[American Institute of Aeronautics and Astronautics 22nd Atmospheric Flight Mechanics Conference - New Orleans,LA,U.S.A. (11 August 1997 - 13 August 1997)] 22nd Atmospheric Flight

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

A97-3726S AIAA 97-3573

COMPUTATION AND VALIDATION OF DELTA WING PITCHINGUP TO 90°AIMPLITUDE

Osama A. Kandil1 and Yahia A. Abdelhamid2

Aerospace Engineering DepartmentOld Dominion University, Norfolk, Virginia 23529

ABSTRACT

The unsteady, three dimensional Navier Stokesequations (NS) are solved time accurately tosimulate and study the aerodynamic responseof a delta wing undergoing pitching motion upto 90° amplitude. The governing equations aresolved using the implicit, upwind, Roe flux-difference splitting, finite-volume scheme. Theprimary model under consideration consists ofa 76° swept, sharp-edged delta wing of zerothickness initially at zero angle of attack. Thefreestream Mach number and Reynolds num-ber are 0.3 and 0.45 x 106; respectively. Thewing is forced to pitch through a ramp func-tion of 0.024i around an axis located at twothird of the root chord length. The effect of theramp motion on the vortex breakdown behav-ior and overall aerodynamic response is studiedto provide understanding of the flow physics atextremely large pitch amplitude. The compu-tational results are validated and are in a goodagreement with the experimental data.

INTRODUCTION

The ability of modern fighter aircraft to flyand maneuver at high angles of attack is ofprime importance for aircraft designer. Al-though there are several experimental inves-tigations in the literature dealing with highangles of attack maneuvers at extremely large

values, there is a lack of computational in-vestigations and solutions. This is one of themotivations for the present study. The com-plicated physics associated with high angle ofattack vortical flows involves massive separa-tion, vortex interaction, and vortex breakdownwhich result in a penalty of undesirable un-steadiness in the flowfield. In order to exploitthese flight regimes and extend current per-formance envelopes, a better understanding ofthese unsteady, vortical flows associated withmaneuvering swept wings must be developed.

Computational fluid dynamics (CFD) playsan important role in the design process byproviding detailed flowfield information at arelatively low cost that is unavailable with ex-periment alone. It helps reduce design cycletime and provides information that is comple-mentary to wind-tunnel and flight-test data.

In Ref. 1, Kandil and Kandil presented a smallamplitude pitching oscillation of a 65° deltawing in transonic flow. The computationswere carried out at a mean angle of attack of20°, and at Mach number and Reynolds num-ber of 0.85 and 3.23 x 106, respectively. Thewing was forced to oscillate in pitch around anaxis at 0.25 root-chord station with an ampli-tude of 4° and a reduced frequency of TT.

Kandil and Menzies (Ref. 2) studied the

1 Professor, Eminent Scholar and Department Chair,Associate Fellow AIAA.

2Graduate Research Assistant, Member AIAACopyright©1997 by Osama A. Kandil. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

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coupled rolling and pitching oscillation intransonic flow. The focus was to analyze theeffects of coupled motion on the wing responseand vortex breakdown flow by varying oscil-lation frequency and phase angle while themaximum pitch and roll amplitude is kept at4°.

In this paper, we consider the flow responseof a 76° swept delta wing undergoing highamplitude pitching motion up to 90°. Compu-tational time-accurate solutions are obtainedfor laminar and turbulent flows. The compu-tational results are validated using the experi-mental data of Ref. 3.

FORMULATION

Governing Equations:

The conservative form of the dimension-less, unsteady, compressible, full Navier-Stokesequations in terms of the time-dependent,body-conformed coordinates ^, £2, and £3, isgiven by:

~at or a?m=l,2,3; s=l,2,3

where£m = r(zi,Z2,z3 ,*) (2)

Q = - = -\p,pui, pu2, pus, pe}* (3)

Boundary And Initial Conditions:

All boundary conditions are explicitly imple-mented. They include inflow-outflow condi-tions, solid-boundary conditions and plane ofgeometric symmetry conditions. At the planeof geometric symmetry, periodic conditionsare enforced. At the inflow boundaries, theRiemann-invariant boundary-type conditionsare enforced. At the outflow boundaries, first-order extrapolation from the interior point is

used.

