A Strongly Coupled Time-Marching Method for Solving the Navier–Stokes and k-g Turbulence Model Equations with Multigrid.pdf

8/13/2019 A Strongly Coupled Time-Marching Method for Solving the NavierStokes and k-g Turbulence Model Equations with Multigrid.pdf

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URNAL OF COMPUTATIONAL PHYSICS 128,289300 (1996)TICLE NO. 0211

A Strongly Coupled Time-Marching Method for Solving theNavierStokes and k- Turbulence Model Equations with Multig

FENGLIU ANDXIAOQINGZHENG*Department of Mechanical and Aerospace Engineering, University of California, Irvine, California 92697

Received May 1, 1995; revised March 18, 1996

to perhaps the fact that the turbulence model equare solved only to obtain the eddy viscosity and aMany researchers use a time-lagged or loosely coupled approach

solving the NavierStokes equations and two-equation turbu- convenience of simply adding separate routines tonce model equations in a time-marching method. The Navier isting NavierStokes code. Consequently, many codokes equations and the turbulence model equations are solved

a time-lagged or loosely-coupled approach in solviparately and often with different methods. In this paper a strongly

NavierStokes and two-equation turbulence modeupled method is presented for such calculations. The Naviertions [7, 8].okes equations and two-equation turbulence model equations, in

rticular, the k- equations, are considered as one single set of In a typical iteration of a loosely coupled approaongly coupled equations and solved with the same explicit time- NavierStokes equations are first solved with fixe

arching algorithm without time-lagging. A multigrid method, to-viscosity and then thek- ork- equations are solve

ther with other acceleration techniques such as local time stepsthe newly updated flow field. Different solution md implicit residual smoothing, is applied to both the Navier

okes and the turbulence model equations. Time step limits due are often used for the NavierStokes and turbulencethe source terms in the k-equations are relieved by treating the equations. To some extent, the model equations loopropriate source terms implicitly. The equations are also strongly pler than the NavierStokes equations, particularlupled in space through the use of staggered control volumes.

the convection velocities are frozen in a loosely coe method is applied to the calculation of flows through cascadesalgorithm. However, this does not appear to yield anwell as over isolated airfoils. Convergence rate is greatly im-

oved by the use of themultigrid method with the strongly coupled task for numerical solution. On the contrary, resultme-marching scheme. 1996 Academic Press, Inc. to show slow convergence or incomplete convergen

to possible reasons such as numerical stiffness formodels, not well-defined boundary conditions, tr

1. INTRODUCTION some source terms, imposed limiters onk, , or, afact that the NavierStokes and turbulence model

Most NavierStokes codes incorporate an algebraic tions are not strongly coupled in the numerical scheodel initially to demonstrate their capability of solving appears that the solution of the turbulence modelgh Reynolds number viscous flows [16]. With the devel- tions has a significant effect on the final converge

pment of efficient numerical methods and powerful com- the complete system. This is particularly true for muters, more complicated turbulence models are being that use very fine grids and integrate the model equed for better simulation of practical flows. Among the to the wall. Not well-solved model equations might

ost used turbulence models today, two-equation eddy slow down the convergence of the NavierStokesscosity models appear to be favored for the reason that tions because of the strong nonlinear interaction beey are more general than algebraic models and af- the two sets of equations. Kunz and Lakshminarayrdable with current available computer resources. had to march up to 10,000 time steps to reduce the reHowever, investigators using two-equation models seem for the NavierStokes and the k- equations by fourhave been more concerned with the solution of the of magnitude and the convergence seems to hang

avierStokes equations. Less attention is paid to the solu- residual level. In [9], Lin et al. were able to reduon method for the turbulence model equations, particu- residual by six orders of magnitude in about 6000rly their coupling with the NavierStokes equations due for 2D transonic flows by using a variant biconjugate

ent method.In this paper, an efficient multigrid algorithm is * Current address: Computational Mathematics, University of Colo-

do, Denver, CO 80217. oped to solve a k- two-equation turbulence mod


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0 LIU AND ZHENG

osed by Wilcox [10]. The NavierStokes and k-turbu-

t()

xj(uj) (/k)ij

ui

xj 2nce model equations are treated as a single set of strongly

oupled equations and solved with the same multistageplicit time-stepping scheme. With proper construction of

xj( T)

xj;e residuals and suitable implicit treatment of the source

rms, a multigrid method is applied to both the Navierokes and the k- equations, giving excellent convergence where tis time, xi is the position vector, ui is the operties. With the multigrid method the residuals of both averaged velocity vector, is density, p is pressur

