further studies of mesh reﬁnement: are shock-free airfoils truly shock...

Further Studies of Mesh Refinement:

Are Shock-Free Airfoils Truly Shock Free?

Antony Jameson∗

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

John C. Vassberg†

Boeing Research & Technology, The Boeing Company, Huntington Beach, CA 92647

Abstract

The Garabedian-Korn airfoil is studied at its inviscid design condition and in the neighborhood

thereof. This airfoil was designed with a hodograph method to be shock-free at M = 0.750 and

Cl = 0.629. However, computational fluid dynamics results show a double-shock pattern appear

at this design point as the resolution of the grid is enhanced. A survey of flow conditions in

the vicinity of the design point has yielded another condition of interest. At this condition

of M = 0.751 and Cl = 0.625, the Garabedian-Korn airfoil becomes shock-free. The work

presented herein verifies that the flow remains shock free through rigorous mesh refinements.

Grids utilized are high-quality analytically-generated O-meshes comprised of aspect-ratio-one

cells and range in size from 16, 384 to 16, 777, 216 quadrilateral elements. Flow solutions on

sequences of meshes are post-processed to establish grid-converged force and moment values,

as well as to provide an estimate on the order-of-accuracy of two widely-used computational

fluid dynamics methods.

Nomenclature

c Camber of an Airfoil

CFD Computational Fluid Dynamics

Cd Drag Coefficient =drag

q∞Cref

Cl Lift Coefficient = lift

q∞Cref

CP Pressure Coefficient = P−P∞

q∞

Cref Airfoil Reference Chord = 1.0

count Drag Coefficient Unit = 0.0001

LE Airfoil Leading Edge

M Mach Number = Va

q Dynamic Pressure = 1

2ρV 2

RED Reduction in Orders-of-Magnitude

t Thickness of an Airfoil

TE Airfoil Trailing Edge

x Streamwise Cartesian Coordinate

y Vertical Physical Coordinate

2D Two Dimensional

α Angle-of-Attack

τ TE Included Angle

ρ Radius of Curvature

∞ Signifies Freestream Conditions

∗Thomas V. Jones Professor of Engineering, Fellow AIAA†Boeing Technical Fellow, Fellow AIAA

Copyright c© 2011 by Jameson & Vassberg.

Published by the American Institute of Aeronautics and Astronautics with permission.

Jameson & Vassberg, AIAA Paper 2011-0000, Honolulu, Hawaii, June 2011 1 of 16

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3983

Copyright © 2011 by Jameson & Vassberg. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

In a recent paper, Vassberg & Jameson1 presented studies of the asymptotic convergence with extrememesh refinement for inviscid subsonic and transonic flows of several widely-used computational fluid dynamics(CFD) methods. These were the NASA Overflow2 and CFL3D3 codes and Jameson’s FLO824 method. Thestudies were performed for the NACA0012 airfoil extended to a closed trailing edge on extremely-smoothanalytically-generated O-meshes comprised of aspect-ratio-one cells.

Since the codes are nominally second-order accurate in smooth regions of the flow, it was anticipatedthat the asympototic convergence rate should be second order for subsonic flows. However, the algorithmsimplemented in these codes fall back to first-order accuracy due to flux limiters when they are applied tonumerically capturing shock waves. In fact it turned out second-order convergence could be obtained forboth nonlifting and lifting subsonic flows with proper treatment of the wall and farfield boundary conditions,including a vortex term in the farfield. In transonic flows, second-order convergence on Cd was observed forthe nonlifting case, but only first-order accuracy was seen for Cl, Cd and Cm for the lifting case.

A principal objective of this study is to find out whether contamination due to shock waves reduces theasymptotic convergence of global quantities such as lift, drag and pitching-moment coefficients (Cl, Cd &Cm) to first order for transonic flows.

In the present work, we examine the asymptotic behavior of transonic flows past a shock-free airfoil.The first question to be addressed is whether airfoils designed to be shock-free are actually shock-free in theasymptotic limit. The second question to be addressed is whether second-order convergence can be recoveredin lifting transonic flows in the absence of contamination by shock waves.

Our study is focused on the well known Garabedian-Korn5 airfoil which was designed to be shock-free atM = 0.750 and Cl = 0.629. It turns out that it is not shock-free when the mesh is sufficiently refined; thishas been observed by many researchers. However, an intensive search has revealed it is shock-free at theslightly-higher Mach and slightly-lower lift condition of M = 0.751 and Cl = 0.625.

