An Efficient Flow Solver with a Transport Equationfor Modeling Turbulence

R. C. Swanson∗, C.-C. Rossow†

∗ Center of Computer Applications in AeroSpace Science and Engineering (C2A2S2E)

Lilienthalplatz 7 D-38108 Braunschweig, Germany

† DLR, Deutsches Zentrum fur Luft- und Raumfahrt

Lilienthalplatz 7 D-38108 Braunschweig, Germany

Abstract

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be aneffective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, thisscheme has been used to solve the compressible Reynolds-averaged Navier-Stokes (RANS) equations for transonic andlow-speed flows. In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulenceare represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equationsand the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approachallows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented withlocal line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the tur-bulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI),symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational re-sults are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that thetwo-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solvedwith an indirectly coupled algorithm that uses RK/implicit schemes.

I. Introduction

Reliable and sufficient convergence for steady-state computations of turbulent flows continues to be a challengein computational fluid dynamics. Here sufficient convergence means that the residuals of the fluid dynamic equationsand the equation set of a turbulence model are reduced to a level below the truncation error of the numerical scheme.In many applications a turbulence model has one or more partial differential equations (PDEs) which have a transportform and represent the effects of turbulence on the flow. When solving the transport-type equations of turbulencemodels, either directly or indirectly coupled to the flow equations, the residuals are frequently reduced only two ordersof magnitude. In addition, the poor convergence of these transport-type equations adversely effects the convergenceof the flow equations. Of course, when adequate convergence is not achieved, there is no assurance that the resultsobtained represent an acceptable approximation of the solution even from an engineering perspective. Thus, thereis a strong need for improved numerical methods for not only obtaining steady-state solutions but also unsteadysolutions when using a dual time-stepping scheme.

When developing an improved numerical method for solving the Reynolds-averaged Navier-Stokes (RANS) equa-tions, a necessary consideration is the coupling of the RANS equations and the equation or equations of the turbulencemodel being applied. If both the fluid dynamic and turbulence equations are directly coupled, then the character-ization of the discrete system can change. That is, with appropriate discretization the fluid dynamic equations arepositive definite (sometimes called a vector positive system1), making them amenable to relaxation, but the directlycoupled system may not be, due to the equation set for the turbulence model.2 The numerical stiffness of the entire

∗Formerly a Senior Research Scientist in the Computational AeroSciences Branch at NASA Langley Research Center; SeniorMember AIAA.

†Director, Institut fur Aerodynamik und Stromungstechnik; Senior Member AIAA.

19th AIAA Computational Fluid Dynamics22 - 25 June 2009, San Antonio, Texas

AIAA 2009-3665

Copyright © 2009 by DLR. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

system is also much higher due to the source terms of the turbulence model. An alternative is to use indirect couplingof the two equation sets. Generally, in an iterative solution process with this approach the flow variables are updatedwhile the turbulence variables are frozen; and then, the turbulence variables are updated while the flow variables aretreated as fixed quantities. By indirectly coupling the equations one can focus on the specific properties of each equa-tion set to obtain the best possible convergence of the two systems of equations. Furthermore, the essential propertiesof an algorithm for efficiently solving the directly coupled system can be identified. There are common design criteriafor the algorithms of both equation sets. These requirements are as follows: (1) high Courant-Friedrichs-Lewy (CFL)limit, (2) convergence with weak dependency on mesh density, (3) suitable for stiff discrete systems. In addition,for the equation set of the turbulence model there must also be appropriate treatment of any source terms so thatconvergence is not adversely affected.

A candidate for the flow solver of the loosely coupled system is the RK/implicit scheme. Previously, Rossow3 andSwanson et al.4 demonstrated that fast convergence can be obtained for both the two-dimensional (2-D) and threedimensional (3-D) RANS equations with the RK/implicit scheme and multigrid when using the Baldwin-Lomax(BL) algebraic eddy viscosity model.5 The underlying three-stage RK scheme of this algorithm is important forclustering of the eigenvalues associated with the error components of the iterative process. Preconditioning with afully implicit operator allows a CFL number of 1000. The implicit operator can be approximately inverted withsymmetric Gauss-Seidel (SGS).