e NavierStokes and the k- model equations can be the molecular viscosity,k is the turbulent mixing educed by 10 to 13 orders of magnitude in a few hun- is the specific dissipation rate. The total energed cycles. enthalpy are E e k uiui/2 and H h k In the following section of this paper we will first outline 2, respectively, withh e p /, ande p /( 1e basic governing equations including the k- turbulence the ratio of specific heats. The other quantities are dodel equations. The numerical method is presented in

ection 3. Section 4 shows the computational results for aw pressure turbine cascade and an airfoil. T *

k

2. GOVERNING EQUATIONSSij

1

2ui

xj

uj

xi

The Favre-averaged NavierStokes equations for a com-essible turbulent flow with a k- model by Wilcox [10]

ij 2TSij 13 ukxk ij2/ 3kij n be summarized as follows:Mass conservation,

ij 2 Sij 13

uk

xkij ij

t

xj(uj) 0; (1)

qj PrL

T

PrT h

xj,

Momentum conservation, where PrL and PrT are the laminar and turbulent P

numbers, respectively. The closure coefficients are

t(ui)

xj(ujui)

p

xi

ji

xj; (2)

, * ,

* 1, , * .

Mean energy conservation,

3. NUMERICAL METHOD

Based on our previous work [11, 12], the abo

t(E )

xj(ujH)

(3) scribed equations are discretized by using a staggere

volume scheme. This scheme strongly couples thek

xj

uiij ( *T)

k

xj qj

; NavierStokes equations and maintains a small ste

the diffusion terms. In this paper, a correction tintroduced to remove the possibility of an oddevcoupling mode that may still be present in the discretTurbulent mixing energy,of the diffusion terms. Through this correction the sfor diffusion terms becomes a compact one. A semi-lcoupled algorithm was used for integrating in tim

t(k)

xj(ujk) ij

ui

xj*k

(4) discrete finite-volume equations in our previous wo

12]. We present here a new strongly coupled approa

xj( *T) k

xj; the NavierStokes and thek- equations with mu

The basic staggered finite volume discretization oriproposed in [11] is outlined in Subsection 1. The corr

for the discretization of diffusion terms is presenSpecific dissipation rate,


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MULTIGRID FOR NAVIERSTOKES ANDk- EQUATIONS

In order to solve the k- equations one coulddefinekand at the cell centers or the cell verticeand were defined at the cell centers, one would hinterpolate the strain tensor calculated at the cell vto the cell center of so that the production terthe control volume can be evaluated. On the otherthe eddy viscosity T calculated from the k and cell centers must be translated to the cell vertices in

to calculate the turbulent stresses there. This doublaging process would broaden the final discretizationfor the coupled NavierStokes and the k-equatiothus reduce the accuracy and increase the likelihuninhibited growing modes.

Alternatively, one can definekand at the cell vand use the staggered control volume to integrk-equations. The discretization is done in a similaion as that for the NavierStokes equations, but

FIG. 1. Staggered Control Volumes for a Mixed Cell-center and Cell- reverse order of using the original and the auxiliarrtex Scheme. Since the variablesk and are defined at the cell v

marked by the crosses, we will no longer need the exaveraging steps for the strain tensor and the eddy visThe production terms are evaluated at exactly the

ubsection 2. Subsection 3 describes the strongly coupledlocations, namely the cell vertices, where the stre

ultigrid time-marching algorithm for solving the Navierstrain tensors are calculated, and the eddy viscosity

okes and the k- equations.lated from the k and at these cell vertices are dused to calculate the turbulent stress tensors. In th

1. Staggered Finite Volume Schemethe NavierStokes equations and the k- equations idiscrete forms are coupled as closely as possible. TThe computational domain is discretized into a number

quadrilateral cells in two dimensions or hexahedral cells cretization of each set of the equations involves a of only nine points: those for the NavierStokes equthree dimensions. Consider a computational mesh in

wo dimensions. The governing equations are applied to are shown by the circles and those for the k- eqare shown by the crosses in Fig. 1. The staggeredach of the cells in integral form. With a cell-centeredheme the flow variables , ui , and E are defined at volume approach proposed here for the conservativ

variables and the k and in the turbulence modele cell centers marked by the circles in Fig. 1. Both thenvective and diffusive fluxes in the NavierStokes equations much resembles the staggering of the pressu

velocities used in the MAC and SIMPLE types of mons have to be estimated over the four cell faces of antrol volume, for example, the cell shown in Fig. 1. [13, 14]. The velocity gradient calculated at the cell v

with the tightest possible stencil directly drives the sohe convective fluxes can be easily estimated by takinge averages of the flow variables on either side of a cell of the turbulence model equations, just as the pr

gradient calculated at a cell face directly drives the mce, yielding a five-point stencil for the total Euler fluxalance. To estimate the diffusion terms, a staggered auxil- tum equation in the MAC and SIMPLE schemes.

