A multi-block ADI finite-volume method for incompressible Navier–Stokes equations in complex geometries

Journal of Computational Physics 230 (2011) 7400–7417

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Journal of Computational Physics

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A multi-block ADI finite-volume method for incompressible Navier–Stokesequations in complex geometries

Satbir Singh, Donghyun You ⇑Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 February 2011Received in revised form 27 May 2011Accepted 3 June 2011Available online 23 June 2011

Keywords:Finite-volume methodAlternating-direction-implicit (ADI) methodComplex geometryMulti-block gridNavier–Stokes equations

0021-9991/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jcp.2011.06.006

⇑ Corresponding author. Tel.: +1 412 268 6808; faE-mail address: [email protected] (D. You).

An efficient second-order accurate finite-volume method is developed for a solution of theincompressible Navier–Stokes equations on complex multi-block structured curvilineargrids. Unlike in the finite-volume or finite-difference-based alternating-direction-implicit(ADI) methods, where factorization of the coordinate transformed governing equations isperformed along generalized coordinate directions, in the proposed method, the discret-ized Cartesian form Navier–Stokes equations are factored along curvilinear grid lines.The new ADI finite-volume method is also extended for simulations on multi-block struc-tured curvilinear grids with which complex geometries can be efficiently resolved. Thenumerical method is first developed for an unsteady convection–diffusion equation, thenis extended for the incompressible Navier–Stokes equations. The order of accuracy and sta-bility characteristics of the present method are analyzed in simulations of an unsteady con-vection–diffusion problem, decaying vortices, flow in a lid-driven cavity, flow over acircular cylinder, and turbulent flow through a planar channel. Numerical solutions pre-dicted by the proposed ADI finite-volume method are found to be in good agreement withexperimental and other numerical data, while the solutions are obtained at much lowercomputational cost than those required by other iterative methods without factorization.For a simulation on a grid with O(105) cells, the computational time required by the presentADI-based method for a solution of momentum equations is found to be less than 20% ofthat required by a method employing a biconjugate-gradient-stabilized scheme.

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1. Introduction

Research in the numerical simulation of fluid flows has been focused primarily on the development of accurate and stablenumerical methods and on the reduction of computational cost. For large-scale simulations involving complicated geome-tries resolved with a huge number of grid points, the computational cost and memory requirement often exceeds the capa-bility of mainframe computers. Although the continuing advance in the computing power has enabled increasingly largerscale computational simulations, further enhancements in numerical methods associated with computational efficiencyare highly necessary to extend the practicable limit.

Reduction of computational cost can be achieved with proper exploitation of computing resources such as parallel com-puting and efficient use of computer memory [1,2], and with adoption of numerical methods that produce a solution faster.For example, on a structured grid, numerical methods based on alternating-direction-implicit (ADI) factorization [3,4], whencompared to methods necessitating non-factored broad-band or sparse matrix solvers, provides greatly enhanced computa-tional efficiency by converting a multi-dimensional problem into a series of one-dimensional problems which can be solved

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with tridiagonal matrix algorithms. ADI factorization has been mainly used in conjunction with finite-difference methodswith various levels of spatial and temporal orders of accuracy [5–16].

Geometric complexity often hinders the use of ADI factorization or necessitates precursory processes prior to ADI factor-ization. Finite-difference methods, when employed for simulations on curvilinear grids to resolve complex geometries, re-quire a coordinate transformation from Cartesian to generalized coordinates as illustrated in Fig. 1(a). A coordinatetransformation greatly increases the number of derivative terms in the governing equations, thereby increasing the memoryrequirement and operation counts significantly [17,18].

For complex geometries, finite-volume methods are often preferred because they can avoid a coordinate transformation,while providing flexibility of using unstructured grids with various polyhedral cell shapes. However, using an unstructuredgrid with a finite-volume method necessitates the use of expensive sparse matrix solvers, which also require large memoryspaces for storing sparsely distributed variables. It is also well known that numerical accuracy can be often compromisedwith the use of irregular unstructured cell shapes.

A few researchers have combined ADI factorization with finite-volume discretization schemes [19–22]. For example,Wang [19] presented second- and third-order-accurate finite-volume ADI methods for computing two-dimensional para-bolic equations on uniform orthogonal Cartesian grids. Yang and Cheng [20] developed an ADI quadratic finite-volume meth-od for second-order hyperbolic equations also on uniform orthogonal Cartesian grids. Itagi [21] combined a finite-volumemethod with ADI factorization to analyze the Fourier thermal transport in layered media on a non-uniform orthogonal polargrid. The method proposed by Itagi [21] was only first-order accurate in time. Jameson and Yoon [22] applied an ADI-basedfinite-volume method to compute two-dimensional steady Euler equations in a generalized coordinate system for flow overan airfoil. In their formulation, ADI factorization was performed along transformed generalized coordinate directions [22].