The main purpose of this work was to initiate an effort to satisfy the need to significantly augment the effectiveness(as measured by reliability and efficiency) of algorithms for solving the RANS equations and the PDEs of turbulencemodels. In this effort we assess the performance of an efficient RANS solver (i.e., RK/implicit scheme) when theturbulent viscosity field is generated by solving a transport-type equation.

To represent the effects of turbulence we use the Spalart-Allmaras (SA) model, which is a transport-type equationmodel that is frequently used in solving a variety of fluid dynamics problems. In the first section of this paper thisturbulence model is described, and the specifics of its implementation are given. Then the numerical schemes forsolving the main flow and turbulence equations are presented and briefly discussed. Three approaches for solvingthe SA equation are considered. These methods are as follows: diagonally dominant alternating direction implicit(DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme (RK2) with implicit preconditioning. Inthe results section the convergence behavior of the algorithm for solving the RANS equations is examined, revealingthe influence of the three methods for solving the turbulence equation.

II. Spalart-Allmaras Turbulence Model

Here we provide a sufficient description of the SA model to allow implementation. A detail discussion explainingthe modeling of the physical terms in the single transport-type equation is given in the paper by Spalart and Allmaras.7

Let νt be the eddy viscosity, which is defined by

νt = ν fv1, fv1 =χ3

χ3 + C3v1

, χ ≡ ν

ν, (1)

where ν is the kinematic viscosity. The transport-type equation for ν given in Ref. 7 is written as

∂ν

∂t+ uj

∂ν

∂xj= Cb1(1− ft2)Sν +

1

σ

j∂ν

∂xj

»(ν + ν)

∂ν

∂xj

–+ Cb2

∂ν

∂xj

∂ν

∂xj

ff(2)

−„

Cw1fw − Cb1

κ2ft2

« „ν

d

«2

+ S,

where t is time, xj and uj are Cartesian coordinates and velocity components, respectively, and

S ≡ S +ν

κ2d2fv2, fv2 = 1− χ

1 + χfv1, (3)

with S being the magnitude of the vorticity (|Ω|), d delineating the distance to the closest wall boundary, and κdenoting the von Karman constant. The first, second, and third terms on the right-hand side of Eq. 2 represent theproduction, diffusion, and destruction terms, respectively. The last term is a source term, which is defined by

S = ft1 ΔU2, (4)

where ΔU is the norm of the difference between the velocity at the transition location and that at a field point beingconsidered. The function fw in Eq. 2 is given by

fw = g

„1 + C6

w3

g6 + C6w3

«1/6

, (5)

where g and r are defined by

g = r + Cw2 (r6 − r), r ≡ ν

Sκ2d2. (6)

For large values of r the function fw goes to a constant, and a value of 10 is appropriate. The function ft2 is definedas

ft2 = Ct3 exp(−Ct4χ2) (7)

Spalart includes the transition function given by

ft1 = Ct1 gt exp

»−Ct2

ω2t

ΔU2(d2 + g2

t d2t )

–, (8)

where dt is the distance from the field point to the boundary-layer trip (where trip refers to a known location fortransition), ωt is the wall vorticity at the trip,

gt ≡ min [0.1, ΔU/(ωtΔxt)]. (9)

and Δxt is the grid spacing along the wall at the trip.In the present implementation of the model we do not include the trip function, which is usually neglected when

applying the model (e.g., Ref. 8). In addition, for convenience, after some algebra and rearranging of terms, werewrite Eq. 2 as

∂ν

∂t+ uj

∂ν

∂xj= Cb1(1− ft2)|Ω|ν +

˘Cb1[(1− ft2)fv2 + ft2]κ

−2 − Cw1fw

¯ „ν

d

«2

(10)

+1

σ

∂

∂xj

»(ν + (1 + Cb2)ν)

∂ν

∂xj

–− Cb2

σν

∂2ν

∂x2j

.

The constants of the model are as follows:

Cb1 = 0.1355, σ =2

3, Cb2 = 0.622, κ = 0.41, Cw1 =

Cb1

κ+

1 + Cb2

σ, (11)

Cw2 = 0.3, Cw3 = 2, Cv1 = 7.1, Ct1 = 1, Ct2 = 2, Ct3 = 1.2, Ct4 = 0.5.