The scheme as presented above reduces to a cery control volume was formed by connecting the cell-nters A, B, C, D and the mid-points of the cell faces a, difference scheme for the convective terms in bo

NavierStokes equations and the k- equations. In rc, and d as shown in Fig. 1. Since the flow variables areefined at the vertices of this auxiliary cell, Gausss formula outside the boundary layer where the grid size is to

to render the physical viscosity effective, dissipationn then be applied as in a vertex scheme to calculate thelocity and temperature gradients at the center of the of fourth-order differences need to be added to eli

odd-and-even decoupling modes for the convectiveuxiliary cell, which in fact is the vertex of the original cell Once the stresses are known at the cell vertices of , and a second-difference dissipation is needed for cap

shocks. The blended second- and foruth-order diffe diffusive fluxes in the NavierStokes equations canen be easily evaluated over the cell faces by trapezoidal formulation by Jameson [15] is used for the Navier

equations. For subsonic flow the second-difference dle as in a vertex scheme. This yields a discretizationencil involving nine points with minimum spatial extent tion is turned off completely.

The staggered finite-volume approach may also bshown by the circles in Fig. 1.


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2 LIU AND ZHENG

tions shown by the circles in Fig. 1 does not comprule out the possibility of an oddeven decoupled Consider the Laplacian 2u 2u/x2 2u/y2, wthe viscous diffusion term in the x-momentum eqfor an incompressible fluid. On a uniform cartesiashown in Fig. 2, u/x are calculated by using Gaumula over the staggered finite volume around thcircles (points A and B). In order to calculate the v

flux through the cell interface for the shaded finite vcentered at point (l, m), u/x at the center of tinterface (l 1/2, m) (point E) is obtained through aing the values of u/x at the cell vertices (l , mand (l ,m ) (points A and B), resulting in a 6stencil shown in Fig. 2 by the open circles. The nubeside the circles stand for the coefficients for thosein the discretization. If we use the subscript averagthis u/x so obtained, we get

FIG. 2. Finite-volume discretization stencil for u/x.

u

xE,averaged

1

4

ul1,m1 ul,m1

xned with upwind-type schemes. For instance, schemesing second- or third-order MUSCL interpolation [16]

nd Roes approximate Riemann solver [17] are imple- 1

2

ul1,m ul,m

x

1

4

ul1,m1 ul,m1

x

ented by the present authors in [12]. The flow over anrfoil presented later in this paper is done with this upwindpe scheme for the NavierStokes equations. Notice that the right-hand side of Eq. (14) is, in In principle, the artificial dissipation formulation of weighted averaging of the finite difference formuended second- and fourth-order differences may also be u/x at (l , m 1), (l , m), and (l , med for the k-equations. However, it is noted that the This value ofu/xis then used to calculate the difand equations have very simple wave structures which flux through the cell face at (l , m), which is insist of essentially the flow convective velocities in the used to calculate the total flux balance for the cell

ree coordinate directions. Therefore, upwind schemes of If this is carried out for all the cell faces and also arious orders can be easily formed, based on the local we get a discretization stencil for 2u shown in

nvective velocity at the interface of the control volume which can be written as. For instance, if the estimated normal convective veloc-y on the interface AaB in Fig. 1 is positive, a second-der upwind interpolation formula may be used to obtaine values kand at the interface mid-point a:

(k)a [3(k)l1,m (k)l2,m] (12)

()a [3()l1,m ()l2,m]. (13)

hese values are then used to form the convective fluxesrough the interfaceAaB. This is similar to the MUSCL-pe scheme for the Euler equations used by Anderson,homas, and Van Leer [16]. In this way no explicit artificialssipation is required for the k- equations, since the

pwinding is simply a way of introducing dissipation im-icitly.

2. Improvement on the Discretization of theDiffusion Terms

As pointed out by Liu and Zheng [11], the 9-pointFIG. 3. Finite-volume discretization stencil for 2u.heme for the diffusion terms in the NavierStokes equa-


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2u)averaged(15)

(ul1,m1 ul1,m1 ul1,m1 ul1,m1) 2ul,m

h2 ,

here h x y.If we apply this to a Fourier component u eI(lxmy),

hereI 1and is the wave frequency, the Fourier

mbol of the finite-difference operation is

Z 1

h212[e(IxIy) e(IxIy)

e(Ix Iy) e(IxIy)] 2 u

2

h2[1 cos(x) cos(y)]u.

