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An efficient solver for the RANS equations and a one-equation turbulence model

R.C. Swanson ,1, C.-C. Rossow

DLR, Deutsches Zentrum fr Luft- und Raumfahrt, Lilienthalplatz 7, D-38108 Braunschweig, Germany

a r t i c l e i n f o

Article history:

Received 23 October 2009

Received in revised form 13 July 2010Accepted 19 October 2010

Available online 28 October 2010

Keywords:

Runge-Kutta

Implicit preconditioner

Multigrid

Reynolds-averaged NavierStokes equations

SA turbulence model

a b s t r a c t

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to

be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin

and Lomax, this scheme has been used to solve the compressible Reynolds-averaged NavierStokes(RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the

RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart

and Allmaras. With the present scheme the RANS equations and the partial differential equation of the

turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior

of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is

used to approximatethe inverse of theimplicit operator of the RANS solver. To solve theturbulence equa-

tion 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. Compu-

tational results are presented for airfoil flows, and comparisons are made with experimental data. We

demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence

modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Reliable and sufficient convergence for steady-state computa-

tions of turbulent flows continues to be a challenge in computa-

tional fluid dynamics. Here sufficient convergence means that the

residuals of the fluid dynamic equations and the equation set of

a turbulence model are reduced to the level of the truncation error

of the numerical scheme. In many applications a turbulence model

has one or more partial differential equations (PDEs) which have a

transport form and represent the effects of turbulence on the flow.

When solving the transport-type equations of turbulence models,

either directly or indirectly coupled to the flow equations, the

residuals are frequently reduced only two orders of magnitude.

In addition, the poor convergence of these transport-type equa-

tions adversely affects the convergence of the flow equations. Of

course, when adequate convergence is not achieved, there is no

assurance that the results obtained represent an acceptable

approximation of the solution even from an engineering perspec-

tive. Thus, there is a strong need for improved numerical methods

for not only obtaining steady-state solutions but also unsteady

solutions when using a dual time-stepping scheme.

When developing an improved numerical method for solving

the Reynolds-averaged NavierStokes (RANS) equations, a neces-

sary consideration is the coupling of the RANS equations and the

equation or equations of the turbulence model being applied. If

both the fluid dynamic and turbulence equations are directly cou-

pled, then the characterization of the discrete system can change.

That is, with appropriate discretization the fluid dynamic equa-

tions are positive definite (sometimes called a vector positive sys-

tem [1]), making them amenable to relaxation, but the directly

coupled system may not be, due to the equation set for the turbu-

lence model [2]. The numerical stiffness of the entire system is also

much higher due to the source terms of the turbulence model. An

alternative is to use indirect coupling of the two equation sets.

Generally, in an iterative solution process with this approach the

flow variables are updated while the turbulence variables are fro-

zen; and then, the turbulence variables are updated while the flow

variables are treated as fixed quantities. Strategies for implement-

ing indirect coupling depend on the algorithm being used. For

example, when applying multigrid methods, there are two princi-

pal strategies for indirect coupling. The first approach [3,4] is to

solve the mean flow and turbulence equations in sequence on each

grid level. This can augment the coupling effects, which may be

beneficial for certain types of problems. The second strategy [5,6]

is to use the eddy viscosity determined on the finest grid on all

coarser grids in the multigrid procedure for the mean flow

equations. Then, the turbulence model equation set is solved with

multigrid or another algorithm. This method can provide an

0045-7930/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2010.10.010

Corresponding author.

E-mail addresses: [email protected](R.C. Swanson), [email protected]

(C.-C. Rossow).1 Corresponding author was visiting scientist at the Center for Computer Applica-

tions in AeroSpace Science and Engineering (C2A2S2E), DLR, Braunschweig, Germany.

Computers & Fluids 42 (2011) 1325

Contents lists available at ScienceDirect

Computers & Fluids

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advantage in flexibility for independent evaluations of the schemes

for solving the two equation sets. In addition, since there is latitude

in choosing the iterative scheme and update procedure for the tur-

bulence quantities, one can also investigate methods to enhance

the coupling effects.

By indirectly coupling the equations one can focus on the

specific properties of each equation set to obtain the best possible

convergence of the two systems of equations. Furthermore, the

essential properties of an algorithm for efficiently solving the di-

rectly coupled system can be identified. There are common design

criteria for the algorithms of both equation sets. These require-

ments are as follows: (1) high CourantFriedrichsLewy (CFL) lim-

it, (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 treat-

ment of any source terms so that convergence is not adversely

affected.

A candidate for the flow solver of the loosely coupled system is

an RK/Implicit scheme with three stages and three evaluations of

the numerical dissipation. Subsequently, this scheme is designated

as the RK3/Implicit scheme. Previously, Rossow [7] and Swanson

et al. [8] demonstrated that fast convergence can be obtained for

both the two-dimensional (2-D) and three dimensional (3-D) RANS

equations with the RK3/Implicit scheme with multigrid when

using the Baldwin-Lomax (BL) algebraic eddy viscosity model [9].

Although there is some slowdown in the convergence rate of the

3-D scheme relative to the 2-D scheme, Swanson et al. demon-

strate that this scheme is more than 10 times faster than a well-

tuned standard RK scheme with scalar implicit residual smoothing

and multigrid. The underlying three-stage RK scheme of this algo-

rithm is important for clustering of the eigenvalues associated with

the error components of the iterative process. Preconditioning with

a fully implicit operator, which allows a CFL number of 1000, treats

the discrete stiffness problemassociated with viscous-layer resolu-

tion. The operator of the discrete implicit system can be approxi-

mately inverted with symmetric Gauss-Seidel (SGS).

The main purpose of this work was to initiate an effort to satisfythe need to significantly augment the effectiveness (as measured

by reliability and efficiency) of algorithms for solving the RANS

equations and the PDEs of turbulence models. Since establishing

a highly effective scheme in two dimensions is a prerequisite for

constructing an efficient scheme in three dimensions, the focus

of the present effort is on a 2-D scheme. Moreover, we assess the

performance of an efficient RANS solver (i.e., RK3/Implicit scheme

with multigrid) when the turbulent viscosity field is generated by

solving a transport-type equation.

To represent the effects of turbulence we use the SpalartAllm-

aras (SA) model, which is a transport-type equation model that is

frequently used in solving a variety of fluid dynamics problems.