On a solid boundary ν = 0. Originally, the free-stream ν was set to 1.342 ν∞, where ν∞ is the free-stream kinematicviscosity. In order to avoid the possibility of a delayed transition, the free-stream value of ν is set to 3 ν∞, as suggestedby Rumsey.9

III. Numerical Schemes

To solve the RANS equations we use the RK/implicit scheme. Complete details of the scheme are presented in thepapers of Rossow3 and Swanson et al.4 The SA turbulence model requires the solution of one transport-type equation.In the present work we do not directly couple the solution of the fluid dynamic equations with the additional equationof the turbulence model. To solve the transport-type equation of the SA turbulence model we use the DDADI, SLGS,and RK2/implicit schemes. In the first part of this section the essential elements of the RK/implicit scheme arepresented. Then the three methods for solving the SA equation are described.

A. RK/Implicit Scheme

We apply a finite-volume approach to discretize the fluid dynamic equations and use the approximate Riemann solverof Roe10 to obtain a second-order discretization of the advection terms. The viscous terms are discretized witha second-order central difference approximation. To obtain an explicit update to the solution vector for the flowequations we use a three-stage RK scheme. The update for the q-th stage of a RK scheme is given by

W(q) = W(0) + δW(q), (12)

where the change in the solution vector W is

δW(q) = W(q) −W(0) = −αqΔt

V LW(q−1), (13)

θx

g

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

ε = 1.0ε = 0.8ε = 0.6ε = 0.5ε = 0.4

Figure 1: Effect on amplification factor of RK/implicit scheme (applied to 1-D Euler equations) due tovariation of implicit parameter ε (3-stage scheme).

and L is the complete difference operator for the system of equations. Here αq is the RK coefficient of the q-th stage,Δt is the time step, and V is the volume of the mesh cell being considered. To extend the support of the differencescheme we consider implicit residual smoothing. Applying the smoothing technique of Ref. 11 we have the following:

Li δW(q)

= δW(q), (14)

where Li is an implicit operator. By approximately inverting the operator Li we obtain

δW(q)

= −αqΔt

V P LW(q−1) = −αqΔt

V PX

all faces

F(q−1)n S, (15)

where P is a preconditioner defined by the approximate inverse eL−1i , Fn is the normal flux density vector at the cell

face, and S is the area of the cell face. The change δW(q)

replaces the explicit update appearing in Eq. (12). Thus,each stage in the RK scheme is preconditioned by an implicit operator.

A first order upwind approximation based on the Roe scheme is used for the convection derivatives in the implicitoperator. To derive this operator one treats the spatial discretization terms in the flow equations implicitly andapplies linearization. For a detailed derivation see Rossow.3 Substituting for the implicit operator in Eq. (14), weobtain for the q-th stage of the RK scheme"

I + εΔt

VX

all faces

An S

#δW

(q)= −αq

Δt

VX

all faces

F(q−1)n S = bR(q−1), (16)

where the matrix An is the flux Jacobian associated with Fn at a cell face, bR(q−1) represents the residual functionfor the (q − 1)-th stage, and ε is an implicit parameter. Figure 1 shows the amplification factor g of the schemedetermined from the one-dimensional Fourier analysis of Swanson et al.4 For stability, good damping of the highestfrequencies, and best convergence the parameter ε is taken to be 0.5.

The matrix An can be decomposed into A+n and A−

n , which are associated with the positive and negativeeigenvalues of An and defined by

A+n =

1

2(An + |An|), A−

n =1

2(An − |An|). (17)

If we substitute for An in Eq. (16) using the definitions of Eq. (17), then the implicit scheme can be written as"I + ε

Δt

VX

all faces

A+n S

#δW

(q)i,j = bR(q−1)

i,j − εΔt

VX

all faces

A−n δW

(q)NB S, (18)

where the indices (i, j) indicate the cell of interest, and NB refers to all the direct neighbors of the cell beingconsidered.

