Simulations of Viscous Transonic Flows over Lifting

Airfoils and Wings

Mohamed Hafez∗ and Essam Wahba∗

Department of Mech. and Aero. Eng., University of California, Davis CA 95616, USA

In this paper, the hierarchical formulation for steady viscous transonic flow simulationsintroduced previously by the authors is reviewed and a simplified version for the calculationof the vortical velocity components is presented. The results are compared to availablesolutions of standard Navier-Stokes equations for laminar flows.

Keywords: Hierarchical formulation, Viscous transonic flows, Navier-Stokes equations

I. Introduction

In Ref. 1 and 2, the authors present a hierarchical formulation for steady inviscid and viscous transonicflows over airfoils and wings. The main idea is to use a Helmholtz type velocity decomposition, where thevelocity vector is split into a gradient of a potential function plus vortical components. The latter are onlycalculated in limited regions, where potential flow is not an adequate model. The advantages of this approachhave been discussed in the above references and can be summarized as follows.

The formulation can be viewed as a viscous/inviscid interaction procedure (Ref. 3-6) where the outerflow field is governed by a potential equation. This way, the calculations of the vortical flow componentsin the outer region, where the grid is stretched to the far field, are eliminated. The artificial entropy andvorticity, usually generated in the standard methods in the outer region, are therefore avoided. The potentialflow calculations provide proper boundary conditions for the limited regions of rotational flows, typical ofexternal aerodynamic applications. The present natural decomposition does not suffer from the difficultiesassociated with the interface boundaries of heterogeneous zonal methods and at the same time it is notlimited to boundary layer approximations and the coupling between the viscous and inviscid regions is builtin automatically.

Another advantage is the simple implementation of upwind schemes for the convection/diffusion scalarequations of the entropy and vorticity components. Moreover, the separation of the acoustic mode repre-sented by the augmented potential equation facilitates the application of multigrid acceleration techniques.

Finally, the formulation has a built-in low Mach number preconditioning on the differential level. Indeed,incompressible flow (zero Mach number) is recovered without loss of accuracy or efficiency.

The present paper deals with the evaluation of the vortical velocity components in the viscous flow regionfor the airfoil and wing problems. A simple and flexible method is discussed and the results are comparedto those obtained in Ref. 1 and 2 as well as available data in the literature. In the following, the governingequations and boundary conditions are derived and the numerical methods and results are discussed.

II. Present Formulation

As discussed in Ref. 1 and 2, the hierarchical formulation is based on Helmholtz-type velocity decompo-sition where

~q = ∇φ + ~q∗ (1)

Substituting in the continuity equation yields an augmented potential equation

∇ · (ρ∇φ) = −∇ · (ρ~q∗)

(2)

∗e-mail: mhafez@ucdavis.edu

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with the boundary condition ∂φ∂n = 0 at a solid surface. The calculation of ~q∗ in the region of inviscid

rotational flow is the same as in the previous work. In the viscous flow region, ~q∗ is calculated by twomethods. In Ref. 1 and 2, the component of ~q∗ in the main flow direction is updated from the tangentialmomentum equation, while the normal momentum equation is used to update the pressure. A more generaland flexible approach is used here, namely, the cartesian components of ~q∗ are updated from the cartesiancomponents of the momentum equation

ρu(δu∗)x + ρv(δu∗)y + ρw(δu∗)z − 1Re∇2(δu∗) = −RX−Mom (3)

ρu(δv∗)x + ρv(δv∗)y + ρw(δv∗)z − 1Re∇2(δv∗) = −RY−Mom (4)

ρu(δw∗)x + ρv(δw∗)y + ρw(δw∗)z − 1Re∇2(δw∗) = −RZ−Mom (5)

where RX−Mom, RY−Mom and RZ−Mom are the residuals of the momentum equations in the x, y and zdirections respectively. The pressure is still updated using the normal momentum equation

∂(δP )∂n

= −~n ·(∇ · (ρ~q ⊗ ~q) +∇P − 1

Re

[∇2~q +

13∇ (∇ · ~q)

])(6)

The boundary condition for ~q∗ at a solid surface is simply

u∗ = −∂φ

∂x, v∗ = −∂φ

∂y, w∗ = −∂φ

∂z(7)

Since we use ~n · ∇φ = 0, hence

~n · ~q∗ = 0 , ~t · ~q∗ = −~t · ∇φ (8)

With this more flexible approach, simulations can be performed on general grids, not necessarily struc-tured, and there is no need to identify the main flow direction. This strategy has been tested for two andthree dimensional flows and the results are compared to those obtained by the first method. Accuracyand convergence of the two calculations are comparable. The extra degree of freedom introduced by thedecomposition is treated via repeated use of the momentum equations.