In the first section of this paper this turbulence model is described,

and the specifics of its implementation are given. Next the numer-ical schemes for solving the mean flow and turbulence equations

are presented and discussed. Modifications of the RK3/Implicit

scheme that have produced improved efficiency and robustness

are emphasized. Then three approaches for solving the SA equation

are considered. These methods are as follows: diagonally dominant

alternating direction implicit (DDADI), symmetric line Gauss-Sei-

del (SLGS), and a two-stage RK scheme (RK2/Implicit) with implicit

preconditioning. In the results section the convergence behavior of

the methods for solving the RANS equations and the SA equation is

examined. The effectiveness of the loosely coupled algorithm at

high Reynolds numbers and low Mach number is presented. Rapid

evolution of global quantities such as lift and drag coefficients is

demonstrated. Furthermore, we show that the convergence of

the RK3/Implicit scheme with the SA model is similar to that ob-tained with the BL model.

2. SpalartAllmaras turbulence model

Here we provide a sufficient description of the SA model to al-

low implementation. A detail discussion explaining the modeling

of the physical terms in the single transport-type equation is given

in the paper by Spalart and Allmaras [10]. Let mt be the eddy viscos-ity, which is defined by

mt ~mfv1; fv1 v3

v3 C3v1

; v ~mm; 2:1

where m is the kinematic viscosity. The transport-type equation forem given in Ref. [10] is written as

@~m@t

uj@~m@xj

Cb11 ft2eS~m

1

r@

@xjm ~m

@~m@xj

! Cb2

@~m@xj

@~m@xj

& ' Cw1fw

Cb1j2

ft2

~md

2 S; 2:2

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

eS S ~mj2d2

fv2; fv2 1 v

1 vfv1; 2:3

with S being the magnitude of the vorticity (jXj), d delineating the

distance to the closest wall boundary, and j denoting the vonKrmn constant. The first, second, and third terms on the right-

hand side of Eq. (2.2) represent the production, diffusion, and

destruction terms, respectively. The last term is a source term,

which is defined by

S ft1DU2; 2:4

where DU is the norm of the difference between the velocity at thetransition location and that at a field point being considered. The

function fw in Eq. (2.2) is given by

fw g1 C6w3

g6 C6w3

1=6; 2:5

where g and r are defined by

g r Cw2r6 r; r

~meSj2d2 : 2:6For large values ofrthe functionfw goes to a constant, and a value of

10 is appropriate. The function ft2 is defined as

ft2 Ct3 expCt4v2: 2:7

Spalart includes the transition function given by

ft1 Ct1gtexp Ct2x2tDU2

d2

g2td2t

!; 2:8

where dt is the distance from the field point to the boundary-layer

trip (where trip refers to a known location for transition), xt is thewall vorticity at the trip,

gt min 0:1;DU=xtDxt : 2:9

andDxt 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. [11]). In addition, for the purposes of grouping

terms similar in form and numerical implementation, we rewrite(after some algebra and rearranging of terms), Eq. (2.2) as

14 R.C. Swanson, C.-C. Rossow / Computers & Fluids 42 (2011) 1325


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@~m@t

uj@~m@xj

SP1 ~m SP2 ~m SD~m

D~m; 2:10

where SP1 and SP2 are the two contributions to the production term,

SD is the destruction term, and D is the diffusion term, and they are

given by

SP1 ~m Cb11 ft2jXj~m;

SP2 ~m SD~m Cb11 ft2fv2 ft2j2 Cw1fw

~md

2;

D~m 1

r@

@xjm 1 Cb2~m

@~m@xj

!

Cb2r

~m@2~m@x2j

:

The constants of the model are as follows:

Cb1 0:1355; r 2

3; Cb2 0:622; j 0:41; 2:11

Cw1 Cb1j

1 Cb2

r;

Cw2 0:3; Cw3 2; Cv1 7:1; Ct1 1; Ct2 2;

Ct3 1:2; Ct4 0:5:

On a solid boundary ~m 0. Originally, the free-stream ~m was set to1.342m1, where m1 is the free-stream kinematic viscosity. In orderto avoid the possibility of a delayed transition, the free-stream va-

lue of ~m is set to 3m1, as suggested by Rumsey [12].

3. Numerical schemes

To solve the two-dimensional RANS equations we use the RK3/

Implicit scheme. Complete details of the scheme are presented in

the papers of Rossow [7] and Swanson et al. [8]. The SA turbulence

model requires the solution of one transport-type equation. In the

present work we do not directly couple the solutionof the fluid dy-

namic equations with the additional equation of the turbulencemodel. To solve the transport-type equation of the SA turbulence

model we consider the DDADI, SLGS, and RK2/Implicit schemes.

In the first part of this section the essential elements of the RK3/

Implicit scheme are presented. Recent enhancements of the origi-

nal RK3/Implicit scheme are also introduced. Then the three meth-

ods considered for solving the SA equation are described.

3.1. RK/implicit scheme

We apply a finite-volume approach to discretize the fluid

dynamic equations and use the approximate Riemann solver of

Roe [13] to obtain a second-order discretization of the convective

terms. The viscous terms are discretized with a second-order cen-

tral difference approximation. To obtain an explicit update to the

solution vector for the flow equations we use a three-stage RK

scheme. The update for the qth stage of the RK scheme is given by

Wq W0 dWq; 3:1

where the change in the solution vector W is

dWq Wq W0 aq

Dt

VLW

q1; 3:2

and L is the complete difference operator for the system of equa-

tions. Here aq is the RK coefficient of the qth stage, Dt is the timestep, and V is the volume of the mesh cell being considered. For

the three-stage scheme we use the coefficients

a1; a2; a3 0:15; 0:4; 1:0

from Ref. [14].

To extend the support of the difference scheme we consider

implicit residual smoothing. Applying the smoothing technique

of Ref. [15] we have the following:

LidWq dWq; 3:3

where Li is an implicit operator. By approximately inverting the

operator Li we obtain

dWq aqDt

VPL; Wq1 aq

Dt

VP

Xall faces

Fq1n S; 3:4

where P is a preconditioner defined by the approximate inverseeL1i , Fn is the normal flux density vector at the cell face, and S isthe area of the cell face. The change dWq replaces the explicit

update appearing in Eq. (3.1). 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 convective derivatives in the implicit operator. To

derive this operator one treats the spatial discretization terms in

the flow equations implicitly and applies linearization. For a

detailed derivation see Rossow [7]. Substituting for the implicit

operator in Eq. (3.3), we obtain for the qth stage of the RK scheme

I eDt

V

Xall faces

AnS" #

dWq aqDt

V

Xall faces

Fq1n S

bRq1; 3:5where the matrix An is the flux Jacobian associated with Fn at a cell

face, bRq1 represents the residual function for the (q1)th stage,and e is an implicit parameter, which will be defined later in thissection.