This paper is organized in the following manner. Section II provides a description of the Garabedian-Kornairfoil geometry. Section III provides the results of the study. Section IV summarizes our conclusions. Forreference, an appendix is included that tabulates the detailed coordinates of the Garabedian-Korn airfoil.Tables of data are embedded within the text, while all figures are appended to the end of the paper.

II. Garabedien-Korn Airfoil

This section provides a description of the shock-free airfoil utilized in this study. The Garabedian-Kornairfoil was developed in the early 1970’s using the hodograph method. Unfortunately, as originally pub-lished,5 the coordinates of this airfoil (designated 75-06-12) include a small discontinuity at the sonic line.However, from the original hodograph results, very accurate values for surface slope and arc-length are avail-able. In order to fix the discontinuity defect, new coordinates of the airfoil have been reconstructed using aconformal mapping based on surface slope and arc-length. The reconstructed contour was then nondimen-sionalized by its maximum-length chordline and rotated to place the true leading-edge point at the originand the trailing-edge point at (XTE , YTE) = (1.0, 0.0). For reference, a high-definition tabulation of theseimproved coordinates are included in the appendix of this paper. Table I includes some pertinent geometriccharacteristics of the Garabedian-Korn airfoil. Here, ρLE is the LE radius, τ is the TE included angle, andTE-Slope is the slope of the camber-line at the TE. Also, tmax is the maximum thickness of the airfoil,located at xtmax, and cmax the maximum camber, occuring at xcmax. Note that the Garabedian-Korn airfoilis about 11.6% thick, has almost 2% camber, and incorporates a cusped TE since τ = 0◦.

Table I: Garabedian-Korn Airfoil.

ρLE τ TE-Slope tmax xtmax cmax xcmax

0.02139 0◦ −0.12614 0.115893 0.331698 0.019399 0.737957

The next section describes the O-mesh grids utilized in this work.


III. Results

In the present work, we examine the asymptotic behavior of transonic flows past the Garabedian-Kornshock-free airfoil. The first question addressed is whether airfoils designed to be shock-free are actuallyshock-free in the asymptotic limit. The second question addressed is whether second-order convergence canbe recovered in transonic lifting flows in the absence of contamination by shock waves.

The Garabedian-Korn airfoil was designed to be shock-free at M = 0.750 and Cl = 0.629. In fact it isnot shock-free when the mesh is sufficiently refined, as indicated below. However, a search has revealed it isshock-free a the slightly-higher Mach and slightly-lower lift condition of M = 0.751 and Cl = 0.625.

The 2D grids utilized in this study are consistent with the O-meshes of Vassberg & Jameson.1 These high-quality analytically-generated grids are comprised of aspect-ratio-one cells. The circular farfield boundaryresides at about 150 chordlengths from the airfoil. Figure 1 is a near-field view of a (128x128)-cell O-mesh.Close-up views of the LE and TE regions of this mesh are provided in Figure 2. To give the reader withan appreciation of how extreme our mesh-refinement studies are, we note that our finest mesh inserts 1,024cells into each cell of the mesh depicted in Figures 1-2.

Results for the two flow conditions of interest are discussed in the following two subsections.

Design-Point: M = 0.750, Cl = 0.629

The design point for the Garabedian-Korn airfoil is reviewed in this subsection. Since most of the remainingfigures are of the same class, a brief description of these is in order. Refer to Figure 3. The left-hand-sideof this figure provides the pressure distribution at the airfoil surface; its legend tabulates flow conditions,force and moment coefficients, grid size and convergence properties. Here, RED is the reduction of residualsin orders-of-magnitude. The right-hand-side depicts Mach contours in the near-field surrounding the airfoil;its legend indicates the minimum and maximum values of Mach number in the discrete solution, as well asthe ∆M between contours. The bold contour always represents the sonic line; keying off of this contour andknowing ∆M , one can surmise the level-set of other contours.

Figures 3-6 illustrate the progression of FLO82 solutions on a sequence of meshes, starting with a resolutionof (128x128) cells and completing with a robustly-fine mesh of (1024x1024) elements. Note that the flowappears to be shock-free on the O-mesh of (128x128) cells, depicted in Figure 3. However, as the meshis continually refined, the flowfield becomes characterized with a double-shock pattern that increases instrength with increasing grid resolution. This double-shock pattern is evident in both the surface pressuredistributions as well as in the coalescence of Mach contours off the airfoil surface.

