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A DUAL-DISSIPATION SCHEMEFOR PRESSURE-VELOCITYCOUPLINGM. M. Rahman & T. SiikonenPublished online: 30 Nov 2010.

To cite this article: M. M. Rahman & T. Siikonen (2002) A DUAL-DISSIPATIONSCHEME FOR PRESSURE-VELOCITY COUPLING, Numerical Heat Transfer, Part B:Fundamentals: An International Journal of Computation and Methodology, 42:3,231-242, DOI: 10.1080/10407790260233547

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TECHNICAL NOTE

A DUAL-DISSIPATION SCHEME FORPRESSURE± VELOCITY COUPLING

M. M. Rahman and T. SiikonenHelsinki University of Technology, Department of Mechanical Engineering,

Laboratory of Applied Thermodynamics, Espoo, Finland

Within the framework of the SIMPLE algorithm, a dual-dissipation scheme is proposed onnonorthogonal collocated grids for incompressible � uid � ow problems, using a cell-centered� nite-volume approximation. The dissipative mechanism employs dynamic limiters to con-trol the amount of dissipation, preserving expediences of greater � exibility and increasedaccuracy in a way similar to the MUSCL approach. The arti� cial density concept is com-bined with the pressure Poisson equation, facilitating an avoidance of pressure under-relaxation. To account for the � ow directionality in the upwinding, a rotational matrix isevoked to evaluate the convective � ux.

1. INTRODUCTION

Considerable research is devoted to constructing the pressure-based methodattributed to the pressure±velocity coupling for incompressible ¯uid ¯ows [1±6]. Inprinciple, the physical requirement herein is to achieve a precluded decoupling of thepressure±velocity ®elds. Using the collocated grid accompanied by a linear inter-polation to evaluate the cell face velocities may result in undesirable grid-scaleoscillations, re¯ecting pressure checkerboarding. Consequently, the approach con-fronts some critical issues that provoke arti®cial damping terms or a special cell faceinterpolation strategy to eliminate the nonphysical wavy pressure=velocity ®elds.Therefore, a more rational approach is essentially to ®nd a mechanism for intro-ducing just enough cell face dissipation to extenuate the destabilizing e� ect arisingfrom the pressure±velocity decoupling.

The present solution algorithm appears with recourse to the cell-centered,nonorthogonal , fully collocated ®nite-volume method designed for incompressible¯uid ¯ow problems. The improvement herein springs principally from the modelingof interactive dissipations ascribed to the cell face pressure/velocity to suppress theodd±even decoupling. The scheme is conceived such as not to provide local extremaat the interface. The pressure Poisson equation, adhering to the SIMPLE method, is

Received 12 September 2001; accepted 14 March 2002.

Address correspondence to M. M. Rahman, Helsinki University of Technology, Laboratory of

Applied Thermodynamics, SaÈ hkoÈ miehentie 4, FIN-02015 HUT, Finland. E-mail: mizanup.rahman@hut.®

Numerical Heat Transfer, Part B, 42: 231±242, 2002

Copyright # 2002 Taylor & Francis

1040-7790 /02 $12.00 + .00

DOI: 10.1080=10407790190053923

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linked to the arti®cial compressibility that presumably is conducive to amplifying thediagonal dominance of the in¯uence coe� cients and forming conservative velocitycorrections. A cursory examination shows that this uni®cation is an expedientstrategy, since it enhances a faster convergence by avoiding any explicit pressureunderrelaxation . Finally, a comparative evaluation of the present and Rhie-Chow [3]interpolation methods is prosecuted within the same framework of a SIMPLE-basedalgorithm.

2. GOVERNING EQUATION

The unsteady convection-di � usion equation dealing with the conservation of ascalar quantity f can be written in Cartesian coordinates for a two-dimensional caseas follows:

q

qt…rf† ‡ qFx

qx‡ qFy

qyˆ Q …1†

with

Fx ˆ ruf ¡ Gfqf

qxFy ˆ rvf ¡ Gf

qf

qy…2†

where r is the density, and u and v are the Cartesian velocity components in the xand y directions, respectively. The quantities Gf and Q represent the exchange co-e� cient and the source term, respectively.