FIG. 4. Five-point compact finite-difference stencil for

his implies that the scheme is insensitive to the Fourierodeu eI(lm) corresponding to the highest frequency We here present a simple alternative, which is ana

x y , which is an oddeven decoupled mode to the approach used by Jameson and Caughey [18] iat can be easily identified in Fig. 3. The reason that this finite-volume method for the transonic potential eqheme is insensitive to this oddeven decoupled mode is We continue evaluating and storing the stress ten

ue to the averaging of the terms u/xin Eq. (14). If we cell vertices as we do in our staggered finite-volumrectly take proach, but we add a correction term to Eq. (14) to r

the compact form given by Eq. (16). This correctiocan be easily identified asu

xE

ul1,m ul,m

x , (16)

1

4

u

x

l,m1

2

u

x

l,m

u

x

l,m1

e will then obtain the usual five-point finite differenceencil for 2u 2u/x2 2u/y2 shown in Fig. 4. We 14

2

y2u

xl,m

y2.ill call it the compact form for 2u

In other words, we can write(2u)compact(17)

ul1,m ul1,m ul,m1 ul,m1 4ul,m

h2 . u

x

compact u

x

averaged

1

4

2

y2u

xl,m

y2.

he Fourier symbol of this operator is With a curvilinear grid as shown in Fig. 5, we ha

u

x

u

x

u

x . Z

2

h2[(1 cos(x)) (1 cos(y))]u

This is equivalent to using Gauss formula, providhich does not allow any oddeven decoupled modes.metric coefficient x and xare appropriately interpearly, in order to get the compact scheme (17), one mustThe problem with our staggered finite-volume appaluate ui/xjat the center of the cell interfaces in acomes from the direction averaging of the /nite-volume method. Since there are twice as many (threetives. A correction term is needed to make u/comes in 3D) cell faces as cell vertices for typical quadrilat-

al grids, directly evaluating and storing the stress tensorthe center of cell faces will require extra storage and

u

compact

u

averaged

1

4

2

2

u

l,m

2

mputational time.


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4 LIU AND ZHENG

the two cells on either side, they preserve the contiveness of the overall finite-volume scheme. Their fuis to convert our 9-point finite-volume discretizatioapproximately a compact 5-point finite difference sfor the diffusion terms so that no oddeven decomodes will occur. It must also be pointed out thcorrection terms do not change the second-order acof the scheme and there are no free parameters inv

If we consider the cell-interface in the direction,

u

x u

x

averaged

1

4 2u2l,m1/ 2 Sx

Vol2

u

y u

x

averaged

1

4

2u

2l,m1/ 2

Sy

Vol2

FIG. 5. Finite-volume discretization stencil on a curvilinear grid.and similar equations for v/xand v/y.

Similar corrections are also formed for the di

terms in the k- equations. As mentioned in Liu andNote that the u/term does not cause oddeven de- [11], our experience shows that in most cases such upling on the computational domain. Consequently, no tion terms are not needed. It seems that the added arrrection is needed. Thus we use the following to estimate dissipation for the convective terms or the intrinsic de partial derivatives at the cell surfaces in order to calcu- tion in an upwind scheme is enough to damp out the te the diffusive flux balance over a control volume, ble oddeven decoupled mode for the diffusive

However, it appears that problems may arise in regsmall shear where oscillating shearing forces may au

x u

x

averaged

1

4

2u

2l1/2,m

S x

Vol2, (21) This situation was found in our RAE airfoil test c

the wake of the airfoil right after the trailing edge, oobserve sawtooth-like distribution of the shearing

hereS xis thexcomponent of the cell surface area vector

Because of the use of the trapezoidal rule in evalVol is the average volume of the cells on either side of the diffusive fluxes, such sawtooth variations of she cell interface.

forces were undetected by the viscous diffusivitySimilarly, we get

implementing the correction terms the saw-toothtions were completely eliminated.

u

y u

y

averaged

1

4

2u

2l1/2,m

S y

Vol2 (22)