Corresponding results of Figures 3-6 are tabulated in Table II. Also included in this table are extrapolatedestimates (NC = ∞) for α, Cd and Cm, as well as associated orders-of-accuracy (P ). The process used todetermine limiting values and order-of-accuracy is documented in Vassberg & Jameson.1 Note that a trulyshock-free inviscid flow (exact solution) yields a drag of zero. Here, the limiting value of drag is approximately1 count. Unexpectedly, drag and pitching-moment appear to be second-order accurate in this sequence.

Table II: FLO82 Results.

M = 0.750, Cl = 0.629

NC α◦ Cd Cm Red

128 0.160641 0.00058089 -0.14528177 -14.6

256 0.144654 0.00006767 -0.14542121 -14.7

512 0.141870 0.00009170 -0.14559448 -10.8

1,024 0.140732 0.00009571 -0.14563748 -9.5

∞ 0.139945 0.00009651 -0.14565167 -

P 1.291 2.583 2.011 -

Figures 7-9 illustrate the progression of Overflow solutions on a sequence of meshes, starting with a reso-lution of (256x256) cells and completing with a robustly-fine mesh of (1024x1024) elements. This sequenceholds lift constant at Cl = 0.629. Note that the flow appears to be shock-free on the O-mesh of (256x256)cells, depicted in Figure 7. However, as the mesh is refined, the flowfield becomes characterized with a


double-shock pattern that increases in strength with increasing grid resolution. This double-shock patternis evident in the coalescence of Mach contours off the airfoil surface, as shown in Figures 8-9.

Table III tabulates the results of the Overflow grid-convergence study. Note that the limiting value of α

is higher for the Overflow calculations than it is for those of FLO82. The cause of this is due to Overflownot imposing a point-vortex term on the farfield boundary conditions, whereas FLO82 does. Note that thelimiting value of drag for Overflow is about 4 counts. Also note that Overflow exhibits second-order-accurateconvergence for all force and moment calculations.

Table III: Overflow Results.

M = 0.750, Cl = 0.629

NC α◦ Cd Cm Red

256 0.08009 0.0013155 -0.1505656 -6.1

512 0.14943 0.0006423 -0.1463823 -5.9

1,024 0.16565 0.0004663 -0.1454846 -5.8

∞ 0.17060 0.0004040 -0.1452393 -

P 2.096 1.936 2.220 -

Next we study the shock-free condition.

Shock-Free Condition: M = 0.751, Cl = 0.625

The shock-free condition for the Garabedian-Korn airfoil is reviewed in this subsection. Figures 10-15illustrate the progression of FLO82 solutions on a sequence of meshes, starting with a resolution of (128x128)cells and completing with an extremely-fine mesh of (4096x4096) elements. Note that the flow appears toremain shock-free as the mesh is continually refined. This shock-free character is evident in both the surfacepressure distributions as well as in the smooth separation of Mach contours off the airfoil surface. However,it is clear that the scheme/limiter, specifically at the sonic line, is causing the flow to become less smoothas the grid is refined. This can be seen in the surface pressure distributions, as well as in the contour of thesonic line. Aside from the immediate vicinity of the sonic line, the flowfield is otherwise well behaved.

Corresponding results of Figures 10-15 are tabulated in Table IV. Also included in this table are extrapo-lated estimates for α, Cd and Cm, as well as associated orders-of-accuracy. Again, a truly shock-free inviscidflow (exact solution) yields a drag of zero. Here, the limiting value of drag is approximately two orders-of-magnitude smaller than that for the design point. In fact, the FLO82 computed drag levels are sufficientlysmall that they confirm that this is a shock-free condition for the Garabedian-Korn airfoil. This set of dataalso answers our second question regarding the order-of-accuracy of FLO82 for transonic flows that are notcontaminated with shock waves. For transonic lifting flows void of shock waves, the H-CUSP scheme ofFLO82 is only first-order accurate.

The calculations have been performed with a version of FLO82 which uses the H-CUSP scheme.6 Thisscheme captures stationary shocks with a single interior point, like the Roe scheme.7 However, flux formulasare not smooth across sonic lines. Moreover, oscillations in the neighborhood of shock waves are prevented bya limiter which reduces the scheme to its first-order-accurate form at non-smooth extrema, and this limiteralso distracts from the smoothness of the scheme. It is conjectured that the first-order-accurate asymptoticconvergence may be a consequence of this lack of smoothness in the discretization formulas.


Table IV: FLO82 Results.

