studies of wings supporting non-unique solutions in transonic...

2014

Studies of Wings Supporting Non-unique Solutions in

Transonic Flows

Kui Ou, ∗

Flight Sciences Department, Honda Aircraft Company, Greensboro, NC 27410

Antony Jameson,†

Aeronautics and Astronautics Department, Stanford University, Stanford, CA 94305

John C. Vassberg, ‡

Advanced Concepts Design Center, Boeing Commercial Airplanes, Long Beach, CA 90846

Non-unique numerical solutions of transonic flows over airfoils have been found, first for

potential flow equation, and later for Euler and RANS equations. The studies have also

been extended to unsteady flow simulations, and non-unique numerical solutions continue

to be demonstrated. The question of whether three-dimensional effect can have a further

influence on the uniqueness of the transonic flow solution remain an important one. Hith-

erto wings supporting non-unique solutions in transonic solutions have not been studied.

Research in this direction will further our understanding of the behavior of non-unique

transonic flows. This paper studied a set of four wings based on recently designed symmet-

rical airfoils that have been found to support non-unique transonic solutions in both steady

and unsteady flows in a narrow band of transonic Mach numbers. The aspect ratios of the

wings have been varied as a way to control the extend of the three-dimensional effect. For

certain of these airfoils, the non-unique solutions cease to exist when extended to a full

wing, while for others, non-unique solutions continue to exist depending on the choice of

the aspect ratios of the wings. The flow conditions that support non-unique solutions also

tend to change when the airfoils are extended to wings of different aspect ratios. The scope

of the study is, at present, limited to Euler solutions.

I. Background

Non-unique solutions of the transonic potential flow equation were discovered by Steinhoff and Jameson1

(1981), who obtained lifting solutions for a symmetric Joukowski airfoil at zero angle of attack in a narrowrange of Mach numbers in the neighborhood of Mach 0.85. This non-uniqueness could not be duplicated withthe Euler equations and it was conjectured by Salas et al2 (1983) that the non-uniqueness was a consequenceof the isentropic flow approximation. Subsequently, however, Jameson3 (1991) discovered several airfoilswhich supported non-unique solutions of the Euler equations in a narrow Mach band. These airfoils werelifting.

The question of non-unique transonic flows was re-examined by Hafez and Guo4–6 (1999) who formedboth lifting and non-lifting solutions for a 12 percent thick symmetric airfoil with parallel sides from 25 to75 percent chord in a Mach range from 0.825 to 0.843. The question was further pursued in detail in a seriesof studies by Kuz’min and Ivanova7–11 (2004,2006) who confirmed the results of Hafez and Guo, and alsoshowed that airfoils with positive curvature everywhere could support non-unique solutions.

Recently, a set of four symmetrical airfoils was designed by Jameson, Vassberg, and Ou17 (2011) whichwere found to support non-unique transonic solutions. The NU4 airfoil was the result of aggressive shapeoptimization to minimize drag of a 12 percent thick symmetrical airfoil. The JF1 airfoil is an extremelysimple parallel sided airfoil. The JB1 airfoil is also parallel sided but has continuous curvature over the entireprofile. The JC6 airfoil is convex and C∞ continuous. It was found that in non-lifting transonic flow these

∗Flight Sciences Department, Honda Aircraft Company, AIAA Member†Thomas V Jones Professor, Aeronautics and Astronautics Department, Stanford University, AIAA Fellow.‡Boeing Technical Fellow, The Boeing Company, AIAA Fellow.