3.3. Multigrid Algorithm for the k- Equations

After discretized in space, the governing equatiov

x v

x

averaged

1

4

2v

2l1/2,m

S x

Vol2 (23)

reduced to a set of ordinary differential equations iwhich can be solved by using a hybrid multistage schproposed by Jameson, Schmidt, and Turkel [19]. Rev

y

v

y

averaged

1

4

2v

2

l1/2,m

S y

Vol

2

. (24) smoothing and multigrid acceleration can be appthe NavierStokes equations as described in JamesoMartinelli and Jameson [2], and Liu and Jameson [6Since the correction terms involve only simple differ-time integration for thek-equations needs some nces on the computational plane, minor computationalattention. The semi-discrete k- equations can bfort is required. In the computer program, the 2u/2

ten asthe above equations is calculated by central differencingthe cell centers first, then their first differences normalthe cell face, /, are calculated and appended to the

t(k) Rk(k,) 0 ffusion fluxes during the assemblage of the diffusion

rms. Notice that the correction terms are only neededevaluating the diffusive fluxes through the cell faces.

t() R(k,) 0, nce they are uniquely defined at each cell interface for


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hereRkand Rare the residuals for the k and , respec- the negative contribution of the source terms in the equations can be moved to the left-hand side of Evely:and (28) to form an implicit time-marching formulaeach stage of the multistage scheme. Thus, we hav

Rk(k,) 1

(Ck Dk) Sk (29)

[1 t][(k)n1 (k)n] t*

[(k)n1()

R(k,) 1

(C D) S . (30)

(k)n()n] Rnk

t

k and C are the discrete forms of the convective terms[1 t ][()n1 ()n] t

*

[()n1(

thek and equations, respectively, andDkand Daree corresponding diffusive terms; Sk and S are the discrete

()n()n] Rn t. rms of the source terms which can be written in a lesseronlinear form than in Eqs.(4) and (5) as

There are several ways to solve the above two nonequations. Notice that Eq. (35) is independent of (and can be solved exactly by using the root formSk tPd

2

3 u (k) *

()(k) (31)

quadratic equations. After obtaining ()n1, (k)be obtained by Eq. (34). Alternatively one may lin

Eq. (35) to avoid solving the quadratic equatiS *Pd 2

3

u

()

()2

. (32) ()n1. Yet, another method, which is used in the ccalculations, is to linearize both Eqs. (34) and (3write the solution in the following delta formhere

Pd (211

222

233)

212

213

223 ()n t

Rn

1 t( 2n)

ij 2Sij 13 ukxk ij; (k)n tRnk *(k)n ()n1 t( *n)

,

is the velocity strain rate as defined in Eq. (7).

whereThetPd and*Pd terms are the major parts of produc-on forkand and are always positive. The ( u)(k)

(k)n (k)n1 (k)nnd ( u)() terms are two minor parts for the pro-uction ofkand , which, however, may be either positive ()n ()n1 ()n.

negative. When the flow is undergoing an expansion, u 0, they dissipate kor . Conversely, when the flow In a loosely coupled approach, the NavierStokesundergoing compression, they produce k or.The (*/ tions and the k- equations would be marched i()(k) and the (/)()2 are the dissipation terms separately. When the NavierStokes equationshich are always negative and thus annihilate k and . marched in time, the values ofk,, and twould be he larger these terms, the faster kand decay, but the When thek-equations were marched, the flow vastem, however, becomes more stiff because of the larger , ui , and E would be fixed. In Liu and Zheng

egative eigenvalues. The explicit time-marching formula semi-loosely coupled approach was used. A five-stagr thekand equations within each stage of a multistage stepping scheme with three evaluations of the viscou

me-stepping scheme can be modified to treat parts of was employed for the NavierStokes equations. Te source terms implicitly so that time steps are not too equations were marched separately, one or morverely restricted due to the stiffness of the k- equations. steps, with the same five-stage scheme at the first

his in general will affect the time accuracy of a multistage and fifth stages of the five-stage time stepping for tme-stepping scheme. But since we are interested in reach- vierStokes equations when the viscous terms wereg a steady state solution, the time accuracy is of lesser ated. Multigrid and residual smoothing were appncern than obtaining a scheme with faster convergence the NavierStokes equations (see [6, 15, 20]) but steady state. If we define the k-equations. As such, it was found that the c

gence of thek-equations usually lagged behind t

vierStokes equations. Consequently, Liu and Zhe Max(0, u), (33)


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6 LIU AND ZHENG

arched the k- equation four time steps for each of the physical reasons. This is expressed in the followingtion taken from [12]:ree updates of the k- equations within each time step

r the NavierStokes equations in order to obtain goodnvergence for the overall system. ()min *Pd. Just as in the case of spatial discretization, the coupling

etween the NavierStokes equations and the two-equa- In a single grid application, this limit can be direcon turbulence model equations has significant effect on posed on at every time step. In a multigrid applie convergence of the complete system. It is anticipated direct application of the above limit appears to hin

at a strongly coupled approach would result in faster effectiveness of multigrid. To avoid this problem, thnvergence. Therefore, the NavierStokes equations and is imposed by limiting the residuals of the eqe k- equations are here marched in time simultaneously calculated on the fine grids before they are passedth the same five-stage time-stepping scheme. Neither the coarse grids as

and values, nor the flow variables in the NavierStokesquations are frozen in calculating the residuals of theavierStokes and the turbulence model equations within R* R Max0,()min ()E

t

e multistage time stepping. Residual smoothing andultigrid are applied uniformly to both the NavierStokes