M = 0.751, Cl = 0.625

NC α◦ Cd Cm Red

128 0.131462 +0.00057645 -0.14597562 -13.1

256 0.115407 +0.00004447 -0.14607142 -12.8

512 0.111177 -0.00000741 -0.14615033 -12.3

1,024 0.109804 -0.00000406 -0.14620200 -12.4

2,048 0.108998 -0.00000108 -0.14624022 -11.9

4,096 0.108657 +0.00000016 -0.14625707 -4.1

∞ 0.107851 +0.00000102 -0.14627035 -

P 1.241 1.276 1.182 -

Figures 16-18 illustrate the progression of Overflow solutions on meshes with (256x256), (512x512) and(1024x1024) cells, respectively. Note that the flow appears to remain shock-free, no matter the mesh; theMach contours do not coalesce in the field. These results further verify that the Garabedian-Korn airfoil isshock-free at this flow condition.

Table V tabulates the complete set of results of this grid-convergence study using Overflow, holdinglift constant, on the sequence of three meshes. Again, the limiting value of α is higher for the Overflowcalculations than it is for those of FLO82. Note that the limiting value of drag for Overflow is about 2.6counts. Since Overflow does not include the point-vortex terms in the farfield boundary conditions, a spuriousfarfield drag is present for 2D lifting flows. This phenomenon has been studied extensively by Destarac.8 Itappears, however, that Overflow does exhibit second-order-accurate convergence to the shock-free solution.

Table V: Overflow Results.

M = 0.751, Cl = 0.625

NC α◦ Cd Cm Red

256 0.05341 0.00132237 -0.15116540 -6.9

512 0.11959 0.00058234 -0.14698620 -6.1

1,024 0.13465 0.00035850 -0.14603230 -6.1

∞ 0.13909 0.00026143 -0.14575019 -

P 2.136 1.725 2.131 -

IV. Conclusions

The authors examine the asymptotic behavior of transonic flows past the Garabedian-Korn shock-free

airfoil in order to answer two questions. The first question addressed is whether airfoils designed to beshock-free are actually shock-free in the asymptotic limit. At its design point, the Garabedian-Korn airfoilis in fact not shock-free. However, it is found to be shock-free at another flow condition in the neighborhoodof its design point. The work presented herein verifies that the flow remains shock-free through a rigorousmesh refinement. The grid-sequence utilized is a set of high-quality O-meshes comprised of aspect-ratio-onecells and range in size from 16, 384 to 16, 777, 216 quadrilateral elements. The second question addressed iswhether second-order convergence can be recovered in transonic lifting flows in the absence of contaminationby shock waves. While our results are somewhat mixed in this regard, Overflow did recover second-order-accurate grid-convergence under these conditions, whereas the H-CUSP scheme of FLO82 did not. Somewhatunexpected, both CFD methods exhibited second-order accuracy for drag and pitching-moment at the designpoint, characterized with a double shock.


References

1J. C. Vassberg and A. Jameson. In pursuit of grid convergence, Part I: Two-dimensional Euler solutions. AIAA paper 2009-

4114, 27th AIAA Applied Aerodynamics Conference, San Antonio, TX, June 2009.2P. G. Buning, D. C. Jespersen, T. H. Pulliam, W. M. Chan, J. P. Slotnick, S. E. Krist, and K. J. Renze. OVERFLOW

user’s manual, version 1.8l. NASA Report , NASA Langley Research Center, Hampton, VA, 1999.3S. L. Krist, R. T. Biedron, and C. L. Rumsey. CFL3D user’s manual, version 5.0. NASA Report , NASA Langley Research

Center, Hampton, VA, November 1996.4A. Jameson. Solution of the Euler equations for two dimensional transonic flow by a multigrid method. Applied Mathematics

and Computation, 13:327–356, 1983.5F. Bauer, P. Garabedian, D. Korn, and A. Jameson. Supercritical Wing Sections II. Springer Verlag, New York, 1975.6A. Jameson. Analysis and design of numerical schemes for gas dynamics 2: Artificial diffusion and discrete shock structures.

International Journal of Computational Fluid Dynamics, 5:1–38, 1995. RIACS Report 94.16.7P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics,

43:357–372, 1981.8D. Destarac. Far-field / near-field drag balance and applications of drag extraction in CFD. VKI Lecture Series CFD-based

Aircraft Drag Prediction and Reduction, National Institute of Aerospace, Hamption, VA, February 2003.