The components of ~q∗ vanish outside the rotational flow regions. The outer boundary condition for thepotential flow calculations depends on the free stream Mach number. For supersonic two dimensional flow,a non reflecting boundary condition, using Riemann invariants along the characteristic directions, is used toprovide a relationship between φx and φy at the boundary points. For subsonic two dimensional flows, auniform flow plus an irrotational vortex is enforced there, where the strength of the vortex, Γ, is related tothe jump in φ across a cut extending from the trailing edge all the way to the outer boundary. Notice thatΓ is related to the total lift for both inviscid and viscous flow simulations, where the contour of the outerboundary is chosen to be normal to the wake.

In this regard, one should mention the work of Thomas and Salas7 and Pulliam and Steger8 for Eulercalculations where they showed that the domain of integration can be reduced dramatically if a vortex isaccounted for in the far field boundary condition. On the other hand, if only uniform flow is used in theEuler calculations, the far field boundary must be hundred of chords away from the airfoil to obtain accuratesurface pressure distributions. The present work extends the idea of Ref. 7 and 8 by solving the potentialequation in the far field and restricting the rotational inviscid and viscous flow calculations to limited regions.

Extension of the work in Ref. 7 and 8 to three dimensional flows should be useful, contrary to theargument that the behavior of the three dimensional flow in the far field approaches uniform flow fasterthan the two dimensional one. Indeed, very far from the wing, hundred of spans, the flow is uniform. Tocut the domain of integration to several chords away from the wing, the two dimensional vortex, stripwise,along the wing is needed. In Ref. 2, an argument based on Prandtl lifting line theory (Ref. 3-5, 10-12) isdiscussed. Considering an elliptic wing of similar cross-sections, the three dimensional problem is reducedto a two dimensional one because in this case, the downwash is uniform along the span of the wing. Thefar field solution can be obtained analyzing a potential flow over a horseshoe. Asymptotic expansion of this

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solution provides a two dimensional vortex behavior with a corrected Γ, reflecting the effect of downwash.With this boundary condition, the outer boundary can be placed several chords from the wing. If a uniformflow is used instead, the outer boundary should be many spans away from the wing.

The lifting line theory, modified with compressibility effects, is useful for high aspect ration wings. Forlow aspect ratio, one can use a far field solution based on slender body theory, see Klunker.13

III. Numerical Methods

The same numerical methods used in Ref. 1 and 2 in the inviscid flow regions are implemented inthe present work. In the viscous flow region, the components of ~q∗ are governed by convection/diffusionequations and are solved in a segregated manner. An artificial time dependent term is added to eachequation. Upwind differences are used for the convection terms. The artificial viscosity introduced via thisdiscretization affects the accuracy of the calculations but it is necessary for numerical stability, particularlyfor high Reynolds number flows. Line relaxation, marching with the main flow direction, is employedto calculate the corrections. A tridiagonal solver is needed to solve for the unknowns along a grid lineas normal to the flow direction as possible. The pressure calculation is the same as in Ref. 1 and 2.Convergence acceleration techniques based on multigrid, restricted to the augmented potential equation,proved to be useful as demonstrated before in Ref. 9. Finally, the incompressible flow limit (M∞ = 0) isrecovered without problems.

IV. Numerical Results

A. Two Dimensional Flows

Viscous flows over NACA0012 airfoil are considered. A (281× 81) structured C-grid is generated around theairfoil using algebraic methods, see figure 1. A supersonic viscous flow case at M∞ = 1.5, Re = 1× 104 andα = 0o is simulated first. The Mach and pressure contours obtained using the second approach are given infigures 3-6. Both approaches of vorticity evaluation inside the viscous layer give similar surface pressure andskin friction distributions as shown in figures 7 and 8.

