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A Novel Semi-Explicit Spatially Fourth Order Accurate Projection Methodfor Unsteady Incompressible Viscous FlowsN. Sekarapandian a; Y. V. S. S. Sanyasiraju b; S. Vengadesan a

a Department of Applied Mechanics, Indian Institute of Technology, Madras, Chennai, India b

Department of Mathematics, Indian Institute of Technology, Madras, Chennai, India

Online publication date: 07 December 2009

To cite this Article Sekarapandian, N., Sanyasiraju, Y. V. S. S. and Vengadesan, S.(2009) 'A Novel Semi-Explicit SpatiallyFourth Order Accurate Projection Method for Unsteady Incompressible Viscous Flows', Numerical Heat Transfer, Part A:Applications, 56: 8, 665 — 684To link to this Article: DOI: 10.1080/10407780903423932URL: http://dx.doi.org/10.1080/10407780903423932

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A NOVEL SEMI-EXPLICIT SPATIALLY FOURTH ORDERACCURATE PROJECTION METHOD FOR UNSTEADYINCOMPRESSIBLE VISCOUS FLOWS

N. Sekarapandian1, Y. V. S. S. Sanyasiraju2, and S. Vengadesan11Department of Applied Mechanics, Indian Institute of Technology, Madras,Chennai, India2Department of Mathematics, Indian Institute of Technology, Madras,Chennai, India

This article describes a simple and elegant compact higher order finite-difference based

numerical solution technique to the primitive variable formulation of unsteady incompress-

ible Navier Stokes equations (UINSE) on staggered grids. The method exploits the advan-

tages of the D’yakanov ADI-like scheme and a non-iterative pressure correction based

fractional step method. Spatial derivatives are discretized to fourth order accuracy and

the time integration is realized through the Euler explicit method. The fast and efficient

iterative solution to the discretized momentum and pressure Poisson equations is achieved

using a variant of conjugate gradient method. Spatial accuracy and robustness of the solver

are tested through its application to relevant benchmark problems.

1. INTRODUCTION

The higher order compact schemes (HOCS) suitable to the discretization ofconvection diffusion equations (CDE) can be derived using polynomial or exponen-tial series expansions. In the context of Navier Stokes equations (NSE), the vorticitytransport equation (VTE) pertaining to the vorticity-stream function (VS) formu-lation can be exactly classified as a CDE; whereas, the analogous momentumequations in primitive variable (PV) formulation only resembles the CDE becauseof the presence of pressure gradient term. It is due to this obvious reason, thatdeveloped HOC polynomial [1–10] and exponential schemes [11] are straightforwardin their implementation to the VS formulation. However, the requirement ofthree-dimensional simulations makes the consideration of extending these schemesto the primitive variable form of UINSE inevitable. A fully implicit ADI algorithmapplying higher order Pade schemes to momentum equations in CDE sense has beenreported in reference [12]. This approach is feasible through the usage of thefractional step based projection method which was first proposed in reference [13];wherein, the pressure gradient term is eliminated while solving momentum equationsto compute pseudovelocity field.

Received 12 August 2009; accepted 17 September 2009.

Address correspondence to Dr. S. Vengadesan, FluidMechanics Laboratory, Department ofApplied

Mechanics, Indian Institute of Technology, Madras, Chennai 600036, India. E-mail: [email protected]

Numerical Heat Transfer, Part A, 56: 665–684, 2009

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780903423932

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Generally, the higher order Pade schemes when applied over a set of discretepoints are conceived in matrix form and can be broadly classified as centraldifference-based compact schemes (CDCS) and upwind difference-based compactschemes (UDCS), as proposed in references [14–16] and [17–19], respectively.Usually, the particular choice of the compact scheme is determined by the physicsof the problem to be simulated. The central difference (CD)-based compact schemesare non-dissipative in nature but may cause spurious oscillations in the convectiondominated flows. This drawback can be subdued by inducing artificial viscosity intothese schemes [16]. But the aliasing error that may occur due to the excess in numeri-cal viscosity has to be controlled using compact filtering schemes, and so, CD-basedcompact schemes are suitable for LES applications [20, 21]. In the case of numericalsimulation of wave propagation or acoustic problems, the higher order upwindschemes are indispensable [18, 19, 22].