To solve Eq. (18) for the changes in conservative variables δW(q)i,j , the 4×4 matrix on the left-hand side of Eq. (18)

must be inverted. It is sufficient to approximate the inverse of the implicit operator. An adequate approximate inverseis obtained with two pointwise symmetric Gauss-Seidel (SGS) sweeps, with each complete sweep followed by one local(boundary layer and near wake) symmetric line sweep. To initialize the iterative process the unknowns are set tozero.

Due to the upwind approximation used for the implicit operator, the coefficients for the RK scheme are also basedon an upwind scheme. For computational efficiency we apply a three-stage RK scheme with the coefficients

[α1, α2, α3] = [0.15, 0.4, 1.0]

from Ref. 12. To augment the parabolic stability limit of this scheme the numerical dissipation is weighted on thesecond and third stages by a factor of 0.5 (see Swanson and Turkel6). In the application of the three-stage RK/implicitscheme as the smoother of a full approximation storage (FAS) multigrid method, the CFL number is increased to1000 after 10 multigrid cycles. A hyperbolic tangent function is used to smoothly increase the CFL number. AW-type cycle (see Trottenberg et al.13) is used to execute the multigrid. Details of the multigrid method are givenin the paper by Swanson et al.4

B. Schemes for SA Equation

After discretizing Eq. 10, we consider the implicit form

(I + Lx + Ly + S) Δν = −R, (19)

where Lx and Ly are the linear discrete operators for the terms of the transport-type equation, S is the source termcontaining the production and destruction of turbulence contributions, and R is the residual function. The operatorsfor the two coordinate directions are as follows:

Lx =Δt

V θ [δux − (δxβ1 + β2δx)δx] , Ly =

Δt

V θˆδu

y − (δyβ1 + β2δy)δy

˜, (20)

where δu is a first-order upwind operator for the convection term, δ is a standard central difference operator, and thecoefficients β1, β2 are defined by the diffusion term of the turbulence model. The parameter θ indicates the temporalaccuracy. If θ = 1/2, then the time derivative is approximated by a central difference, which is second-order accurate(i.e., Crank-Nicolson scheme). When θ = 1 the approximation is a first-order backward difference, and we have afully (an Euler) implicit scheme. The parameter θ may also be viewed as a measure of implicitness with 0 < θ < 1and θ > 1 indicating under-relaxation and over-relaxation, respectively. The source term and the residual functionare defined by

S =Δt

V θS, R =Δt

V R. (21)

For the advection and diffusion terms of the residual function we use first-order upwind difference and central dif-ference approximations, respectively. A first-order approximation of advection terms is frequently applied in theimplementation of turbulence models to promote positivity of the turbulence variables. In general, this is not suffi-cient to ensure positivity, so usually there is also limiting (clipping) of the turbulence quantities and/or certain terms(e.g., production term) in the set of turbulence field equations.

An appropriate linearization of the source term is extremely important to allow the use of large CFL numbers.One approach for solving Eq. 19 is to factor the implicit (left-hand side) operator and apply the DDADI scheme.Define the diagonal contribution in Eq. 19 as

D = I + Dx + Dy + S, (22)

where Dx and Dy are the diagonal parts of Lx and Ly, respectively. Then, after factoring out D, we factor theresulting operator, obtaining ˆ

I + Ly + Dx + S ˜D−1 ˆ

I + Lx + Dy + S ˜Δν = −R. (23)

To invert this implicit operator, we solve the sequence of one-dimensional systems corresponding to the two coordinatedirections. To prevent deterioration in the allowable CFL number and damping behavior of the DDADI scheme dueto the factorization error and possible boundary condition lagging error, we use the subiterative procedure describedby Klopfer et al.14 Currently we use 4 subiterations when performing one iteration. Convergence with iteration andsubiteration can be enhanced by choosing an appropriate implicit parameter, just as we do for the fluid dynamicequations.

With the SLGS scheme the implicit operator of Eq. 19 is approximately inverted in each iteration with twosymmetric Gauss-Seidel line relaxation sweeps (line solves performed in radial direction only). The RK2/implicitscheme involves two RK stages and an implicit preconditioner, and it is the same type of scheme used to solve the

Table 1: Flow conditions for RAE 2822 airfoil.

