chew shu niu (ijmpc) 2002

8/14/2019 Chew Shu Niu (Ijmpc) 2002

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International Journal of Modern Physics C, Vol. 13, No. 6 (2002) 719738c World Scientific Publishing Company

SIMULATION OF UNSTEADY INCOMPRESSIBLE

FLOWS BY USING TAYLOR SERIES EXPANSION- AND

LEAST SQUARE-BASED LATTICE BOLTZMANN METHOD

Y. T. CHEW, C. SHU, and X. D. NIU

Department of Mechanical Engineering, National University of Singapore

10 Kent Ridge Crescent, Singapore 117576

Received 4 December 2001Revised 31 January 2002

In this work, an explicit Taylor series expansion- and least square-based lattice Boltz-mann method (LBM) is used to simulate two-dimensional unsteady incompressible vis-cous flows. The new method is based on the standard LBM with introduction of theTaylor series expansion and the least squares approach. The final equation is an explicitform and essentially has no limitation on mesh structure and lattice model. Since theTaylor series expansion is only applied in the spatial direction, the time accuracy of the

new method is kept the same as the standard LBM, which seems to benefit for unsteadyflow simulation. To validate the new method, two test problems, that is, the vortexshedding behind a circular cylinder at low Reynolds numbers and the oscillating flow ina lid driven cavity, were considered in this work. Numerical results obtained by the newmethod agree very well with available data in the literature.

Keywords: Lattice Boltzmann equation; explicit method; Taylor series expansion; leastsquare approach; unsteady flow; incompressible.

1. Introduction

The study of unsteady fluid systems has been a major interest for fluid dynamicresearchers over almost the last whole century due to its great relevance to en-

gineering applications in reality. As the need to account for the effect of time-

dependence presents a considerable difficulty of analysis, numerical simulation plays

an important role in this field.

Conventional methods for simulating viscous unsteady flow include, macroscop-

ically, numerical integration of the NavierStokes equations, and, microscopically,

molecular dynamics simulation. However, the former has particular difficulty on

the implementation of complex geometries while the latter is extremely intensive in

computation. Recently, the lattice Boltzmann method (LBM),1,2

which originatedfrom the lattice gas cellular automata (LGCA), has provided an alternative ap-

proach for solving continuum problems on various physical systems. The LBM is

based on gas-kinetic representations of fluid flow in a strongly reduced particle ve-

locity space, in which flow is described through the evolution of the discrete particle

719


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720 Y. T. Chew, C. Shu & X. D. Niu

distribution functions on uniform lattices. Hydrodynamic variables are computed

at the lattice nodes as moments of the discrete distribution functions.

One of the advantages of the LBM is that it can recover the unsteady in-compressible NavierStokes equations with second-order of accuracy in space and

time at low Knudsen number and low frequency limit through Taylor series and

ChapmanEnskog expansions.3,4 Thus it is expected that the LBM can simulate

the unsteady fluid problems in well accuracy. In fact, under the low frequency limit,

the characteristic timeTof the LBM is in the order of (M a)1t, whereM ais the

global Mach number, is the Knudsen number and t is the particle streaming

time. Therefore, the LBM can be applied to the unsteady fluid systems with a

range of Strouhal numbers ofStr = L/UT = t/TMa O (1) (L and U is the

characteristic length and velocity, respectively). Many numerical experiments58have confirmed this conclusion.

The major advantage of LBM is its algebraic form and ease for application.

