a class of high-resolution algorithms for incompressible flows

11
A class of high-resolution algorithms for incompressible flows Long Lee * Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA article info Article history: Received 2 March 2009 Received in revised form 20 August 2009 Accepted 18 January 2010 Available online 1 February 2010 Keywords: High-resolution Finite volume methods Incompressible Navier–Stokes equations Finite-amplitude density variation Liddriven cavity flow Rayleigh–Taylor instability Rayleigh–Bénard convection abstract We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed by the incompressible Navier–Stokes equations. The algorithms use high-resolution finite volume methods developed in LeVeque (SIAM J Numer Anal 1996;33:627–665) for the advective terms and finite differ- ence methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advects the cell-centered velocities using the divergence-free cell-edge velocities. The resulting cell-centered velocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robust for constant-density flows at high Reynolds numbers via an example of lid-driven cavity flow. With a slight modification for the projection operator in the constant-density solvers, the algorithms also solve incompressible flows with finite-amplitude density variation. The strength of such algorithms is illus- trated through problems like Rayleigh–Taylor instability and the Boussinesq equations for Rayleigh– Bénard convection. Numerical studies of the convergence and order of accuracy for the velocity field are provided. While simulations for two-dimensional regular-geometry problems are presented in this study, in principle, extension of the algorithms to three dimensions with complex geometry is feasible. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Chorin, in a series of papers [8,7,9] introduced a practical frac- tional-step method for solving viscous incompressible Navier– Stokes equations. Similar ideas were independently introduced by Temam [29]. This fractional-step method, or projection method, computes an intermediate velocity without regard to the diver- gence constraint and then projects this velocity onto the diver- gence-free subspace. Many other different forms of projection methods were developed after those of Chorin and Temam, making it impossible to report an exhaustive reference list. We review the following two projection methods that are closely related to the proposed high-resolution algorithms. Consider the dimensionless incompressible Navier–Stokes equations with an external force u t þðu rÞu þ rp ¼ 1 Re r 2 u þ F ; r u ¼ 0; u ¼ b on @X: ð1:1Þ The first projection method we consider is a finite difference method introduced by Kim and Moin [15]. This method is similar to the first-order scheme of Chorin. The method first computes the intermediate velocity without the gradient pressure term. Then it imposes the incompressibility by solving a Poisson equation. The method can be written in a semi-discrete form: u u n Dt þðu nþ1=2 rÞu nþ1=2 ¼ 1 2Re ðr 2 u þ r 2 u n Þþ F ; ð1:2Þ u ¼ b þ Dtr/ n on @X; ð1:3Þ u nþ1 u Dt ¼r/ nþ1 ; ð1:4Þ r u nþ1 ¼ 0; ð1:5Þ n u nþ1 ¼ n b on @X; ð1:6Þ where n is the unit normal. The nonlinear convection term ðu nþ1=2 rÞu nþ1=2 can be approximated, for example, by an explicit Adams–Bashforth formula 3 2 ðu n rÞu n 1 2 ðu n1 rÞu n1 . The pres- sure p nþ1 can be obtained from / nþ1 through the relationship [2] p nþ1 ¼ / nþ1 Dt 2Re r 2 / nþ1 : ð1:7Þ To compute / nþ1 , we apply the divergence operator r to (1.4) and use (1.5). We then have the Poisson equation r 2 / nþ1 ¼ r u Dt : ð1:8Þ We refer to this projection method as Kim and Moin’s scheme. The second method we consider is the pressure-correction scheme first introduced by Van Kan [31], and later Bell, Colella, and Glaz (BCG) modified the scheme by using Godunov’s method- ology to compute the advection term [1]. We outline this projec- tion method in the semi-discrete form as follows: 0045-7930/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2010.01.012 * Tel.: +1 307 7664368; fax: +1 307 7666838. E-mail address: [email protected] Computers & Fluids 39 (2010) 1022–1032 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid

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Computers & Fluids 39 (2010) 1022–1032

Contents lists available at ScienceDirect

Computers & Fluids

journal homepage: www.elsevier .com/locate /compfluid

A class of high-resolution algorithms for incompressible flows

Long Lee *

Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA

a r t i c l e i n f o

Article history:Received 2 March 2009Received in revised form 20 August 2009Accepted 18 January 2010Available online 1 February 2010

Keywords:High-resolutionFinite volume methodsIncompressible Navier–Stokes equationsFinite-amplitude density variationLiddriven cavity flowRayleigh–Taylor instabilityRayleigh–Bénard convection

0045-7930/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compfluid.2010.01.012

* Tel.: +1 307 7664368; fax: +1 307 7666838.E-mail address: [email protected]

a b s t r a c t

We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed bythe incompressible Navier–Stokes equations. The algorithms use high-resolution finite volume methodsdeveloped in LeVeque (SIAM J Numer Anal 1996;33:627–665) for the advective terms and finite differ-ence methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advectsthe cell-centered velocities using the divergence-free cell-edge velocities. The resulting cell-centeredvelocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robustfor constant-density flows at high Reynolds numbers via an example of lid-driven cavity flow. With aslight modification for the projection operator in the constant-density solvers, the algorithms also solveincompressible flows with finite-amplitude density variation. The strength of such algorithms is illus-trated through problems like Rayleigh–Taylor instability and the Boussinesq equations for Rayleigh–Bénard convection. Numerical studies of the convergence and order of accuracy for the velocity fieldare provided. While simulations for two-dimensional regular-geometry problems are presented in thisstudy, in principle, extension of the algorithms to three dimensions with complex geometry is feasible.

� 2010 Elsevier Ltd. All rights reserved.

� n

1. Introduction

Chorin, in a series of papers [8,7,9] introduced a practical frac-tional-step method for solving viscous incompressible Navier–Stokes equations. Similar ideas were independently introducedby Temam [29]. This fractional-step method, or projection method,computes an intermediate velocity without regard to the diver-gence constraint and then projects this velocity onto the diver-gence-free subspace. Many other different forms of projectionmethods were developed after those of Chorin and Temam, makingit impossible to report an exhaustive reference list. We review thefollowing two projection methods that are closely related to theproposed high-resolution algorithms.