()E () tR

1 t( 2), nd the turbulence model equations. In this way the gov-

ning equations (1)(5) are treated truly as a single system

coupled equations. where R* is the limited residual, ()E is the preThere are, however, two modifications. First, one may, based on the original residual R .ave the option to update the eddy viscosity by Equation

During the multigrid cycle, the residuals on a fin) only at the end of each time step although the flow

are passed down to the next coarse grid as forcingriables , ui , E, and k and are updated within each that are used to drive the update on the coarse grid

age of the time step. Second, even though the Navierk and are defined at cell vertex, their values on the

okes and thek- equations are marched simultaneously,grid are transferred directly from the corresponding

ne may still have the option to use different time stepson the fine grid. As discussed in the previous sectio

reach steady state. The time step limit for the k-equa-production termPdhas an important role on the soons can be estimated by the following equations with theofkand . As such it is only calculated on the fines

ourant number CFL also possibly different from that forto preserve its accuracy. The values calculated on th

e NavierStokes equations, grid are then passed down to the coarse grids and usource terms to drive the solution ofk and . The c

t CFL

u S (S 2( t)/, (40) tion calculated on each grid is passed back to the nex

grid by bilinear interpolation. To further accelerasolution, implicit residual smoothing is also used f

here is the cell volume of the staggered cell shown ink-equations. With the help of this technique, the

g. 1, S is the face area vector of the cell in each coordinateable time steps are increased significantly. CFL va

rection,Sis the cell face area, and the summation is overaround 7 can be used.

l three directions in a three-dimensional problem.In a very recent paper, Mohammadi and Pironneau [21]

4. BOUNDARY CONDITIONSesented an implicit treatment of the source terms of thecompressible k- equation in a multistep method in which

H-type meshes are used for cascade flows and e convection and diffusion operators are split andmeshes are used for airfoil flows. Boundary conditio

arched in time separately. They also proved that theirthe NavierStokes equations are set in the same m

mplicit treatment guarantees positivity ofk and with aas outlined in [11]. Appropriate boundary conditio

agrangian finite element method under certain condi-thek-equations must be imposed in the far field a

ons. It is likely that the implicit treatment of the sourcesolid walls. In the far field a small value of the tur

rms in the compressible k- equations proposed in thiskinetic energy is specified. In our calculations k 10

aper may also preserve positivity under certain condi-freestream values of is estimated by using the fol

ons. However, even if the numerical scheme may guaran-equation as Menter proposed [22]

e the positivity ofk and , the computation may lead tow levels of that is not physical, resulting in excessivelyrge values of eddy viscosity. In order to prevent this,

O

10

U

L

. heng and Liu [12] derived a lower limit for based on


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t solid walls k 0. The specific dissipation rate doesot have a natural boundary condition. Its asymptotic be-avior is specified as

6wy2

as the wall distancey 0. (45)

our calculations, the boundary condition of is imposed

the first point away from the wall by using the abovequation. Theoretically, is infinite at the wall. One couldmply set a large value there. However, since what weally need is to impose the asymptotic behavior specified

y Eq. (45) and the value at the wall is really not usefulcept to obtain the value of at the cell interface for theird grid point from the wall through the use of Eq. (13),e value at the wall can be set to ensure that the interpo-ted value ()a at the cell interface also satisfies the asymp-tic solution specified in Eq. (45) for a positive normalnvective velocity from the wall. Thus, the at wall is

t to be

0 19

9

6wy21

, (46)

herey1is the distance from the wall of the first grid point.In the near wall region, the equation is dissipation

ominant. As shown in Eq. (45), decays very rapidly asne moves away from the wall. During the multigrid cycle, FIG. 6. Isentropic Mach Number Distribution over a Turbinee coarse grids cannot resolve such great variation. There- at an Off-design Condition.re values at the wall and the first grid point are passed

own without updating on each grid level.On outlet boundaries where the flow velocity in the by assuming a constant total pressure equal to the up