However, due to the use of uniform lattice, the broad application of the LBM in

engineering has been greatly hampered, especially for the problems with curved

boundaries or with high Reynolds numbers, which need to use a nonuniform mesh

to obtain high-resolution results in the very thin boundary layer. Theoretically, the

feature of lattice-uniformity is not necessary to be kept because the distribution

functions are continuous in physical space. Currently, there are two ways to re-

move the difficulty of the standard LBM for application to complex problems withnonuniform meshes. One is the so-called interpolation-supplemented LBM (ISLBM)

proposed by He and his colleagues.79 They successfully applied this approach to

simulate flows past an impulsively started cylinder. The other is based on the so-

lution of a differential lattice Boltzmann equation (LBE). For complex problems,

the differential LBE can be solved by the finite difference (FDLBE) method with

the help of coordinate transformation10 or by the finite volume method.11,12 These

methods have been successfully applied to solve quite a number of complex prob-

lems. However, the ISLBE has an extra computational effort for interpolation at

every time step, and it also has a strict restriction on the selection of interpo-lation points. For the FDLBE and FVLBE methods, one need to select efficient

approaches such as upwind schemes to do numerical discretization in order to get

the stable solution. As a consequence, the computational efficiency greatly depends

on the selected numerical scheme. In addition, the time accuracy of FDLBE and

FVLBE methods is reduced due to truncation error of discretization in time as

compared to the standard LBM.

In the present work, an explicit Taylor series expansion- and least square-based

LBM is proposed. The new method is based on the standard LBM with introduction

of Taylor series expansion in the spatial direction and least squares optimization.The final form of the method is an algebraic formulation, in which the coefficients

depend only on the coordinates of mesh points and lattice velocity, and are com-

puted in advance. The new method can be easily applied to different other lattice

models. Since the Taylor series expansion is applied only in the spatial direction, the


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Simulation of Unsteady Incompressible Flows 721

time accuracy of the new method is kept the same as the standard LBE. Thus, it

is expected that the new method can well simulate the unsteady flow problems. To

validate this, two unsteady flows problems are considered in this work. One is thevortex shedding flow behind a circular cylinder. The investigation of vortex shed-

ding in the wake of a circular cylinder at low Reynolds number is very important

because it can provide some basic insights into the vortex shedding mechanism.

The other case is the oscillating flow in a lid driven cavity. This problem is often

chosen to test new numerical schemes for simulation of unsteady flows. There are

many publications8,1322 on these two cases in the literature. So, it is ideal to use

them as unsteady benchmarks to verify the present method.

2. Taylor Series Expansion- and Least Squares-Based LBM

The method developed in this work is based on the fact that the distribution func-

tion is a continuous function in physical space and can be well defined in any mesh

system. Let us start with the standard LBM. The two dimensional, standard LBE

with BGK approximation can be written as:

f(x+ext,y+eyt,t+t)

=f(x,y,t) +feq (x,y,t) f(x,y,t)

, = 0, 1, . . . , M , (1)

where is the single relaxation time; f is the density distribution function along

the direction; feq is its corresponding equilibrium state, which depends on the

local macroscopic variables such as a density and velocity U(u, v); t is the time

step and e(ex, ey) is the particle velocity in the direction; M is the number

of discrete particle velocities. Obviously, the standard LBE consists of two steps:

collision and streaming. The macroscopic density and momentum density Uare

defined as:

=

M=0

f, U=

M=0

fe. (2)

Suppose that a particle is initially at the grid point (x,y,t). Along the direc-

tion, this particle will stream to the position (x+ ext,y+eyt,t+ t). For a

uniform lattice, x = ext, y = eyt. So, (x+ ext, y+ eyt) is on the grid

point. In other words, Eq. (1) can be used to update the density distribution func-

tions exactly at the grid points. However, for a nonuniform grid, (x+ext, y+eyt)

is usually not at the grid point (x+ x, y+ y). In the numerical simulation, we

are interested only in the density distribution function at the mesh point for all

the time levels. So, the macroscopic properties such as the density, flow velocitycan be evaluated at every mesh point. To get the density distribution function at

the mesh point (x+ x, y +y) and the time level t+ t, we need to apply the

Taylor series expansion or other interpolation techniques such as the one used by

He et al.79 In this work, the Taylor series expansion is used. Note that the time


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P

A

B

C

D

E

A'

B'

C'

D'

P'

E'

Fig. 1. Configuration of particle movement along the direction.

level for the position (x + ext, y + eyt) and the grid point (x + x,y + y) is the

same, that is, t+t. So, the expansion in the time direction is not necessary. As

shown in Fig. 1, we let point A represent the position (xA, yA) point A represent

the position (xA+ ext, yA+ eyt), and point Prepresent the position (xP, yP).