3. SPATIAL DISCRETIZATION

A cell-centered ®nite-volume scheme is applied to solve the ¯ow equationshaving an integral form

d

dt

Z

8rf d 8 ‡

Z

SF ¢ d S ˆ

Z

8Q d 8 …3†

NOMENCLATURE

C arti®cal sound speed

CFL Courant number

F; G ¯ux vectors in x and y directions_M* ®ctitious mass source

S cell face area

p static pressure

p0

pressure correction

p¤

tentative pressure

t time

T rotational matrix

u; v velocity components in x and y directions

u0; v0

velocity corrections

u¤; v¤

tentative velocity

U; V contravariant or dimensionless velocity

components

x; y Cartesian coordinatesG di� usion coe� cienty dimensionless temperaturen kinematic viscosityr densityf scalar transport variable8 cell volume

Subscripts

ref reference value

nb neighboring grid point

232 M. M. RAHMAN AND T. SIIKONEN

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for an arbitrary ®xed region 8 with a boundary S. Performing the integration for acomputational cell i provides

r8idfi

dtˆ

X

faces

¡SF ‡ 8iQi …4†

where the sum is taken over the faces of the computational cell. Each face has a unitnormal vector n de®ned by

n ˆ nxi ‡ nyj ˆ Sx

Si ‡ Sy

Sj …5†

and the corresponding expression for the cell face ¯ux becomes

F ˆ nxFx ‡ nyFy …6†

Here Fx and Fy are the ¯uxes de®ned by Eq. (2) in the x and y directions, respec-tively. To account for directional in¯uences in the upwinding process, the inviscidmomentum ¯ux on the cell face (i ‡ 1

2) is evaluated as

Fi‡1=2ˆ T

¡1i‡1=2

rU

rUu ‡ p

rUv

0

BB@

1

CCA

i‡1=2

T ˆ

1 0 0

0 nx ny

0 ¡ny nx

0

BB@

1

CCA …7†

where the overbar indicates that a linear interpolation between grid points is used toobtain this quantity. The contravariant velocity U ˆ unx ‡ vny, and u ˆ unx ‡ vny

and v ˆ vnx ¡ uny are the velocity components, normal and parallel to the cell face.The velocity components u and v are calculated utilizing a fully upwinded second-order (FUS) scheme and p is approximated by a simple averaging with the inclusionof cell face dissipation. The rotational matrix T [6] transforms the dependentvariables to a local coordinate system with the principal direction being perpendi-cular to the cell face. After multiplying by the rotational matrix, the ¯ux has thefunctional form akin to the Cartesian coordinate system and can be split into itscorresponding contributions. A second-order central di� erencing in conjunctionwith a thin-layer approximation is applied while evaluating the derivatives in theviscous ¯uxes.

Allowing a ®rst-order backward di� erence to the temporal variation of f, theimplicit pseudo-time integration pertaining to Eq. (4) yields a discretized system ofalgebraic equations [6]:

r8i

Dti

‡X

nb

Afnb

‡ j _MijÁ !

Dfiˆ

X

nb

Afnb Dfnb

‡ Rfi

…8†

where _Mi is the mass residual induced by the corrected velocity ®elds and nb standsfor a run over the neighboring nodes (i ‡ 1), (i ¡ 1), ( j ‡ 1), and ( j ¡ 1). Thein¯uence coe� cient A consists of the di� usion and mass ¯uxes. The residual R

fi is

de®ned by the right-hand side of Eq. (4), and the spatially varying time step Dt is

DUAL-DISSIPATION SCHEME FOR PRESSURE±VELOCITY COUPLING 233

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evaluated by employing the relation in [6]. After the implicit stage, the solutionvector f is updated from

fn‡1i

ˆ fni

‡ Dfi…9†

4. SIMPLE WITH ARTIFICIAL COMPRESSIBILITY

To achieve satisfaction of both the mass and momentum conservation laws, thevelocity and pressure ®elds are corrected in the SIMPLE method as [1]

u ˆ u¤ ‡ u

0 …10†

p ˆ p¤ ‡ p

0 …11†

where u0

and p0

are the incremental velocity and pressure, respectively. The super-script asterisk used on u and p indicates tentative values. The velocity correction canbe linked to the pressure correction by [5].

u0iˆ ¡ Dt

0i

r

qp0

qx

³ ´

i

Dt0iˆ r8iP

nb Aunb

…12†

where Dt0i signi®es quantitatively a weighting function evaluated using Eq. (8).