Cases M∞ α (deg.) Rec xtr/c

Case 1 0.676 1.93 5.7 × 106 0.11Case 9 0.730 2.79 6.5 × 106 0.03Case 10 0.750 2.81 6.2 × 106 0.03

main flow equations. One point SGS sweep and one local (boundary layer + near wake) symmetric line relaxationsweep are applied twice to obtain an approximate inversion of the implicit preconditioner. The coefficients for thetwo-stage scheme are

[α1, α2] = [0.25, 1.0] .

Subsequently this method is designated as the RKI-SGS scheme.To achieve favorable convergence rates the turbulence equation is solved on each stage of the RK/implicit smooth-

ing scheme for the main flow equations. The CFL number for all solvers of the SA equation is 1000. When solving themain flow equations, solution of the turbulence equation is performed on the fine mesh only, and the eddy viscosity isfrozen on the coarser meshes. For additional enhancement of efficiency and robustness when solving Eq. 2 the threedifferent solution strategies, namely DDADI, SLGS, and RKI-SGS, are supported by a V-cycle multigrid algorithm.The multigrid algorithm is called at each stage of the fine mesh RK3/implicit scheme when solving the main flowequations.

During the course of this work we have observed the following convergence behavior, which is similar to theobservations reported by Walsh and Pulliam.16 The rate of development of the turbulence field can signficantlyaffect the convergence of the flow solver. Conversely, how well the flow solver converges can have an impact on theeffectiveness of the scheme for solving the equation set of the turbulence model. Moreover, when the RANS andturbulence equations are being solved in a loosely coupled manner, an essential requirement for an effective totalalgorithm is that the numerical solution vector of each equation set exhibits a similar evolution rate.

IV. Computational Results

A series of computations for turbulent, viscous flow over the RAE 2822 airfoil were performed to evaluate theconvergence behavior of the RK/implicit scheme when applying the SA turbulence model. In solving the flowequations structured meshes with a C-type topology were used. We primarily considered two mesh densities, onewith 320 cells around the airfoil (256 cells on the airfoil) and 64 cells in the normal direction, and the other with twiceas many cells in each coordinate direction. To investigate the RANS solver for a range of Reynolds (Re) numbers weused a set of meshes (adapted to the Re of the flow17) containing 368× 88 cells. The airfoil solutions were calculatedwith the Case 1, Case 9, and Case 10 flow conditions (see Table 1) from the experimental investigation of Cook,McDonald and Firmin.18 For Case 1 the flow is primarily subsonic with a relatively small region of supersonic flow.The other two cases are transonic. In Case 9 there is a shock wave occurring on the upper surface of the airfoil, butin Case 10 there is a sufficiently strong shock on the upper surface to cause flow separation behind the shock.

In all the applications the same boundary conditions were imposed for the fluid dynamic equations. On thesurface the no-slip condition was applied. At the outer boundary Riemann invariants were used. A far-field vortexeffect was included to specify the velocity for an inflow condition at the outer boundary. A detailed discussion of theboundary conditions is given in Ref. 6. The calculations were started on the solution grid with the initial solutiongiven as the free-stream conditions. All computations were performed on a Dell XPS 630 computer, which has twoquad-core processors at 2.66 GHz.

Figures 2 and 3 show convergence histories for Case 9 of the schemes for the RANS and turbulence equations.For these results the SA equation was solved either with DDADI or RKI-SGS. The L2 norm of the residual of thecontinuity equation is used as a measure of convergence for the flow equations. For both sets of results the residualof the main flow equations is reduced 13 orders of magnitude in less than 75 multigrid cycles. Moreover, as seen inFig. 4a, the convergence behavior is similar to that obtained with the BL turbulence model. With both schemes forsolving the SA equation there is a residual reduction of about 8 orders of magnitude. In addition, each method hasa similar rate of convergence on the two grids, indicating that there is only a weak effect of mesh density. We shouldemphasize that the RKI-SGS scheme has the advantage that with appropriate ordering for Gauss-Seidel it can beimplemented in an unstructured algorithm. The computational (CPU) times and convergence rates of the flow solverwhen using these schemes as well as the SLGS scheme are presented in Table 2. These results indicate that there isno deterioration in convergence when performing local line solves (RKI-SGS) rather than global line solves (SLGS).