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American Institute of Aeronautics and Astronautics Paper 2014

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7th AIAA Theoretical Fluid Mechanics Conference

16-20 June 2014, Atlanta, GA

AIAA 2014-2928

Copyright © 2014 by __________________ (author or designee). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA Aviation

airfoils exhibit a transition from a solution with two supersonic zones on each surface below a certain criticalMach number to a situation with one supersonic zone on each surface above the critical Mach number. Inthe region of instability solutions with positive lift are found in which there is a single supersonic zone onthe top surface and two supersonic zones on the lower surface. Solutions with negative lift are the mirrorimages of the solutions with positive lift. The CL − α plots of these airfoils exhibit three branches at zeroangle of attack corresponding to a P-branch with positive lift, a Z-branch with zero lift, and a N-branchwith negative lift. In the most recent work by the same authors,18 unsteady RANS solutions have beenperformed for the same set of airfoils in the transonic regime. Non-unique transonic solutions continue to bedemonstrated for all four airfoils under very similar flow conditions as the Euler cases.

II. Introduction

Given that the equations governing steady inviscid compressible flow are nonlinear, one can anticipate thepossibility of non-unique solutions. A familiar example is the case of supersonic flow past a wedge at an angleθ, where there are two solutions with different shock angles β corresponding to the strong and weak branchesof the β − θ diagram. Supersonic flows past a three-dimensional cone are qualitatively similar in that for agiven cone angle and Mach number, there are two possible oblique shock waves corresponding to the strong-and weak-shock solutions. In going from the wedge to the cone, the maximum angle beyond which the shockbecomes detached changes, as a result of three-dimensional relieving effect. The same three-dimensionaleffect that influences the supersonic flows over the cone can influence transonic flows over the wing. As aresult, the conditions supporting non-unique flow solutions might change in going from a two-dimensionalairfoil to a three-dimensional wing; alternatively, the non-unique solutions might cease to exist altogether.The purpose of the present study is to investigate the three-dimensional effect on the non-unique transonicflows.

III. Overview of Airfoils Supporting Non-unique Transonic Flows

A set of four airfoils of very different characteristics were designed and found to share the property thatin non-lifting transonic flow they exhibit a transition from a solution with two supersonic zones on eachsurface below a certain critical Mach number to a situation with one supersonic zone on each surface abovethe critical Mach number. In the region of instability solutions with positive lift are found in which there isa single supersonic zone on the top surface and two supersonic zones on the lower surface, and also solutionswith negative lift which are the mirror images of the solutions with positive lift. For further details relatedto the Euler solutions, please refer to the work by Jameson, Vassberg, and Ou17 (2011); for details of theunsteady RANS simulations, please refer to the work by Ou, Jameson, and Vassberg18 (2014); for details ofthe numerical schemes, please refer to the work by Jameson.13–15

The geometric definitions of the airfoils are described below. The non-unique transonic solutions com-puted using steady Euler simulations are discussed briefly below for completeness.

III.A. NU4 Airfoil

III.A.1. NU4 Airfoil Geometry

The NU4 airfoil was a consequence of a shape optimization study for symmetric airfoils in transonic flow,12

in which an attempt was made to find a 12 percent thick airfoil with a shock free solution at Mach 0.84. Theresulting NU4 airfoil has an almost shock free solution at its design Mach number, but also allows a liftingand non-lifting solution at zero angle of attack.

III.A.2. Non-unique Solutions for NU4 Airfoil at M.840

The non-unique solutions for NU4 airfoils at M.840 are shown in Figure 1 and 2. In Figure 1, the CL versus α

curve shows the presence of three distinctive solutions at zero angle of attack. In Figure 2, the correspondingCp distribution and pressure contour at zero angle of attack are plotted. Figure 2(b) corresponds to thesolution in the Z-branch. Figure 2(a) corresponds to the solution in the P-branch. The solution in theN-branch is the mirror image of the solution in the P-branch, hence is not plotted.

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Figure 1. Plot of CL-α sweep for flows over the NU4 airfoil at M.840, showing the presence of the positive, negative,

and zero branches.

(a) Symmetrical solution on the Z-branch (b) Asymmetrical solution on the P-branch

NU4 AIRFOIL MACH 0.8400 ALPHA 0.0000CL 0.000000 CD 0.000743 CM 0.000000GRID 640X 128 NCYC 2000 RES 0.384E-07

1.