Inserting Eq. (12) into Eq. (10) supplies a velocity correction equation as

ui ˆ u¤i

¡ Dt0i

r

qp0

qx

³ ´

i

…13†

Extension of Eq. (13) to the curvilinear coordinate system is cited in [6].Within the framework of the traditional arti®cial compressibility approach,

the continuity equation is modi®ed by incorporating an arti®cial time-dependentpressure term [7]:

1

C2

qp

qt‡ r

qu

qx‡ r

qv

qyˆ 0 …14†

where C is the arti®cial speed of sound optimized as

C ˆ l

����������������������������������������������

max …u2 ‡ v2†;1

2U2

ref

µ ¶s

…15†

on the basis of numerical experiments, where Uref represents a reference velocity.Typical values of l in the range of 2 to 10 are recommended for better convergenceto the steady state.

With attention focused, a pressure-linked equation can be devised on a cur-vilinear coordinate as [6]

8i

C2i Dti

‡X

nb

Bnb

Á !

p0iˆ

X

nb

Bnb p0nb

¡ _M¤i

…16†

234 M. M. RAHMAN AND T. SIIKONEN

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where Bnb and _M¤i are the discretization coe� cient and ®ctitious mass source,

respectively. In principal, the cross-di � usion ¯uxes in Eq. (16) are neglected. Thisform is frequently appreciated due to its simplicity and having diagonally dominantcoe� cient matrices. Apparently, the physical relevance of the ®rst term on the left-hand side in Eq. (14) is to add more weight on the diagonal of the in¯uence co-e� cients.

5. DUAL-DISSIPATION SCHEME

The elimination of odd±even point decoupling can be accelerated by introdu-cing dissipations in both the cell face pressure and velocity. The purpose herein is toconstruct a numerical dissipation model with greater ¯exibility and increased accu-racy in a way similar to the MUSCL (monotone upstream-centered schemes forconservation laws) approach of Van Leer [8]. To facilitate the subsequent develop-ment, the calculation commences with resorting to one-dimensional linearizations of… p; u† in x:

~pi‡1=2 ˆ p ‡ Dxqp

qx

³ ´

i‡1=2

~ui‡1=2 ˆ u ‡ Dxqu

qx

³ ´

i‡1=2

…17†

Ostensibly, the interface pressure and velocity can be regarded as consisting ofcentral di� erence and di� usive contributions. Restricting attention to the steadystate, in the absence of source term with the pressure gradient extracted, the non-conservative momentum equation reduces to

ruqu

qx

³ ´

i‡1=2

ˆ ¡ qp

qx

³ ´

i‡1=2

…18†

The viscous term disappears with a supposition that the velocity at the cell face isevaluated from a linear interpolation between grid nodes, essentially elucidating

mq2u

qx2

³ ´

i‡1=2

ˆ 4 mi‡1=2

ui ¡ 2ui‡1=2‡ ui‡1

Dx2i‡1=2

ui‡1=2ˆ ui ‡ ui‡1

2

where m is the viscosity and Dxi‡1=2 is the distance between cell centers. CombiningEqs. (17) and (18) together with an upwind de®nition demonstrates that

~pi‡1=2 ˆ 1

2… pi ‡ pi‡1† ¡ ru

2

± ²

i‡1=2

…uR ¡ uL† …19†

~ui‡1=2 ˆ 1

2…ui ‡ ui‡1† ¡ 1

2ru

³ ´

i‡1=2

… pR ¡ pL† …20†

where …u; p†L and …u; p†

R are evaluated on the left and right sides of the cell face usingthe MUSCL approach. Consequently, dissipation terms comprised of second andhigher di� erences are possible. For instance, when the state variables are determinedusing the FUS scheme, a third-di � erence dissipation is embraced at the cell face

DUAL-DISSIPATION SCHEME FOR PRESSURE±VELOCITY COUPLING 235

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(fourth-di � erence dissipation at the node). The factor …1=2† is introduced with thedissipation terms so that a full upwinding is achieved.