Since the wing is undergoing pitching motion,the grid is moved with the same angular mo-tion as that of the body. The grid speed, ^-,and the metric coefficient, |^—, are computedat each time step of the computational scheme.Consequently, the kinematical boundary con-ditions at the inflow-outflow boundaries and atthe wing surface are expressed in terms of therelative velocities. The dynamical boundarycondition, |̂ , on the wing surface is no longerequal to zero. This condition is modified forthe oscillating wing as:

dn \wing = —pa • n (4)

where a is the acceleration of a point on thewing flat surface; n, the unit normal to thewing surface. The acceleration is given by:

(5)—*

where Q is the angular velocity. Notice thatfor a rigid body, the position vector r*, is not afunction of time and hence, r = r = 0. Finally,the boundary condition for the temperature isobtained from the adiabatic boundary condi-tion and is given by

dT_.dn \wtng = 0 (6)

The initial conditions correspond to zero angleof attack for which the wing surface is parallelto the streamwise direction, and the boundaryconditions are imposed on the wing surface.

COMPUTATIONAL SCHEME

The implicit, upwind, flux-difference split-ting, finite-volume scheme is used to solvethe unsteady, compressible, full Navier-Stokesequations. This scheme uses the flux-differencesplitting of Roe and a smooth flux limiter isused to eliminate oscillation at locations oflarge flow gradients. The viscous and heatflux terms are linearized in time and the crossderivative terms are eliminated in the implicit

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operator and retained in the explicit terms.The viscous terms are differenced using sec-ond order accurate central differencing. Theresulting difference equation is approximatelyfactored to solve the equations in three sweepsin the (f1,^2, and £3, directions. The compu-tational scheme is coded in the computer pro-gram "FTNS3D".

RESULTS AND DISCUSSION

The delta wing model used in the presentcomputational study consists of a 76° sweptback, sharp-edged wing with zero thicknessand an aspect ratio of one (similar to that usedin Ref. 3.). The three-dimensional grid topol-ogy used in the calculations is shown in Fig. 1with a cross section at the trailing edge. In ourcomputations we used a relatively coarse gridto minimize the computational cost needed toperform very low reduced frequency maneu-vers in order to compare with the availableexperimental data. Even with our coarse grid,84x65x43 in the axial, wrap-around, and out-ward directions; respectively, each case took82 hours on Sabre (CRAY-Ymp computer atNASA Langley Research Center) to completea ramp amplitude of 90°. The pitch axis islocated at two third of the root chord station,as measured from the wing vertex. The wingis forced to move in pitch through a rampfunction shown in Fig. 2 and is described bya = 0.024£, which is related to the reduced fre-quency. In our analysis, the reduced frequencyis equivalent to k=0.04. The freestream Machnumber and Reynolds number are 0.3 and0.45 x 106, respectively. The NS Eqs. are inte-grated time accurately with A£ = 0.001. Thistranslates into 65,450 time steps to completeramp motion flow response up to a = 90°.

In this study, two cases are investigated thor-oughly. The first one uses the laminar solu-tion of the governing equations, whereas thesecond one uses Baldwin-Lomax turbulencemodel along with the Schiff and Degani mod-ification to study the turbulent effects on the

solution. Next, we discuss the laminar casefollowed by the turbulent case.

Laminar Flow Case

Figure 2 shows the variation of the angle ofattack a with the time. The angle of attack,a varies from 0° to 90° through this function.The slope of this curve is propertional to thepitch rate, and hence the reduced frequency.

Figure 3 shows the variations of CL andCD with a of the present computational re-sults and the corresponding values of thoseof the experimental data of Ref. 3. Figure3 shows an excellent agreement between CDobtained from the present study and that ob-tained experimentally until a reaches 60°. Af-ter a = 60°, the computed results overesti-mates the experimental data. This differencemay be attributed to the absence of turbulencemodeling which is needed for the massive flowseparation at very high angles of attack afterthe onset of the vortex breakdown. Anothersource might be the coarse grid fineness to cap-ture the massive flow separation and the vortexbreakdown regions. The CL curve shows a verygood agreement until a reaches 40°. For angleof attack greater than 40°, the CL obtainedfrom the present study overestimates the ex-perimental data by about 12%. The predictedpeak of the CL curve slightly underestimatesthe experimental value. Again this could beattributed to the effects of turbulence at highangles of attack and the grid resolution in thevortex breakdown region. The present studypredicted accurately the angle of attack atwhich the breakdown passes through the trail-ing edge, which is in our case about 39°.