A subsonic flow case at (M∞ = 0.5, Re = 1 × 104 and α = 0o) is considered next where the Mach andpressure contours are given in figures 9-11. The comparison of the surface pressure distributions given infigure 12 further confirms that both approaches of vorticity evaluation yield similar results.

Next, transonic viscous flow at M∞ = 0.8, Re = 500 and α = 0o is simulated. The Mach and pressurecontours are given in figures 13-16. Again, both approaches of vorticity evaluation result in similar surfacepressure and skin friction distributions as shown in figures 17 and 18.

A lifting transonic case at M∞ = 0.8, Re = 500 and an angle of attack α = 10o is also simulated. TheMach and pressure contours are given in figures 19-22. The comparison of surface pressure and skin frictiondistributions is given in figures 23 and 24.

A second more difficult transonic case, with extended shocks in the flow field, is considered at M∞ = 0.9,Re = 500 for α = 0o and α = 10o. The results of the simulations are given in figures 25-33. The applicationof multigrid for this case results in an order of magnitude reduction in works units as compared to singlegrids computations as shown in figures 34 and 35 for both potential and viscous flow simulations. Onlythree grids are used; coarse, intermediate and fine in these calculations, with bilinear interpolation for theprolongation operator and full weighting for the restriction operator.

B. Three Dimensional Flows

Three dimensional inviscid and viscous flows over a NACA0012 wing are simulated next. The NACA0012wing is derived from ONERA M6 wing where the cross sections are replaced by NACA0012 airfoils. Astructured C-H grid of (140× 40× 40) points is generated around the wing using algebraic methods, wherea C-grid is wrapped around each wing cross-section and a H-grid is used in the spanwise direction whichcollapses into a single plane after the wing tip, see figure 2. The results are compared with those of Ref. 2and those of Overflow code (Ref. 14, 15).

First, potential subsonic flows for a lifting wing at M∞ = 0.3 and α = 2o are calculated. Comparisonwith Euler codes show good agreement as expected, see figure 36. The surface pressure contours for upperand lower surfaces are plotted in figures 37 and 38.

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For M∞ = 0.84 and α = 3o, potential flow calculation has a shock wave on the upper surface. Theshock location and strength are different if entropy and vorticity effects are included. The surface pressurecontours and the surfaces pressure distributions for both calculations are shown in figures 39 to 43.

For M∞ = 0.9 and α = 1o, the potential flow solution exhibit strong shocks on both upper and lowersurfaces of the wing as shown in figures 44 and 45. The Mach contours at the symmetry plane is plottedin figures 48. On the other hand, in the viscous flow calculations, at Re = 1000, no shocks appear. Thesurface pressure contours and the Mach contours are plotted in figures 46, 47 and 49. Finally, the presentcalculations are in good agreement with results of Ref. 2 and with Overflow results based on standardNavier-Stokes equations as shown in figure 50.

V. Concluding Remarks

The hierarchical formulation of Ref. 1 and 2 has been reconsidered, where a more general approach tocalculate the rotational velocity components is presented. Two and three dimensional laminar transonicflows are simulated and the results are encouraging. In principle, the method ia applicable to the simulationof turbulent flows as well, using available turbulence models.

References

1M. Hafez and E. Wahba: Numerical Simulations of Transonic Aerodynamic Flows, AIAA paper 03-3564, 20032M. Hafez and E. Wahba: Viscous/Inviscid Interaction Procedures for Compressible Aerodynamic Flow Simulations, to

appear3L. Prandtl: Essentials of Fluid Dynamics, Blackie, London, 19524L. Prandtl and O. G. Tietjens: Fundamentals of Hydro and Aeromechanics, Dover publications, 19575L. Prandtl and O. G. Tietjens: Applied Hydro and Aeromechanics, Dover publications, 19576H. Schlichting and K. Gersten: Boundary Layer Theory, Springer, 19997J. L. Thomas and M. D. Salas: Far-Field Boundary Conditions for Transonic Lifting Solutions to the Euler Equations,

AIAA J., vol 24, pp 1074-1080, 19868T. H. Pulliam and J. L. Steger: Recent Improvements in Efficiency, Accuracy and Convergence for Implicit Approximate