Equations (19) and (20) clarify that the cell face pressure and velocity,respectively, are directly linked to the velocity and pressure at the adjacent nodalvalues, leading to a nonlinear cell face interpolation scheme. This ensures that astrong coupling between the pressure and velocity ®elds is retained. Extension ofthe cell face interpolation technique to curvilinear coordinates results in the fol-lowing expressions for the pressure and velocity at the computational cell face…i ‡ 1

2†:

~pi‡1=2ˆ 1

2… pi ‡ pi‡1† ¡ r U

2

³ ´

i‡1=2

uR ¡ uL… † …21†

~Ui‡1=2 ˆ 1

2…Ui ‡ Ui‡1† ¡ 1

2rU

³ ´

i‡1=2

… pR ¡ pL† …22†

Nevertheless, the professed interest lies in implementing the pressure±velocity cor-rection method in conjunction with the pressure Poisson equation. The prerequisiteherein is to a� ord appropriate signs with the pressure and velocity dissipations informing the mass imbalance in the Poisson equation and estimating the pressuregradients in the momentum equations. To preserve the requirements, Eqs. (21) and(22) are rearranged to yield

~pi‡1=2 ˆ 1

2… pi ‡ pi‡1† ¡ r jUj

2

³ ´

i‡1=2

uR ¡ uL… † …23†

~U¤i‡1=2

ˆ 1

2…U ¤

i‡ U

¤i‡1

† ¡ Dt

2r Dn

³ ´

i‡1=2

… pR ¡ pL† …24†

where U and U¤

are, respectively, the corrected and tentative contravariant velocitycomponents in the i direction, and Dni‡1=2 ˆ …8i ‡ 8i‡1†=2Si‡1=2. The choice com-prising jUj in Eq. (23) and U ˆ Dn=Dt in Eq. (24) warrants positivity in the cell facedissipations.

To recover particularly the desirable attributes pertaining to the cell faceinterpolation method that minimize the continuity error and provide a physicallymeaningful smooth pressure ®eld, the relation (23) is reconstructed with an assis-tance of [6]:

pi‡1=2 ˆ 1

2… pi ‡ pi‡1† ¡ Cp

i‡1=2

r jUj2

³ ´

i‡1=2

…uR ¡ uL† …25†

with

Cpi‡1=2

ˆ 1:0 ‡ CFL¤ jDpj

jpj ‡ E

³ ¡1

i‡1=2

D p ˆ r jUj2

…uR ¡ uL†

236 M. M. RAHMAN AND T. SIIKONEN

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where CFL¤ ˆ max…1:0; CFL† and E is a small number (¹10

¡6) included to preventthe singularity. The corresponding reasonable selection for the cell face velocityU

¤i‡1=2 becomes

U¤i‡1=2

ˆ 1

2…U ¤

i‡ U

¤i‡1

†Cui‡1=2

Dt

2 r Dn

³ ´

i‡1=2

pR ¡ pL… † …26†

with

Cui‡1=2

ˆ 1:0 ‡ CFL¤ jDuj

jU¤j ‡ E

³ ¡1

i‡1=2

Du ˆ Dt

2r Dn… pR ¡ pL†

where the sign …~:† is dropped out for notational convenience. The monitor functionC ˆ …Cu

; Cp† provides insight into the manner by which the scheme is susceptibleto controlling the degree of biasing the dissipation. To avoid pressure checker-boarding (i.e., zero pressure force=pressure±velocity decoupling), Eqs. (25) and (26)are interactive whose strength depends on the scalings of dissipation terms yieldedby the limiter functions. In this way, the scheme suppresses local extrema into thecell face velocity=pressure. Alternatively, the magnitude of dissipation is boundedand it remains a certain fraction of the magnitude of the actual cell face pressure=velocity. The interface velocity=pressure in the j direction can be de®ned corre-spondingly.