2

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0.

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NU4 AIRFOIL MACH 0.8400 ALPHA 0.0000CL 0.038750 CD 0.004184 CM -0.012122GRID 640X 128 NCYC 2000 RES 0.320E-13

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Figure 2. Pressure contours showing non-unique solutions of NU4 airfoil at α = 0◦, M=.840

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III.B. JF1 Airfoil

III.B.1. JF1 Airfoil Geometry

The JF1 airfoil has a shape that is very simple, consisting of a parallel sided slab closed by a semi-circularnose and two parabolic arcs at the rear. Depending on the extent of the parabolic arcs a Mach range existsin which lifting solutions can be found at zero angle of attack.

III.B.2. Non-unique Solutions for JF1 Airfoil at M.835

The CL versus α curve at M.835 is plotted in Figure 3. The corresponding Cp distributions and pressurecontours at zero angle of attack are shown in Figure 4(a) for the symmetrical Z-branch solution, and inFigure 4(b) for the asymmetrical P-branch solution.

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Figure 3. Plots of CL-α sweep for flows over the JF1 airfoil at M.840, showing the presence of the positive, negative,

and zero branches.


JF1 AIRFOIL MACH 0.83500 ALPHA 0.00000CL 0.000000 CD 0.046047 CM 0.000000GRID 384X 64 NDES 0 RES0.568E-13 GMAX 0.100E-05

1.

2

0.8

0.

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4 -

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2 -

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JF1 AIRFOIL MACH 0.83500 ALPHA 0.00000CL 0.056998 CD 0.051628 CM -0.024147GRID 384X 64 NDES 0 RES0.210E-05 GMAX 0.100E-05

1.

2

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Figure 4. Pressure contours showing non-unique solutions of NU4 airfoil at α = 0◦, M=.840

III.C. JB1 Airfoil

III.C.1. JB1 Airfoil Geometry

The JB1 airfoil also has a parallel center section but the nose and tail are closed by higher order curveswhich maintain continuity of the curvature at the junction points. The nose section is defined by a Beziercurve with the control points

x1 = 0, y1 = 0

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x2 = 0, y2 = 1

x3 = 1, y3 = 1

x4 = 1, y4 = 1

scaled to a length of .125 and a height of .0625. The upper surface curve is defined by

x = .125(3t2 − 2t3)

y = .0625(3t− 3t2 + t3), 0 ≤ t ≤ 1

The trailing curve from x = .625 to 1 is

y = .0625

[

1 −

(

1 −

(

1 − x

.375

))3]

III.C.2. Non-unique Solutions for JB1 Airfoil at M.827

The CL versus α curve at M.827 is plotted in Figure 5. The corresponding Cp distributions and pressurecontours at zero angle of attack are shown in Figure 6(a) for the symmetrical Z-branch solution, and inFigure 6(b) for the asymmetrical P-branch solution.

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Figure 5. Plots of CL-α sweep for flows over the JF1 airfoil at M.827, showing the presence of the positive, negative,

and zero branches.


JB1 AIRFOIL MACH 0.82700 ALPHA 0.00000CL 0.000000 CD 0.020558 CM 0.000000GRID 384X 64 NDES 0 RES0.109E-06 GMAX 0.100E-05

1.

2

0.8

0.

4

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4 -

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-1.

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JB1 AIRFOIL MACH 0.82700 ALPHA 0.00000CL 0.101583 CD 0.024709 CM -0.033702GRID 384X 64 NDES 0 RES0.291E-10 GMAX 0.100E-05

1.