Although, ADI-based finite-volume or finite-difference methods solving the governing equations in a generalized coordi-nate system on a single-block curvilinear grid, have been successful in many applications, the geometric complexity involvedin the applications was relatively mild [2,13,15–17,22]. Higher geometric complexity can be effectively treated by decom-posing the complex domain with multiple blocks of curvilinear grid. The multi-block grid topology is advantageous not onlyfor the easiness in grid generation, but also for the control of mesh quality in terms of skewness and aspect ratio.

To date, the use of ADI factorization in conjunction with finite-volume methods for a solution of the Cartesian form Na-vier–Stokes equations on a curvilinear grid without a coordinate transformation, is not found in the literature. In addition,the application of an ADI scheme has been mostly limited to simulations on single block structured grids. The objective of thepresent research is to develop a new numerical method, which combines a finite-volume method and ADI factorization, for

Fig. 1. Schematic illustration of (a) the conventional ADI factorization along transformed coordinates (n and g) for transformed velocity components (q1 andq2) and (b) the proposed ADI factorization along non-transformed grid lines (i and j) for Cartesian velocity components (u1 and u2).

7402 S. Singh, D. You / Journal of Computational Physics 230 (2011) 7400–7417

simulations of the incompressible Navier–Stokes equations on multi-block structured curvilinear grids. Unlike in the con-ventional finite-volume or finite-difference alternating-direction-implicit (ADI) methods, where factorization is performedalong transformed generalized coordinate directions, in the proposed method, the Cartesian form Navier–Stokes equations,are factorized along curvilinear grid lines as illustrated in Fig. 1(b). Therefore, in the present method, a coordinate transfor-mation from a physical domain to a computational domain is not required.

As will be discussed in the following sections, the present method requires significantly lower computational cost andmemory space than other structured-grid methods treating the Navier–Stokes equations on generalized coordinates andunstructured-grid methods. Furthermore, the present method extends the utility of ADI factorization for a solution on a mul-ti-block curvilinear grid, which significantly enhances the capability for resolving complex geometries.

In the following section, the mathematical formulation of the present method is derived first. The temporal and spatialorders of accuracy are verified in an unsteady convection–diffusion problem. The method is then extended for the incom-pressible Navier–Stokes equations on single-block and multi-block curvilinear structured grids. The accuracy and efficiencyof the proposed method for a computation of the incompressible Navier–Stokes equations are demonstrated in simulationsof decaying vortices, flow in a lid-driven cavity, flow around a circular cylinder, and turbulent flow through a straightchannel.

2. Numerical method

2.1. An ADI finite-volume method

An unsteady convection–diffusion equation for a variable / in a two-dimensional space can be written as follows:

@/@tþ ux @/

@xþ uy @/

@y¼ m

@2/@x2 þ m

@2/@y2 ; ðx; yÞ 2 X� ð0; T�; ð1Þ

/ð0; x; yÞ ¼ /0ðx; yÞ; ðx; yÞ 2 X; ð2Þ

aðt; x; yÞ/þ bðt; x; yÞ @/@n¼ f ðt; x; yÞ; ðx; yÞ 2 @X; t 2 ð0; T�; ð3Þ

where X � R2 is a rectangular domain, (0,T] is the time interval, and /0 and f are the initial and boundary conditions. Thecoefficients a and b define the boundary condition as one of the Dirichlet, Neumann, and Robin type boundary conditions.ux(y) is the convection velocity in the x(y)-direction and m is the viscosity. The convection–diffusion equation is encounteredin many physical problems involving transport of scalar and vector quantities.

Integrating Eq. (1) in time using the Crank–Nicolson method and assuming a solenoidal velocity field results in

/lþ1 � /l

Dtþ 1

2@

@xux/lþ1 � m

@/lþ1

@x

!þ @

@yuy/lþ1 � m

@/lþ1

@y

!" #

¼ �12

@

@xux/l � m

@/l

@x

!þ @

@yuy/l � m

@/l

@y

!" #þ OðDt2Þ; ð4Þ

where l is the time-step index and Dt is the time-step size. Then, discretizing Eq. (4) in space using the divergence theoremover a computational cell of volume V leads to

/lþ1 � /l

Dtþ 1

2V

IS

uxnx þ uyny� �

� m@

@n

� �/lþ1dS ¼ � 1

2V

IS

uxnx þ uyny� �

� m@

@n

� �/ldS; ð5Þ

where nx and ny are the components of surface outward-normal vectors in x- and y-directions, respectively. @/@n is the nor-mal derivative operator at the surface. dS is the cell surface area.