The matrix An can be decomposed into An and A

n , which are

associated with the positive and negative eigenvalues of An and

defined by

An

1

2An jAnj; A

n

1

2An jAnj: 3:6

If we substitute for An in Eq. (3.5) using the definitions of Eq. (3.6),

then the implicit scheme can be written as

I eDt

V

Xall faces

An S

" #dW

qi;j

bRq1i;j eDtV Xall faces

An dW

qNBS; 3:7

where the indices (i,j) indicate the cell of interest, and NB refers to

all the direct neighbors of the cell being considered.

To solve the implicit system of Eq. (3.7) for the changes in con-

servative variables dWq

i;j , the implicit operator must be inverted. It

is sufficient to approximate the inverse of the implicit operator.

Based on analysis and numerical testing, an adequate approximate

inverse is 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. Previously

[16], we have demonstrated that line relaxation can be used to effi-ciently approximate the inverse. By using local line relaxation, we

make the scheme amenable to application on unstructured grids.

To initialize the iterative process the unknowns are set to zero.

The choice of the implicit parameter e in Eq. (3.5) affects thehigh-frequency damping of the scheme, and thus, its effectiveness

as a smoother for multigrid. Fig. 1a shows the amplification factorg

of the original RK3/Implicit scheme [7,8] with variation ofe. The gwas determined with the one-dimensional Fourier analysis of

Swanson et al. [8] and is a function of the phase angle hx, which

is proportional to frequency. At e = 0.5 there is a significant in-crease ingat the highest frequencies, and when e = 0.4, the schemeis unstable. Thus, for the original scheme we chosee = 0.6. By intro-ducing non-standard weighting of the explicit numerical dissipa-

tion on stages of the RK scheme, the lower bound on e can bedecreased, which results in faster convergence. Here, weighting

R.C. Swanson, C.-C. Rossow / Computers & Fluids 42 (2011) 1325 15


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of the numerical dissipation means that the dissipation in the

residual function on a given RK stage q (q > 1) is weighted with that

from a previous stage. In general, there is weighting of both the

numerical dissipative and physical diffusive terms, and the opera-

tor L in Eq. (3.2) is defined by

LWq

1

VLcW

q Xqr0

cqrLvWr

Xqr0

cqrLdWr

" #; 3:8

withPcqr 1 for consistency. The operatorsLc;Lv, andLd relate to

the convective, viscous, and numerical dissipative terms. The coef-

ficients cqr are the weights of the viscous and dissipative terms oneach stage, and for the 3-stage scheme,

c00 c1; c10 1 c2; c11 c2; c20 1 c3;

c21 0; c22 c3: 3:9

When the weights c1; c2; c3 are [1,1, 1], this is called standardweighting. Based upon analysis and numerical testing we have

determined that the modified weights [1,0.5,0.5] lead to improved

robustness of the smoother. With this weighting there is a shift in

the intersection of the locus of the residual eigenvalues to the left

along the negative real axis of the complex plane, which increases

the parabolic stability limit of the basic RK scheme by more than

a factor of two. A detailed discussion of dissipation weighting is

given in Jameson [17] and Swanson and Turkel [18]. Fig. 1b shows

the effect on g due to weighting of the dissipation on the second

and third stages by a factor of 0.5. The modified scheme is now

stable when e = 0.4. However, in practice, for stability, good damp-ing of the highest frequencies, and best convergence the parameter

e is taken to be 0.5.

In the application of the RK3/Implicit scheme as the smoother ofa full approximation storage (FAS) multigrid method, the CFL num-

ber is increased to 1000 after 10 multigrid cycles. The hyperbolic

tangent function defined in

CFL 1000 tanh0:0052N1

n o; 3:10

where N is the number of cycles, is used to smoothly increase the

CFL number. A W-type cycle (see Ref. [19]) is used to execute the

multigrid. Coarse meshes are created by eliminating every other

mesh line in each coordinate direction (i.e., full coarsening). Details

of the multigrid method are given in the paper by Swanson et al. [8].

Although we have used lexicographic ordering in applying

Gauss-Seidel, alternative ordering strategies can also be used. For

example, the lexicographic ordering of the solution points can bereplaced with red-black ordering. Roberts and Swanson [20] have

applied eigensystem analysis to show that the RK/ Implicit scheme

with a Gauss-Seidel preconditioner and red-black ordering is an

effective smoother for multigrid. With a different data structure

such as that employed for unstructured grids, an alternative order-

ing can be more convenient to implement and even lead to a more

robust iterative method. Due to the larger stencils that are pro-

duced on unstructured grids when approximating the spatial

derivatives of the governing equations, the red-black ordering

must be replaced by a multicolor ordering to ensure that each

point of a particular color only directly connects to points of a dif-

ferent color, and thus, obtain a Gauss-Seidel type scheme.

By using multicoloring the algorithm can be highly parallelized

(see Refs. [21,22]), and the convergence rate is the same as for

sequential processing. We can define each color in the multicolor-

ing ordering as a member of an independent set. Then, each solu-

tion point is only directly connected to points with a different

color. First, all points of a particular color on all subdomains areupdated in parallel. Next, another set of points of a different color

is updated, using the latest information available. This procedure is

continued until all points have been updated. In updating the solu-

tion points of the initial color, Jacobi relaxation is used since no up-

dated points are available. However, this would also be true for

red-black (odd-even) Gauss-Seidel on a structured mesh. The num-

ber of colors required to build independent sets is a function of the

number of points in the stencil.

3.2. Schemes for SA equation

After discretizing Eq. (2.10), we consider the implicit form

I Lx Ly SJD~m R~m; 3:11

where Lx and Ly are the linear discrete operators for the terms of the

transport-type equation, SJ is a Jacobian of the source term contain-

ing the production and destruction of turbulence contributions, and

R~m is the residual function. The operators for the two coordinatedirections are as follows:

Lx Dt

Vh dux dxb1 b2dxdx

;

Ly Dt

Vh duy dyb1 b2dydy

h i; 3:12

where du is a first-order upwind operator for the convective term, d

is a standard central difference operator, and the coefficients b1, b2

are defined by the diffusion term of the turbulence model. Theparameter h indicates temporal accuracy. If h = 1/2, then the time

x

g

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

= 1.0

= 0.8

= 0.6

= 0.5

= 0.4

x

g

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

= 1.0

= 0.8

= 0.6

= 0.5

= 0.4

(a) (b)

Fig. 1. Effecton amplification factorof RK3/Implicit scheme (applied to 1-D Euler equations) due to variation of implicit parametere (3-stage scheme). (a) Standard weightingof numerical dissipation, (b) modified weighting of dissipation.