6. PHYSICAL VELOCITY FIELD

The inclusion of arti®cial compressibility in the pressure-correction equationassists in constructing conservative corrections, enhancing the residual smoothingproperties. The arti®cial change in density is de®ned by

Dr ˆ r0 ˆ p

0

C2…27†

which leads to the following linearized relation:

Drw¤ ˆ r Dw

¤ ‡ w¤

Dr ˆ rw0 ‡ r

0w

¤ …28†

where Dw¤ ˆ w

0and w ˆ …u; v†T

. Subsequently, the corrected velocity ®eld isobtained as follows:

w ˆ w¤ ‡ Drw

¤

rˆ w

¤ ‡ w0 ‡ r0

w¤

r…29†

Intuitively, r0can be interpreted as density preconditioning=perturbation to the

incompressible limit. It takes implicitly the e� ect of existing mass imbalanceinto account, provoking a transformation between conservative and primitivevariables. Comparing Eq. (10) with Eq. (29) reveals that the tentative velocity®eld is perturbed by an amount of …r0

w¤=r† to correct the primitive variable.

This aspect results in a signi®cant improvement in robustness of the solutionmethod [6]. As the solution converges to the steady state, density perturbationsdisappear.

DUAL-DISSIPATION SCHEME FOR PRESSURE±VELOCITY COUPLING 237

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7. SOLUTION ALGORITHM

The system of equations is solved using a tridiagonal matrix algorithm(TDMA). The solution procedure consists of the following steps:

1. Guess the velocity and pressure ®elds.2. Evaluate pressure gradients using relations like Eq. (25).3. Solve the momentum equations and update using Eq. (9) to obtain u

¤

and v¤.

4. Calculate U¤i§1=2, etc., using relations like Eq. (26) to form the mass im-

balance _M¤i , and solve Eq. (16) for p

0.

5. Update the velocity ®eld using Eq. (29), and the pressure ®eld using Eq. (11),to provide the physical ®elds.

6. Solve other f equations and update properties, coe� cients, etc.7. Repeat steps 2±6 until convergence is achieved.

The characteristic boundary conditions for the solution algorithm can be obtainedfrom [6].

8. TEST COMPUTATIONS

Computational experiments consisting of viscosity and buoyancy-driven ¯owsare designed to detect whether the proposed dissipation model causes deteriorationof numerical accuracy=convergence in comparison with the Rhie-Chow (RC)method within the similar algorithmic framework. Since the consequence of dis-sipation=arti®cial compressibility is presumably perceived well for coarser grids=low-speed laminar ¯ows, relatively coarse grids together with relevant ¯ow parametersare considered.

The calculations that follow utilize CFL ˆ 3, compressibility parameter l ˆ 2with no explicit pressure underrelaxation. The state variables, for instance,uR ¡ uL; pR ¡ pL, associated with Eqs. (25) and (26), are determined using the FUSscheme, which generates third-di� erence dissipations, convincingly comparable tothe RC approach. The root-mean-square mass residual M

¤is used as a measure for

the convergence. In every coordinate direction, the implicit stage performs typicallytwo sweeps for the f equation and three sweeps for the pressure-correction equation.

8.1. Viscosity-Driven Cavity Flow

A schematic of the test problem con®guration is shown in [6], with the top wallmoving to the right at a velocity Uref ˆ uw while the three sides are at rest. A non-uniform 32 £ 32 grid, containing ®ner grid points near the walls than in the core toresolve the sharp velocity gradients, is employed for the computation. Plots of thehorizontal velocity pro®le at the vertical centerline of the cavity are shown forReynolds number Re ˆ 400 in Figure 1. The result of Ghia et al. [9] is also includedfor comparison. As is evident, the quality of both the present and RC solutions isessentially equivalent to that of Ghia et al. Figure 2 displays the convergencehistories of the mass residuals with the same parameters mentioned above, starting

238 M. M. RAHMAN AND T. SIIKONEN

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from the same initial conditions. It seems likely that both formulations presumeidentical behaviors, since no appreciable changes are observed in the plots. Thisunambiguously con®rms the conclusion that, like the RC scheme, the present cellface dissipation system works e� ectively toward eliminating the nonphysicaloscillations.