Numerically, the surface integrals are replaced with summation over all faces of a computational cell as follows:

/lþ1 � /l

Dtþ 1

2V

Xk

uxnx þ uyny� �

� m@

@n

� �/lþ1S

� �k

¼ � 12V

Xk

uxnx þ uyny� �

� m@

@n

� �/lS

� �k

; ð6Þ

where k is the index of faces for a cell and S is the face area. In Eq. (6), ux(y), /, and @//on need to be evaluated at each cell face.In the present study, these quantities are evaluated from neighboring cell-center values using non-weighted central differ-encing as follows:

Ujk ¼ux

c þ uxnbr

2nx þ

uyc þ uy

nbr

2ny; ð7Þ

/jk ¼/c þ /nbr

2; ð8Þ

@/@n

��k

¼ /nbr � /c

dN; ð9Þ


where U is the face-normal velocity, c is the index of the cell where / is evaluated, nbr is the index of the neighboring cell,and dN is the distance between c and nbr cell centers, projected on the cell face normal, n.

The interpolation, /jk = (/c + /nbr)/2 is second-order accurate on uniform grids. On non-uniform grids, the interpolantcould be weighted by the distances between cell faces and neighboring cell centers. However, such weighted interpolationdoes not conserve discrete kinetic energy [23]. The spatial-order of accuracy can be degraded by the non-weighted interpo-lation scheme on highly stretched non-uniform meshes. However, the second-order accuracy in space is found not to benoticeably affected by the level of mesh non-uniformity considered in this paper.

Substituting Eqs. (7)–(9) into Eq. (6) results in

/lþ1c þ Dt

2V

Xk

/lþ1c � /lþ1

nbr

2

!UkSk �

Xk

m/lþ1

nbr � /lþ1c

dN

!Sk

" #

¼ /lc �

Dt2V

Xk

/lc � /l

nbr

2

!UkSk �

Xk

m/l

nbr � /lc

dN

!Sk

" #þ OðDt2;Dx2Þ; ð10Þ

or

1þ Dt2V

Xk

US2þ mS

dN

� �k

" #/lþ1

c � Dt2V

Xk

US2þ mS

dN

� �k

" #/lþ1

nbr

¼ 1� Dt2V

Xk

US2þ mS

dN

� �k

" #/l

c þDt2V

Xk

US2þ mS

dN

� �k

" #/l

nbr þ OðDt2;Dx2Þ; ð11Þ

where the outward face normal, n, points from cell c to cell nbr leading to opposite signs of convective flux contributions forcells c and nbr.

On a two-dimensional structured grid with quadrilateral cells, each cell is surrounded by four neighbors, i.e., k = 4. Thisproduces an algebraic system with an N � N penta-diagonal matrix, where N is the total number of computational cells. Inthe proposed method, the contribution of neighbors is separated along grid lines (i and j directions) as illustrated in Fig. 1(b)to obtain the following matrix–vector equation:

½I � Ai � Aj�Ulþ1 ¼ ½I þ Ai þ Aj�Ul; ð12Þ

where I is the identity matrix, and Ai and Aj are tridiagonal matrices in i and j directions, respectively, with entries given asfollows:

Ai ¼ ½ai�1;j; �ðai�1;j þ aiþ1;jÞ; aiþ1;j�;Aj ¼ ½ai;j�1; �ðai;j�1 þ ai;jþ1Þ; ai;jþ1�;

where

ai;j ¼Dt2V

US2þ mS

dN

� � i;j:

ADI factorization of Eq. (12) leads to

½I � Ai�½I � Aj�Ulþ1 ¼ ½I þ Ai�½I þ Aj�Ul þ OðDt2Þ: ð13Þ

Then, the following set of equations is solved sequentially using a tridiagonal matrix solution algorithm:

½I � Ai�U� ¼ ½I þ Ai�½I þ Aj�Ul; ð14Þ½I � Aj�Ulþ1 ¼ U�: ð15Þ

The proposed method is unconditionally stable as can be proven using the von Neumann stability analysis. Substituting/ij ¼ /0eik1xeik2y into Eq. (13) by assuming an uniform orthogonal grid where hx = hy = h and the distance between centers oftwo neighboring cells, dN = h, leads to

/lþ10 1þ k�1� �

1þ k�2� �

¼ /l0 1� k�1� �

1� k�2� �

; ð16Þ

where

k�1 ¼aDt2Vð1� cos k1hÞ;

k�2 ¼aDt2Vð1� cos k2hÞ;

a ¼ US2þ mS

h:


The amplification factor thus becomes

jrj ¼ /lþ10

/l0

�� ¼ 1� k�1

� �1� k�2� �

1þ k�1� �

1þ k�2� �

�� 6 1 ð17Þ

for all values of modified wavenumbers. Therefore, the proposed ADI method is unconditionally stable.

2.2. Application to the incompressible Navier–Stokes equations

The incompressible momentum and continuity equations in Cartesian coordinates are as follows:

@ui

@tþ @uiuj

@xj¼ � @p

@xiþ m

@

@xj

@ui

@xj; ð18Þ

@ui

@xi¼ 0; ð19Þ

where ui, p, m are the velocity, pressure, and kinematic viscosity, respectively. Note that the density is constant and absorbedin the pressure. In the present study, a collocated arrangement of variables on a structured grid is employed. The Cartesianvelocity components and pressure are all stored at centroids of the computational cells. In addition to the cell-center veloc-ity, the face normal velocity, un = (ui)face.ni, is treated as an independent variable and stored at cell faces.