5/13

derivative is approximated by a central difference, which is second-

order accurate (i.e., CrankNicolson type scheme). When h = 1 the

approximation is a first-order backward difference, and we have a

fully (an Euler) implicit scheme. The parameter h may also be

viewed as a measure of implicitness withh > 1 a n d 0 < h < 1 indicat-

ing under-relaxation and over-relaxation, respectively (to see this

consider the effect of h for large Dt). The source term Jacobian

and the residual function are defined by

SJ Dt

Vh

@

@~mSP2 ~m SD~m

; R~m Dt

VR~m: 3:13

For the convective and diffusive terms of the residual function we

use first-order upwind difference and central difference approxima-

tions, respectively. A first-order approximation of convective terms

is frequently applied in the implementation of turbulence models to

promote positivity of the turbulence variables. In general, this is not

sufficient to ensure positivity, so usually there is also limiting (clip-

ping) of the turbulence quantities and/or certain terms (e.g., pro-

duction 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. (3.11) is to factor the implicit (left-hand side) oper-

ator and apply the DDADI scheme. Define the diagonal contribu-tion in Eq. (3.11) as

D I Dx Dy SJ; 3:14

where Dx and Dy are the diagonal parts of Lx and Ly, respectively.

Then, after factoring out D, we factor the resulting operator,

obtaining

I Ly Dx SJ

D1 I Lx Dy SJ

D~m R~m: 3:15

To invert this implicit operator, we solve the sequence of one-

dimensional systems corresponding to the two coordinate direc-

tions. To prevent deterioration in the allowable CFL number and

damping behavior of the DDADI scheme due to the factorization

error and possible boundary condition lagging error, we use the

subiterative procedure described by Klopfer et al. [23]. Using Fou-rier analysis and some applications of the iterative DDADI scheme

(also called the modified approximate factorization (MAF) scheme)

MacCormack and Pulliam [24] and Walsh and Pulliam [25] have

demonstrated that a few subiterations (e.g., two to four) makes

the DDADI scheme unconditionally stable and improves the damp-

ing properties. In Pulliam et al. [26] best performance for a diago-

nalized DDADI was obtained with three to six iterations, and

three iterations were recommended. Certainly, the effectiveness

and reliability of the DDADI scheme depends on the convergence

behavior of the subiterative process and on the magnitude of the

implicit parameter h. By numerical testing we have found that four

subiterations produces reliable convergence and best performance

when DDADI is used by itself or as a smoother for multigrid. Cur-

rently we use four subiterations when performing one outer itera-tion. Convergence with iteration and subiteration can be

enhanced by choosing an appropriate implicit parameter. Over a

range of mesh densities a h between 1.2 and 2.0 works well. Addi-

tional discussion of the present implementation is given in Swanson

and Rossow [16].

With the SLGS scheme the implicit operator of Eq. (3.11) is

approximately inverted in each iteration with two symmetric

Gauss-Seidel line relaxation sweeps (line solves performed in ra-

dial direction only). The RK2/Implicit scheme involves two RK

stages and an implicit preconditioner, and it is the same type of

scheme used to solve the mean flow equations. One point SGS

sweep and one local (boundary layer + near wake) symmetric line

relaxation sweep are applied twice to obtain an approximate inver-

sion of the implicit preconditioner. The coefficients for the two-stage scheme are

a1; a2 0:25; 1:0 :

Subsequently this method is designated as the RKI-SGS scheme.

Due to the strong nonlinearities of the source terms, we have

employed numerical evaluation to determine an appropriate num-

ber of relaxation sweeps for the SLGS and RKI-SGS schemes. In the

evaluation we also considered the effect of mesh density on the

number of relaxation sweeps for solving the turbulence equation.

To achieve favorable convergence rates the turbulence equationis solved on each stage of the RK/Implicit smoothing scheme for

the mean flow equations. The CFL number for all solvers of the

SA equation is 1000. When solving the mean flow equations, solu-

tion of the turbulence equation is performed on the fine mesh only,

and the eddy viscosity is frozen on the coarser meshes. For addi-

tional enhancement of efficiency and robustness when solving

Eq. (2.2) the three different 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 RK/

Implicit scheme when solving the mean flow equations.

One issue that can arise in solving the turbulence equation is

the unbounded growth of the solution. Exponential growth of ~mcan occur when there is a sufficiently large imbalance of the pro-

duction and destruction terms so as to produce an instability. Aspointed out by Allmaras [27], this is a consequence of the Jacobian

of the production and destruction terms becoming positive. Such a

behavior can become a significant problem especially when using

multistage (e.g., RK) relaxation and only updating the precondi-

tioner on the zeroth stage. In the current formulationwe do not ob-

serve this type of problem. There are several possible reasons for

this. One is the form of the weak coupling used, where the multi-

grid scheme for the turbulence equation is separate from that of

the mean flow equations. Another reason is that the preconditioner

is updated on each stage of the RK scheme. There is also the poten-

tial benefit from reducing the positive contribution to the Jacobian

of the production and destruction terms (see Eq. (3.13)). These fac-

tors have contributed to the increased reliability of the solver.

During the course of this work we have made the followingconvergence behavior observations, which are similar to the ones

reported by Walsh and Pulliam [25]. The rate of development of

the turbulence field can signficantly affect the convergence of the

flow solver. Conversely, how well the flow solver converges can

have an impact on the effectiveness of the scheme for solving the

equation set of the turbulence model. Moreover, when the RANS

and turbulence equations are being solved in a loosely coupled

manner, an essential requirement for an effective total algorithm

is that the numerical solution vector of each equation set exhibits

a similar evolution rate.

4. Computational results

Computations for turbulent, viscous flow over the RAE 2822 air-foil were performed to evaluate the convergence behavior of the

RK3/Implicit scheme when applying the SA turbulence model.

The airfoil solutions were primarily calculated with the Cases 1

and 9 flow conditions given in Table 1 from the experimental

investigation of Cook et al. [28]. In the table M1 is the free-stream

Mach number, a denotes the angle of attack, Rec represents theReynolds number based on chord length, and xtr/cis the transition

Table 1

Flow conditions for RAE 2822 airfoil.