Using Eq. (1), we can get the density distribution function at the position A

f(A, t+t) =f(A, t) +

feq (A, t) f(A, t)

. (3)

For the general case, A

may not coincide with the mesh point P. In this case, weneed to obtain the density distribution function at the mesh point P. This can be

done by applying the Taylor series expansion in the spatial direction only. With

Taylor series expansion, f(A, t+t) can be approximated by the corresponding

function and its derivatives at the mesh point P as:

f(A, t+t) =f(P, t+t) + xA

f(P, t+t)

x + yA

f(P, t+t)

y

+1

2(xA)

2 2f(P, t+t)

x2 +

1

2(yA)

2 2f(P, t+t)

y2

+ xAyA2f(P, t+t)

xy +O[(xA)

3, (yA)3] , (4)

where xA = xA + ext xp, yA = yA +eyt yp. Note that the above

approximation has a truncation error of the third order. Substituting Eq. (4) into

Eq. (3) gives

f(P, t+t) + xAf(P, t+t)

x

+ yAf(P, t+t)

y

+1

2(xA)

2 2f(P, t+t)

x2 +

1

2(yA)

2 2f(P, t+t)

y2

+ xAyA2f(P, t+t)

xy =f(A, t) +

feq (A, t) f(A, t)

. (5)


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It is indicated that Eq. (5) is a differential equation, which involves only two

mesh pointsAand P. Solving Eq. (5) can provide the density distribution functions

at all the mesh points. Equation (5) can be considered as a new version of differentialLBE, which can give very accurate numerical results. In this work, we go further

to develop a new solution procedure. In fact, our new development is inspired from

the RungeKutta method. As we know, the RungeKutta method is developed to

improve the Taylor series method in the solution of ordinary differential equations

(ODEs). Like Eq. (5), Taylor series method involves evaluation of different orders of

derivatives to update the functional value at the next time level. For a complicated

expression of given ODEs, this application is very difficult. To improve the Taylor

series method, the RungeKutta method evaluates the functional values at some

intermediate points and then combines them (through the Taylor series expansion)to form a scheme with the same order of accuracy. With this idea in mind, we look

at Eq. (5). We know that at the time level t + t, the density distribution function

and its derivatives at mesh pointPare all unknowns. So, Eq. (5) has six unknowns

in total. To solve for the six unknowns, we need six equations. However, Eq. (5)

just provides one equation. We need additional five equations to close the system.

As shown in Fig. 1, we can see that along the direction, the particles at five mesh

points P, B, C, D, E at the time level t will stream to the new positions P, B,

C,D, E at the time level t + t. The density distribution functions at these new

positions can be computed through Eq. (1), which are given below

f(P, t+t) =f(P, t) +

feq (P, t) f(P, t)

, (6)

f(B, t+t) =f(B, t) +

feq (B, t) f(B, t)

, (7)

f(C, t+t) =f(C, t) +

feq (C, t) f(C, t)

, (8)

f(D

, t+t) =f(D, t) +

feq (D, t) f(D, t)

, (9)

f(E, t+t) =f(E, t) +

feq (E, t) f(E, t)

. (10)

Using Taylor series expansion, f(P, t + t), f(B, t + t), f(C, t + t),

f(D, t+t), f(E, t+t) in above equations can be approximated by the func-

tion and its derivatives at the mesh point P. As a result, Eqs. (6)(10) can be

reduced to:

f(P, t+t) + xPf(P, t+t)

x

+ yPf(P, t+t)

y

+1

2(xP)

2 2f(P, t+t)

x2 +

1

2(yP)

2 2f(P, t+t)

y2

+ xPyP2f(P, t+t)

xy =f(P, t) +

feq (P, t) f(P, t)