uter normal of the boundary is positive, only the pressure total pressure. The experimental Reynolds numberspecified. All other variables are extrapolated. on exit velocity and blade chord length for this teFor airfoil flows, one-dimensional Riemann invariants is 2.9 105. Although the blade is linear, the side used to form nonreflection boundary conditions if the have a 6 divergence. Therefore, a purely two-dimen

ow is subsonic (see Jameson, Schmidt, and Turkel [19] calculation would underpredict the isentropic Machnd Jameson [15]). For supersonic flows, all the flow quan- ber on the forward part of the blade for the sam

ies are set to the free stream values at the inflow. They Mach number. To avoid that, the stream tube thie extrapolated from the interior at the outflow part of correction as described in Liu and Zheng [11] is use

oundary. computational mesh, which contains 161 49 grid and all flow conditions used in the current calculati

5. COMPUTATIONAL RESULTS the same as in [11]. The only difference is that wuse the new strongly coupled time integration

1. Cascade Flowsmultigrid and residual smoothing for both the NStokes and turbulence model equations.To demonstrate the efficiency of the multigrid algorithm

r the k-equations, we recalculate the turbine cascade A difficult test condition for this cascade is whincoming flow has a negative incidence angle of 20.3ows that we did in Ref. [11]. The cascade was tested by

odson and Dominy [2325]. At its design condition this tive to the design condition. In this case there is separation bubble on the pressure surface. Usuascade has an exit isentropic Mach number of 0.7 and an

cidence angle of 38.8. The isentropic Mach number, causes slow convergence. Figure 6 shows the iseMach number distribution over the cascade. The flatten used by the turbomachinery community, is defined

the Mach number calculated from local static pressure of the isentropic Mach number distribution on the pr


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8 LIU AND ZHENG

FIG. 7. Comparison of Convergence histories against CPU time fore semi-loosely coupled and the strongly coupled methods. FIG. 8. Convergence history for the off-design cascade c

de of the blade signifies the large separation bubble ine flow. The computed flow field as shown in Fig. 6 has nofference from that obtained by the semi-loosely coupledethod in [11]. Figure 7 shows the comparison of thenvergence history against CPU time by the loosely cou-

ed algorithm and that by the strongly coupled algorithm.he implicit treatment of the source terms and the correc-on of the diffusive operators are used in both computa-ons. It is seen that the computational time is reduced byore than half with the strongly coupled multigrid method.the calculations are continued, the residuals keep going

own continuously. As shown in Fig. 8, the residuals ofass conservation, k and equations are driven downore than 11 orders of magnitude in less than 1000ork units.At the design conditions, the flow through this cascade

oes not have the large separation on the pressure side ofe blade. Figure 9 shows the computed isentropic Mach

umber against experimental data. The flat isentropicach number distribution on the pressure surface is nonger present. Because of the absence of the large separa-

on, the convergence of the computation is even better.s shown in Fig. 10, the residuals of each equation is drivenmachine zero within 700 work units.

2. Transonic Airfoil Flow

This method is extended to compute the airfoil flows. FIG. 9. Isentropic Mach number distribution over a turbinat its design condition.he flow over the RAE airfoil is calculated with an upwind


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G. 10. Convergence history for the cascade case at design condition.

rsion of our program using a MUSCL type second-orderterpolation [16] and Roes approximate Riemann solver FIG. 11. Pressure Distribution for the RAE2822 airfoil ca7] for the convective terms (see Zheng and Liu [12]).

est data of this airfoil were reported by Cook, McDonald,nd Firmin in [26]. Computation of the test case numberin [26] is presented here. The free stream mach number

the flow is 0.725 for this case. The Reynolds number is5 106 based on chord length. The nominal experimentalngle of attack is 2.92 but is adjusted to 2.4 to accountr wall interference. This adjustment is the same as used

y Martinelli and Jameson [2]. Figure 11 shows the calcu-ted pressure coefficient compared with experimental

ata. The shock wave is captured almost exactly, showingry good promise of the k- model in the simulationtransonic airfoil flows. The experimental normal forceefficient, pitching moment coefficient around 0.25 chord,

nd the drag coefficient arecN 0.743,cm 0.095, and 0.0127, respectively. The computational results give

0.770, cm 0.098, and cd 0.0163 which includesoth wave and skin-friction drag.

Figure 12 shows the convergence history for 300 timeeps with three levels of multigrid for both the semi-osely coupled approach and the strongly coupled ap-oach. Again it is seen that the latter approach yields a

etter convergence rate. However, the overall convergencete for this case is slower, compared to the cascade flowlculations. This is due to the higher grid aspect and

retching ratios at the specified Reynolds number and therger extent of the computational domain for this case.