Time integration of Eqs. (18) and (19) is performed using the fractional-step method of Kim and Moin [24]. The Crank–Nicolson time-integration scheme is employed for both convective and diffusive fluxes in Eq. (18). The semi-discretized algo-rithm of the fractional-step method is as follows:

ui � uli

Dt¼ � @pl

@xi� 1

2@uiuj

@xjþ@ul

iulj

@xj

!þ m

2@2ui

@x2j

þ @2ul

i

@x2j

!; ð20Þ

u�i � ui ¼ Dt@pl

@xi; ð21Þ

@2plþ1

@x2i

¼ 1Dt

@u�i@xi

; ð22Þ

ulþ1i � u�i

Dt¼ � @plþ1

@xi; ð23Þ

where ui and u�i are intermediate cell-center velocities. The boundary condition for the intermediate velocity is set ui ¼ ulþ1i ,

which allows to maintain the second-order accuracy in time [25]. The nonlinear momentum equations (Eq. (20)) are solvedusing a Newton-iterative method.

Applying a Newton-iterative method to Eq. (20) leads to

@FiðuÞ@uj

r

durþ1j ¼ �FiðurÞ þ Rl

i; ð24Þ

where

@FiðuÞ@uj

¼ dij þ12

Dt@

@uj

@uiuj

@xj� m

@2ui

@x2j

!; ð25Þ

and

durþ1j ¼ urþ1

j � urj ; ð26Þ

FiðurÞ ¼ uri þ

12

Dt@ur

i urj

@xj� m

@2uri

@x2j

!; ð27Þ

Rli ¼ �Dt

@pl

@xiþ ul

i �12

Dt@ul

iulj

@xj� m

@2uli

@x2j

!: ð28Þ

Eq. (24) is recast as follows:

1þ 12

Dt@uj

@xjþ @ui

@xi� m

@2

@x2j

!( )r

durþ1i ¼ �1

2Dt

@uri

@xjdu�j

� �i–j

� Fi uri

� �þ Rl

i: ð29Þ

where r is the iteration index and j = 1, 2, and 3. du�j is updated during the iteration step.


Integrating Eq. (29) over a computational cell of volume V and applying the divergence theorem lead to

1þ Dt2V

Xk

Q k

" #r

durþ1i;c �

Dt2V

Xk

Q k

" #r

durþ1i;nbr ¼ RHSi; ð30Þ

where Q k ¼ US2 þ ui:niSþ mS

dN

� �k. U is an interpolated face-normal velocity obtained from Eq. (7). ui is the Cartesian velocity at

the face center obtained from Eq. (8) and RHSi is the entire right-hand-side terms in Eq. (29) calculated using quantities atiteration step r and time step l.

As derived in Section 2.1 for a convection–diffusion equation, ADI factorization is performed to advance Eq. (30). The pres-ent ADI method is distinct from the conventional ADI-based methods in that the discretized Eq. (30) is factorized along cur-vilinear grid lines without a coordinate transformation.

The solution algorithm to advance the simulation from time step l to l + 1 can be summarized as follows:

Step 1: Start with an initial condition u0i or a solution from the previous time step ul

i.Step 2: Solve the discretized nonlinear momentum equations using a Newton-iterative method coupled with the present

ADI finite-volume method.Step 2.1: Start with ur

i ¼ uli and dur

i ¼ duli.

Step 2.2: Solve Eq. (30) for i = 1 with du�2 ¼ dur2 and du�3 ¼ dur

3 to obtain durþ11 .

Step 2.3: Solve Eq. (30) for i = 2 with du�1 ¼ durþ11 and du�3 ¼ dur


Step 2.4: Solve Eq. (30) for i = 3 with du�1 ¼ durþ11 and du�2 ¼ durþ1


Step 2.5: Update urþ1i .

Step 2.6: Repeat steps 2.2–2.5 until urþ1i is converged. At convergence, ulþ1

i ¼ urþ1i .

Step 3: Solve Eq. (21) to obtain u�i .Step 4: Solve the Poisson equation (Eq. (22)) to obtain pl+1.Step 5: Obtain ulþ1

i using Eq. (23).

3. Verification and validation

3.1. Unsteady convection–diffusion of a Gaussian pulse

To verify the accuracy of the present ADI finite-volume method, unsteady convection–diffusion of a Gaussian pulse in asquare domain [0,2] � [0,2] is considered with the following initial condition [10]:

/ð0; x; yÞ ¼ exp �ðx� 0:5Þ2

m� ðy� 0:5Þ2

m

!: ð31Þ

Fig. 2. A skewed grid with 80 cells in each x- and y-direction.