In this work, we have adopted CD compact schemes keeping the focus on LESapplications in the future. Similarly, an essential comparison on the performance ofcompact schemes in the staggered and collocated grids, made in reference [20], showsthat the former has better conservation properties of mass and momentum. Inaddition, it is pointed out in references [23, 24], when the skew symmetric form of

NOMENCLATURE

A,B computational matrices

Axx, Ayy coefficient matrices of the second

derivatives in x, y directions

Bxx, Byy coefficient matrices of the function

values in x, y directions

ci,j storage for RHS defined in Eq. (11)

f column vectors of the function

values

F column vectors of the derivatives

values

h height of the backward facing step

H height of the downstream channel

I identity matrix

K computational matrix

lc characteristic length

L length of the downstream channel

L2 error norm

N number of nodes along x or y

direction

p pressure

q assumption for pressure

Re Reynolds number

Rþ non-negative time increment in real

space

Rd dimension of the real space

t time

Txx, Tyy computational matrices in x, y

directions

Uc characteristic velocity

u, v instantaneous velocity components

uavg, umax average inflow and maximum inflow

velocities, respectively

x, y streamwise and transverse

coordinates

x1 length of the recirculation zone

x2 point of separation

x3 point of reattachment

zi,j storage for RHS defined in Eq. (13)

D Laplacian operator

n kinematic viscosity

/ pressure correction

Dx, Dy grid spacing

uk, wk symbols to denote respective

computational nodes along a

particular grid line

X computational domain

qX computational domain boundary

Dt time step

r gradient operator

r2 Laplacian

Subscripts

i, j Cartesian coordinate directions

k index for numbering the node along

a particular grid line

Superscripts

d dimension

þ positive increment

666 N. SEKARAPANDIAN ET AL.


the convection term is used on a staggered grid, the unsteady incompressible flowsimulation will be stable and free of numerical dissipation, due to the satisfactionof conservation of kinetic energy. These advantages substantiate the choice of stag-gered grid and the skew symmetric form of the convection term in the computations.

The next vital issue to be discussed is the time step restriction while using thecompact schemes. Previous works [23, 25], signify that the higher order spatial accu-racy pays penalty in terms of shrinking the CFL limit. Keeping note of this fact, wehave adopted a semi-explicit form of discretization to the momentum equations.This selection allows a larger CFL limit than that for a fully explicit form, andincreases the sparsity of the iterative matrix which is denser in the case of fullyimplicit discretization.

In this work we have introduced a new approach of using higher order Padeschemes other than the fully implicit scheme given by reference [12] to solve theUINSE. In this direction, some of the recent works [24, 26–28] show the common-ality in using fractional step or a pressure correction-based method to solve theUINSE, each of which differs in the procedure to satisfy the incompressibilityconstraint.

It is well known that in the projection method approach, in order to satisfy theincompressibility constraint, it is imperative to solve a Poisson equation for pressureor pressure correction. In the literature, several methods have been reported to solvefor the pressure or pressure correction. One of the approaches is to convert thePoisson equation, which is of an elliptic kind, to a parabolic problem [26] usingpseudo transient. In another approach [27], the local cell by cell pressure correctionand global pressure correction strategies are followed. The comparison shows thatthe former performs better for low Reynolds number, but the latter is the attractivealternative for high Reynolds number flows. To reduce the computational cost, thematrix diagonalization technique has been recommended in reference [28] to solvethe Poisson Neumann problem. In the work of reference [24], it is pointed out thatusing a compact operator to the Laplacian either for the second derivative or in theform of divergence of a gradient may lead to the singularity of the discretematrix. But the fourth order convergence can be compensated by using a conven-tional second order operator and performing inner iterations to the point wherethe source term of the Poisson equation vanishes towards zero [29]. On the collo-cated grids, the Laplacian is discretized first using a compact second derivativeapproximation on a finer grid, and then with two time application of first derivativeapproximation on coarser grids. The reason for such an implementation is explainedin reference [30].