Cases M1 a (deg.) Rec xtr/c

Case 1 0.676 1.93 5.7 106 0.11

Case 9 0.730 2.79 6.5 106 0.03



6/13

location divided by the chord length. For Case 1 the flow is

primarily subsonic with a relatively small region of supersonic

flow. For Case 9 the flow is transonic, with a shock wave occurring

on the upper surface at approximately the 55% chord location. In

addition, an incompressible (M1 = 0.001) airfoil flow calculation

was made with a = 2.79 and Rec = 6.5 106.

In solving the flow equations structured meshes with a C-type

topology were used. We primarily considered three mesh densi-

ties, with the finest having 1280 cells around the airfoil and 256

cells in the radial direction. Successively coarser grids (640 128

and 320 64) were generated with half as many cells as the next

finer grid in each coordinate direction. There is clustering of the

grids at the leading and trailing edges of the airfoil and also at

the surface in the radial direction. The finest mesh has 1024 cells

on the airfoil and a minimum normal mesh spacing of 3 106.

On the airfoil surface the maximum cell aspect ratio is 2032. The

outer boundary is located at 20 chords away from the airfoil. To

investigate the RANS solver for a range of Reynolds (Re) numbers

we used a set of meshes (adapted to the Re of the flow [29]) con-

taining 368 88 cells.

In all the applications the same boundary conditions were im-

posed for the fluid dynamic equations. On the surface the no-slip

condition was applied. At the outer boundary Riemann invariants

were used. A far-field vortex effect was included to specify the

velocity for an inflow condition at the outer boundary. A detailed

discussion of the boundary conditions is given in Ref. [18]. In the

computations two types of initial conditions were considered.

One type uses the free-stream values of the dependent variables.

The other one uses an initial solution determined by applying

full multigrid (FMG). With FMG a grid sequencing process is

used to generate an initial solution on successively finer meshes.

Multigrid is used to solve the discrete problem on each grid in

the sequence. All computations were performed on a Fujitsu

computer with an Intel core two duo CPU 6750 processor at

2.66 GHz.

When comparing the computational performance of the RK3/

Implicit scheme for different turbulence model solvers and for dif-ferent mesh densities, the computational time is included. These

computing times provide a reasonable estimate of performance

since all solvers were programmed in Fortran 77 by the same per-

son using the same coding practices. This also applies to compari-

sons that are made with a frequently used scheme for solving the

RANS equations. Furthermore, by providing a description of the

processor used, the computational times required on other com-

puters can be determined.

Fig. 2 shows convergence histories for Case 9 of the schemes for

the RANS and turbulence equations. For these three results on the

320 64 grid the SA equation was solved with DDADI, SLGS, and

RKI-SGS schemes. The L2 norm of the residual of the continuity

equation is used as a measure of convergence for the flow equa-

tions. With each scheme the residual of the mean flow equations

is reduced 13 orders of magnitude in less than 75 multigrid cycles

(for an average reduction rate of about 0.65). In fact, the residual

histories for the mean flow equations essentially coincide. This is

not surprising since for all schemes the residuals of the turbulence

equation are reduced between seven and eight orders. The DDADI

scheme requires less CPU time than the other two schemes, as seen

in Table 2. However, the residual is reduced about an order of mag-

nitude more with the RKI-SGS scheme than the DDADI scheme. The

RKI-SGS scheme has the advantage of being compatible with the

solver of the mean flow equations, allowing the possibility of con-

structing a fully coupled solver. In addition, it is amenable to appli-

cation in an unstructured flow solver, since it does not require lines

across the entire domain for the solution algorithm. For these rea-

sons we use the RKI-SGS scheme to solve the SA equation.

4.1. Essentially subsonic flow

For Case 1 we first consider the effect of the approximation

order for the convective terms of the mean flow equations on the

coarse grids in the multigrid method. Usually, only first-order

accurate spatial discretization is used for these terms. In Fig. 3

the effect on convergence behavior when using a second-order

approximation is shown. Clearly there is a significant improvement

in convergence with the second order, as the number of multigrid

cycles to reduce the residual of the flow equations 13 orders is de-

creased from 82 cycles to 65 cycles. The residual of the SA equation

(using the RKI-SGS scheme) is reduced to almost the same level

(exceeding nine orders) in both calculations. In all subsequent re-

sults for subcritical flows the second-order approximation is used.

Furthermore, to elimnate the possibility of convergence effects due

to limiting, no limiter is applied.The effect of mesh refinement on convergence for Case 1 is

shown Fig. 4. The finest mesh (1280 256) contains over

300,000 cells. Similar convergence behavior is obtained on all grids

for both the mean flow and turbulence equations. As revealed in

Table 3 the convergence rate in solving the mean flow equations

is approximately 0.6, and the CPU time is increased by about a fac-

tor of four as the number of mesh points is doubled in each coor-

dinate direction, indicating convergence without mesh

Cycles

Log(||Res||

2)

CL

0 20 40 60 80 100 1 20-14

-12

-10

-8

-6

-4

-2

0

0.2

0.4

0.6

0.8

1

DDADI

CL

SLGS

CL

RKI-SGS

CL

RK3/Implicit, SA Model, RAE 2822: Case 9

M

= 0.73, = 2.79o, Re = 6.5 x 10

6

Grid: 320 x 64

Cycles

Log(||Restur|

|2)

0 20 40 60 80 100 120-12

-10

-8

-6

-4

-2

0

2

4

DDADI

SLGS

RKI-SGS

SA Model, RAE 2822: Case 9, 320 x 64

M

= 0.73, = 2.79o, Re - 6.5 x 10

6(a) (b)

Fig. 2. Convergence histories for solvers of flow and turbulence equations (Case 9, grid: 320 64). SA equation solved with three different methods: DDADI, SLGS, RKI-SGS.(a) Flow equations, (b) SA equation.



7/13

dependency. In Table 4 a comparison is made of the computational

efficiency of the current RK3/Implicit scheme (denoted by RK3/I)

with the SA and BL turbulence models. In addition, the computa-

tional effort required by a highly tuned standard five stage RK

scheme (RK5/S) with three evaluations of numerical dissipative

and physical diffusive terms, scalar implicit residual smoothing,

and multigrid is given. The BL model was used when applying

the RK5/S algorithm. With the BL model the RK3/Implicit scheme

is about four times faster than the RK5/S scheme. Convergence his-

tories with the BL model are given in Fig. 5. Even with the addi-

tional computing time required by the SA model, the RK3/

Implicit scheme is still about two times faster than the RK5/S

scheme. The computing time with the SA model is increased by

roughly a factor of 1.6 relative to that with the BL model.