The analytical solution to the problem is

Fig. 3.directio

/ðt; x; yÞ ¼ 14t þ 1

exp �ðx� uxt � 0:5Þ2

mð4t þ 1Þ � ðy� uyt � 0:5Þ2

mð4t þ 1Þ

!: ð32Þ

Dirichlet boundary conditions are imposed using the analytical solution. The viscosity is fixed as m = 0.01. Two cell Rey-nolds numbers of 2 and 200 are chosen by setting convection velocities ux = uy = 0.8 and ux = uy = 80, respectively. An uniformCartesian grid and a non-uniform skewed grid with 80 cells in both x- and y-directions, as shown in Fig. 2, are employed. Inaddition to the present ADI-based method, a finite-volume method coupled with the strongly implicit procedure (SIP) ofStone [26] for solving the non-factorized algebraic equations given by Eq. (12), is also employed to identify the influenceof ADI factorization on numerical solutions.

Contour plots of a pulse in the region 1.0 6 x,y 6 1.6 for Re = 2 at t = 1.0 on (a) uniform Cartesian and (b) skewed grids with 80 cells in x- and y-ns, respectively. Solid lines, exact; dashed lines, the present ADI finite-volume method.


Fig. 3 shows a comparison of the exact and numerical solutions obtained using the present ADI finite-volume method onboth Cartesian and skewed grids at t = 1.0 for Re = 2. It appears that at t = 1.0, numerical solutions on both Cartesian andskewed grids, are quite similar to each other, while slight phase errors are observed in numerical solutions on both grids.This is an expected feature for a second-order accurate method as discussed by You [12].

Fig. 4 shows L2-norm errors with respect to the exact solution for the proposed ADI finite-volume method and the SIPfinite-volume method at Re = 2 and 200 on both Cartesian and skewed grids. The numerical error obtained on the skewedgrid is found to be either larger or smaller than that on the Cartesian grid. This is because, on the skewed grid, the travelingGaussian pulse experiences a variation in the computational cell sizes and skewness, which affect the local accuracy of thesolution. The ADI finite-volume method produces nearly the same error as the SIP finite-volume method on both grids, ver-ifying that the present ADI finite-volume method is applicable to both Cartesian and non-orthogonal grids without notice-able loss of accuracy.

In order to verify the spatial and temporal orders of accuracy of the ADI finite-volume method, additional simulations areperformed with various levels of grid refinement and time step sizes. Fig. 5 shows L2-norm errors at Re = 2 and 200 as a func-tion of grid spacing and time step size. Grid convergence tests are performed with the time step size fixed at Dt = 2.5 � 10�4

for Re = 2 and Dt = 2.5 � 10�5 for Re = 200. To assess the temporal accuracy, the grid spacing is fixed at Dx(y) = 0.002. For gridconvergence tests, the Courant–Friedrichs–Lewy (CFL) number is kept smaller than 1 for all cases, while for temporal accu-racy tests, the CFL number is varied in the ranges of 1–100 for Re = 2 and 1–40 for Re = 200. As shown in Fig. 5, the presentADI finite-volume method is second-order accurate in both space and time at both Reynolds numbers. Higher absolute errorsare obtained at Re = 200. As discussed in You [12], this is an expected result for a scheme with a second-order spatial accu-racy. More accurate predictions at Re = 200 can be obtained with higher-order spatial discretization schemes [12].

3.2. Decaying vortices

Two dimensional decaying vortices are simulated to identify the spatial and temporal accuracy of the present ADI finite-volume method for a solution of the incompressible Navier–Stokes equations. The numerical solution obtained with the ADI

Fig. 4. L2-norm errors as a function of time on uniform Cartesian and skewed grids at (a) Re = 2 and (b) Re = 200. Solid lines, SIP finite-volume method;symbols, the present ADI finite-volume method.

10−3

10−2

10−1

10−6

10−5

10−4

10−3

10−2

10−1

dx

L2 −n

orm

err

or

10−5

10−4

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

dt

L2 −n

orm

err

or

Fig. 5. L2-norm errors produced by the present ADI finite-volume method as a function of (a) grid spacing and (b) time step size. Solid lines, second-orderslope; dashed lines, at Re = 2; dotted lines, at Re = 200. Errors are shown at t = 1.0 for Re = 2 and t = 0.01 for Re = 200.


finite-volume method is compared to that obtained with a method employing a biconjugate gradient stabilized (BCGSTAB)scheme [27], which is one of widely used iterative methods for solving sparse matrices. The ADI finite-volume method and aBCGSTAB finite-volume method are employed for computations of discretized momentum equations, while a BCGSTABmethod is employed for the pressure Poisson equation (Eq. (22)) in both finite-volume methods.