The purpose of this work is to test the utility of a semi-explicit fractional stepmethod, suggested by reference [31], when combined with advantages of matrix formof compact finite-difference schemes. There are two main unique implementations inthis article. First, instead of using conventional Peaceman-Rachford type ADIschemes, the multidimensional PDE is solved using the locally one-dimensionalapproach suggested by D’yakanov [32]. The main advantage of this factorizationis that the tridiagonal nature of the coefficient matrix for the derivatives can beretained at all times so that the matrix inversion is cost effective. The second uniquefeature is a new approach of deriving a discrete equation for pressure correctionusing the continuity equation. Moreover, owing to the elliptic nature of the discrete

SEMI-EXPLICIT PROJECTION METHOD FOR INCOMPRESSIBLE VISCOUS FLOWS 667


momentum and Poisson equations, the efficient iterative procedure of BICGSTAB(2), discussed in reference [33], has become viable to obtain their solutions. Theorganization of the rest of this article is as follows. The governing equations andthe numerical scheme are explained in section 2. Section 3 elucidates the implemen-tation of the D’yakanov factorization procedure into the discrete momentumequations, and discusses the details of the compact finite-difference (FD) approxima-tions (along with the appropriate staggered grid schematics), used to discretize thespatial derivatives contained in these equations. The distinct derivations ofthe discrete pressure Poisson equation of second and fourth order accuraciesand their implementation is discussed in section 4. In section 5, the performanceof the developed numerical scheme is validated against a few benchmark problems.The conclusions are drawn in the last section.

2. OUTLINE OF THE NUMERICAL SCHEME

2.1. Governing Equations

Generally, the incompressible Navier-Stokes equations denote the continuityand the momentum equations. These equations in vector form in Cartesian coordi-nates are given by

Ut þrp ¼ �ðU � rÞU þr2U

Reð1Þ

r �U ¼ 0 ð2Þ

on X Rd�Rþ, where d is the dimension of the problem, U is the velocity vector, p is

the pressure, Re is the Reynolds number defined by Re ¼ Uclcn , Uc is the characteristic

velocity, lc is the characteristic length, and n is the kinematic viscosity. SolvingEqs. (1) and (2) require some initial and boundary conditions. Such generalconditions are represented by U(X, 0)¼U0 and UjqX¼Ub, respectively.

2.2. Discrete Formulation of Governing Equations

The second order, time-discrete semi-explicit form of Eqs. (1) and (2) can bewritten as

Unþ1 �Un

Dtþrpnþ

12 ¼ �ðU � rÞU½ �nþ

12þ 1

Rer2ðUnþ1 þUnÞ ð3Þ

r �Unþ1 ¼ 0 ð4Þ

subjected to the boundary condition Unþ1jqX ¼ Unþ1b . The spatially discretized

version of the coupled system Eqs. (3) and (4) is cumbersome to solve directly. There-fore, a fractional step procedure is used to approximate the solution of the coupledsystem by first solving an analog to Eq. (3), (excluding the divergence constraint) for



an intermediate quantity U�. Then U� is projected onto the space of divergence freefield to get Unþ1. The sequence of steps of the algorithm is as follows.

Step 1: Computation of U�

U� �Un

Dtþrq ¼ �ðr �UÞUnþ1

2 þ 1

2Rer2 U� þUnð Þ ð5Þ

U�jqX ¼ Unþ1b ifrq ¼ rpn�

12 ð6Þ

and

ðr �UÞUnþ12 ¼ 3

2ðr �UÞUn � 1

2ðr �UÞUn�1 ð7Þ

In Eq. (5), the convection and diffusion terms are discretized using Adams-Bashforthand Crank-Nicholson schemes, respectively.

Step 2: Projection

U� ¼ Unþ1 þ Dtr/nþ1 ð8Þ

where / is the pressure correction, computed by enforcing Unþ1¼ 0.Step 3: Updating of the pressure

pnþ12 ¼ qþ K/nþ1 ð9Þ

where K ¼ I � Dt2Rer2

� �if q ¼ pn�

12:

3. IMPLEMENTATION

3.1. Compact Schemes in Matrix Form

In this section, our approach to develop a compact scheme in matrix form isbriefly described. In general, in the compact schemes a linear combination of deriva-tives is equated with a linear combination of function values. In the case of a struc-tured Cartesian grid, if a line by line marching approach is considered, then on everyline enclosing the boundary nodes there will be several interior nodes. Therefore, theapproximation of the derivatives using compact schemes, on all nodes along a parti-cular line, involves an implicit solution procedure and is represented in the matrixequation as,

½A�F ¼ ½B�f ð10Þ

where [A] and [B] are the coefficient matrices and F and f are the column vectors ofthe derivative and function values, respectively. The decision of either including orexcluding the boundary nodes in the system Eq. (10) has to be made dependingon the requirement of imposing Dirichlet, Neumann, periodic, or nonperiodicboundary conditions. In the next subsection, we demonstrate the utilization of the



compact FD schemes, in the form of Eq. (10), to factorize the discrete momentumequations given by Eq. (5).