, (11)


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f(P, t+t) + xBf(P, t+t)

x + yB

f(P, t+t)

y

+1

2(xB)

2 2f(P, t+t)

x2 +

1

2(yB)

2 2f(P, t+t)

y2

+ xByB2f(P, t+t)

xy =f(B, t) +

feq (B, t) f(B, t)

, (12)

f(P, t+t) + xCf(P, t+t)

x + yC

f(P, t+t)

y

+

1

2(xC)2

2f(P, t+t)

x2 +

1

2(yC)2

2f(P, t+t)

y2

+ xCyC2f(P, t+t)

xy =f(C, t) +

feq (C, t) f(C, t)

, (13)

f(P, t+t) + xDf(P, t+t)

x + yD

f(P, t+t)

y

+1

2(xD)

2 2f(P, t+t)

x2 +

1

2(yD)

2 2f(P, t+t)

y2

+ xDyD2f(P, t+t)

xy =f(D, t) +

feq (D, t) f(D, t)

, (14)

f(P, t+t) + xEf(P, t+t)

x + yE

f(P, t+t)

y

+1

2(xE)

2 2f(P, t+t)

x2 +

1

2(yE)

2 2f(P, t+t)

y2

+ xEyE2f(P, t+t)

xy =f(E, t) +

feq

(E, t) f(E, t)

, (15)

where

xP =ext , yP =eyt ,

xB =xB+ ext xP, yB =yB+ eyt yP,

xC =xC+ext xP, yC =yC+eyt yP,

xD =xD+ext xP , yD =yD+eyt yP,

xE

=xE

+ex

t xP

, yE

=yE

+ ey

t yP

.

Equations (5), (11)(15) form a system to solve for six unknowns. Now, we

define

gi=f(xi, yi, t) +feq (xi, yi, t) f(xi, yi, t)

, (16)


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{s}T =

1, xi, yi,

(xi)2

2 ,

(yi)2

2 , xiyi

, (17)

{V} =

f,

fx

, f

y ,

2fx2

,2f

2y ,

2fxy

T, (18)

wheregi is the post-collision state of the distribution function at the ith point and

the time level t, {si}T is a vector with six elements formed by the coordinates of

mesh points, {V} is the vector of unknowns at the mesh point P, which also has

six elements. Our target is to find its first element V1 =f(P, t+t). With above

definitions, Eqs. (5), (11)(15) can be written as:

gi= {si}T{V} =

6j=1

si,jVj, i= P, A, B,C,D, E , (19)

where si,j is the jth element of the vector {si}T and Vj is the jth element of the

vector{V}.

Equation system (19) can be put into the following matrix form

[s]{V} = {g} , (20)

where

{g} = {gP, gA, gB, gC, gD, gE}T

,

[S] = [si,j ] =

{sP}T

{sA}T

{sB}T

{sC}T

{sD}T

{sE}T

=

1 xP yP(xP)2

2

(yP)2

2 xPyP

1 xA yA(xA)

2

2

(yA)2

2 xAyA

1 xB yB(xB)

2

2

(yB)2

2 xByB

1 xC

yC

(xC)2

2

(yC)2

2 x

Cy

C

1 xD yD(xD)

2

2

(yD)2

2 xDyD

1 xE yE(xE)

2

2

(yE)2

2 xEyE

.

Note that when lattice velocity is specified, the matrix [S] depends only on

the coordinates of mesh points, which can be computed once and stored for the

application of Eq. (20) at all time levels. In practical applications, it was found

that matrix [S] might be singular or ill conditioned. To overcome this difficulty andmake the method be more general, we propose the following least squares-based

LBM.