The flow field consists of an array of decaying vortices with following analytical expressions for velocity and pressure:

u ¼ � cosðpxÞ sinðpyÞe�2p2t=Re; ð33Þv ¼ sinðpxÞ cosðpyÞe�2p2t=Re; ð34Þ

p ¼ �14½cosð2pxÞ þ cosð2pyÞ�e�4p2t=Re: ð35Þ

Computations are carried out in a square domain of �1 6 x 6 1 and �1 6 y 6 1 at Re = 10, where Re = U1L/m. U1 is the initialmaximum velocity and L is the size of a vortex. The initial condition at t = 0 is set with the exact solution, while periodicboundary conditions are imposed in all directions.

Fig. 6 shows L2-norm errors as a function of grid spacing and time-step size. For a grid convergence study, the time stepsize is fixed at Dt = 0.001, while for a temporal accuracy measurement, the grid spacing is fixed at Dx(y) = 0.004. For gridconvergence tests, the CFL number is kept smaller than 1 for all cases, while for temporal accuracy tests, the CFL numberis varied in the range of 1–25. As shown in Fig. 6, second-order accurate solutions in both space and time are obtained bythe present ADI finite-volume method which combines a Newton iteration method and ADI factorization.

3.3. Flow in a lid-driven cavity

A flow configuration and a curvilinear grid for a simulation of flow in a lid-driven square cavity are shown in Fig. 7. Twodifferent grids: an uniform Cartesian grid and a curvilinear grid, each with 128 � 128 cells, are employed. In Fig. 7, only thecurvilinear grid, which was created using a combination of straight lines and sine functions, is shown. The Reynolds number

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

dx

L2 −n

orm

err

or

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

dt

L2 −n

orm

err

or

Fig. 6. L2-norm errors produced by the present ADI finite-volume method as a function of (a) grid spacing and (b) time step size for decaying vortices atRe = 10. Solid lines, second-order slope; dashed lines, the present ADI finite-volume method. Errors are shown at t = 0.5.


is defined as Re = UL/m, where U is the speed of the top lid and L is the length of each side of the cavity. Computations areperformed at Re = 100 with a constant time-step size of Dt = 0.01L/U, which corresponds to the CFL number of 0.6.

The steady-state solution at t = 100L/U is compared with the numerical solution of Ghia et al. [28], on the same uniformCartesian grid with 128 � 128 cells as considered in the present study. Fig. 8 shows the normalized mean velocity compo-nents u(y)/U and v(x)/U along the vertical and horizontal centerlines, respectively, on both uniform Cartesian and curvilineargrids. The present numerical solutions are found to agree very well with those of Ghia et al. [28] on both grids. These sim-ulations clearly demonstrate that the present ADI finite-volume method where factorization is performed along grid lines,provides equally good solutions on both uniform Cartesian and curvilinear grids.

3.4. Flow over a circular cylinder

Fig. 9(a) shows a flow configuration and a C-type curvilinear grid for simulations of flow over a circular cylinder. The com-putational domain size is �50d < x < 20d and �50d < y < 50d, where d is the cylinder diameter. The coordinate origin (x/d = 0,y/d = 0) is located at the center of the cylinder. A C-type curvilinear grid with 326 cells in the azimuthal direction and 120cells in the radial direction is created using a hyperbolic grid generation technique. Sixty four grid points are allocated on thecylinder surface. The current grid and domain size have been previously determined to be adequate to produce accurate pre-dictions for the range of Reynolds numbers considered in the present study [29]. A Dirichlet boundary condition, u = U, v = 0,where U is the free-stream velocity, is imposed at the far-field boundary, and the no-slip condition is applied on the cylinderwall. A Neumann boundary condition is used at the outflow boundary. For all cases, the computational time-step size, Dt U/L,is set to 0.01, which corresponds to the CFL number of 0.7.

The present ADI finite-volume and BCGSTAB finite-volume methods are used to solve a system of discretized momentumequations. In the ADI-based method, the tridiagonal matrix inversion is first conducted along the azimuthal (i) direction andthen along the radial (j) direction of the C-type grid, with periodic boundary conditions imposed at the branch cut.

Fig. 7. (a) Flow configuration and (b) a curvilinear grid for a simulation of flow inside a lid-driven cavity. Every other grid line are shown for clarity.


Computations are performed at three different Reynolds numbers: Re = Ud/m = 100, 130, and 160. Fig. 9(b) shows contourlines of the instantaneous streamwise velocity over a cylinder obtained with the present ADI finite-volume method atRe = 100. Fig. 10(a) shows a variation of the lift force as a function of time at Re = 100, while a variation of the Strouhal num-ber (St) is shown in Fig. 10(b) as a function of Reynolds number. The computed Strouhal number is found to agree quite wellwith the experimental data of Williamson [30]. As shown in Table 1, lift and drag forces predicted by the ADI-based and BCG-STAB-based methods, are found to match each other very well, and also to agree with other numerical results reported byYou et al. [29].