2

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Figure 6. Pressure contours showing non-unique solutions of JB1 airfoil at α = 0◦, M=.827

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III.D. JC6 Airfoil

III.D.1. JC6 Airfoil Geometry

The JC6 airfoil is a fully convex airfoil defined by a simple algebraic formula

y(x) = Cx1

n (1 − xn), 0 ≤ x ≤ 1

where the constant C, with a value of 0.06817, is adjusted to give the specified maximum thickness, 12percent of the chord. The choice n=6 results in a very blunt-nosed airfoil with maximum thickness at about55 percent of the chord, which has positive curvature everywhere and is C∞ continuous.

III.D.2. Non-unique Solutions of JC6 Airfoil at M.847

The CL versus α curve at M.847 is plotted in Figure 7. The corresponding Cp distributions and pressurecontours at zero angle of attack are shown in Figure 8(a) for the symmetrical Z-branch solution, and inFigure 8(b) for the asymmetrical P-branch solution. In Figure 7, results of OVERFLOW16 simulations forthe same flow conditions are also plotted and compared with the FLO82 results. Very similar results wereobtained using both methods.

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Figure 7. Plots of CL-α sweep for flows over the JC6 airfoil at M.847, showing the presence of the positive, negative,

and zero branches.


JC6 AIRFOIL MACH 0.847 ALPHA 0.0000000CL 0.000000000 CD 0.009175197 CM 0.000000000 GRID 512x 512 NCYC 4000 RED -14.45

1.2

0.8

0.4

0.0

-0.4

-0.8

-1.2

-1.6

-2.0

CP

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MACH: MIN 0.0012867 MAX 1.3963696 CONTOURS 0.050

JC6 AIRFOIL MACH 0.847 ALPHA 0.0000000CL 0.105210511 CD 0.016071270 CM-0.023791015 GRID 512x 512 NCYC 12000 RED -6.18

1.2

0.8

0.4

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-0.8

-1.2

-1.6

-2.0

CP

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MACH: MIN 0.0022379 MAX 1.4407564 CONTOURS 0.050

Figure 8. Pressure contours showing non-unique solutions of JC6 airfoil at α = 0◦, M=.847

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IV. Three-Dimensional Results

A set of four wings were designed based on the above airfoils that were found to exhibit non-uniquetransonic solutions. Flow simulations based on Euler equations (FLO88)13 were performed on meshes thatcontain 128 cells in the clockwise direction, 64 cells in the normal direction, and 48 cells in the spanwisedirection. An example of the mesh used for JF1 wing is shown in Figure 9. The aspect ratios of the wingsare varied as a way to control the extend of the three-dimensional effect.

JF1 WING

GRID 128 X 64 X 48

K = 26

JF1 WING

GRID 128 X 64 X 48

K = 49

JF1 WING

GRID 128 X 64 X 48

(a) Nearfield (b) Farfield (c) SurfaceFigure 9. Mesh for JF1 wing

The flow simulations were obtained by first perturbing the wing to a finite angle of attack from zero. Oncethe flow has sufficiently converged, the perturbation was removed so that the wing attitude returned to zerodegree, and the flow simulation was reconverged starting from the previously converged state. A non-uniquesolution will be obtained if the reconverged flow is asymmetrical. The other solution is the symmetrical zerolift solution, as is expected for flow over a symmetrical wing at zero angle of attack.

For wings with very high aspect ratio, the three-dimensional effect is small, and one expects the non-unique behaviors of the airfoils to carry over to the case of the wings. If non-unique solution can be replicatedfor high aspect ratio wings, the aspect ratio can be gradually decreased to investigate the non-unique flowcharacteristics. Because the three-dimensional effect can affect the conditions for certain flow characteristics,the Mach numbers at which the airfoils exhibit non-unique solutions can be different for wings of differentaspect ratios. As a result, as the aspect ratio changes, the Mach number might need to be varied to checkthe existence of non-unique solutions. This is the methodology adopted for the following study.