Consider the dimensionless incompressible Navier–Stokesequations with an external force

ut þ ðu � rÞuþrp ¼ 1Rer2uþ F;

r � u ¼ 0;u ¼ b on @X:

ð1:1Þ

The first projection method we consider is a finite differencemethod introduced by Kim and Moin [15]. This method is similarto the first-order scheme of Chorin. The method first computesthe intermediate velocity without the gradient pressure term. Thenit imposes the incompressibility by solving a Poisson equation. Themethod can be written in a semi-discrete form:

ll rights reserved.

u � uDt

þ ðunþ1=2 � rÞunþ1=2 ¼ 12Reðr2u� þ r2unÞ þ F; ð1:2Þ

u� ¼ bþ Dtr/n on @X; ð1:3Þ

unþ1 � u�

Dt¼ �r/nþ1; ð1:4Þ

r � unþ1 ¼ 0; ð1:5Þ

n � unþ1 ¼ n � b on @X; ð1:6Þ

where n is the unit normal. The nonlinear convection termðunþ1=2 � rÞunþ1=2 can be approximated, for example, by an explicitAdams–Bashforth formula 3

2 ðun � rÞun � 12 ðun�1 � rÞun�1. The pres-

sure pnþ1 can be obtained from /nþ1 through the relationship [2]

pnþ1 ¼ /nþ1 � Dt2Rer2/nþ1: ð1:7Þ

To compute /nþ1, we apply the divergence operator r� to (1.4) anduse (1.5). We then have the Poisson equation

r2/nþ1 ¼ r � u�

Dt: ð1:8Þ

We refer to this projection method as Kim and Moin’s scheme.The second method we consider is the pressure-correction

scheme first introduced by Van Kan [31], and later Bell, Colella,and Glaz (BCG) modified the scheme by using Godunov’s method-ology to compute the advection term [1]. We outline this projec-tion method in the semi-discrete form as follows:

L. Lee / Computers & Fluids 39 (2010) 1022–1032 1023

u� � un

Dtþ ðunþ1=2 � rÞunþ1=2 þrpn�1=2

¼ 12Reðr2u� þ r2unÞ þ F; ð1:9Þ

u� ¼ b on @X; ð1:10Þunþ1 � u�

Dt¼ �rpnþ1=2 �rpn�1=2

c¼ �r/nþ1; ð1:11Þ

r � unþ1 ¼ 0; ð1:12Þn � unþ1 ¼ n � b on @X: ð1:13Þ

From the first equality of (1.11), we have

unþ1 � u� � ðDt2Þc

@rp@t

: ð1:14Þ

Eq. (1.14) implies u� ¼ bþ OðDt2Þ on @X. Discussion on the choice ofc can be found in [13,27]. We refer to this project method as theBCG scheme. The projection method introduced for the high-resolu-tion algorithms can be put into either the framework of Kim andMoin’s scheme or the BCG scheme.

2. Conservative algorithms for advection

The class of high-resolution algorithms for incompressible flowsproposed in this study is motivated by the high-resolution wavepropagation algorithms for advection in incompressible flows,developed by LeVeque [18]. A brief introduction for the wave-propagation algorithm is described as follows. We consider thescalar advection equation in a specified velocity field uðx; tÞ inone, two, or three space dimensions

qt þr � ðuqÞ ¼ 0; ð2:1Þ

where qðx; tÞ is a conservative quantity which can be the scalar con-centration or the density function. If the flow is incompressible,r � u ¼ 0, the conservative form (2.1) can be written in a advectiveform

qt þ u � rq ¼ 0: ð2:2Þ

Although (2.1) and (2.2) are mathematically equivalent forincompressible flow, numerical algorithms based on the two maybehave differently. In two dimensions, the staggered grid as shownin Fig. 2.1 is used in [18] so that the numerical algorithms modeledeither from the conservative form or from the advective form arethe same. The notations in Fig. 2.1 are described as follows: LetCij be the ði; jÞ grid cell ½xi�1=2; xiþ1=2� � ½yj�1=2; yjþ1=2�. The ‘‘edgevelocities” ui�1=2;j and v i;j�1=2 are the velocities at the midpoints of

Fig. 2.1. Cell-variable locations for the conservative algorithm.

the interfaces ðxi�1=2; yjÞ and ðxi; yj�1=2Þ, giving rise to the wavesbeing propagated. They should satisfy the discrete divergence-freerelationship

uiþ1=2;j � ui�1=2;j

Dxþ v i;jþ1=2 � v i;j�1=2

Dy¼ 0: ð2:3Þ

Qi;j represents an approximation to the cell average at the currenttime level tn,

Qi;j �1

DxDy

ZCi;j

qðx; y; tÞdxdy: ð2:4Þ

In two-dimensional space, we consider the scalar color equation(2.2) in the component form,

qt þ uðx; yÞqx þ vðx; yÞqy ¼ 0: ð2:5Þ

The unsplit finite volume wave propagation algorithm for solvingthe above equation takes the form

Qnþ1i;j ¼ Q i;j �

DtDx

AþDQ i�1=2;j þA�DQ iþ1=2;j

� �� Dt

DyBþDQ i;j�1=2 þB�DQ i;jþ1=2

� �� Dt

DxeF iþ1=2;j � eF i�1=2;j

� �� Dt

DyeGi;jþ1=2 � eGi;j�1=2

� �: ð2:6Þ

The term AþDQi�1=2;j represents the first-order Godunov update tothe cell value Qi;j resulting from the Riemann problem at the edgeði� 1=2; jÞ. The other three similar terms are the Godunov updatesresulting from the Riemann problems at the other three edges. eFand eG are the correction fluxes. The Riemann problem at each inter-face leads to a single wave with speed given by the edge velocity. Inthe x-direction we have

wave : Wi�1=2;j ¼ Q i;j � Q i�1;j;

speed : si�1=2;j ¼ ui�1=2;j;ð2:7Þ

and in the y-direction

wave : Wi;j�1=2 ¼ Q i;j � Q i;j�1;

speed : si;j�1=2 ¼ v i;j�1=2:ð2:8Þ

The first-order Godnov updates then take the following forms

A�DQi�1=2;j ¼ s�i�1=2;jWi�1=2;j ¼ u�i�1=2;j Q i;j � Qi�1;j

� �;

B�DQi;j�1=2 ¼ s�i;j�1=2Wi;j�1=2 ¼ v�i�1=2;j Q i;j � Q i;j�1

� �;

ð2:9Þ

where u� and v� mean the positive and negative part of thevelocity.