Table 2: Comparison for Case 9 of solution strategies for solving the turbulence equation of the SA model.

Method Grid CPU Time (sec.) MG Cycles Convergence RateDDADI 320 × 64 63 70 0.651DDADI 640 × 128 264 65 0.629

SLGS 320 × 64 65 70 0.651SLGS 640 × 128 263 64 0.626

RKI-SGS 320 × 64 76 70 0.651RKI-SGS 640 × 128 318 64 0.628

Table 3: Comparison for Case 10 of solution strategies for solving the turbulence equation of the SA model.

Method Grid CPU Time (sec.) MG Cycles Convergence RateDDADI 320 × 64 79 88 0.710DDADI 640 × 128 333 82 0.693

SLGS 320 × 64 81 88 0.710SLGS 640 × 128 332 81 0.691

RKI-SGS 320 × 64 95 88 0.711RKI-SGS 640 × 128 398 81 0.691

Furthermore, when using the three different strategies for solving the turbulence equation there is almost no effecton the convergence of the main flow equations.

The next results are for Case 10, which frequently causes convergence difficulties for numerical methods. Figures 5and 6 show the convergence histories when applying the DDADI and and RKI-SGS schemes for solving the turbulenceequation. Here we observe similar convergence behavior to that for Case 9. There is some slowdown in convergence,but still the residual of the main flow equations is reduced 13 orders in less than 90 multigrid cycles. Again theconvergence behavior is similar to that obtained with the BL model (see Fig. 4b). Figures 5b and 6b indicate thatthe RKI-SGS scheme is somewhat more effective than the DDADI scheme in reducing the residual of the turbuelnceequation. This difference is partially due to the more rapid initial development of the eddy-viscosity field with theRKI-SGS approach. Table 3 gives the CPU times and convergence rates of the RANS solver when using the threemethods. In Fig. 7 a comparison of the computed surface pressure distributions and experimental data for Cases 10and 9 is displayed.

To further investigate the RKI-SGS scheme we also consider Case 1, where the flow is primarily subsonic. Forthis case the effect of further mesh refinement is presented. The finest mesh contains 1280 × 256 mesh cells (over300,000 cells). This mesh is essentially the 640× 128 grid with the mesh spacing halved in each coordinate direction.For these computations there was no limiter, eliminating the possibility of convergence effects due to the limiter.Figure 8 displays the convergence histories for the main flow and turbulence equations. Almost the same convergencebehavior is obtained on all grids. For the main flow equations, the residual is reduced 13 orders in less than 66 cycles,and the average residual reduction rate is about 0.6.

So far we have presented results for grids with moderately high aspect ratio cells. Figure 9 shows the residualhistories for Case 1 when the Re number is varied by more than an order of magnitude (from 5.7× 106 to 100× 106).The SA equation was solved with the RKI-SGS scheme. Even at a Re = 100× 106 a good convergence rate (0.751)is still obtained for the RK/implicit scheme. Despite a Reynolds number increase exceeding an order of magnitude,there is only a factor of about two increase in computational effort.

Since the numerical dissipation matrix of the present scheme is written as a function of Mach number (see

Refs. 3, 4), the dissipation can be scaled apprropriately for low-speed flows. For the final test case we consider anincompressible airfoil flow. Except for the free-stream Mach number of M∞ = 0.001, the flow conditions are the sameas for Case 9. Figure 10 exhibits the residual histories. Here the density residual is decreased by only nine orders ofmagnitude to avoid round-off errors.3

V. Concluding Remarks

In this work the fluid dynamic equations and the transport-type equation of the SA turbulence model have beensolved in a loosely coupled manner. The fluid dynamic equations have been solved with the RK/implicit schemeand multigrid. Three different methods have been considered for solving the SA equation: diagonally dominantalternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and Runge-Kutta/implicit with localline solves (RKI-SGS). To enhance efficiency and robustness of these schemes multigrid acceleration has also beenapplied. For both the fluid dynamic and turbulence equations a CFL of 1000 has been used.