3.4.1. Turbulent channel flowIn order to evaluate the performance of the proposed ADI finite-volume method for high Reynolds number flows, turbu-

lent flow in a planar channel is computed. Numerical simulations without a subgrid-scale model are performed on a meshconsisting of 64 � 64 � 64 cells at Res = 180. The Reynolds number, Res, is based on the wall-friction velocity, us, and thehalf-height of the channel, d. The computational domain size is 4pd � 2d � 4pd/3. As shown in Fig. 11, two different grids,a Cartesian orthogonal grid and a skewed grid are used. For the skewed grid, the vertical grid lines are slanted at an angleof 45o with the walls of the channel. The grid distribution is uniform in the streamwise and spanwise directions, while it isdescribed by a hyperbolic tangent function in the wall-normal direction:

Fig. 8. Profiles of the mean velocity components (u/U,v/U) along the centerlines for flow in a lid-driven cavity at Re = 100 on (a) a Cartesian grid and (b) acurvilinear grid. Solid lines, BCGSTAB finite-volume method; dashed lines, the present ADI finite-volume method; symbols, numerical data of Ghia et al.[28].


yðjÞ ¼ � tanhðcð1� 2j=N2ÞtanhðcÞ ; j ¼ 0;1;2 . . . ;N2; ð36Þ

where N2 is the number of grid points in the wall-normal direction and c is a grid stretching parameter, which is fixed at 2.4.Periodic boundary conditions are imposed on the streamwise and spanwise directions. The flow is driven by a fixed pres-

sure gradient in the streamwise direction. Computations are performed for a fixed CFL number of 1.0. The flow field is al-lowed to evolve for 100d/us. Thereafter, each case is integrated for another 50d/us to produce a statistically convergedsolution. Results obtained by using the present ADI finite-volume method and the BCGSTAB finite-volume method are com-pared with the DNS data of Moser et al. [31].

Fig. 12 shows the mean streamwise velocity and root-mean-squared fluctuating velocity components as a function of thewall-normal distance on both Cartesian and skewed grids. Good agreement of the mean and fluctuating velocities with theDNS data of Moser et al. [31] is obtained on the Cartesian grid. Although a slight deterioration in the accuracy is observed onthe skewed grid, the present ADI finite-volume method, in which factorization is performed along the skewed grid lines, isfound to produce a similar solution to that obtained by the BCGSTAB-based non-factorized finite-volume method.

4. Multi-block formulation

For a complex flow configuration, it is advantageous to discretize the computational domain using multiple blocks ofstructured grid to accommodate complex geometries and domain-decomposition-based parallelization. For a single proces-sor computation, it is relatively easy to logically combine multiple blocks while maintaining the tridiagonal matrix structurefor the entire computational domain. However, for a parallel computation, utilizing multiple blocks requires extensive com-munication among computer processors, severely affecting the computational efficiency.

Fig. 9. (a) Curvilinear grid and flow configuration for a simulation of flow over a circular cylinder and (b) the instantaneous streamwise velocity contourspredicted by the present ADI finite-volume method.


To overcome the difficulty, the present ADI finite-volume method is recast to be capable of multi-block grids. Eq. (37)represents a set of algebraic equations on a two-block computational domain. The block interfaces are represented by thedashed lines. Matrix elements (A) and (C) need to be communicated between the blocks for a tridiagonal matrix algorithmconsisting of a forward elimination and a backward sweep. In the present ADI method, Eq. (37) is recast such that the influ-ence of matrix elements (A) and (C) is approximated as in Eq. (38). With the approximation, each block can be solved inde-pendently of neighboring blocks while communication among the blocks occurs only after the completion of each Newtoniteration.

ð37Þ

ð38Þ

Fig. 10. (a) Lift force as a function of time at Re = 100 and (b) the Strouhal number (St) as a function of Reynolds number for flow over a circular cylinder.Solid lines, BCGSTAB finite-volume method; dashed lines, the present ADI finite-volume method; symbols, experimental data excerpted from Williamson[30]. Dashed lines are nearly overlapped with the solid lines.

Table 1Mean drag and lift fluctuations as a function of Reynolds number predicted by the present ADI finite-volume method and a BCGSTAB finite-volume method.

Reynolds number Mean drag RMS Lift

100 130 160 100 130 160

Present ADI method 0.675 0.680 0.687 0.164 0.227 0.280BCGSTAB method 0.675 0.680 0.687 0.164 0.227 0.280You et al. [29] 0.680 � 0.690 0.165 � 0.280

Fig. 11. (a) Cartesian and (b) skewed grids used for simulations of turbulent channel flow.


Fig. 12. Profiles of (a) the mean streamwise velocity and (b) the root-mean-square of velocity fluctuations on a Cartesian grid, and (c) the mean streamwisevelocity and (d) the root-mean-square of velocity fluctuations on a skewed grid, as a function of the wall-normal distance for turbulent channel flow atRes = 180. Solid lines, BCGSTAB finite-volume method; dashed lines, the present ADI finite-volume method; symbols, DNS data of Moser et al. [31].