Appendix: Garabedian-Korn Airfoil Coordinates

i x y-upper y-lower thick camber

1 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.000222 0.003186 -0.002875 0.006061 0.000156

3 0.000887 0.005997 -0.005510 0.011507 0.000243

4 0.001995 0.008472 -0.007982 0.016454 0.000245

5 0.003544 0.010667 -0.010334 0.021001 0.000167

6 0.005534 0.012761 -0.012602 0.025364 0.000080

7 0.007963 0.014862 -0.014758 0.029620 0.000052

8 0.010828 0.016995 -0.016799 0.033794 0.000098

9 0.014127 0.019164 -0.018770 0.037934 0.000197

10 0.017857 0.021354 -0.020697 0.042051 0.000328

11 0.022015 0.023556 -0.022595 0.046151 0.000480

12 0.026597 0.025761 -0.024470 0.050231 0.000646

13 0.031599 0.027956 -0.026322 0.054278 0.000817

14 0.037017 0.030129 -0.028150 0.058279 0.000989

15 0.042846 0.032268 -0.029951 0.062218 0.001159

16 0.049080 0.034361 -0.031720 0.066081 0.001321

17 0.055715 0.036397 -0.033451 0.069848 0.001473

18 0.062744 0.038356 -0.035140 0.073496 0.001608

19 0.070162 0.040210 -0.036781 0.076991 0.001714

20 0.077961 0.041927 -0.038370 0.080297 0.001779

21 0.086136 0.043487 -0.039900 0.083387 0.001793

22 0.094677 0.044916 -0.041367 0.086283 0.001775

23 0.103579 0.046256 -0.042765 0.089021 0.001745

24 0.112834 0.047541 -0.044091 0.091632 0.001725

25 0.122432 0.048781 -0.045341 0.094122 0.001720

26 0.132367 0.049977 -0.046510 0.096487 0.001733

27 0.142628 0.051130 -0.047597 0.098727 0.001767

28 0.153207 0.052243 -0.048598 0.100840 0.001823

29 0.164095 0.053314 -0.049511 0.102825 0.001901

30 0.175282 0.054342 -0.050336 0.104678 0.002003

31 0.186758 0.055326 -0.051069 0.106395 0.002128

32 0.198514 0.056266 -0.051712 0.107977 0.002277

33 0.210538 0.057161 -0.052262 0.109423 0.002449

34 0.222820 0.058008 -0.052720 0.110728 0.002644

35 0.235350 0.058809 -0.053082 0.111891 0.002863

36 0.248116 0.059561 -0.053350 0.112911 0.003106

37 0.261107 0.060264 -0.053522 0.113785 0.003371

38 0.274311 0.060916 -0.053596 0.114511 0.003660

39 0.287718 0.061516 -0.053571 0.115087 0.003973

40 0.301315 0.062064 -0.053447 0.115511 0.004309

41 0.315090 0.062558 -0.053221 0.115779 0.004669

42 0.329032 0.062999 -0.052891 0.115890 0.005054

43 0.343127 0.063384 -0.052457 0.115840 0.005463

44 0.357363 0.063712 -0.051916 0.115629 0.005898

45 0.371729 0.063985 -0.051269 0.115253 0.006358

46 0.386211 0.064199 -0.050513 0.114712 0.006843

47 0.400796 0.064353 -0.049649 0.114002 0.007352

48 0.415473 0.064448 -0.048677 0.113124 0.007886

49 0.430227 0.064481 -0.047596 0.112077 0.008443

50 0.445045 0.064452 -0.046408 0.110859 0.009022


i x y-upper y-lower thick camber

51 0.459916 0.064358 -0.045112 0.109471 0.009623

52 0.474825 0.064199 -0.043713 0.107912 0.010243

53 0.489759 0.063973 -0.042212 0.106185 0.010881

54 0.504706 0.063678 -0.040612 0.104290 0.011533

55 0.519653 0.063310 -0.038919 0.102229 0.012195

56 0.534585 0.062867 -0.037137 0.100005 0.012865

57 0.549490 0.062346 -0.035272 0.097618 0.013537

58 0.564355 0.061743 -0.033329 0.095071 0.014207

59 0.579167 0.061052 -0.031317 0.092369 0.014868

60 0.593913 0.060269 -0.029246 0.089515 0.015512

61 0.608579 0.059391 -0.027125 0.086516 0.016133

62 0.623153 0.058412 -0.024967 0.083378 0.016722

63 0.637622 0.057330 -0.022785 0.080115 0.017273

64 0.651973 0.056147 -0.020594 0.076741 0.017777

65 0.666193 0.054862 -0.018410 0.073272 0.018226

66 0.680271 0.053478 -0.016252 0.069730 0.018613

67 0.694194 0.052000 -0.014138 0.066139 0.018931

68 0.707948 0.050433 -0.012089 0.062521 0.019172