3.2. D’yakanov ADI Procedure for Discrete Momentum Equation

In Eq. (5), the convection and diffusion terms have been subjected to anexplicit and implicit treatment, respectively, and so its solution involves a matrixinversion. In particular, the coefficient matrix of the unknown vector in Eq. (5) isof the form (I�D), where I is the identity matrix and D is the Laplacian. Forexample, the second order discretization of this matrix yields a banded pentadiago-nal matrix which should be iteratively inverted using methods like the stronglyimplicit procedure (SIP). Furthermore, the compact fourth order discretizationyields a full matrix, whose inversion is computationally very expensive. Therefore,in this work we have factorized the implicit part in a similar fashion to that ofreference [12], which is explained below. Considering for example, the u momentumequation from Eq. (5) and factorizing it in two dimensions yields,

1� Dt2Re

q2

qx2

!1� Dt

2Re

q2

qy2

!u�i;j ¼ � qp

qx

n�12

� 3

2r �Uð Þun þ 1

2r �Uð Þun�1

þ 1

2Re

q2uqx2

þ q2uqy2

!n ð11Þ

The calculation of the pseudo velocity u� from Eq. (11) involves the matrix manip-ulations in its LHS. Herein, we apply the D’yakanov ADI-like procedure making useof Eq. (10). Now, the compact approximations of the derivatives in the RHS ofEq. (11) are estimated a priori due to their explicit treatment and stored in say ci,j.Then, Eq. (11) can be written as

A�1xx TxxA

�1yy Tyyu

�i;j ¼ ci;j ð12Þ

where Txx ¼ Axx � Dt2ReBxx

� �and Tyy ¼ Ayy � Dt

2ReByy

� �. The symbols Axx, Ayy

represent the coefficient matrices of the second derivatives and Bxx, Byy representthe coefficient matrices of the function values in x and y directions, respectively.Eq. (12) is solved in two stages which are analogous to LU decomposition and isgiven by

A�1yy Tyyu

�i;j ¼ zi;j ð13Þ

A�1xx Txxzi;j ¼ ci;j ð14Þ

The first stage involves two steps. In the first step, the quantity zi,j is calculated usingEq. (13) and Eq. (6) for the boundary values of u� along y direction at the physical xboundaries of the computational domain. The calculations in this step only involvethe tridiagonal matrix inversion, which is computationally cost effective. Subse-quently, in the second step Eq. (14) is solved along the constant y lines, to compute



zi,j for all interior nodes, making use of zi,j already computed along boundary nodesin the first step. In the second stage, the pseudo velocities are calculated for allinterior nodes in the computational domain by solving Eq. (13) along the constantx lines. Here it is worth mentioning that in the second step of the first stage andin the second stage the bandwidth of the coefficient matrix may slightly increase fromthat of a tridiagonal matrix. Therefore, in these calculations the BICGSTAB (2)algorithm has been used to obtain a faster converged solution.

3.3. Details of the Compact System for Various Spatial Derivatives

The schematic of the staggered Cartesian grid used in the simulations is givenin Figure 1. The horizontal and vertical stencil arrangements to deduce the compactsystems of form Eq. (10) for all x and y partial derivatives of Eq. (11), have to beidentified from this figure. In the subsequent discussion of this subsection, the detailsof the compact system used to discretize the convection, diffusion, and the pressuregradient terms of Eq. (11) are elaborated.

3.3.1. Convection term. Convection terms can be represented in four differ-ent forms: advection, divergence, rotational, and skew-symmetric. The conservationproperties of these forms in terms of mass, momentum, and kinetic energy have beendiscussed in reference [23]. Among these four forms the skew-symmetric form satis-fies the conservation property in all aspects and has been proved reliable in reference[24]. The mathematical expression for the skew-symmetric convection term in itstensor form is given by,

1

2

qujuiqxj

þ ujquiqxj

� �ð15Þ

where i, j¼ 1, 2 for the two-dimensional case. It is clear from Eq. (15), that theskew-symmetric form of the convection term is equal to the average of its divergenceand the advection forms. The respective discretized system of equations usingcompact FD schemes for each of these terms in two dimensions is given below.

Figure 1. Schematic of the staggered Cartesian grid.