Fig. 13 shows L2-norm errors as a function of the CFL number for decaying vortices computed with the single-block formu-lation (Eq. (37)) and the multi-block formulation (Eq. (38)) of the present ADI method. In the multiple block case, a two-dimensional computational domain is divided into 16 (4 � 4) blocks with 64 � 64 computational cells in each block. Bothsingle- and multi-block formulations are found to produce almost the same error and order of accuracy for a wide rangeof CFL number. The present multi-block scheme has been found to be stable and to maintain globally the second-order accu-racy even at a high CFL number (�10) in the simulations of decaying vortices and flow inside a lid-driven cavity.

A comparison of the ensemble-averaged streamwise velocity and root-mean-squared fluctuating velocity componentsobtained on a single- and multi-block grids for turbulent channel flow, is shown in Fig. 14. For multi-block simulations,

Fig. 13. L2-norm errors produced by the present ADI-based method as a function of the CFL number on single-block (256 � 256) and multi-block (16 blocksof 64 � 64 mesh) meshes for decaying vortices at Re = 10. Solid line, second-order slope; dashed line, on a single-block mesh; dotted line, on a multi-blockmesh. Errors are shown at t = 0.5.

Fig. 14. Profiles of (a) the mean streamwise velocity and (b) the root-mean-square of velocity fluctuations as a function of the wall-normal distance forturbulent channel flow at Res = 180. Solid lines, on a multi-block mesh (16 blocks of 16 � 64 � 16 mesh); dashed lines, on a single-block mesh (64 � 64 � 64mesh); symbols, DNS data of Moser et al. [31].

Fig. 15. Ratio of the wall-clock time required by the present ADI finite-volume method to the wall-clock time required by the BCGSTAB finite-volumemethod for the solution of the momentum equations.


the computational domain is decomposed of 16 blocks of 16 � 64 � 16 computational cells. Nearly identical velocity profiles,which also agree well with the DNS data of Moser et al. [31], are obtained with the single- and multi-block formulations.

5. Computational efficiency

Simulations of decaying vortices on a number of different grid resolution are performed to quantify the reduction in com-putational cost due to the use of the present ADI factorization, when compared to the BCGSTAB finite-volume method. InFig. 15, the ratio of the wall-clock time required by the present ADI finite-volume method to the wall-clock time requiredby the BCGSTAB finite-volume method for a solution of momentum equations, is shown as a function of the number of com-putational cells. A rapid reduction in computational time is achieved with the present ADI method. For example, on a600,000 cell grid, the computational time required by the present ADI finite-volume method is found to be less than 20%of that required by the BCGSTAB finite-volume method.

In addition to the reduction in computational time, the present ADI finite-volume method also significantly lower thememory requirement, when compared to unstructured grid methods. The present ADI finite-volume method requires mem-ory spaces only on the order of 3N (tridiagonal matrix) for the momentum equations and 7N (heptadiagonal matrix in 3D) forthe Poisson equation.

6. Conclusions

A new second-order-accurate multi-block alternating-direction-implicit (ADI) finite-volume method for time-accuratesimulations of incompressible flows in complex flow configurations, has been developed. Unlike in the conventional ADI fi-


nite-volume or finite-difference methods, where factorization of the coordinate transformed governing equations is per-formed along generalized coordinate directions, in the proposed method, the discretized Cartesian form Navier–Stokes equa-tions are factored along curvilinear grid lines. Robustness and stability of the present numerical method are ensured by theconservation of discrete mass, momentum, and kinetic energy not by the addition of ad-hoc artificial or numerical dissipa-tion. The proposed method is especially suitable for domain-decomposition-based parallel computations.

The spatial and temporal orders of accuracy were verified for an unsteady convection–diffusion equation on both regularCartesian and skewed grids. The method has been extended to the incompressible Navier–Stokes equations on single-blockand multi-block curvilinear structured grids. The accuracy and efficiency of the proposed method for computation of theincompressible Navier–Stokes equations have been demonstrated in predictions of decaying vortices, flow in a lid-drivencavity, flow around a circular cylinder, and turbulent flow through a straight channel.

Numerical accuracy and computational cost of the present ADI finite-volume method were compared with those of abiconjugate gradient stabilized (BCGSTAB) finite-volume method. For the flow configurations investigated, predictions ofthe present ADI method were found to agree well with published experimental and numerical data, and with those predictedby a BCGSTAB-employing method. The present ADI finite-volume method significantly lower computational cost requiredfor a solution of momentum equations and the overall memory requirement. For a simulation on a grid with O(105) cells,the computational time for a solution of momentum equations is found to be less than 20% of the cost required by a methodemploying a BCGSTAB scheme.

Acknowledgments

Authors are thankful to A. Sifounakis and T. Kim for fruitful technical discussion.

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A multi-block ADI finite-volume method for incompressible Navier–Stokes equations in complex geometries