Equation (19) has six unknowns (elements of the vector {V}). If Eq. (19) is

applied at more than six mesh points, then the system is over-determined. For this


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case, the unknown vector can be decided from the least square method. For simplic-

ity, let the mesh pointPbe represented by the indexi= 0, and its adjacent points

be represented by index i = 1, 2, . . . , N , where N is the number of neighboringpoints around P and it should be larger than 5. At each point, we can define an

error in terms of Eq. (19), that is,

erri =gi 6

j=1

si,jVj, i= 0, 1, 2, . . . , N . (21)

The square sum of all the errors is defined as:

E=

N

i=0 err

2

i =

N

i=0

gi

6

j=1 si,jVj

2

. (22)

To minimize the error E, we need to set E/Vk = 0, k = 1, 2, . . . , 6, which

leads to:

[S]T[S]{V} = [S]T{g} , (23)

where [S] is a (N+ 1) 6 dimensional matrix, which is given as:

[S] =

1 x0 y0(x0)2

2

(y0)2

2 x0y0

1 x1 y1(x1)2

2

(y1)2

2 x1y1

1 xN yN(xN)2

2

(yN)2

2 xNyN

(N+1)6

and {g} = {g0, g1, . . . , gN}T

.The x and y values in the matrix [S] are given as:

x0 =ext, y0=eyt , (24a)

xi =xi+ext x0, yi =yi+eyt y0, for i= 1, 2, . . . , N . (24b)

Clearly, when the coordinates of mesh points are given, and the particle velocity

and time step size are specified, the matrix [S] is determined. Then from Eq. (23),

we obtain

{V} = ([S]

T

[S])

1

[S]

T

{g} = [A]{g} . (25)Note that [A] is a 6 (N+ 1) dimensional matrix. From Eq. (25), we can have

f(x0, y0, t+t) =V1

N+1k=1

a1,kgk1, (26)


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01

23

4

5 6

Fig. 2. Schematic plot of D2Q7 model on a solid boundary (thick black line).

where a1,k are the elements of the first row of the matrix [A], which are pre-

computed before the LBM is applied. Therefore, little computational effort is intro-

duced as compared with the standard LBE. Note that the function g is evaluated

at time level t. So, Eq. (26) is actually an explicit form to update the distribution

function at time level t+ t for any mesh point. In the above process, there is

no requirement for the selection of neighboring points. In other words, Eq. (26) is

nothing to do with the mesh structure. It needs only to know the coordinates of

the mesh points. Thus, we can say that Eq. (26) is basically a meshless form.

It can be seen that Eq. (26) is applied along the direction. Here can be

any direction. This implies that Eq. (26) can be uniformly applied to the different

lattice models. In this work, we use the D2Q7 model. The configuration of this

model is shown in Fig. 2. The discrete velocity of this model is defined as:

e=

(0, 0) , = 0 ,

cos

( 1)

3 , sin

( 1)

3 c , = 1, 2, . . . , 6 .

(27)

The parameter c is the particle streaming speed. The fluid kinetic viscosity isgiven by:

= 2 1

8 c2t, (28)

and the equilibrium density distribution feq is chosen to be

feq =

1

2+

1

6

2

e U

c2 + 4

e U

c2

2

U2

c2

. (29)

The speed of sound of this model is cs = c/2, and the equation of state isP = c2s for an ideal gas. Although the proposed method has meshless feature,

it is recommended to use a structured grid. This is because in our method, only

the coordinates of mesh points are involved. When a structured grid is used, it is

easy for us to define the coordinates of mesh points. In our application, we use a


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1 1,

i j+ +1

,

i j+1 1

,

i j +

1,i j

1 1,i j 1,i j 1 1,i j+

1,i j+,i j

Fig. 3. Schematic plot of neighboring point distribution around the point (i, j).

structured grid, and takeNas 8 for convenience. As shown in Fig. 3, for an internal

mesh point (i, j) [noted as 0 in Eq. (26)], the eight neighboring points are taken

as (i 1, j 1); (i 1, j); (i 1, j+ 1); (i, j 1); (i, j+ 1); (i + 1, j 1); (i + 1, j);

(i + 1, j+ 1). Therefore, at each mesh point, we need only to store nine coefficients

a1,k, k = 1, 2, . . . , 9 before Eq. (26) is applied. Note that the configuration of nine

mesh points as shown in Fig. 3 is applied in all lattice directions ( = 1, 2, . . . , 6).