69 0.721524 0.048779 -0.010122 0.058901 0.019329

70 0.734907 0.047048 -0.008255 0.055303 0.019396

71 0.748087 0.045245 -0.006505 0.051750 0.019370

72 0.761053 0.043379 -0.004885 0.048264 0.019247

73 0.773791 0.041456 -0.003405 0.044861 0.019025

74 0.786292 0.039485 -0.002069 0.041554 0.018708

75 0.798545 0.037475 -0.000881 0.038356 0.018297

76 0.810538 0.035436 0.000162 0.035274 0.017799

77 0.822261 0.033379 0.001065 0.032313 0.017222

78 0.833703 0.031310 0.001835 0.029475 0.016572

79 0.844856 0.029242 0.002477 0.026765 0.015860

80 0.855708 0.027186 0.003001 0.024185 0.015094

81 0.866250 0.025152 0.003416 0.021736 0.014284

82 0.876473 0.023150 0.003730 0.019419 0.013440

83 0.886369 0.021190 0.003952 0.017237 0.012571

84 0.895927 0.019283 0.004090 0.015193 0.011686

85 0.905141 0.017439 0.004151 0.013288 0.010795

86 0.914001 0.015665 0.004142 0.011523 0.009904

87 0.922500 0.013969 0.004072 0.009897 0.009021

88 0.930630 0.012357 0.003945 0.008411 0.008151

89 0.938385 0.010835 0.003770 0.007065 0.007302

90 0.945757 0.009406 0.003552 0.005854 0.006479

91 0.952740 0.008073 0.003297 0.004776 0.005685

92 0.959328 0.006837 0.003011 0.003826 0.004924

93 0.965516 0.005699 0.002701 0.002998 0.004200

94 0.971296 0.004659 0.002372 0.002287 0.003516

95 0.976665 0.003717 0.002030 0.001687 0.002873

96 0.981618 0.002871 0.001680 0.001191 0.002276

97 0.986151 0.002121 0.001329 0.000792 0.001725

98 0.990258 0.001460 0.000981 0.000480 0.001220

99 0.993938 0.000890 0.000641 0.000249 0.000766

100 0.997186 0.000407 0.000314 0.000093 0.000361

101 1.000000 0.000000 0.000000 0.000000 0.000000


Figure 1. O-Mesh (128x128) about the Garabedian-Korn Airfoil.

Figure 2. Close-Up Views of O-Mesh (128x128) near the Leading Edge and Trailing Edge.


GARABEDIAN-KORN AIRFOILMACH 0.750 ALPHA 0.1606409CL 0.629000000 CD 0.000580885 CM-0.145281774 GRID 128x 128 NCYC 1000 RED -14.58

1.2

0.8

0.4

0.0

-0.4

-0.8

-1.2

-1.6

-2.0

CP

++++++++++++++++++++++

++

++

++

+++++++++++++++++++++++

++

+++++

+

+

+

+++

+

+

+

+

+++++++

+

+++++++++++++++++++++++++

++

++++

++

++

++

+++++++++++++++

+

MACH: MIN 0.0930992 MAX 1.2036681 CONTOURS 0.040

Figure 3. FLO82 Solution, Garabedian-Korn O-Mesh (128x128), M = 0.750 and Cl = 0.629.


1.2

0.8

0.4

0.0

-0.4

-0.8

-1.2

-1.6

-2.0

CP

+++++++++++++++++++++++++++++++++++++++

+++++

++++

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++++

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++++++

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++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

++++

+

+

+

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+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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++++++++++++++++++++++++++++++++++++++++++++++





1.2

0.8

0.4

0.0

-0.4

-0.8

-1.2

-1.6

-2.0

CP

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++

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Figure 7. Overflow Solution, Garabedian-Korn O-Mesh (256x256), M = 0.750 and Cl = 0.629.





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GARABEDIAN-KORN AIRFOILMACH 0.751 ALPHA 0.1111766CL 0.625000000 CD-0.000007412 CM-0.146150334 GRID 512x 512 NCYC 1000 RED -12.29

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further studies of mesh reﬁnement: are shock-free airfoils truly shock...

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