For the correction terms, at the beginning of each step, we setthe terms to be zeros for all i; j:eF i�1=2;j :¼ 0 and eGi;j�1=2 :¼ 0: ð2:10Þ

After each Riemann problem is solved in the x-direction, at theinterface ði� 1=2; jÞ;A�DQi�1=2;j is set as in (2.9), and then the near-by correction fluxes are updated by

eGi�1;j�1=2 :¼ eGi�1;j�1=2 �12

DtDx

v�i�1;j�1=2u�i�1=2;j Q i;j � Q i�1;j

� �;

eGi�1;jþ1=2 :¼ eGi�1;jþ1=2 �12

DtDx

vþi�1;jþ1=2u�i�1=2;j Q i;j � Q i�1;j

� �;

eGi;j�1=2 :¼ eGi;j�1=2 �12

DtDx

v�i;j�1=2uþi�1=2;j Q i;j � Q i�1;j

� �;

eGi;jþ1=2 :¼ eGi;jþ1=2 �12

DtDx

vþi;jþ1=2uþi�1=2;j Q i;j � Q i�1;j

� �:

ð2:11Þ

Similarly, after solving the Riemann problem in the y-direction atinterface ði; j� 1=2Þ, we set B�DQi;j�1=2 as in (2.9) and then updatethe nearby fluxes by

Fig. 4.1. Cell variables of the projection method.

1024 L. Lee / Computers & Fluids 39 (2010) 1022–1032

eF i�1=2;j�1 :¼ eF i�1=2;j�1 �12

DtDy

u�i�1=2;j�1v�i;j�1=2 Q i;j � Q i;j�1

� �;

eF iþ1=2;j�1 :¼ eF iþ1=2;j�1 �12

DtDy

uþiþ1=2;j�1v�i;j�1=2 Q i;j � Q i;j�1

� �;

eF i�1=2;j :¼ eF i�1=2;j �12

DtDy

u�i�1=2;jvþi;j�1=2 Q i;j � Q i;j�1

� �;

eF iþ1=2;j :¼ eF iþ1=2;j �12

DtDy

uþiþ1=2;jvþi;j1=2 Qi;j � Q i;j�1

� �:

ð2:12Þ

For practical implementation of (2.11) and (2.12), we refer readersto [18,19].

The algorithms discussed so far are first-order accurate. For sec-ond-order accurate methods, in each direction we can replace thefirst-order upwind approximation by a Lax–Wendroff approxima-tion in that direction. To achieve this, we make the following up-dates to the correction fluxes already defined:eF i�1=2;j :¼ eF i�1=2;j þ

12jui�1=2;jj 1� Dt

Dxjui�1=2;jj

� �fWi�1=2;j;

eGi;j�1=2 :¼ eGi;j�1=2 þ12jv i;j�1=2j 1� Dt

Dyjv i;j�1=2j

� �fWi;j�1=2:

ð2:13Þ

Here fWi�1=2;j represents a limited version of the single waveWi�1=2;j, obtained by comparing this wave Wi�1=2;j with the wavein the upwind direction. For example, if ui�1=2;j > 0 and v i;j�1=2 < 0,then Wi�1=2;j is compared to Wi�3=2;j while Wi;j�1=2 is compared toWi;jþ1=2. It is worth noting that if we apply the algorithms to con-stant-coefficient advection equation with no limiter, these sec-ond-order corrections are exactly the same as the correspondingterms in the standard Lax–Wendroff method [18,19]. A flux-limiteris normally applied for problems with steep gradients or disconti-nuities in qðx; y; tÞ to avoid spurious numerical oscillations. For de-tailed explanation of the high-resolution algorithms and limiterfunctions, we refer readers to [18,19].

3. The advection–diffusion equation

Adding a diffusive term to (2.2), obtains the advection-diffusionequation. The Eqs. (1.2) and (1.9) are the semi-discrete forms of theadvection-diffusion equations. For the high-resolution algorithms,one way to tackle the advection–diffusion equation is to use frac-tional-step methods [4]. We consider the semi-discrete advec-tion-diffusion equation

Q nþ1 � Q n

Dtþ ðun � rÞQnþ1 ¼ l

2ðr2Q nþ1 þr2Q nÞ; ð3:1Þ

where the spatial discretization of un satisfies (2.3) and l is the vis-cosity. The first-order fractional-step method, sometimes called theGodunov splitting, can be written as follows:

Q � � Q n

Dtþ ðun � rÞQ � ¼ 0; ð3:2Þ

Q nþ1 � Q �

Dt¼ l

2ðr2Q nþ1 þr2Q �Þ: ð3:3Þ

We first compute Q � in (3.2) using the high-resolution method de-scribed in the previous section. Eq. (3.3) is a Crank–Nicolson dis-cretization for the diffusion. It results in a Helmholtz typeequation. This splitting method is only first-order accurate. Usingthe same notation, we can write a formally second-order method,the Strang splitting, as follows:

Q � � Q n

Dt=2¼ l

2ðr2Q � þ r2Q nÞ; ð3:4Þ

Q �� � Q �

Dtþ ðun � rÞQ �� ¼ 0; ð3:5Þ

Q nþ1 � Q ��

Dt=2¼ l

2ðr2Q nþ1 þr2Q ��Þ: ð3:6Þ

In order to reduce numerical diffusion and computational cost, (3.4)from one step and (3.6) from the following step can be combinedinto a single step of length Dt. By doing so, Strang splitting usesDt=2 only in the first and last time step. In between, Strang splittingis identical to Godunov splitting. In practice (see [4]), one can obtainbetter than first-order accuracy using Godunov splitting.

4. A projection method: conservative algorithm approach

We begin this section with the definitions of our discrete diver-gence and discrete gradient operators. In order to use the high-res-olution algorithms from Section 2, the discrete values of the cellinterface velocities have to satisfy (2.3). In two-dimensional space,let

ui�1=2;j�1=2 ¼ ðui�1=2;j;v i;j�1=2Þ

denote the horizontal and vertical components of the discretevelocity field at the edges of the ði; jÞ cell Ci;j and let

U i;j ¼ ðUi;j;Vi;jÞ

denote the approximation to the cell average

U i;j �1

DxDy

ZCi;j

Uðx; y; tÞdxdy:

Assuming that / is a scalar function, we consider a staggered grid asshown in Fig. 4.1 and define the discrete gradient and divergenceoperators as follows:

ðDuÞi;j ¼uiþ1=2;j � ui�1=2;j

Dxþ v i;jþ1=2 � v i;j�1=2

Dy; ð4:1Þ

ðG/Þi;j ¼/i;j � /i�1;j

Dx;/i;j � /i;j�1

Dy

� �; ð4:2Þ

ðDUÞi;j ¼Uiþ1;j � Ui�1;j

2Dxþ Vi;jþ1 � Vi;j�1

2Dy; ð4:3Þ

ðG/Þi;j ¼/iþ1;j � /i�1;j

2Dx;/i;jþ1 � /i;j�1

2Dy

� �: ð4:4Þ

Using these operators, we introduce the algorithm of the new pro-jection method in two-dimensional space. Let U be the divergence-free velocity field with respect to the continuous divergence opera-tor. U i;j is the discrete value of U at the cell center of Ci;j. We find thediscrete edge velocity by taking the average the cell-centered val-ues. For example, the left edge value of cell Ci;j is

ui�1=2;j ¼12ðUi�1;j þ Ui;jÞ ð4:5Þ

and the bottom edge value is

v i;j�1=2 ¼12ðVi;j�1 þ Vi;jÞ: ð4:6Þ

L. Lee / Computers & Fluids 39 (2010) 1022–1032 1025

If the edge values (4.5) and (4.6) satisfy (2.3) initially, the edgevelocities are discrete divergence-free with respect to the operator(4.1). Otherwise, we need to apply one projection step so that (2.3)is satisfied. We will explain the projection step in Step P2. The algo-rithm then takes the following steps:

Step P1. Using the edge values uni�1=2;j and vn

i;j�1=2, we advect anddiffuse the cell-centered velocities over a time step Dtusing the algorithm in Section 2, with the conservativequantity Qi;j to be the centered value Un

i;j or Vni;j. This

results in the intermediate velocity field U�i;j ¼ ðU�i;j;V

�ijÞ.

Step P2. We average the discrete centered values to get the edgevalues

u�i�1=2;j ¼12ðU�i�1;j þ U�i;jÞ; ð4:7Þ

v�i;j�1=2 ¼12ðV�i;j�1 þ V�i;jÞ: ð4:8Þ

To compute divergence-free edge values that satisfy (2.3) fromu�i�1=2;j and v�i;j�1=2, we need to find the scalar field / by solving

DGðDt/Þnþ1i;j ¼ Du�i;j; ð4:9Þ

where the D;G operators and Du�i;j are defined as (4.1) and (4.2). TheDG operator is a compact five-point Laplacian operator and (4.9) isthe discrete version of the Poisson Eq. (1.8).Step P3. Let

GðDt/Þnþ1i;j ¼ ðG1;ði;jÞ;G2;ði;jÞÞ: ð4:10Þ

We update the interface values by

unþ1i�1=2;j ¼ u�i�1=2;j � G1;ði;jÞ; ð4:11Þ

vnþ1i;j�1=2 ¼ v�i;j�1=2 � G2;ði;jÞ; ð4:12Þ

so that Dunþ1i;j ¼ 0 and (2.3) is satisfied.

Step P4. Let

G0ðDt/Þnþ1i;j ¼ ðG0

1;ði;jÞ;G02;ði;jÞÞ: ð4:13Þ

We update the cell-centered values by

Unþ1i;j ¼ U�i;j � G0

1;ði;jÞ; ð4:14Þ

Vnþ1i;j ¼ V�i;j � G0

2;ði;jÞ: ð4:15Þ

The exactly divergence-free edge values are obtained by sub-tracting G1;ði;jÞ and G2;ði;jÞ from the intermediate edge velocities. Itis easy to show

G01;ði;jÞ ¼

12ðG1;ði;jÞ þ G1;ðiþ1;jÞÞ; ð4:16Þ

G02;ði;jÞ ¼

12ðG2;ði;jÞ þ G2;ði;jþ1ÞÞ; ð4:17Þ

by the definition of Eqs. (4.2) and (4.4). So it is natural to updatecell-centered values Unþ1 using (4.14) and (4.15). One can expectUnþ1 to be divergence-free up to second-order, since the edge valuesare exactly divergence-free. This completes one time step. Go toStep P1 for the next time step.

4.1. Discussion

1. The boundary conditions used for the elliptic Eq. (4.9) are theNeumann boundary conditions discussed in [10].

2. From (4.1), (4.3), (4.7), and (4.8), it is easy to show

Du�i;j ¼ D0U�i;j: ð4:18Þ

Combining (4.9) and (4.18) gives

DGðDt/Þnþ1i;j ¼ D0U�i;j: ð4:19Þ

If we apply the operator D0 on (4.14) and (4.15) and compare theresulting equation with (4.19), we see that the cell-centered Unþ1

is not discrete divergence-free with respect to the discrete diver-gence operator D0, but it is divergence-free up to second-order.Minion [21] used a similar approach in his BCG version of adap-tive projection methods to avoid the decoupled stencil arisingfrom the projection step.

3. Although in Step P1 we do not need the pressure p to advancethe velocity, we can obtain the pressure pnþ1 from (1.7) withineach time step. The projection method stated in the previoussection is parallel to the finite difference scheme of Kim andMoin.

4. vIf we update the pressure gradientrpnþ1=2 using (1.11) and addrpn�1=2 in the advection-diffusion equation in Step P1, the mod-ified projection method is parallel to the pressure-correction(BCG) scheme. The termrpn�1=2 in the advection-diffusion equa-tion is treated as an external force and is solved together with thediffusive term using the fractional-step method in Section 3.

5. Since the high-resolution method used in Step P1 to advect theintermediate velocity field is explicit, a CFL condition must beimposed to ensure stability. The method requires

maxi;jDtDxjui�1=2;jj;

DtDyjv i;j�1=2j

� �6 1: ð4:20Þ

6. It was pointed out in [20] that if we use an interpolant, such as thearithmetic average, to interpolate the divergence-free cell-edgevalues back to the cell centers, instead of using the approximateprojection, i.e. (4.14) and (4.15), it will introduce a diffusive terminto the discretized equation which resembles a one-dimen-sional Laplacian with a magnitude which scales like the grid sizeh. Even if higher order interpolation is used, a similar diffusiveterm will result, making this approach undesirable for flows athigh Reynolds numbers.

7. The authors in [1] indicated that 76% of the computational timefor their projection method is used to solve the elliptic potentialequation (the projection linear algebra). While the high-resolu-tion wave propagation algorithms are robust for treating flowsat high Reynolds numbers, potentially they could be moreexpensive than traditional finite difference methods, due tothe cost of solving Riemann problem at each interface, but weexpect that this is not the dominant cost for the proposed pro-jection methods. Since the computational cost heavily dependson the choice of elliptic solvers for projection methods, weexpect that the efficiency of the proposed algorithms shouldbe at least comparable to traditional finite difference methods.