Figure 4 shows the surface pressure coefficient,Cp, distribution with the spanwise directionfor three axial stations of x=0.3, 0.6, and 0.9.Also, Cp distribution is shown for four valuesof the angle of attack; 30°, which is below thevortex breakdown critical angle, and 40°, 50°,and 70°, which are over the vortex breakdowncritical angle. From this figure we observe the

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decrease in Cp with the x direction for all an-gles of attack. Also, we observe the increase ofCp with Q; before the occurrence of the break-down and the decrease of Cp after the occur-rence of the vortex breakdown. The Cp curvesalso show flow asymmetry at very large angles.

Figure 5 shows the cross-flow instantaneousstreamlines pattern at the trailing edge fora — 30° and 50°. This figure shows how thevortex breakdown develops over the wing sur-face. At low angles of attack, a tight vortexcore develops at the leading edge of the wing.As the angle of attack increases, the vortexcore grows up until it breaks down at thetrailing edge around 39° due to the adverseaxial pressure gradient. Then it moves up-stream expanding the size of the vortex core,due to the axial momentum loss, until it en-compasses all the wing surface. Figure 6 showstwo snapshots of the flow before the onset ofthe breakdown at a = 30° and 38°, whereasFig. 7 shows two snapshots after the onset ofthe breakdown at a = 40° and 60°.

At a — 70° the breakdown moves more up-stream to encompass a larger area of the wingmaking this area a non-lifting area and the re-maining area of the wing generates a lift force.At a = 90°, the whole wing is no longer gen-erating any lift. Figure 8 gives four differentviews of the wing at an angle of attack of 70°showing the vortex core shape and flow. Afterthe angle of attack reaches a = 40°, the break-down moves upstream of the trailing edge andas the angle of attack increases the breakdownmoves more upstream. In the same time, thevortex pair core flows expand and coalesce asshown in Fig. 8 (see back and top view).

Turbulent Flow Case

Figure 9 shows variations of the lift and dragcoefficients with a. Again there is an excellentagreement between the computed CL and CDand the corresponding experimental Data ofRef. 3. For angles of attack less or greaterthan 40°, CL shows a very close agreement

with the experimental data. For angles of at-tack around 40°, the difference between thecomputed CL and the experimental value isappreciable. This may be attributed to thegrid resolution at the trailing edge where thevortex breakdown crosses and moves over thewing surface. Using Baldwin-Lomax turbu-lence model enhanced the computed CL values.On the other hand, the computed CD valuesare slightly enhanced. The good agreementwith the experimental data has been improvedup to a = 65°.

The spanwise surface pressure coefficient dis-tributions at three axial stations of x=0.3,0.6, 0.9; for three angles of attack; 20°, 40°and 70°, are shown in Figs. 11 and 12 alongwith a comparison with the corresponding val-ues obtained using the laminar NS equations.The differences between the two solution areappreiable. Also, the values obtained usingBaldwin-Lomax model underestimate the lam-inar solution.

Figure 11 shows snapshots of the stream-lines over the delta wing for various view an-gles for an angle of attack, a = 70°.

CONCLUDING REMARKS

The unsteady, three dimensional Navier Stokesequations are solved to simulate and study theaerodynamic response of a delta wing under-going pitching motion up to 90° amplitude.The governing equations are solved time ac-curately using the implicit, upwind, Roe flux-difference splitting, finite-volume scheme. Twocases are studied: the first one deals with thelaminar solution whereas the second one usesthe Baldwin-Lomax turbulence model. For thedrag coefficient, the laminar results show goodagreement with the corresponding experimen-tal data except at very high angles of attack(a > 60°) where the difference between thepredicted values and the experimental valuesreach about 24% maximum. The Baldwin-Lomax model introduced some improvementsof the computational results in comparison

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with the experimental data up to an angleof attack (about 65°). As for the lift coef-ficient, the Baldwin-Lomax model gives verygood agreement with the experimental dataexcept around a = 40°. The differences arebelieved to be due to the grid resolution par-ticularly at the trailing edge and in the vortexbreakdown location. Moreover, a higher-ordermodel such as the Sparlart and Alamaras andthe k-u; models may improve the predictions atvery large angles of attack. This investigationof the unsteady flow over a wide range of an-gles of attack shows the variation of the vortexcore and its breakdown behavior at very highangles of attack. It also shows that computa-tional solutions and results in the very high-angle-of-at tack range can be obtained.

ACKNOWLEDGMENT

This research work is supported under GrantsNo. NAG-1-648 by the NASA Langley Re-search Center. The authors would like to rec-ognize the computational resources providedby the NAS facilities at Ames Research Cen-ter and the NASA Langley Research Center.