To provide an additional perspective on the efficiency of the

RK/Implicit algorithm Table 4 also includes a comparison with

the SLGS scheme when used to solve both the mean flow and SA

Table 2

Comparison for Case 9 of solution strategies for solving the turbulence equation of the

SA model (grid: 320 64).

Method CPU time (s) MG cycles

DDADI 63 69

SLGS 65 70

RKI-SGS 75 69

Cycles

L

og(||Res||

2)

CL

0 20 40 60 80 100 120-14

-12

-10

-8

-6

-4

-2

0

0.2

0.3

0.4

0.5

0.6

0.7

1st order

CL

2nd order

CL


M

= 0.676, = 1.93o, Re = 5.7 x 10

6

Grid:3 20 x 64

Cycles

Log(||Restur|

|2)

0 20 40 60 80 100 1 20-12

-10

-8

-6

-4

-2

0

2

4

1st order

2nd order

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

M

= 0.676, = 1.93o, Re - 5.7 x 10

6

Grid: 320 x 64

(a) (b)

Fig. 3. Effect on convergence of approximation order of convective terms in themean flowequations on coarse grids of themultigrid method (SAmodel, Case 1, grid density:

320 64). (a) Flow equations, (b) SA equation.

Cycles

Log(||Res||

2)

CL

0 20 40 60 80 100 120-14

-12

-10

-8

-6

-4

-2

0

0.2

0.3

0.4

0.5

0.6

0.7

320 x 64

CL

640 x 128

CL

1280 x 256

CL


M

= 0.676, = 1.93o, Re = 5.7 x 10

6

Cycles

Log(||Restur|

|2)

0 20 40 60 80 100 120-12

-10

-8

-6

-4

-2

0

2

4

320 x 64

640x 128

1280 x 256

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

M

= 0.676, = 1.93o, Re - 5.7 x 10

6(a) (b)

Fig. 4. Convergence histories for solvers of flow and turbulence equations for Case 1 on three grids. SA equation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

Table 3

Effect of mesh density on convergence of RK3/Implicit scheme (Case 1). SA model

solved with RKI-SGS scheme.

Mesh size CPU time (s) MG cycles Convergence rate

320 64 71 65 0.629

640 128 299 63 0.619

1280 256 1242 60 0.604

Table 4

Comparison of computational efficiency of RK3/Implicit scheme with that of SLGS and

tuned RK5/S schemes. Case 1 on the 320 64 grid.

Scheme Turb. model CPU time (s) MG cycles Convergence rate

RK3/I SA 71 65 0.629

SLGS SA 152 344 0.917

RK3/I BL 44 64 0.624

SLGS BL 128 351 0.918

RK5/S BL 181 1792 0.983



8/13

equations. As a multigrid smoother for the mean flow, the SLGSscheme, due to stability, required a reduced CFL of 100 and under-

relaxation on the coarse grids. When solving the SA equation, the

RK3/Implicit scheme requires less than half the computer time of

the SLGS scheme.

In Fig. 6 convergence plots with FMG, starting on a 160 32

grid, are displayed. Four grids were used on all levels of grid refine-

ment, except for the 1280 256 level, which used five grids. The

benefit of the FMG in accelerating the convergence of a global

quantity such as lift coefficient (CL) is evident. In Table 5 the com-

puted CL and drag coefficient (CD) are presented for each grid level.

The two contributions to the total drag coefficient, pressure drag

(CD)p and skin-friction drag (CD)f coefficients, are also given. The

development of these coefficients after three, five, and 10 multigrid

cycles on each level are included in the table as well. In 10 cyclesthe CL and CD are obtained to at least four significant digits. Fur-

thermore, with just three cycles on each level of the FMG the error

in these quantities is less than 0.1%.

Fig. 7 shows a comparison of the computed surface pressure

and skin-friction distributions on the 1280 256 grid with exper-

imental data. In general, there is very good agreement with the

data. The computational pressure distribution does exhibit a weak

shock on the upper surface of the airfoil in the transition region

(11% chord location). There is insufficient data in the region to ver-

ify this behavior.

So far we have presented results for grids with moderately high

aspect ratio cells. Fig. 8 shows the residual histories 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 RK3/Implicit scheme. Despite a Rey-

nolds number increase exceeding an order of magnitude, there is

only a factor of about two increase in computational effort. A com-

parison of the RK3/Implicit and RK5/S schemes (with the SA and BL

models, respectively) reveals that the RK3/Implicit method is more

than five times faster when Re = 100 106.

4.2. Incompressible flow

Since the numerical dissipation matrix of the present scheme is

written as a function of Mach number (see Refs. [7,8]), the dissipa-

tion can be scaled appropriately for low-speed flows. To demon-

strate the effectiveness of the present algorithm at a low Mach

number we consider an incompressible airfoil flow. Except for

the free-stream Mach number of M1 = 0.001, the flow conditions

are the same as for Case 9. Fig. 9 exhibits the residual histories.

Here the density residual is decreased by only nine orders of mag-

nitude to avoid round-off errors [7]. Removal of round-off errors at

lowMach number canbe achieved by introducing a gauge pressure

[30].

4.3. Transonic flow

In Fig. 10 the convergence histories on three grids is presented

for Case 9. For these results the limiter was activated, and first-

order differencing was used for coarse-grid convective terms. As

for Case 1, similar convergence behavior is obtained on all grids.

Cycles

Log(||Res||

2)

CL

0 20 40 60 80 100 1 20-14

-12

-10

-8

-6

-4

-2

0

0.2

0.3

0.4

0.5

0.6

0.7

320 x 64

CL

640 x 128

CL

RK3/Implicit, BL Model, RAE 2822: Case 1

M

= 0.676, = 1.93o, Re = 5.7 x 10

6

Fig. 5. Convergence histories with BL model for Case 1. Second-order approxima-

tion of convective terms on coarse grids.

Cycles

Log(||Res||

2)

CL

0 50 100 150 200

-14

-12

-10

-8

-6

-4

-2

0

0.2

0.3

0.4

0.5

0.6

0.7

Residual

CL


160 x 32 320 x 64 640 x 128 1280 x 256

Cycles

Log(||Restur|

|2)

0 50 100 150 200-12

-10

-8

-6

-4

-2

0

2

4RKI-SGS, SA Model, RAE 2822:Case 1

160 x 32 320 x 64 640 x 128 1280 x 256

(a) (b)

Fig. 6. Convergence histories with FMG for solvers of flow and turbulence equations (Case 1). SA equation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

Table 5

Effect of mesh density on computed lift and drag coefficients for Case 1.