5. A projection method for variable-density flows

In this section, we extend the high-resolution algorithms forsolving the variable-density incompressible Navier–Stokes equa-tions. Only a slight modification of the projection operator is re-quired. We consider the unsteady incompressible Navier–Stokesequations for flows with finite-amplitude density variation. Con-servation of mass is described by an advection equation. We writethe system of equations as follows:

ut þ ðu � rÞu ¼1qr � lðruþ ðruÞTÞ � rpþ F� �

;

qt þ ðu � rÞq ¼ 0;r � u ¼ 0;

ð5:1Þ

where q is the density and l is the viscosity that is allowed to de-pend on density. Function F is an external force while u and p are

Fig. 5.1. Staggered grid for the Poisson problem with variant coefficients.

1026 L. Lee / Computers & Fluids 39 (2010) 1022–1032

the velocity and hydrodynamic pressure respectively. In two-dimensional space, let u ¼ ðu;vÞ and q ¼ ðu; v;qÞ ¼ ðu;qÞ. For sim-plicity, we assume l is a constant. Again for simplicity, we considerthe Dirichlet boundary conditions. The projection method based onthe high-resolution conservative algorithm for the above systemcan be stated as follows:

Step V1. Use the centered values Q n ¼ ðUn;Vn;qnÞ and edge val-ues un ¼ ðun;vnÞ to solve the advection equation

Q � Q n

Dtþ un � rQ ¼ 0: ð5:2Þ

This is solved on a finite volume grid using the explicit second-orderhigh-resolution algorithm developed by LeVeque [19]. The resultingsolution is Q y ¼ ðUy;qyÞ ¼ ðUy;V y;qyÞ.Step V2. Use a Crank–Nicolson discretization for the diffusion

U� � Uy

Dt¼ l

2qyðr2U� þ r2Uy þ FÞ: ð5:3Þ

This gives the intermediate velocity U� at cell center.Step V3. Obtain the edge velocity unþ1 by

unþ1 ¼ u� � 1q̂GðDt/nþ1Þ; ð5:4Þ

where u� is the average of the adjacent U� and q̂ is the average ofthe adjacent qy. The update (5.4) requires the value /nþ1, whichcan be obtained by solving a discrete Poisson problem: Taking thediscrete divergence of (5.4) and using the fact that we wantDunþ1 ¼ 0, we obtain the variable-coefficient Poisson problem

D1q̂G/nþ1

� �¼ 1

DtDu�: ð5:5Þ

Consider the staggered grid shown in Fig. 5.1. If we denote

bi�1=2;j ¼1

q̂i�1=2;j; ð5:6Þ

then a finite difference discretization for (5.5) is/iþ1;jbiþ1=2;j � /i;jðbiþ1=2;j þ bi�1=2;jÞ þ /i�1;jbi�1=2;j

ðDxÞ2

þ/i;jþ1bi;jþ1=2 � /i;jðbi;jþ1=2 þ bi;j�1=2Þ þ /i;j�1bi;j�1=2

ðDyÞ2

¼ 1Dt

u�iþ1;j � 2u�i;j þ u�i�1;j

ðDxÞ2þ

v�i;jþ1 � 2v�i;j þ v�i;j�1

ðDyÞ2

!

:/iþ1;jbiþ1=2;j � /i;jðbiþ1=2;j þ bi�1=2;jÞ þ /i�1;jbi�1=2;j

ðDxÞ2

þ/i;jþ1bi;jþ1=2 � /i;jðbi;jþ1=2 þ bi;j�1=2Þ þ /i;j�1bi;j�1=2

ðDyÞ2

¼ 1Dt

u�iþ1;j � 2u�i;j þ u�i�1;j

ðDxÞ2þ

v�i;jþ1 � 2v�i;j þ v�i;j�1

ðDyÞ2

!: ð5:7Þ

The resulting linear system from (5.7) is

A/ ¼ f ; ð5:8Þ

where the coefficient matrix A is symmetric and positive definite[12].Step V4. In the final step, we update the cell-centered values by

Unþ1 ¼ U� � Dt2

1q̂i�1=2;j

ð/i;j � /i�1;jÞ þ1

q̂iþ1=2;jð/iþ1;j � /i;jÞ

� �; ð5:9Þ

Vnþ1 ¼ V� � Dt2

1q̂i;j�1=2

ð/i;j � /i;j�1Þ þ1

q̂i;j�1=2ð/i;jþ1 � /i;jÞ

� �; ð5:10Þ

and set qnþ1 ¼ qy. This completes one time step. Go to Step V1 fornext time step.

It is worth noting that if the density q 1 is constant, then thealgorithm is identical to the projection method in Section 4. We re-mark that the proposed algorithm becomes unstable if the densityratio is over a threshold (between 100 and 200).

6. A high-resolution method for the Boussinesq flow

We consider the convection of a Boussinesq fluid in a two-dimension rectangular cavity on x–z plane. The flow is governedby the dimensionless Boussinesq equations [11]:

Pr�1½ut þ ðu � ruÞu� ¼ �rpþr2uþ RaTez;

Tt þ ðu � rÞT ¼ r2T;

r � u ¼ 0;u ¼ 0 on z ¼ Za and z ¼ Zb;

T ¼ Ta on z ¼ Za; T ¼ Tb on z ¼ Zb;

ð6:1Þ

where T is the temperature, Pr the Prantal number, and Ra the Ray-leigh number. Here ez is the unit vector in the vertical direction. Thedimensionless parameter Ra is given by

Ra ¼ agðTb � TaÞd3

mj; ð6:2Þ

where a is the thermal expansion coefficient, Ta and Tb the temper-atures of the top and bottom plates, d ¼ Za � Zb the height of thecavity, g the gravity acceleration, m the kinematic viscosity, and jthe thermal diffusion. The other parameter Pr is given by

Pr ¼ mv ; ð6:3Þ

where v is the thermal conductivity.The Boussinesq equation is a good approximation for studying

Rayleigh–Bénard convection, in which a viscous fluid in a cavityis heated from the bottom plate and the top plate is maintainedat a lower temperature. When the temperature between the topand bottom plates is a linear function of the height of the cavityand the initial velocity is zero everywhere, the linear stability the-orem shows that a static solution exists to the problem (6.1) [5].Increasing the temperature of the hot plate until the Rayleigh num-ber is above a critical value, Rac , the static solution becomes unsta-ble to any small disturbance. The system then turns fromconduction to convection. Some properties of non-linear thermalconvection are analyzed and compared with experimental obser-vations in the paper by Busse [3] .