3.3.1.1. Divergence part of the convection term. This part contains thecomputation of the partial derivatives of the form qu2

qx ,qv2qy ,

qvuqy ,

quvqx . Among these four,

the partial derivatives with respect to x have to be obtained over horizontal lines,and the other over vertical lines. The stencil arrangement given in Figure 2a,considered along the pressure nodes with w0 and wN representing values of theunknown at the physical boundary, is used to compute the derivatives of the squareterms; whereas, partial derivatives of the cross product terms are computed overvelocity nodes as given in Figure 2b, with /0 and /N representing the boundarylocations wherein the cross product of velocity has to be interpolated. The whitecircles in Figure 2a correspond to the pressure nodes and the filled circles inFigure 2b represent the vertices, wherein the velocity values have to be interpolated.For example, the compact system for qu2

qx with Dirichlet boundary conditions isgiven by

qu2

qx

��w1

� qu2

qx

��w2

¼ 1

Dx�u2u0

þ 2u2u1� u2u2

� �þ oðDx3Þ ð16Þ

qu2

qx

��wi�1

þ 22qu2

qx

��wi

þ qu2

qx

��wiþ1

¼ 24

Dxu2uiþ1

� u2ui�1

� �þ oðDx4Þi ¼ 2; 3; . . . ;N � 2 ð17Þ

qu2

qx

��wN�1

� qu2

qx

��wN�2

¼ 1

Dxu2uN

� 2u2uN�1þ u2uN�2

� �þ oðDx3Þ ð18Þ

Thus, the system of equations will have the tridiagonal form of the coefficient matrixon the LHS. Similar set of equations, as given in Eqs. (16)–(18), have been derivedfor other terms.

3.3.1.2. Advection part of the convection term. This part containsthe computation of the partial derivatives of the form u qu

qx, v qvqy, v qu

qy, and u qvqx.

Among these four, once again the partial derivatives with respect to x have to beobtained over horizontal lines and the remaining terms over vertical lines. The stencil

Figure 2. Stencil arrangement for divergence part of the convection term. (a) qu2=qx or qv2=qy; and(b) quv=qx or qvu=qy.



arrangement given in Figure 3a, considered along the pressure nodes, is used to com-pute u qu

qx, vqvqy with w0 and wN representing the values of the unknown at the physical

boundary; whereas, v quqy and u qv

qx are computed over velocity nodes, as given inFigure 3b, with w0 and wN representing the values of the unknown at the fictitiousboundary point. It must be mentioned here that in the calculation of v qu

qy and u qvqx,

the convective velocities v and u at their corresponding derivative locations are esti-mated using a four-point interpolation. The white circles in Figure 3a correspond tothe pressure nodes, and the filled circles in Figure 3b represent the vertices of thecomputational cells. For example, the compact system for qu

qx with Dirichlet boundaryconditions is as follows.

quqx

��w1

þ quqx

��w2

¼ 1

2Dx�uw0

� 4uw1þ 5uw2

� �þ o Dx3

� �ð19Þ

quqx

��w1

þ quqx

��w2

¼ 1

6Dx�uw0

� 9uw1þ 9uw2

þ uw3

� �þ o Dx4

� �ð20Þ

quqx

��wi�1

þ 4quqx

��wi

þ quqx

��wiþ1

¼ 3

4Dxuwiþ1

� uwi�1

� �þ o Dx4

� �i ¼ 2; 3; . . . ;N � 2 ð21Þ

quqx

��w1

þ quqx

��w2

¼ 1

2DxuwN

þ 4uwN�1� 5uwN�2

� �þ o Dx3

� �ð22Þ

quqx

��wN�1

þ quqx

��wN�2

¼ 1

6Dxuw0

þ 9uw1� 9uw2

� uw3

� �þ o Dx4

� �ð23Þ

Equations (19) and (22) or (20) and (23) have to be used for the immediate neighborof the boundary nodes. Thus, the system of equations will have the tridiagonalform of the coefficient matrix on the LHS. Similar set of equations, as given inEqs. (19)–(23), have been derived for other terms.

3.3.2. Diffusion terms. Again, the schematic shown in Figure 3a illustratesthe grid arrangement for the discretization of the diffusive terms q2u

qx2 or q2vqy2, and

Figure 3. Stencil arrangement for advection part of the convection term. (a) qu=qx or qv=qy; and (b) qv=qxor qu=qy.