Factorization Algorithms, AIAA paper 85-0360, 19859M. Hafez and E. Wahba: Multigrid Acceleration of Transonic Aerodynamic Flow Simulations based on a Hierarchical

Formulation, ICCFD3 Conference Proceedings, 200410H. Schlichting and E. Truckenbrodt: Aerodynamics of the Airplane, McGraw-Hill, 197911R. T. Jones: Wing Theory, Princeton Univ Pr, 199012H. Ashley and M. Landahl: Aerodynamics of Wings and Bodies, Dover Publications, 198513E. B. Klunker: Contribution to Methods for Calculating the Flow about Thin Lifting Wings at Transonic Speeds - Analytic

Expressions for the Far Field, NASA TMD-6530, 197114C. Tang and M. M. Hafez: Numerical simulation of steady compressible flows using a zonal formulation. Part I:Inviscid

Flows, Comp. & Fluids, pp 898-1002, Vol. 30, 200115C. Tang and M. M. Hafez: Numerical simulation of steady compressible flows using a zonal formulation. Part II:Viscous

Flows, Comp. & Fluids, pp 1003-1016, Vol. 30, 2001

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Figure 1. Structured C-Grid for NACA0012 airfoil

Figure 2. Structured C-H Grid for NACA0012 wing

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Figure 3. Mach contours

(M∞ = 1.5, Potential Flow, α = 0o)

Figure 4. Mach contours

(M∞ = 1.5, Re = 10000, α = 0o)

Figure 5. Cp contours

(M∞ = 1.5, Potential Flow, α = 0o)

Figure 6. Cp contours

(M∞ = 1.5, Re = 10000, α = 0o)

−1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

−C

p

x/c

Tangential MomentumX and Y Momentum

Figure 7. Cp on the body

(M∞ = 1.5, Re = 10000, α = 0o)

0 0.2 0.4 0.6 0.8 1−0.02

−0.01

0

0.01

0.02

0.03

0.04

Cf

x/c

Tangential MomentumX and Y Momentum

Figure 8. Cf on the body

(M∞ = 1.5, Re = 10000, α = 0o)

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Figure 9. Mach contours

(M∞ = 0.5, Potential Flow, α = 0o)

Figure 10. Mach contours

(M∞ = 0.5, Re = 10000, α = 0o)

Figure 11. Cp contours

(M∞ = 0.5, Re = 10000, α = 0o)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

−C

p

x/c

Tangential MomentumX and Y Momentum

Figure 12. Cp on the body

(M∞ = 0.5, Re = 10000, α = 0o)

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Figure 13. Mach contours

(M∞ = 0.8, Potential Flow, α = 0o)

Figure 14. Mach contours

(M∞ = 0.8, Re = 500, α = 0o)

Figure 15. Cp contours

(M∞ = 0.8, Potential Flow, α = 0o)

Figure 16. Cp contours

(M∞ = 0.8, Re = 500, α = 0o)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

−C

p

x/c

Tangential MomentumX and Y Momentum

Figure 17. Cp on the body

(M∞ = 0.8, Re = 500, α = 0o)

0 0.2 0.4 0.6 0.8 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Cf

x/c

Tangential MomentumX and Y Momentum

Figure 18. Cf on the body

(M∞ = 0.8, Re = 500, α = 0o)

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Figure 19. Mach contours

(M∞ = 0.8, Potential Flow, α = 10o)

Figure 20. Mach contours

(M∞ = 0.8, Re = 500, α = 10o)

Figure 21. Cp contours

(M∞ = 0.8, Potential Flow, α = 10o)

Figure 22. Cp contours

(M∞ = 0.8, Re = 500, α = 10o)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

−C

p

x/c

Tangential MomentumX and Y Momentum

Figure 23. Cp on the body

(M∞ = 0.8, Re = 500, α = 10o)

0 0.2 0.4 0.6 0.8 1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Cf

x/c

Upper Surface (Tangential Momentum)Lower Surface (Tangential Momentum)Upper Surface (X and Y Momentum)Lower Surface (X and Y Momentum)

Figure 24. Cf on the body

(M∞ = 0.8, Re = 500, α = 10o)

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Figure 25. Mach contours

(M∞ = 0.9, Potential Flow, α = 0o)

Figure 26. Mach contours

(M∞ = 0.9, Re = 500, α = 0o)