Similar to Section 5, in two-dimensional space, let u ¼ ðu; vÞand define q ¼ ðu; v; TÞ ¼ ðu; TÞ. The high-resolution algorithm forthe Boussinesq approximation can be described in the following:

L. Lee / Computers & Fluids 39 (2010) 1022–1032 1027

Step B1. Use the centered values Q n ¼ ðUn;Vn; TnÞ and edge val-ues un ¼ ðun;vnÞ to solve the advection equation

Q � Q n

Dtþ un � rQ ¼ 0: ð6:4Þ

Similar to Step V1, this is solved by the high-resolution algorithmdeveloped in [19], and the resulting solution isQ y ¼ ðUy; TyÞ ¼ ðUy;V y; TyÞ.Step B2. Use a Crank–Nicolson discretization for the thermal dif-

fusion and the diffusive term.

Tnþ1 � Ty

Dt¼ 1

2ðr2Tnþ1 þr2T yÞ; ð6:5Þ

and

U� � Uy

Dt¼ Pr

2ðr2U� þ r2UyÞ þ PrRaTnþ1ez: ð6:6Þ

Both Step B1 and Step B2 are subject to boundary conditionsaccordingly.Step B3. Obtain the edge velocity unþ1 by

unþ1 ¼ u� � PrGðDt/nþ1Þ; ð6:7Þ

and update the centered velocity

Unþ1 ¼ U� � PrG0ðDt/nþ1Þ: ð6:8Þ

Again u� is the average of the adjacent U�, and /nþ1 is obtained bysolving (4.9) with Neumann boundary condition (1.8). G and G0

are defined as before.

7. Numerical results

In this section, we present several examples that validate theconvergence properties of the methods developed previously. Wedemonstrate the performance of the methods via numerical resultsand show their potential for solving more realistic problems.

Example 7.1. The first example is the stationary Taylor’s vortices[28]. We use this example to demonstrate the rate of convergence

Table 7.1Convergence rate, u-component velocity.

32 � 32 Rate 64 � 64

ku� uexactk2 2.066E�3 2.69 3.196E�3ku� uexactk1 4.526E�2 2.73 6.825E�3

0 0.5 10

0.5

1

(a) t=0

Fig. 7.1. u-Component velocity contour of the Taylor’s vortex. Tw

of our methods. We consider a stationary inviscid flow to showthat our method is suitable for fluids at high Reynolds numbers.

In the absence of viscosity, an exact steady solution of theincompressible Navier–Stokes equations in a periodic unit squareis given by

uðx; yÞ ¼ � cosð2mpxÞ sinð2mpyÞ; ð7:1Þvðx; yÞ ¼ sinð2mpxÞ cosð2mpyÞ; ð7:2Þ

where m is some integer. We consider the case m ¼ 2. Using (7.1)and (7.2) as initial data, we show the results for time t ¼ 1. Ideallythe solution would be unchanged. Table 7.1 shows the l2 and infin-ity norms for the error of the u-component velocity. The l2 norm isdefined by

kuk2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiPjuj2

mxmy

s; ð7:3Þ

where mx and my are the numbers of grid cells in x and y directions,respectively.

The CFL number is fixed at 0.9. It appears that the method is for-mally second-order accurate. We show that the method used tocompute the results has the same formulation as the BCG scheme.The algorithm is stable for 0 < c 6 2=3. We choose c ¼ 2=3 and nolimiter is used for this simulation. A fixed time step Dt ¼ 0:01 is usedfor the 32 � 32 grid. The time step is reduced to a half when the gridsize is reduced to a half. The projection method that uses the formu-lation resembling Kim and Moin’s scheme has similar results.

Fig. 7.1 shows the u-component velocity contours. The compu-tation is done using a 256 � 256 grid. The interval between plottedcontour lines is 0.1. There are 20 contour lines with equally spacedvalues between ±1.

Example 7.2. The second example is the bench-mark lid-drivencavity flow, which has been intensively studied. It is a hard testproblem, particularly at high Reynolds numbers, due to the twosingular points at the upper corners. In the paper by Gresho andChan [13], several finite difference schemes based on projection

Rate 128 � 128 Rate 256 � 256

2.83 4.506E�4 2.74 6.724E�52.84 9.534E�4 2.64 1.531E�4

0 0.5 10

0.5

1

(b) t=1

enty contour lines with equally spaced values between ±1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q(1) at time t = 1000

8

76 5

10

9

8

7

6

54

6

d

ef

g

h

i

j

k

Fig. 7.2. Stream function contour.

1028 L. Lee / Computers & Fluids 39 (2010) 1022–1032

methods were tested and compared with the stream-function-vorticity formulation [6]. They all delivered weaker flows thandesired. Compared with the results by Gresho and Chan [13], ourmethods give a stronger primary vortex. For Fig. 7.2 and Table 7.2,we use a uniform 256 � 256 grid and the Reynolds number isRe = 5000. The CFL number is 0.9. Fig. 7.2 is almost identical toFig. 3 in [6], except for the top left vortex cf. Fig. 7.2 and Fig. 3 in[6]. The primary vortex is slightly weaker than the one in [6], butthe difference is in a satisfactory range. For comparison, we use thesame set of contour values as that in [6]. Stronger primary vorticesthan those in Fig. 7.2 can be achieved by decreasing the CFLnumber. No limiter is used for this problem. The Poisson problem issolved by a Fast Poisson Solver.

Example 7.3. The example shown here is the Rayleigh–Taylorproblem first documented by Tryggvason [30] for inviscid fluidsusing boundary integral methods. Later, Quartapelle et al. [25,24]repeated the same calculation for viscous fluids by means of finiteelements and finite volumes. The problem consists of two layers offluid initially at rest in a gravity field. A heavy fluid is put on top ofthe light fluid, and the heavy fluid accelerates into the light fluidunder the action of gravity. The domain for the fluid isð�d=2; d=2Þ � ð2d;2dÞ, and the initial position of the interface ofthe layers is given by

Table 7.2Values for stream function contour for Fig. 7.2.