IV.A. Wings based on JB1 airfoil

A slender wing based on JB1 airfoil was constructed. The wing has a high aspect ratio of 16 and an aspectratio of 0.6. For the two-dimensional JB1 airfoil, non-unique solutions were found at M=.827. When thethree-dimensional simulations were performed for the high aspect ratio wing in the neighborhood this Machnumber, non-unique solutions were again found to exist at M=.827. For the same flow conditions, the wingassumes a lifting solution and a zero-lift solution. The exact non-unique solution depends on the initialcondition of the flow simulation. The Cp distributions showing the symmetrical and asymmetrical flowsolutions at zero angle of attack are plotted in Figure 10.

The CL versus α sweep of the simulation solutions shows the presence of the P-branch and N-branch,corresponding to the positive lift and negative lift solutions. The P- and N-branches can be extended pastzero degree of angle of attack, with a narrow overlapping region. The Z-branch cannot be obtained otherthan the single point corresponding to the zero lift solution at zero angle of attack. The result is plotted inFigure 11.

However, as the aspect ratio of the wing is further decreased, the non-unique solutions cannot be foundfor the wing, even after different Mach numbers were tried in the neighborhood of the original condition. For

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JB3 WING Mach: 0.827 Alpha: 0.000

CL: 0.000 CD: 0.03482 CM: 0.0000

Design: 0 Residual: 0.2337E-05

Grid: 129X 65X 49

Cl: 0.000 Cd: 0.03508 Cm: 0.0000

Root Section: 6.2% Semi-Span

Cp = -2.0

Cl: 0.000 Cd: 0.03145 Cm: 0.0000

Mid Section: 49.2% Semi-Span

Cp = -2.0

Cl: 0.000 Cd: 0.04063 Cm: 0.0000

Tip Section: 92.3% Semi-Span

Cp = -2.0

JB3 WING Mach: 0.828 Alpha: 0.000

CL: 0.077 CD: 0.02595 CM:-0.0444


Grid: 385X 65X 49

Cl: 0.096 Cd: 0.02346 Cm:-0.0551


Cp = -2.0

Cl: 0.079 Cd: 0.02339 Cm:-0.0486


Cp = -2.0

Cl: 0.027 Cd: 0.03627 Cm:-0.0118


Cp = -2.0

(a) Z-branch solution, M=.827, α = 0◦ (b) P-branch solution, M=.827, α = 0◦

Figure 10. Euler solutions for wing based on JB1 airfoil at M.827, and α = 0◦. Wing aspect ration=16. Taper ratio=0.6.

−0.1 −0.05 0 0.05 0.1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Angle of Attack (deg)

CL

JB1 Wing, M.827, Aspect Ratio=16, Taper Ratio=0.6

Figure 11. CL versus α sweep for wing based on JB1 airfoil at M.827.

the wing based on JB1 airfoil, non-unique solutions are only supported when the aspect ratio is sufficientlylarge.

IV.B. Wings based on NU4 airfoil and JC6 Airfoil

When similar analyses were carried out for wings based on the NU4 and JC6 airfoils, non-unique solutionscannot be found even when very high aspect ratio wings were used. The results indicate that the non-unique

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behaviors over these airfoil shapes are very sensitive to the influence of three-dimensional effects.

IV.C. Wings based on JF1 airfoil

Wings based on JF1 airfoil, on the other hand, were found to support non-unique solutions over a muchwider range of aspect ratios. Non-unique solutions have been found for aspect ratios of 16, 14, and 12 atM=.835. The corresponding Cp distributions for the wing with aspect ratio of 12 are plotted in Figure 12.