Figure 27. Cp contours

(M∞ = 0.9, Potential Flow, α = 0o)

Figure 28. Cp contours

(M∞ = 0.9, Re = 500, α = 0o)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

−C

p

x/c

Tangential MomentumX and Y Momentum

Figure 29. Cp on the body

(M∞ = 0.9, Re = 500, α = 0o)

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Figure 30. Mach contours

(M∞ = 0.9, Potential Flow, α = 10o)

Figure 31. Mach contours

(M∞ = 0.9, Re = 500, α = 10o)

Figure 32. Cp contours

(M∞ = 0.9, Re = 500, α = 10o)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

−C

p

x/c

X and Y MomentumTangential Momentum

Figure 33. Cp on the body

(M∞ = 0.9, Re = 500, α = 10o)

0 1 2 3 4 5

x 104

10−6

10−5

10−4

10−3

10−2

10−1

Work Units

Res

idua

l

SGMG

Figure 34. Convergence History

(M∞ = 0.9, Potential Flow, α = 10o)

0 1 2 3 4 5 6

x 104

10−6

10−5

10−4

10−3

10−2

10−1

100

Work Units

Res

idua

l

SGMG

Figure 35. Convergence History

(M∞ = 0.9, Re = 500, α = 10o)

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0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.2

−C

p

x/c

Present MethodEuler

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.44

−C

p

x/c

Present MethodEuler

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.65

−C

p

x/c

Present MethodEuler

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.8

−C

p

x/c

Present MethodEuler

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.9

−C

p

x/c

Present MethodEuler

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.95

−C

p

x/c

Present MethodEuler

Figure 36. Surface pressure distributions for inviscid flow over NACA0012 wing

(M∞ = 0.3, α = 2o)

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Figure 37. Pressure contours on

upper surface of NACA0012 wing(M∞ = 0.3,α = 2o)

Figure 38. Pressure contours on

lower surface of NACA0012 wing (M∞ = 0.3,α = 2o)

Figure 39. Pressure contours on

upper surface of NACA0012 wing (M∞ = 0.84,α = 3o, Potential Flow)

Figure 40. Pressure contours on

lower surface of NACA0012 wing (M∞ = 0.84,α = 3o, Potential Flow)

Figure 41. Pressure contours on

upper surface of NACA0012 wing (M∞ = 0.84,α = 3o, Inviscid Rotational Flow)

Figure 42. Pressure contours on

lower surface of NACA0012 wing (M∞ = 0.84,α = 3o, Inviscid Rotational Flow)

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0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.2

−C

p

x/c

Potential FlowRotational FlowEuler

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.44

−C

p

x/c

Potential FlowRotational FlowEuler

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.65

−C

p

x/c

Potential FlowRotational FlowEuler

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.8

−C

p

x/c

Potential FlowRotational FlowEuler

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.9

−C

p

x/c

Potential FlowRotational FlowEuler

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5y/b=0.95

−C

p

x/c

Potential FlowRotational FlowEuler

Figure 43. Surface pressure distributions for inviscid flow over NACA0012 wing

(M∞ = 0.84, α = 3o)

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Figure 44. Pressure contours on

upper surface of NACA0012 wing(M∞ = 0.9, α = 1o, Potential Flow)

Figure 45. Pressure contours on

lower surface of NACA0012 wing(M∞ = 0.9, α = 1o, Potential Flow)

Figure 46. Pressure contours on

upper surface of NACA0012 wing(M∞ = 0.9, Re=1000, α = 1o)

Figure 47. Pressure contours on

lower surface of NACA0012 wing(M∞ = 0.9, Re=1000, α = 1o)

Figure 48. Mach contours at

symmetry plane of NACA0012 wing(M∞ = 0.9, α = 1o, Potential Flow)

Figure 49. Mach contours at

symmetry plane of NACA0012 wing(M∞ = 0.9, Re=1000, α = 1o)

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0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.2

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.44

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.65

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.8

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.9

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1y/b=0.95

−C

p

x/c

Present Method (Body−fitted)Present Method (Cartesian)Navier−Stokes

Figure 50. Surface pressure distributions for viscous flow over NACA0012 wing

(M∞ = 0.9, Re=1000, α = 1o)

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