Contour letter Values Contour number Values

d �1.0E�4 5 1.0E�4e �1.0E�2 6 2.5E�4f �3.0E�2 7 5.0E�4g �5.0E�2 8 1.0E�3h �7.0E�2 9 1.5E�3i �9.0E�2 10 2.7E�3j �1.0E�1k �1.1E�1

gðxÞ ¼ �0:1d cos2px

d

� �:

The density ratio of heavy fluid to light fluid is 3, which makesthe Atwood number 0.5. The Atwood number At is defined asAt ¼ ðqmax � qminÞ=ðqmax þ qminÞ. In order to compare this with theresults in [25,24], we regularize the transition between the two flu-ids by a tanh profile:

qðx; y; t ¼ 0Þqmin

¼ 2þ tanhy� gðxÞ

0:01d

� �:

The governing equations are made dimensionless by using thefollowing references: qmin for density, d for length, d1=2g�1=2 fortime, where g is the magnitude of gravity field, and g for gravity.The reference velocity is d1=2g1=2, and the Reynolds number is de-fined by Re ¼ qmind2=3g1=2=l. No-slip boundary conditions are ap-plied to the top and bottom walls while periodic boundaryconditions are imposed on the two vertical sides. The output timet̂ is in the scale of Tryggvason. t̂ is related to our dimensionlesstime scale t by t̂ ¼ t

ffiffiffiffiffiAtp

.Fig. 7.3 shows the results for Re ¼ 1000 using 128 � 512 cells. In

order to compare the positions of the falling jet and the uprisingbubble with those in [25,24], we only plot half of the domain,which is composed of 32,768 cells. Both the positions and thedevelopment of the roll-up structure of our results are similar tothose in [25,24], for which 30,189 P2 nodes are used. There areno noticeable differences between the two sets of results, exceptthat our results reveal more detailed roll-up structure. It is worthnoting that in [25,24], a rather small time step Dt̂ ¼ 5� 10�4 ischosen for the spatial discretization. Our CFL number is fixed at0.9 and time step can be chosen by Dt̂ � 6� 10�3. A monotonizedcentered (MC) limiter is used for both velocity and density field,and the linear system (5.8) is solved by an incomplete Choleskyconjugate-gradient method.

0 0.25 0.5−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2t = 1

(a)

0 0.25 0.5

t = 1.5

(b)

0 0.25 0.5

t = 1.75

(c)

0 0.25 0.5

t = 2

(d)

0 0.25 0.5

t = 2.25

(e)

0 0.25 0.5

t = 2.5

(f)

Fig. 7.3. Density profile of the Rayleigh–Taylor instability. Re = 1000; Atwood number is 0.5 (density ratio = 3). Only a half of the computational domain is shown above. Theinitial amplitude is 10% of the wavelength. Density contours shown are 1:4 6 q 6 1:6.

L. Lee / Computers & Fluids 39 (2010) 1022–1032 1029

Based on their results, the authors in [25,24] indicate that obtain-ing an accurate and detailed prediction of the large-time phenomenafor the viscous Rayleigh–Taylor instability at high Reynolds numbersis very difficult. They report that for t̂ P 1:5 and Re P 5000, thenumerical results of this problem are not only sensitive to the gridresolutions but also to the numerical methods used to computethose results. The solutions computed by either the finite elementmethod or the finite volume method for Reynolds numberRe ¼ 5000 are very different from the inviscid one reported in [30]at or beyond t̂ ¼ 1:5. The solutions of the two methods are also dif-ferent from each other. The authors speculate that this problemhas no smooth inviscid solution for large time, based on Birkhoff’sconjecture. Birkhoff’s conjecture says that the initial-value problemfor inviscid stratified flows might be ill-posed, as a consequence ofthe fact that the growth rate of an infinitely small unstable wave isproportional to the square root of its wave number. Fig. 7.4 showsthe solutions computed by our algorithm for Re ¼ 5000. (a) usesthe MC limiter and (b) uses the superbee limiter to eliminate thenumerical oscillation due to the density discontinuity. In contrastto the results reported in [25,24], our results are compatible withthose in [30] before t̂ ¼ 2, especially for the solution using the MClimiter. However, after t̂ ¼ 2 we did observe that the results are sen-sitive to the limiters used for the computation. We conclude that it ishard to give an accurate and detailed prediction of the large-timephenomena for this problem at high Reynolds numbers, as theauthors in [25,24] indicated.

Because the above example does not consider surface tensionthat is a curvature-dependent effect, no interface tracking schemeis required. When the surface tension is taken into account, differ-ent approaches such as the volume of fluid method [22], the front-tracking algorithm [23], or the immersed interface method [17]will be needed to do this work.

Example 7.4. The last example of this chapter is two-dimensionalRayleigh–Bénard convection. Linear stability theory predicts that ifthe top and bottom of the domain are rigid while the horizontal isinfinite, then at the onset of instability, the critical Rayleighnumber is Ra = 1707.762 and the corresponding dimensionlesscritical wave number is a ¼ 3:117. This implies that the convectionroll develops most readily in cells with an aspect ratio of2p=a ¼ 2:016 (see [5] page 43). In order to compare our resultswith the linear stability theory, we set periodic boundary condi-tions in the horizontal direction for both hydrodynamic andthermal equations. No-slip boundary conditions are imposed onthe top and bottom for the velocity, and the temperature is-0.5 forthe top and 0.5 for the bottom. The channel is chosen with anaspect ratio 2:1.

We compute three different cases with Rayleigh numbers,Ra = 5000; 10,000; and 50,000 respectively. These numbers are allabove the predicted critical Rayleigh number. We expect Rayleigh–Bénard convection to occur. Fig. 7.5 shows the temperature contoursin Rayleigh–Bénard convection for the three cases. For Ra = 5000 and

(a)

−0.5 0 0.5

t = 0

−0.5 0 0.5

t = 1

−0.5 0 0.5

t = 1.5

−0.5 0 0.5

t = 1.75

−0.5 0 0.5

t = 2

−0.5 0 0.5

t = 2.25

−0.5 0 0.5

t = 2.5

(b)

−0.5 0 0.5

t = 0

−0.5 0 0.5

t = 1

−0.5 0 0.5

t = 1.5

−0.5 0 0.5

t = 1.75

−0.5 0 0.5

t = 2

−0.5 0 0.5

t = 2.25

−0.5 0 0.5

t = 2.5

Fig. 7.4. Re = 5000; density ratio 3. The grid is 128 � 512. The interface is shown at times 0, 1, 1.5, 1.75, 2, 2.25, and 2.5 using the density contours 1:4 6 q 6 1:6. The initialamplitude is 10% of the wavelength. (a) Is computed with the MC limiter; (b) is computed with the superbee limiter.