Mesh size Cycles (FMG) CL CD (CD)p (CD)f

160 32 50 0.5840 0.010030 0.004012 0.006023

320 64 50 0.5915 0.008541 0.002669 0.005872

640 128 50 0.5903 0.008298 0.002497 0.005802

1280 256 50 0.5884 0.008261 0.002484 0.005777

640 128 3 0.5900 0.008300 0.002501 0.005798

640 128 5 0.5901 0.008303 0.002503 0.005800640 128 10 0.5903 0.008298 0.002497 0.005802

1280 256 3 0.5887 0.008254 0.002484 0.005770

1280 256 5 0.5885 0.008254 0.002480 0.005775

1280 256 10 0.5884 0.008261 0.002484 0.005777



9/13

The rate of convergence on the three grids is between 0.63 and0.65. From Table 6 we see that the increase in CPU time in going

from the 640 128 grid to the 1280 256grid is slightly greater thana factor of four, which suggests a weak dependence of convergence

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Exp.

1280 x 256

RK3/Implicit, SA Model, Case 1

x/c

Cf

0 0.2 0.4 0.6 0.8 1-0.002

0

0.002

0.004

0.006

0.008

Exp.

1280 x 256

RK3/Implicit, SA Model, Case 1(a) (b)

Fig. 7. Comparison of computed surface pressures and skin friction with experimental data (Case 1, grid: 1280 256). (a) Surface pressures, (b) surface skin friction.

Cycles

Log(||Res||

2)

0 20 40 60 80 100 1 20-14

-12

-10

-8

-6

-4

-2

0RK3/Implicit, SA Model, RAE 2822: Case 1

M

= 0.676, = 1.93 o, Re = 5.7 x 10 6

5 .7 m 2 0 m 5 7 m 1 00 m

Grid:368 x 88

Cycles

Log(||Restur|

|2)

0 20 40 60 80 100 1 20-12

-10

-8

-6

-4

-2

0

2

4

5.7 x 106

20 x 106

57 x 106

100 x 106

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

M = 0.676, = 1.93o, Re = 5.7 x 106(a) (b)

Fig. 8. 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.

Cycles

Log(||Res||

2)

CL

0 20 40 60 80-10

-8

-6

-4

-2

0

0.2

0.3

0.4

0.5

0.6

0.7

320 x 64

CL

640 x 128C

L

RK3/Implicit, SA Model, Incomp.

M

= 0.001, = 2.79o, Re = 6.5 x 10

6

Cycles

Log(||

Restur|

|2)

0 20 40 60 80-10

-8

-6

-4

-2

0

2

4

320 x 64

640 x 128

RKI-SGS, SA Model, Incomp.

M

= 0.001, = 2.79o, Re - 6.5 x 10

6(a) (b)

Fig. 9. Convergence histories for solvers of flow and turbulence equations for incompressible case on two grids. SA equation solved with RKI-SGS. (a) Flow equations, (b) SA

equation.



10/13

on mesh density. The computational efficiency of the RK3/Implicit

scheme with both the SA and BL models is given in Table 7.

For this case the computer time of the RK3/Implicit scheme is

approximately a factor of 3.5 smaller than that of the SLGS scheme.

Again, the RK3/Implicit scheme with the SA model is about two

times faster than the standard scheme RK5/S with the BL model.

The convergence of the RK3/Implicit scheme with the BL model

is similar to that obtained with the SA model, as revealed in the

convergence plots of Fig. 11.

The convergence behavior with FMG for Case 9 is displayed in

Fig. 12. As in Case 1, we observe a rapid evolution of the CL. Table

8 gives the CL and CD after three, five, 10, and 50 multigrid cycles.

Even for this transonic case these coefficients are obtained to four

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 64

CL

640 x 128

CL

1280 x 256

CL


M

= 0.730, = 2.79o, Re = 6.5 x 10

6

Cycles

Log(||Restur|

|2)

0 20 40 60 80 100 120-12

-10

-8

-6

-4

-2

0

2

4

320 x 64

640 x 128

1280 x 256

RKI-SGS, SA Model RAE 2822: Case 9

M

= 0.730, = 2.79o, Re - 6.5 x 10

6(a) (b)

Fig. 10. Convergence histories for solvers of flow and turbulence equations for Case 9 on three grids. SA equation solved with RKI-SGS. (a) Flow equations, (b) SA equation.

Table 6

Effect of mesh density on convergence of RK3/Implicit scheme (Case 9). SA model

solved with RKI-SGS scheme.

Mesh size CPU time (s) MG cycles Convergence rate

320 64 75 69 0.648

640 128 308 64 0.626

1280 256 1307 66 0.632

Table 7

Comparison of computational efficiency of RK3/Implicit scheme with that of SLGS and

tuned RK5/S schemes. Case 9 on 320 64 grid.

Scheme Turb. model CPU time (s) MG cycles Convergence rate

RK3/I SA 75 69 0.648

SLGS SA 268 632 0.954

RK3/I BL 44 62 0.616

SLGS BL 211 599 0.951

RK5/S BL 191 1891 0.984

Cycles

Log(||Res||

2)

CL

0 20 40 60 80 100 1 20-14

-12

-10

-8

-6

-4

-2

0

0.2

0.4

0.6

0.8

1

320 x 64

CL

640 x 128

CL

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

M

= 0.73, = 2.79o, Re = 6.5 x 10

6

Fig. 11. Convergence history for Case 9 using the BL model.

Cycles

Log(||Res||

2)

CL

0 50 100 150 200

-14

-12

-10

-8

-6

-4

-2

0

0.2

0.4

0.6

0.8

1

Residual

CL


160 x 32 320 x 64 640 x 128 1280 x 256

Cycles

Log(||Restur|

|2)

0 50 100 150 200-12

-10

-8

-6

-4

-2

0

2

4RKI-SGS, SA Model, RAE 2822: Case 9

160 x 32 320 x 64 640 x 128 1280 x 256

(a) (b)

Fig. 12. Convergence histories with FMG for solvers of flow and turbulence equations (Case 9). SA equation solved with RKI-SGS. (a) Flow equations, (b) SA equation.