FIG. 12. Convergence history for the RAE2822 airfoil cashe far field boundary is 18 chord lengths away from the


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0 LIU AND ZHENG

rfoil for this calculation. Nevertheless, the convergence REFERENCESte is seen to be comparable to that shown by Martinelli

1. J. L. Steger,AIAA J. 16(7), 679 (1978).nd Jameson [2] with a multigrid code using central differ-2. L. Martinelli and A. Jameson, AIAA Paper-88-0414, Januncing, scalar artificial disspation, and the BaldwinLomax

(unpublished).gebraic turbulence model. Further increase of conver-

3. D. J. Mavriplis,AIAA J. 29(12), 2086 (1991).nce for multigrid methods may hinge on proper precon-4. R. R. Varma and D. A. Caughey, AIAA Paper 91-1571, Jutioning to relieve the stiffness due to grid stretching and

(unpublished).rge cell aspect ratios.

5. M. Giles, J. Turbomach. Trans. ASME 115(1), 110 (1993).

6. F. Liu and A. Jameson,AIAA J. 31(10), 1785(1993).6. SUMMARY

7. M. G. Turner and I. K. Jennions, ASME 92-GT-308, 1992 lished).A strongly coupled multigrid algorithm is developed for

lving the NavierStokes and Wilcoxsk-two-equation 8. R. F. Kunz and B. Lakshminarayana, AIAA J. 30(1), 13 (1rbulence model equations. The NavierStokes and the 9. H. Lin, D. Y. Yang, and C. C. Chieng, AIAA Paper 93-33

1993; 11th AIAA Computational Fluid Dynamics Conferencrbulence model equations are treated as a single system10. D. C. Wilcox,AIAA J. 26(11), 1299 (1988).equations and marched in time by the same multistage

heme with multigrid. Source terms in the turbulence 11. F. Liu and X. Zheng, AIAA J. 32(8), 1589 (1994).quations are treated implicitly within each stage of the 12. X. Zheng and F. Liu,AIAA J. 33(6), 991 (1995).ultistage time stepping. Results for a turbine cascade 13. F. H. Harlow and J. E. Welch,Phys. Fluids 8,2182 (1965).ows that the method greatly increases the computational 14. S. V. Patanka, Numerical Heat Transfer and Fluid Flow (Hemficiency compared to a semi-loosely coupled algorithm. Washington, DC 1980).esiduals of both the NavierStokes equations and thek- 15. A. Jameson, MAE Report 1651, Department of Mechan

Aerospace Engineering, Princeton University, 1983 (unpubequations can be reduced to machine zero in less than00 work units. 16. W. K. Anderson, J. L. Thomas, and B. Van Leer, AIAA

1453 (1986).A correction method for removing potential oddeven17. P. L. Roe, J. Comput. Phys. 43(7), 357 (1981).ecoupled modes in the finite-volume discretization of dif-

sion terms is also presented. This method does not re- 18. A. Jameson and D. A. Caughey, AIAA Paper 77-635, 1977lished).uire storing and calculating the stress tensor for every

19. A. Jameson, W. Schmidt, and E. Turkel, AIAA Paper ll face. Thus, it needs less memory and computer time.1981 (unpublished).et, it can convert a 9-point finite-volume discretization

20. F. Liu, Ph.D. dissertation, Department of Mechanical and Aeencil into approximately a compact 5-point finite differ-

Engineering, Princeton University, June 1991 (unpublishednce stencil. Such corrections may be needed to prevent21. B. Mohammadi and O. Pironneau,Int. J. Numer. Methods Fmputational results of oscillating shear stresses.

819 (1995).

22. F. R. Menter,AIAA J. 30(6), 1657 (1992).ACKNOWLEDGEMENT

23. H. P. Hodson and R. G. Dominy,J. Eng. Gas Turbines PoThis research was supported in part by the California Space Institute (1985).der Grant CS-35-92, the National Science Foundation under grant

24. H. P. Hodson and R. G. Dominy,J.Turbomach. 109(2), 177TS-9410800 and the Academic Senate Committee on Research at UC

25. H. P. Hodson and R. G. Dominy,J. Turbomach. 109(2), 20vine. Computer time was provided on the Convex-C240 computer by

e Office of Academic Computing at UC Irvine and on the Cray-YMP 26. M. A. McDonald,P. H. Cook, andM. C. P. Firmin, AGARD AReport No. 138, May 1979 (unpublished).the San Diego Super Computer Center.

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A Strongly Coupled Time-Marching Method for Solving the Navier–Stokes and k-g Turbulence Model Equations with Multigrid.pdf