Figure 3b for q2uqy2 or q2v

qx2. For example, the compact system for q2uqx2 with Dirchlet

boundary conditions is given by

q2uqx2

��w0

þ 12q2uqx2

��w1

¼ 1

Dx2167

12uw0

� 86

3uw1

þ 31

2uw2

� 2

3uw3

� 1

12uw4

� �þ o Dx3

� �ð24Þ

q2uqx2

��wi�1

þ 10q2uqx2

��wi

þ q2uqx2

��wiþ1

¼ 6

5Dx2uwiþ1

� 2uwiþ uwi�1

� �

þ o Dx4� �

i ¼ 1; 2; . . . ;N � 1

ð25Þ

q2uqx2

��wN

þ 12q2uqx2

��wN�1

¼ 1

Dx2167

12uwN

� 86

3uwN�1

þ 31

2uwN�2

�

� 2

3uwN�3

� 1

12uwN�4

�þ o Dx3

� � ð26Þ

Thus, the system of equations will have the tridiagonal form of the coefficient matrixon the LHS. Similar set of equations, as given in Eqs. (24)–(26) after taking care ofthe necessary grid orientations, are applicable to the remaining diffusive terms.

3.3.3. Pressure gradient term. For the discretization of the terms qpqx or qp

qy,which appear in the momentum equations, the schematic shown in Figure 2a hasbeen used after replacing u2 or v2, wherever present, with pressure p. For example,in the compact system for qp

qx, we have used the same system of equations as givenin Eqs. (16)–(18).

4. PRESSURE POISSON EQUATION

For the fractional step algorithm used in this work, the Poisson equation forthe pressure correction (/) has to be deduced by taking the divergence of Eq. (8)to give

r � r/ ¼ r �U�

Dtð27Þ

In our work, instead of discretizing Eq. (27) straightforward, we follow the conven-tion of deriving the discrete equation for pressure correction through the continuityequation which involves the substitution of Eq. (8) into Eq. (4). Herein, we show thedistinction between the conventional second order and compact fourth order discreteequations for pressure correction for the purpose of comparing the difference in thestructure of the coefficient matrix of the unknown vector. The second order accuratediscrete form of Eq. (4) at the nþ 1 time level obtained at any cell center (i, j) whereinthe pressure node is located, is given by

uiþ12;j� ui�1

2;j

Dxþvi;jþ1

2� vi;j�1

2

Dy¼ 0 ð28Þ



Using the discrete form of Eq. (8) in Eq. (28) gives,

�/i�1;j þ /iþ1;j

� �Dx2

þ 21

Dx2þ 1

Dy2

� �/i;j �

/i;j�1 þ /i;jþ1

� �Dy2

¼ 1

Dt

ui�12;j� uiþ1

2;j

Dxþvi;j�1

2� vi;jþ1

2

Dy

� ð29Þ

Equation (29) involves the conventional five-point stencil arrangement. Similarly, thecompact fourth order approximation of the first derivatives in Eq. (4) gives,

dxui;j

Dx 1þ Dx2d2x24

� �þ dyvi;j

Dy 1þ Dy2d2y24

� � ¼ 0 ð30Þ

where, for example, dxui;j ¼ uiþ12;j� ui�1

2;jand d2xui;j ¼ uiþ1;j � 2ui;j þ ui�1;j.

Using the discrete form of Eq. (8) into Eq. (30) yields,

� Dt24

1

Dx2þ 1

Dy2

� �/iþ1;jþ1 þ /iþ1;j�1 þ /i�1;j�1 þ /i�1;jþ1

� �þ 11Dt

6

1

Dx2þ 1

Dy2

� �/i;j �

Dt12

11

Dx2� 1

Dy2

� �/i�1;j þ /iþ1;j

� �

� Dt12

11

Dy2� 1

Dx2

� �/i;jþ1 þ /i;j�1

� �¼

11 ui�12;j� uiþ1

2;j

� �12Dx

þ11 vi;j�1

2� vi;jþ1

2

� �12Dy

þui�1

2;jþ1 þ ui�12;j�1 � uiþ1

2;jþ1 � uiþ12;j�1

� �24Dx

þviþ1;j�1

2þ vi�1;j�1

2� viþ1;jþ1

2� vi�1;jþ1

2

� �24Dy

ð31Þ

Equation (31) involves the nine-point stencil arrangement. Equations (29) or (31)have been solved after imposing q/

qn ¼ 0 on the boundaries using the BICGSTAB(2) algorithm. Further, in the pressure correction computation we have also adoptedthe inner iteration strategy as recommended by reference [29], which ensures theincompressibility constraint at every discrete time step.