JF1 WING Mach: 0.835 Alpha: 0.000

CL: 0.000 CD: 0.05958 CM: 0.0001


Grid: 129X 65X 49

Cl: 0.000 Cd: 0.06091 Cm: 0.0001


Cp = -2.0

Cl: 0.000 Cd: 0.05445 Cm: 0.0001


Cp = -2.0

Cl: 0.000 Cd: 0.06608 Cm: 0.0000


Cp = -2.0


CL: 0.188 CD: 0.06135 CM:-0.1042


Grid: 129X 65X 49

Cl: 0.212 Cd: 0.05752 Cm:-0.1156


Cp = -2.0

Cl: 0.195 Cd: 0.06159 Cm:-0.1158


Cp = -2.0

Cl: 0.090 Cd: 0.06674 Cm:-0.0358


Cp = -2.0


Figure 12. Euler solutions for wing based on JF1 airfoil at M.835, and α = 0◦. Wing aspect ration=12. Taper ratio=0.6.

It is also found that, for the same aspect ratio, changing the taper ratio does not significantly change thenon-unique behavior of the flow, as is shown in Figure 13. In this case, the aspect ratio is increased from0.6 to 1.

The CL versus α sweep of the simulation solutions also shows the presence of the P-branch and N-branch.The P- and N-branches can extend past zero degree of angle of attack resulting in a narrow overlapping region.The Z-branch cannot be obtained other than the single point corresponding to the zero lift solution at zeroangle of attack. The result is plotted in Figure 14.

The non-unique solution disappeared when the aspect ratio is further reduced to 10. As the three-dimensional effect becomes more significant when the aspect ratio decreases, the flow conditions that supportnon-unique flows also change. When the simulations were performed at M=.840, the non-unique solutionswere rediscovered for the wing with an aspect ratio of 10. The flow solution is plotted in Figure 15. Thecorresponding plot of CL versus α sweep is shown in Figure 16.

When the aspect ratio is further decreased, the non-unique solutions disappeared. Varying the Machnumber did not yield further non-unique solutions.

V. Conclusion

In this work, two-dimensional airfoils that were previously found to exhibit non-unique transonic solutionswere extended to three-dimensional wings to investigate the influence of the three-dimensional effect on non-unique flow behavior. It is found that certain airfoil shapes are more sensitive to the three-dimensional effect

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CL: 0.000 CD: 0.06111 CM: 0.0000


Grid: 129X 65X 49

Cl: 0.000 Cd: 0.05569 Cm: 0.0000


Cp = -2.0

Cl: 0.000 Cd: 0.05906 Cm: 0.0000


Cp = -2.0

Cl: 0.000 Cd: 0.06440 Cm: 0.0000


Cp = -2.0


CL: 0.083 CD: 0.06037 CM:-0.0501


Grid: 129X 65X 49

Cl: 0.131 Cd: 0.05607 Cm:-0.0851


Cp = -2.0

Cl: 0.093 Cd: 0.05719 Cm:-0.0549


Cp = -2.0

Cl: 0.021 Cd: 0.06445 Cm:-0.0081


Cp = -2.0


Figure 13. Euler solutions for wing based on JF1 airfoil at M.835, and α = 0◦. Wing aspect ration=12. Taper ratio=1.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


CL

JF1 Wing, M.835, Aspect Ratio=12, Taper Ratio=1.0

Figure 14. CL versus α sweep for wing based on JF1 airfoil at M.835.

than other airfoils, with the following cases observed. For wings based on NU4 and JC6 airfoils, non-uniquetransonic solutions can not be found even when the wings have very high aspect ratio. For wings based onJB1 airfoil, non-unique transonic solutions were found, but only when the wings have sufficiently high aspectratios. For wings based on JF1 airfoil, non-unique transonic solutions were supported for a wider range ofaspect ratios. The lowest aspect ratio that was achieved so far that still supports the non-unique solutionsis around 10. As another effect, the three-dimensional flows also change the exact condition for which the

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CL: 0.000 CD: 0.05914 CM: 0.0000