11/13

significant digits in just 10 cycles. After five cycles the computed CLand CD, on both the 640 128 and 1280 256 grids, are obtained

to within about 0.25% of their final values. On the finest grid with

three cycles on each refinement level the coefficients have an error

of less than 0.2%. In Fig. 13 a comparison is made of the calculated

surface pressure and skin-friction variations on the finest grid at

three, five, and 10 cycles. With just three cycles on each level there

are bearly discernible differences on the upper airfoil surface and

at the shock. The distributions on the finest grid are compared with

the experimental data in Fig. 14. There is fairly good agreement

with the data.

For all the computations the local line solves of the RKI-SGS

scheme were terminated at the jl/4 location, where jl is the number

of cells in the normal direction to the airfoil. In Fig. 15 the effect of

varying the number of points in the line solves is shown. The con-

vergence is only slightly faster by doubling the number of points in

the normal direction.

5. Concluding remarks

In this work the fluid dynamic (RANS) equations and the trans-

port-type equation of the SA turbulence model have been solved in

a loosely coupled manner. The RANS equations have been solved

with a RK3/Implicit scheme (RK3/I) and multigrid. This scheme

has been enhanced by weighting the numerical dissipative and

physical diffusive terms, a smooth initial increase of the CFL num-

ber, and local line implicit relaxation. Three different methods have

been considered for solving the SA equation: diagonally dominant

alternating direction implicit (DDADI), symmetric line Gauss-Seidel

(SLGS), and a RK2/Implicit with local line solves (RKI-SGS). To

enhance efficiency and robustness of these schemes multigrid

acceleration has also been applied. For both the fluid dynamic

and turbulence equations a CFL of 1000 has been used.

Similar convergence behavior has been observed for the three

schemes evaluated for solving the turbulence model equation.

Although the computational effort required with the RKI-SGS

scheme is somewhat larger than that needed with the other two

schemes, it provides important advantages. With appropriate

ordering for Gauss-Seidel, this method can be implemented in an

unstructured grid algorithm. In addition, it allows for the possibil-

ity to solve the mean flow and turbulence equations in a fully cou-

pled manner. Thus, the present loosely coupled algorithm is based

on RK/Implicit schemes. This algorithm also has the advantage that

it can be readily incorporated into many existing codes that

employ RK smoothers for multigrid methods.

The performance of the loosely coupled algorithm (RK3/I +

RKI-SGS schemes) has been investigated by computing solutions

to subsonic and transonic airfoil flows. We have demonstrated that

there is no significant slowdown in convergence of the RK/Implicit

scheme when the SA model is used instead of the algebraic model

of Baldwin and Lomax. This is quite important since it suggests

that for at least similar 3-D problems, such as wing flows, the

performance of the 3-D scheme for the SA model will be similar

to that observed for the BL model. In addition, even with the SA

model, the loosely coupled algorithm is approximately two to five

times faster, depending on the Reynolds number, than the highly

tuned standard RK scheme (RK5/S) with the BL model. It should

be emphasized that the RK5/S scheme includes three evaluations

of the dissipative and diffusive terms, multigrid, and scalar implicit

residual smoothing. The RK/Implicit algorithm has also been com-

pared to the SLGS scheme when applied to both the mean flow and

SA equations. It is between two and 3.5 times faster than the SLGS

scheme, depending on the flow conditions.

Although the indirectly coupled algorithm uses local line solves

(in boundary layer and wake) rather than line solves extending

across the entire domain, there is no significant deterioration in

convergence. The RK/Implicit schemes applied to the mean flowand turbulence equations have exhibited a low sensitivity to dis-

crete stiffness associated with large aspect ratio mesh cells. Fur-

thermore, it has been shown that the algorithm can also

effectively solve a low-speed flow; and thus, the analytical stiffness

due to disparity in wave speeds has been removed.

By using FMG to generate the initial conditions on the solution

grid, we have observed rapid development of the aerodynamic

Table 8

Effect of mesh density on computed lift and drag coefficients for Case 9.

Mesh size Cycles (FMG) CL CD (CD)p (CD)f

160 32 50 0.7955 0.01748 0.01192 0.005563

320 64 50 0.8185 0.01678 0.01120 0.005576

640 128 50 0.8227 0.01665 0.01113 0.005510

1280 256 50 0.8238 0.01655 0.01108 0.005474

640 128 3 0.8152 0.01650 0.01103 0.005471

640 128 5 0.8213 0.01661 0.01111 0.005500640 128 10 0.8227 0.01665 0.01113 0.005510

1280 256 3 0.8222 0.01654 0.01107 0.005471

1280 256 5 0.8235 0.01655 0.01108 0.005474

1280 256 10 0.8238 0.01655 0.01108 0.005474

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

3 cycles

5 cycles

50 cycles

RK3/Implicit, SA Model, Case 9, 1280 x 256

x/c

Cf

0 0.2 0.4 0.6 0.8 1-0.002

0

0.002

0.004

0.006

0.008

3 cycles5 cycles

50 cycles

RK3/Implicit, SA Model, Case 9, 1280 x 256(a) (b)

Fig. 13. Effect of number of cycles in each level of FMG on computed surface pressures and skin friction (Case 9, grid: 1280 256). (a) Surface pressures, (b) Surface skinfriction.



12/13

coefficients for turbulent viscous flows at both subsonic and tran-

sonic speeds. Moreover, in just three cycles on each refinement le-

vel of the FMG, the lift and drag coefficents for both the subsonic

and transonic cases have an error less than 1.0%.

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x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

Exp.

1280 x 256

RK3/Implicit, SA Model, Case 9

x/c

Cf

0 0.2 0.4 0.6 0.8 1-0.002

0

0.002

0.004

0.006

0.008

Exp.

1280 x 256

RK3/Implicit, SA Model, Case 9(a) (b)

Fig. 14. Comparison of computed surface pressures and skin friction with experimental data. (Case 9, grid: 1280 256). (a) Surface pressures, (b) Surface skin friction.

Cycles

Log(||Res||

2)

CL

0 20 40 60 80 100 1 20-14

-12

-10

-8

-6

-4

-2

0

0.2

0.4

0.6

0.8

1

jl/4

CL

jl/2

CL

jl

CL


M

= 0.73, = 2.79o, Re = 6.5 x 10

6

Cycles

Log(||Restur

||2

)

0 20 40 60 80 100 1 20-12

-10

-8

-6

-4

-2

0

2

4

jl/4

jl/2

jl

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

M

= 0.73, = 2.79o, Re - 6.5 x 10

6(a) (b)

Fig. 15. Effect of number of points in line solves on convergence (Case 9, grid: 640 128). (a) Flow equations, (b) SA equation.



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