1030 L. Lee / Computers & Fluids 39 (2010) 1022–1032

10,000 our results are identical to those in [14]. For Ra = 50,000, how-ever, our result is more symmetric between hot and cold fluids. Forcomparison, our computational grid is 80 � 40, and the Prantal num-ber is Pr ¼ 0:71. Both are the same as [14].

8. Conclusion

We presented a class of high-resolution algorithms for incom-pressible flows that prove to be robust and suitable for incom-pressible flows at high Reynolds numbers. The implementation

of the proposed algorithms for variable-density flows is straight-forward, and the strength of the algorithms is illustrated throughproblems like Rayleigh–Taylor instability and the Boussinesqequations for Rayleigh–Bénard convection. Although only two-dimensional regular-geometry problems are present in thisstudy, extension of the algorithms to three dimensions withcomplex geometry is possible, because the wave-propagationalgorithms on curved manifold has been introduced in [26]and three-dimensional wave-propagation algorithms are avail-able though [16].

(a) −0.5

0

0.5

(b) −0.5

0

0.5

(c)

−0.5

0

0.5

Fig. 7.5. The temperature contour of the Rayleigh–Bénard convection. (a) Ra = 5000, (b) Ra = 10,000, and (c) Ra = 50,000. A total of 21 equally spaced contour lines that are ofthe values between �0.5 and 0.5 (including �0.5 and 0.5) for each plot.

L. Lee / Computers & Fluids 39 (2010) 1022–1032 1031

Acknowledgment

This work is partially supported by NSF through the Grant DMS-0610149.

References

[1] Bell JB, Colella P, Glaz HM. A second-order projection method for theincompressible Navier–Stokes equations. J Comput Phys 1989;85:257–83.

[2] Brown D, Cortez R, Minion ML. Accurate projection methods for theincompressible Navier–Stokes equations. J Comput Phys 2001;168:464–99.

[3] Busse FH. Non-linear properties of thermal convection. Rep Prog Phys1978;41:1929–67.

[4] Calhoun D, LeVeque RJ. A Cartesian grid finite-volume method for theadvection–diffusion equation in irregular regions. J Comput Phys2002;156:1–38.

[5] Chandrasekhar S. Hydrodynamic and hydromagnetic. Oxford University Press;1961.

[6] Ghia U, Ghia KN, Shin CT. High-resolutions for incompressible flow using theNavier–Stokes equations and a multigrid method. J Comput Phys1982;48:387–411.

1032 L. Lee / Computers & Fluids 39 (2010) 1022–1032

[7] Chorin AJ. Numerical solution of incompressible flow problems. Stud NumerAnal 1968;2:64–71.

[8] Chorin AJ. Numerical solution of the Navier–Stokes equation. Math Comput1968;22:742–62.

[9] Chorin AJ. On the convergence of discrete approximations to the Navier–Stokesequations. Math Comput 1968;22:742–62.

[10] E W, Liu J-G. Projection method I: convergence and numerical boundary layers.SIAM J Numer Anal 1995;32:1017–57.

[11] Gelfgat AY. Different modes of Rayleigh–Bénard instability in two-and three-dimensional rectangular enclosures. J Comput Phys 1999;156:300–24.

[12] Greenbaum A. Iterative methods for solving linear system. Frontiers in appliedmathematics. SIAM Publish; 1997.

[13] Gresho PM, Chan ST. On the theory of semi-implicit projection methods forviscous incompressible flow and its implementation via a finite elementmethod that also introduces a nearly consistent mass matrix. Part II:implementation. Int J Numer Methods Fluids 1990;11:621–59.

[14] He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmannmethod in incompressible limit. J Comput Phys 1998;146:282–300.

[15] Kim J, Moin P. Application of a fractional-step method to incompressibleNavier–Stokes equations. J Comput Phys 1985;59:308–23.

[16] Langseth JO, LeVeque RJ. A wave propagation method for three-dimensionalhyperbolic conservation laws. J Comput Phys 2000;165:126–66.

[17] Lee L, LeVeque RJ. An immersed interface method for the incompressibleNavier–Stokes equations. SIAM J Sci Comput 2003;25:832–56.

[18] LeVeque RJ. High-resolution conservative algorithms for advection inincompressible flow. SIAM J Numer Anal 1996;33:627–65.

[19] LeVeque RJ. Finite volume methods for hyperbolic problems. Cambridge textsin applied mathematics, August 26, 2002.

[20] Minion ML. Two methods for the study of vortex patch evolution on localrefined grids. PhD thesis, University of California, Berkeley, CA 94720, May1994.

[21] Minion ML. A projection method for locally refined grids. J Comput Phys1996;127:158–78.

[22] Puckett EG, Almgren AS, Bell JB, Marcus DL, Rider WJ. A high-order projectionmethod for tracking fluid interfaces in variable density incompressible flows. JComput Phys 1997;130:267–82.

[23] Popinet S, Zaleski S. A front-tracking algorithm for accurate representation ofsurface tension. Int J Numer Methods Fluids 1999;30:775–93.

[24] Guermond J-L, Quartapelle L. A projection FEM for variable densityincompressible flows. J Comput Phys 2000;165:167–88.

[25] Frigneau Y, Guermond J-L, Quartapelle L. Approximation of variable densityincompressible flows by means of finite elements and finite volumes. CommunNumer Methods Eng 2001;17:893–902.

[26] Rossmanith JA, Bale DS, LeVeque RJ. A wave propagation algorithm forhyperbolic systems on curved manifolds. J Comput Phys 2004;199:631–62.

[27] Shen J. On error estimates of some higher order projection and penalty–projection methods for Navier–Stokes equations. Numer Math 1992;62:49–73.

[28] Taylor GI. On the decay of vortices in a viscous fluid. Philos Mag1923;46:671–4.

[29] Temam R. Sur l’approximation de la solution des equations de Navier–Stokespar la méthode des fractionaries II. Arc Rational Mech Anal 1969;33:377–85.

[30] Tryggvason G. Numerical simulations of the Rayleigh–Taylor instability. JComput Phys 1988;75:253–82.

[31] Van Kan J. A second-order accurate pressure-correction scheme for viscousincompressible flow. SIAM J Sci Comput 1986;7:870–91.