Grid: 129X 65X 49

Cl: 0.000 Cd: 0.05084 Cm: 0.0001


Cp = -2.0

Cl: 0.000 Cd: 0.05712 Cm: 0.0000


Cp = -2.0

Cl: 0.000 Cd: 0.06495 Cm: 0.0000


Cp = -2.0


CL: 0.060 CD: 0.06002 CM:-0.0380


Grid: 129X 65X 49

Cl: 0.088 Cd: 0.05510 Cm:-0.0614


Cp = -2.0

Cl: 0.069 Cd: 0.05639 Cm:-0.0435


Cp = -2.0

Cl: 0.015 Cd: 0.06498 Cm:-0.0065


Cp = -2.0


Figure 15. Euler solutions for wing based on JF1 airfoil at M.840, and α = 0◦. Wing aspect ration=10. Taper ratio=1.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


CL

JF1 Wing, M.840, Aspect Ratio=10, Taper Ratio=1.0

Figure 16. CL versus α sweep for wing based on JF1 airfoil at M.840.

non-unique solutions are admitted when the aspect ratio decreases.

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References

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2M.D. Salas, R.E. Melnik, and A. Jameson, A comparative study of the non-uniqueness problem of the potential equation,AIAA Paper 83-1888, (1983)

3A. Jameson, Airfoil admitting non-unique solutions to the Euler equations, AIAA Paper 91-1625 (1991)4M.M. Hafez and W.H. Guo, Nonuniqueness of transonic ows, Acta Mech. 138, 177184 (1999a)5M.M. Hafez and W.H. Guo, Some anomalies of numerical simulation of shock waves, Part I: inviscid ows. Comput.

Fluids 28(45), 701719 (1999b)6M.M. Hafez and W.H. Guo, Some anomalies of numerical simulation of shock waves. Part II: effect of articial and real

viscosity, Comput. Fluids 28(45), 721739 (1999c)7A.G. Kuzmin and A.V. Ivanova, The structural instability of transonic flow associated with amalgamation/splitting of

supersonic regions, Theoret. Comput. Fluid Dynamics (2004) 18:335-3448A.G. Kuzmin, Instability and bifurcation of transonic flow over airfoils, AIAA Paper, 20049A.V. Ivanova and A.G. Kuzmin, Non-uniqueness of the transonic flow past an airfoil, Fluid Dynamics, Vol. 39, No.4,

pp.642-648, (2004)10A.G. Kuzmin, Bifurcation of transonic flow over a flattened airfoil, Frontiers of Computational Fluid Dynamics, World

Scientific Publishing Company LTD, (2006)11Kuzmin, Structural instability of transonic flow over an airfoil, Journal of Engineering Physics and Thermophysics, Vol.

77, No.5, pp.1022-1026, (2004)12J. C. Vassberg, N. A. Harrison, D. L. Roman and A. Jameson, Systematic Study on the Impact of Dimensionality for

a Two-Dimensional Aerodynamic Optimization Model Problem, 29th AIAA Applied Aerodynamics Conference, Honolulu, HI,June, 2011

13A. Jameson, W. Schmidt, and E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods UsingRunge-Kutta Time-Stepping Schemes, AIAA Paper 81-1259, AIAA 14th Fluid and Plasma Dynamic Conference, Palo Alto,(1981)

14A. Jameson, .Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings.,AIAA Paper 91-1596, AIAA 10th Computational Fluid Dynamic Conference, Hawaii (1991).

15A. Jameson, Analysis and Design of Numerical Schemes for Gas Dynamics 2 Artificial Diffusion and Discrete ShockStructure, International Journal of Computational Fluid Dynamics, Vol. 5, 1995, pp. 1-38.

16P. G. Buning, D. C. Jespersen, T. H. Pulliam, G. H. Klopfer, W. M. Chan, J. P. Slotnick, S. E. Krist and K. J. Renze,OVERFLOW User’s Manual, Version 1.8L, NASA, July, 1999

17A. Jameson, J. Vassberg, and K. Ou, Further Studies of Airfoils Supporting Non-unique Solutions in Transonic Flow,AIAA Journal, DOI: 10.2514/1.J051713, Vol. 50, No. 12, pp. 2865-2881, December 2012.

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