REFERENCES

'Kandil, 0. A. and Kandil, H. A., "Pitch-ing Oscillation Of a 65-Degree Delta Wing inTransonic Vortex Breakdown Flow", AIAA-94-1426-CP, AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics and Materi-als Conference, Hilton Head, SC- April 18-20,1994, pp. 955-966.

2Kandil, 0. A. and Menzies, M. A., "CoupledRolling And Pitching Oscillation Effects OnTransonic Shock-Induced Vortex-BreakdownFlow Of a Delta Wing", AIAA-96-0828 34thAerospace Sciences Meeting and Exhibit, RenoNV- January 13-18, 1996.

3Jarrah, M. A., "Unsteady Aerodynamics OfDelta Wings Performing Maneuvers To HighAngle Of Attack", Ph. D thesis, Stanford Uni-versity, 1988.

4Menzies, M. A., "Unsteady, Transonic FlowAround Delta Wings Undergoing Coupled AndNatural Modes Response- A MultidisciplinaryProblem", Ph. D thesis, Old Dominion Uni-versity, 1996.

5Kandil, 0. A., Kandil, H. A. and Massey,S. J., "Simulation of Tail Buffet Using DeltaWing-Vertical Tail Configuration", AIAA 93-3688-CP, AIAA Atmospheric Flight MechanicsConference, Monterey, CA, August 1993, pp.566-577.

6Baldwinv B. and Lomax, H., "Thin-LayerApproximation and Algebraic Model for Sep-arated Turbulent Flows", AIAA 78-0257, Jan-uary, 1978.

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(a) Portion of the 3-D grid (b) Cross section at the trailing edge

Figure 1: The three dimensional grid topology

9O

ao7O

eoeo•4-03O

20-i o

O 1O 3O 4O

Figure 2: Forced ramp function time history

— Presento Expm, Ref. 3

0 1 0 2 0 3 0 4 0 S O & 0 7 0 B 0 9 0


0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

Figure 3: Lift and drag coefficients vs. a using laminar NS equations.

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Pressure Coefficient Pressure Coefficient

O.OY/Span

(a) a = 30°

. x=O.3- x=O.6- x=O.9

O.OY/Span

(b) a = 40°

. x=O.3x=O.6X=O.9

Pressure Coefficient

O.O

Y/Span

. x=O.3• x=0.6• x=O.9


-0.5 O.OY/Spar

(c) a = 50° (d) a = 70°

Figure 4: Spanwise pressure coefficient distribution

(a) a = 30° (b) a = 50°

Figure 5: Cross-flow instantaneous streamlines at the trailing edge

. X=O.2- X=O.5

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a=30°

Top a=38°Side

Figure 6: Stagnation pressure and streamlines over delta wing before the onset of the breakdownat a = 30° and 38°

a=40°

Top a=60' Side

Figure 7: Stagnation pressure and streamlines over delta wing after the onset of the breakdownat a = 40° and 60°

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a=70°

Stagnation Pressure

0.58 0.64 0.71 0.77

<x=70'

Stagnation Pressure

0.58 8.64 0.71 0.77

Front Side

a=70°

O.50 O.fr*

Back

O.S* O.M O.71 0.77

Top

Figure 8: Stagnation pressure and streamlines over the delta wing at a. = 70° from different viewangles using laminar NS-equations



a

Figure 9: Lift and drag coefficients using turbulent NS equations

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Pressure Coefficient Pressure Coefficient. X-O.3. x=O.6• X-O.9

-1.0

-O.S

-1.O -O.5

_ X-O.3- XtxO.6- x«0.0


Y/Span

. X-O.3- XaO.6• X=O.9


0.0Y/Span

. x-0.3

. x-O.6• x«0.9

Pressure CoefficientPressure Coefficent

-1.0 -O.5 O.O

Y/Span

Laminar flow solution Turbulent flow solution

Figure 10: Spanwise pressure coefficient distributions for laminar and turbulent flow solutions.

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Stagnation Pressure

0.56 0.<3 0.70 0.77

Side Back

Side Bottom

U.56 0.63

Top

Figure 11: Stagnation pressure and streamlines over delta wing at a = 70° from different viewangles using turbulent NS-equations.

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Documents

[American Institute of Aeronautics and Astronautics 22nd Atmospheric Flight Mechanics Conference - New Orleans,LA,U.S.A. (11 August 1997 - 13 August 1997)] 22nd Atmospheric Flight