5. NUMERICAL EXPERIMENTS

This section is devoted to assess the performance of the developed compacthigher order numerical solver (NS). To start with, the spatial and temporal accuracyof the solver is tested by comparing the numerical and exact solutions of the decay-ing vortex problem governed by the NS equation. Later on, to ensure the robustnessof the solver in simulating the fluid dynamic problems of academic and practicalinterest, two well explored benchmark problems, namely the steady lid driven cavityand backward facing step flows, have been simulated.



5.1. Decaying Vortices

The analytical solution to the velocities and pressure for the two-dimensionalunsteady flow of decaying vortices in a domain bounded by x2 [0, 1] and y2 [0, 1]is given by

u x; y; tð Þ ¼ �e�4p2tRe

� �sin pxð Þ cos pyð Þ

v x; y; tð Þ ¼ �e�4p2tRe

� �cos pxð Þ sin pyð Þ

p x; y; tð Þ ¼ 1

4e

�4p2tRe

� �cos 2pxð Þ þ cos 2pyð Þ½ � ð32Þ

To assess the temporal and spatial accuracy of the solver, a couple of numericalinvestigations have been conducted. The first investigation is the numerical simula-tion of the time evolution of decaying vortices for two different combinations ofReynolds numbers (Re) and nondimensional time (s), viz., Re¼ 1000 until s¼ 10.0and Re¼ 10 until s¼ 0.3. In the evaluation of temporal accuracy, the L2 error normbehavior of both velocities and pressure are compared; whereas, the L2 error normbehavior of the pressure is emphatic enough to comment on the spatial accuracy ofany incompressible flow simulations. Figures 4a and 4b illustrate the ability of thedeveloped solver in producing the second order time accurate solutions for velocityand pressure. The fractional step method recommended by reference [31] isresponsible for generating these high order solutions. These computations have beenmade with the time step ranging between 5� 10�4 and 5� 10�1 over a fixed gridspacing of 1

64.Similarly, in Figures 5a and 5b, the pressure solutions obtained by refining the

grid size gradually from 16� 16 to 256� 256 for a fixed time step of 5� 10�4, areplotted in order to evaluate the spatial accuracy of the developed solver. Here, theabscissa and the ordinate denote the number of nodes (N) and the L2 error-norm,

Figure 4. L2 Errors norms versus time step: (a) Re¼ 10 and s¼ 0.3; and (b) Re¼ 1000 and s¼ 10.



respectively. On comparing the error norm of the pressure patterns obtained for thesame parametric study conducted in reference [24], it is apparent that our solver hasattained the fourth order spatial accuracy.

5.2. Two-Dimensional Lid-Driven Cavity (LDC) Flow

This problem involves the solution of the NS equations in a unit squaredomain where the upper boundary moves with a uniform nondimensional velocity(u¼ 1, v¼ 0). No-slip boundary conditions are applied at all other boundaries(u¼ 0, v¼ 0). An earlier analysis on this problem reported in reference [34] showsthat for Re greater than 8000, the first Hopf bifurcation occurs due to the steadysolution losing its stability and attaining a steady periodic solution. Since ourpurpose is only to validate the developed solver, we restrict ourselves to the steadystate solution regime. Hence, for this problem the numerical solutions have beensimulated at two different Reynolds numbers, viz., Re¼ 400 and 7500. The uand v velocity profiles, respectively, about the geometric vertical centerline (g.v.c)and geometric horizontal centerline (g.h.c) of the cavity are the benchmarkcomparisons usually made on this problem. In our validation, at first forRe¼ 400, the grid independence study is performed using different grid spacings,viz., 1

32,148,

164,

184,

196,

1108, and

1128. The actual and magnified images of the u velocity

profiles about the g.v.c, for the abovementioned range of grids are compared inFigures 6a and 6b, respectively. The discrepancies among the results, in comparis-ing them with Ghia et al. [35], are found to be negligible for the grid spacingbeyond 1

84. Further verification to this assertion is achieved by showing that forRe¼ 400, the same 84� 84 grid is insufficient to produce comparable results withthat of Ghia et al. [35] when the fourth order accurate spatial discretization isreplaced with the second order counterpart. This feature is illustrated throughthe comparative plots of u and v velocity profiles shown in Figures 7a, 7b, 8a,