We have demonstrated that there is no significant slowdown in convergence of the RK/implicit scheme with theSA turbulence model. An effective algorithm for solving the main flow and turbulence equations in a loosely coupledmanner has been presented. Furthermore, we have shown that the turbulence equation can be efficiently solved withan RK/implicit scheme (RKI-SGS) that uses local (boundary layer + near wake) line relaxation. Such a scheme withappropriate ordering for Gauss-Seidel can be implemented in an unstructured algorithm.

Similar computational times have been observed for the three schemes used in solving the turbulence modelequation. We have found that there is no deterioration in convergence when performing local line solves (in boundarylayer and near wake) rather than line solves extending across the entire domain. The RK/implicit scheme applied toboth the main flow and turbulence equations has exhibited a low sensitivity to discrete stiffness associated with largeaspect ratio mesh cells. In addition, it has been demonstrated that the scheme can effectively solve a low-speed flow.

References

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1784–1807.3C.-C. Rossow, Efficient computation of compressible and incompressible flows. J. Comput. Phys. 220 (2007) 879–899.4R. C. Swanson, E. Turkel, and C.-C. Rossow, Convergence acceleration of Runge-Kutta Schemes for solving the Navier-

Stokes equations, J. Comput. Phys. 224 (2007) 365–388.5B. S. Baldwin and H. Lomax, Thin layer approximation and algebraic model for separated flows, AIAA Paper 78-257,

1978.6R. C. Swanson and E. Turkel, Multistage schemes with multigrid for Euler and Navier-Stokes equations, NASA TP 3631,

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(1994) 5–21.8S. L. Krist, R. T. Biedron, and C. L. Rumsey, CFL3D user’s manual, NASA TM 1998-208444, June 1998.9C. L. Rumsey, Apparent transition behavior of widely-used turbulence models, AIAA Paper 2006-3906, June 2006.10P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357–372,

1981.11A. Jameson, The evolution of computational methods in aerodynamics, J. Appl. Mech., 50 (4b), 1052–1070, 1983.12B. Van Leer, C.-H. Tai, and K. G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations,

AIAA Paper 89-1933, 1989.13Trottenberg, U., Oosterlee, C. W., Schuller, A., Multigrid, Academic Press, 2001.14G. H. Klopfer, R. F. Van der Wijngaart, C. M. Hung, J. T. Onufer, A diagonalized diagonal dominant alternating direction

implicit (D3ADI) scheme and subiteration correction, AIAA Paper 98-2824, June 1998.15J. Bardina and C. K. Lombard, Three dimensional hypersonic flow simulations with the CSCM implicit upwind Navier-

Stokes Method, AIAA Paper 87-1114, June 1987.16P. C. Walsh and T. Pulliam, The effect of turbulence model solution on viscous flow problems, AIAA Paper 2001-1018,17J. Faßbender, Improved robustness of numerical simulation of turbulent flows around civil transport aircraft at flight

Reynolds numbers, Ph. D. Thesis, Technical University of Braunschweig, Braunschweig, Germany, 2004.18P. H. Cook, M. A. McDonald, M. C. P. Firmin, Aerofoil RAE 2822 pressure distributions and boundary layer and wake

measurements, AGARD-AR-138, 1979.

Cycles

Log(

||Res

|| 2)

CL

0 20 40 60 80 100 120-14

-12

-10

-8

-6

-4

-2

0

0.2

0.4

0.6

0.8

1

320 x 64CL640 x 128CL

RK3/Implicit, Case 9: RAE 2822, SA Model (DDADI)

M∞ = 0.73, α = 2.79o, Re = 6.5 x 106

(a)

Cycles

Log(

||Res

tur||

2)

0 20 40 60 80 100 120-12

-10

-8

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-2

0

2

4

320 x 64640 x 128

DDADI, Case 9: RAE 2822, SA Model

M∞ = 0.73, α = 2.79o, Re - 6.5 x 106

(b)

Figure 2: Convergence histories for solvers of flow and turbulence equations for Case 9 on two grids. SAequation solved with DDADI. (a) Flow equations, (b) SA equation.

