuracy of the two methods.3.0 CONCLUSIONSTwo methods are described and applied to study the effects of low Reynolds numberand flow unsteadiness on blade boundary layers. The first is based on an interactiveboundary layer scheme in which the inviscid flow is computed by a panel method andthe boundary layer flow by an inverse method that makes use of the Hilbert integral tocouple the solutions of the inviscid and viscous flow equations. The second method is basedon the solution of the compressible Navier-Stokes equations which employs an embeddedgrid technique for accurate boundary layer calculations with small computational cost.Calculated results obtained with both methods for a NACA-0012 airfoil subject to aramp type motion at relatively high Reynolds number also indicate good agreement withexperimental data. These results suggest that unsteady blade boundary layers can becomputed accurately with either method provided the location of transition is computedinteractively with the e'_-method and the transitional region is modelled properly. Futurework will be directed at the systematic study of the effect of Reynolds number, transitionmodeling, reduced frequency and the effect of the airfoil leading edge geometry. In additionupwinding and TVD schemes will be used to enable accurate capturing of possible shockformation at the leading edge.Acknowledgment: This work was supported by the Naval Air Systems Command, andNASA Ames Research Center.181|References1 Cebeci, T., Clark, R. W., Chang, K. C., Halsey, N. D., and Lee, K., "Airfoils withSeparation and the Resulting Wakes_" J. Fluid Mech., No. 163, pp. 323-347, 1986.2 Ekaterinaris, J. A., "Compressible Studies on Dynamic Stall," AIAA Paper 89-0024,27th Aerospace Sciences Meeting, Reno NV, Jan. 1989.3 Cebeci, T., "Essential Ingredients of a Method for Low Reynolds Number Airfoils,"To be published in the AIAA journal 1989.4 Veldman, A. E. P., "New Quasi-Simultaneous Method to Calculate InteractivelyBoundary Layers," AIAA Journal,, No. 19, 1981, p. 769.5 Teng, N. H., "The Development of a Computer Code (U2DIIF) for the NumericalSolution of Unsteady, Inviscid and Incompressible Flow over an Airfoil," M.S. Thesis,Naval Postgraduate School, Montery CA, 1987.6 Hess, J. L., and Smith, A. M. O., "Calculation of Potential Flow about ArbitraryBodies," Progress in Aeronautical Sciences, Pergamon Press, Oxford, Vol. 8, 1966,pp 1-138.7 Basu, B. C., and Hancock, G. J. "The Unsteady Motion of a Two-Dimensional Airfoilin Incompressible Inviscid Flow," Journal of Fluid Mech., Vol. 87, 1987, pp. 157-168.s Jang , H. M., "A Viscous-Inviscid Interactive Method for Unsteady Flow," Ph.D.Thesis, University of Michigan, Ann Arbor, Michigan 1989.9 Cebeci, T, and Smith, A. M. O., "Analysis of Turbulent Boundary Layers," AcademicPress , New York, 1974.10 Beam, R. M. and Warming, R. F., "An Implicit Factored Scheme for the CompressibleNavier-Stokes Equations," AIAA Journal, Vol. 16, No. 4, April 1978, pp. 393-402.11 Steger, J. L, VanDalsem, W. R., Panaras, A. G., and Rao, K. V., "A Formulationfor the Boundary-Layer Equations In General Coordinates," NASA Technical Mem-orandum 100079, June 1988.12 Landon, R. H., "NACA 0012 Oscillatory and Transient Pitching," AGARD ReportNo. 702, 1981.182Figure CaptionsFigure 1.Figure 2.Figure 3.Figure 4.Pressure Coefficient at a = 2.9°,5.8°,8.9°,11.7°,15.5° predicted by the viscous-inviscid interaction method. (Re = 2.7 x 108,k = 0.0127)Pressure Coefficient at a = 2.9 °, 5.8 °, 8.9 °, 11.7 °, 15.5 ° predicted by the Navier-Stokessolution. (Re = 2.7 × 108, k = 0.0127)Comparison of the boundary-layer profiles computed with both methods, at a =2.9°,5.8°,8.9°,11.7°,15.5° for the 50% chord. (Re = 2.7 x 108,k = 0.0127)Comparison of the boundary-layer profiles computed with both methods, at a =2.9 °, 5.8 °, 8.9 °, 11.7 °, 15.5 ° for the 90% chord. (Re = 2.7 x 108, k = 0.0127)183o_.2a _ _teract/ve Method J(.) -- 2.9°_ .t_--_ _ T Tv.__.___----- T wo-24"1- t-O'O"2U -4-6(d) a = 11.7 °-8 _--_ ,_ l-2U-4-6(e) a : 155 °4 0 .2 .4 .6 .8 1.0X/C04L J _ --0 .-4 _(c)_ =$-g'j_ _L_ J J-- F r Ta."2O-4-60-2-6-O (e) o =15.5"0 .2 .4 .6 .8x/c1.0Figure _ pressure coefEdent distributions predicted by theinterac_,ive method: (Re -- 2.'/x 10',k = 0.0127).Figure :2, Pressure coefficient distributions predicted by theNavler-Stokes method: (Re -- 2.'/x 10e,k -- 0.012"/).184-- N.*_., SloL. i,W_4• -. _._,*, U.m¢,#i')_ ._.r .",'"/'J_c)o 4.1" ""IO|J' ,.,o ]J m | .I IoFTgure _ C_mpar{son or the b<_und_)" |a_er pro_ cs ohm-puled w;Ih brdh *n_l},_,ds, for the 50_ chnrd:(Re = 2.7 _ 10ek 001271 ._i).._t' . "/, ,,f",'"(Re = 2.7 :* 10',k : 0.0127)185

Computation of unsteady flows over airfoils

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Computation of unsteady flows over airfoils

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