Figure 5. L2 Errors norms versus grid density: (a) Re¼ 10 and s¼ 0.3; and (b) Re¼ 1000 and s¼ 10.



and 8b, respectively. Similarly, streamlines and vorticity fields for the same Replotted in Figures 9a and 9b, respectively, also strengthens our contention. In orderto check the reliability of the developed numerical method at high Re simulations,an extreme value of 7500, beyond which only steady periodic solution is reported,has been validated on a 160� 160 grid. The streamlines and vorticity contours forthis case, shown in Figures 10a and 10b, respectively, are found to have capturedall essential patterns that are reported in an earlier work [35] for this Re, but witha grid size of 256� 256.

Figure 6. Comparison of versus component of velocity along g.v.c in a LDC at Re¼ 400, for various grid

resolutions. (a) Actual profile and (b) magnified profile.

Figure 7. Comparison of versus component of velocity along g.v.c in a LDC at Re¼ 400, with a grid size

of 84� 84. (a) Actual profile and (b) magnified profile.



5.3. Flow Over a Backward-Facing Step (BFS)

This problem deals with the incompressible laminar channel flow over a BFS ofheight (h). The length (L) and the height (H) of the downstream channel are chosento be (40h) and (2h), respectively. At the inlet of the channel, a fully developedvelocity profile represented by u(0.5� y� 1)¼ 24(1� y)(y� 0.5) has been imposed.This profile yields the maximum umax and average inflow uavg velocities of 1.5 and1.0, respectively. The schematic of the computational domain with details to the

Figure 8. Comparison of versus component of velocity along g.h.c in a LDC at Re¼ 400, with a grid size

of 84� 84. (a) Actual profile and (b) magnified profile.

Figure 9. Streamlines and vorticity contours in a LDC at Re¼ 400, with a grid size of 84� 84. (a) Stream-

lines and (b) vorticity.



geometry and boundary conditions is shown in Figure 11. Here, the Reynoldsnumber is defined as Re ¼ uavg

hn , where n refers to kinematic viscosity. The primeobjective to solve this problem is to verify the applicability of the present solver whenapplied to problems with outflow boundary conditions. The two representative Re of200 and 800 are taken for this purpose and simulated on a computational domainwith a grid spacing of 1

40. The streamline patterns of the converged steady state solu-tions for both the cases are plotted in Figures 12a and 12b, respectively. The lengthof the recirculation zone (x1) behind the step for both the Re, and the point ofseparation (x2) and reattachment (x3) of an upper eddy found at Re¼ 800, similarto reference [36], is given in Table 1. Further, the u velocity profiles at x¼ 7 andx¼ 15 for Re¼ 800 shown in Figure 13a and 13b are found to be in good agreementwith the results of Gartling [37].

Figure 11. Schematic of a BFS.

Figure 10. Streamlines and vorticity contours in a LDC at Re¼ 7500, with a grid size of 160� 160.

(a) Streamlines and (b) vorticity.



Figure 13. Comparison of the versus component of velocity profiles for a flow over a BFS at Re¼ 800 at

two different sections along stream-wise direction of the flow. (a) x¼ 7 and (b) x¼ 15.

Table 1. Comparison of locations of recirculation (x1), separation (x2), and reattachment (x3)

Reynolds number (Re)Re¼ 200 Re¼ 800

Length Present [36] Present [36]

x1 2.62 2.5906 6.002 6.1405

x2 – – 4.85 4.8054

x3 – – 10.3057 10.2202

Figure 12. Streamline pattern for a flow over a BFS at (a) Re¼ 200 and (b) Re¼ 800.



6. CONCLUSION

In this study, a robust method of using higher order compact FD schemes tosolve UINSE on a staggered grid has been analyzed. All of the necessary compactdiscretization procedures to various derivative terms in the governing equations havebeen clearly explained. An efficient factorization strategy, based on D’yakanovADI-like scheme, adopted for the implicit part of the momentum equation has beenhighlighted. The developed scheme has the ease in implementation of Dirichlet orNeumann boundary conditions. Further, a novel approach of deriving fourth orderaccurate pressure Poisson solver on staggered grids has been introduced andcompared with its second order counterpart. Finally, the validation of the developedsolver with the benchmark solutions prove the improvement in the spatial accuracyattained while using compact finite difference schemes in place of conventionalsecond order schemes.

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