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CL

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0

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320 x 64CL640 x 128CL

RK3/Implicit, CASE 9: RAE 2822, SA Model (RKI-SGS)

M∞ = 0.73, α = 2.79o, Re = 6.5 x 106

(a)

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320 x 64640 x 128

RKI-SGS, Case 9: RAE 2822, SA Model

M∞ = 0.73, α = 2.79o, Re - 6.5 x 106

(b)

Figure 3: Convergence histories for solvers of flow and turbulence equations for Case 9 on two grids. SAequation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

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|| 2)

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0

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320 x 64CL640 x 128CL

RK3/Implicit, CASE 9: RAE 2822, BL Model

M∞ = 0.73, α = 2.79o, Re = 6.5 x 106

(a)

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320 x 64CL640 x 128CL

RK3/Implicit, CASE 10: RAE 2822, BL Model

M∞ = 0.75, α = 2.81o, Re = 6.2 x 106

(b)

Figure 4: Convergence histories for Cases 9 and 10 using the BL model. (a) Case 9, (b) Case 10.

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320 x 64CL640 x 128CL

RK3/Implicit, CASE 10: RAE 2822, SA Model (DDADI)

M∞ = 0.75, α = 2.81o, Re = 6.2 x 106

(a)

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320 x 64640 x 128

DDADI, CASE 10: RAE 2822, SA Model

M∞ = 0.75, α = 2.81o, Re - 6.2 x 106

(b)

Figure 5: Convergence histories for solvers of flow and turbulence equations for Case 10 on two grids. SAequation solved with DDADI. (a) Flow equations, (b) SA equation.

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CL

0 20 40 60 80 100 120-14

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0

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320 x 64CL640 x 128CL

RK3/Implicit, Case 10: RAE 2822, SA Model (RKI-SGS)

M∞ = 0.75, α = 2.81o, Re = 6.2 x 106

(a)

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320 x 64640 x 128

RKI-SGS, Case 10: RAE 2822, SA Model

M∞ = 0.75, α = 2.81o, Re - 6.2 x 106

(b)

Figure 6: Convergence histories for solvers of flow and turbulence equations for Case 10 on two grids. SAequation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Exp.320 x 64640 x 128

RK3/Implicit, Case 9, SA model

(a)

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Exp.320 x 64640 x 128

RK3/Implicit, Case 10, SA model

(b)

Figure 7: Surface pressure distributions with SA model for Cases 9 and 10 on 320× 64 and 640× 128 grids.(a) Case 9, (b) Case 10.

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320 x 64CL640 x 128CL1280 x 256CL

RK3/Implicit, CASE 1: RAE 2822, SA Model (RKI-SGS)

M∞ = 0.676, α = 1.93o, Re = 5.7 x 106

(a)

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320 x 64640x 1281280 x 256

RKI-SGS, CASE 1: RAE 2822, SA Model

M∞ = 0.676, α = 1.93o, Re - 5.7 x 106

(b)

Figure 8: Convergence histories for solvers of flow and turbulence equations for Case 1 on three grids. SAequation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

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0RK3/Implicit, CASE 1: RAE 2822, SA Model (RKI-SGS)

M∞ = 0.676, α = 1.93o, Re = 5.7 x 106

5.7 m 20 m 57 m 100 m

Grid: 368 x 88

(a)

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5.7 x 106

20 x 106

57 x 106

100 x 106

RKI-SGS, CASE 1: RAE 2822, SA Model

M∞ = 0.676, α = 1.93o, Re = 5.7 x 106

(b)

Figure 9: Effect of Reynolds number variation on convergence of solvers for RANS and turbulence equations(Case 1, grid: 368× 88). RKI-SGS scheme used to solve SA equation. (a) Flow equations, (b) SA equation.

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320 x 64CL640 x 128CL

RK3/Implicit, Incomp., SA Model (RKI-SGS)

M∞ = 0.001, α = 2.79o, Re = 6.5 x 106

(a)

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320 x 64640 x 128

RKI-SGS, Incomp., SA Model

M∞ = 0.001, α = 2.79o, Re - 6.5 x 106

(b)

Figure 10: Convergence histories for solvers of flow and turbulence equations for incompressible case on twogrids. SA equation solved with RKI-SGS. (a) Flow equations, (b) SA equation.