8.2. Buoyancy-Driven Flow in an Annulus

Attention is drawn to the half-concentric annulus detailed in [5]. The referencevelocity considered herein is Uref ˆ

������Gr

p, where Gr is the thermal Grashof number.

The nonuniform grid employed herein is composed of 38 radial and 28 circumfer-ential line segments. The representative structured grid forming the half-annulus isshown in [6]. Noteworthily, the solution method needs only the Cartesian compo-nents and, therefore, the nonorthogonality e� ect is encountered even for theorthogonal curvilinear grid. Figure 3 exhibits the vertical velocity and temperaturepro®les, respectively, at the horizontal midplane for Gr ˆ 105, where X is measured

Figure 1. Horizontal velocity pro®le at the vertical centerline for viscosity-driven cavity ¯ow.

DUAL-DISSIPATION SCHEME FOR PRESSURE±VELOCITY COUPLING 239

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exactly from the inner surface. Remarkably, for this particular grid re®nement, thecomparison evinces an encouraging qualitative agreement. Figure 4 portrays theconvergence histories of the mass residuals on the same grid with the same para-meters. The plot is self-explanatory .

9. CONCLUSIONS

A cell-centered ®nite-volume approximation with an unfactored pseudo-timeintegration method is developed for solving incompressible ¯uid ¯ow problems oncurvilinear collocated grids. The dissipative mechanism has motivation forstrengthening the pressure±velocity coupling and reducing the numerical dissipationproduced. In principle, the pressure- and velocity-correction equations associatedwith the arti®cial compressibility provide a remarkable approach to alleviating anyexplicit pressure underrelaxation even in the presence of dominant sources in themomentum equations. Results demonstrate that the present formulation comparesfavorably with the RC scheme.

Figure 2. Convergence of mass residuals for viscosity-driven cavity ¯ow.

240 M. M. RAHMAN AND T. SIIKONEN

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Figure 3. Velocity and temperature pro®les on the horizontal midplane for buoyancy-driven ¯ow in an

annulus.

DUAL-DISSIPATION SCHEME FOR PRESSURE±VELOCITY COUPLING 241

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REFERENCES

1. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC,1980.

2. J. P. Van Doormaal and G. D. Raithby, Enhancements of the SIMPLE for Predicting

Incompressible Fluid Flows, Numer. Heat Transfer, vol. 7, pp. 147±163, 1984.3. C. M. Rhie and W. L. Chow, Numerical Study of the Turbulent Flow Past an Airfoil with

Trailing Edge Separation. AIAA J., vol. 21, pp. 1525±1532, 1983.

4. M. M. Rahman, A. Miettinen, and T. Siikonen, Modi®ed SIMPLE Formulation on aCollocated Grid with an Assessment of the Simpli®ed QUICK Scheme, Numer. Heat

Transfer B, vol. 30, pp. 291±314, 1996.

5. M. M. Rahman, T. Siikonen, and A. Miettinen, A Pressure Correction Method for SolvingFluid Flow Problems on a Collocated Grid, Numer. Heat Transfer B, vol. 32, pp. 63±84, 1997.

6. M. M. Rahman and T. Siikonen, An Improved SIMPLE Method on a Collocated Grid,

Numer. Heat Transfer B, vol. 38, pp. 177±201, 2000.7. A. J. Chorin, A Numerical Method for Solving Incompressible Viscous Flow Problems,

J. Comput. Phys., vol. 2, pp. 12±26, 1967.

8. W. K. Anderson, J. L. Thomas, and B. Van Leer, Comparison of Finite Volume FluxVector Splittings for the Euler Equations, AIAA J., vol. 24, pp. 1453±1460, 1986.

9. U. Ghia, K. Ghia, and C. Shin, High-Re Solutions for Incompressible Flow Using the Navier-

Stokes Equations and a Multigrid Method, J. Comput. Phys., vol. 48, pp. 387±411, 1982.

Figure 4. Convergence of mass residuals for buoyancy-driven cavity ¯ow in an annulus.

242 M. M. RAHMAN AND T. SIIKONEN

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