Implementation of boundary conditions is an essiential issue in LBM. In this

work, we found that a complete half-way wall bounceback condition6 is the most

simple and efficient method in implementing the boundary condition on the soild

wall, where the nonslip condition holds. The complete half-way wall bounceback

condition, which originated from LGCA, assigns each f the value of the f in its

opposite direction with no relaxation on the bounceback points. The treament is

independent of the direction, which gives us more conveniences in treating compli-

cated boundary problems. The complete half-way wall bounceback condition has

second order of accuaracy because macroscopic quantities such as stress force is

evaluated on the half-way wall between the bounceback row and the first flow row.

As shown in Fig. 2, the flow field is below the solid boundary represented by the

black thick line. For the D2Q7 model, at a boundary point, f4, f5 and f6 point to

the flow field from the wall, which will be determined from the boundary condition.

f1, f2, f3 are computed by streaming from points inside the flow field. Note that

whenf1, f2, f3 are computed by Eq. (26), all the neighboring points involved must

be inside the flow field. In other words, the configuration as shown in Fig. 3 cannot

be used for this case. We need to select the neighboring points from one side(bottom

part of the wall as shown in Fig. 2). Using the half-way wall bounceback condition,

f4, f5 and f6 are evaluated as:

f4=f1, f5 =f2 , f6=f3. (30)

3. Results and Discussion

3.1. Vortex shedding behind a circular cylinder

As we know, a periodic vortex shedding will occur for flow behind a circular

cylinder when Reynolds number is above the critical value of 49, and the flow


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field is two-dimensional when Reynolds number is below the second critical value

around 149. Above this value, three-dimensional structure of the flow field be-

comes essential.13,18 In this part, our study focuses only on two-dimensional vor-tex shedding behind a circular cylinder. For this reason, the Reynolds numbers

(Re =U D/), based on the upstream velocity Uand the diameter of the cylinder

D, are chosen to be 50, 100 and 150.

In the flow domain, the following coordinate stretching is used in the radial

direction

r= r0+ (r r0)

1

1

tan1[(1 )tan()]

, (31)

where r0 = 1 is the cylinder radius, r is the outer boundary, is the parameter

to control the coordinate stretching and = (j 1)/(jmax 1) with j representingthe index of a mesh point in the radial direction. Uniform grid in the direction is

adopted.

Three boundary conditions are required in our simulation for this problem. One

is at the cylinder surface, where a complete half-way wall bounce back condition

described in the previous section is used. Another is on the cut line in the wake,

where the periodic boundary condition is imposed. The third is at the far field

boundary r where the free stream flow is taken and the density distribution

function is always set to its equilibrium state. A sketch of the problem is shown in

Fig. 4. Initially, an asymmetrical flow field

u= Ur0y

r2 , v= U

r0x

r2 , (32)

is imposed to serve as an artificial initiator for the vortex shedding process. The

far field velocityUand pressure are set to 0.15 and 1.0, respectively. The mesh size

of 241 241 is used for all of the above three Reynolds numbers (typical mesh is

shown in Fig. 5). The far field boundary is set at 25.5 diameters away from the

center of the cylinder and the time step, in units ofD/(2U), is equal to 0.00375,

which corresponds to = 0.72. Our experience showed that this is sufficient to

capture all the details of the flow at these three Reynolds numbers.

Periodic BC

y

x

q

R

U

Fig. 4. A sketch of the flow past a circular cylinder.


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Fig. 5. Computational mesh for flow around a circular cylinder.

The most attractive feature of the vortex shedding behind a circular cylinder is

the periodic variation of the flow field. This periodicity can be illustrated by the

time evolution of two characteristic parameters, the drag coefficient CD and the lift

coefficientCL. They are defined as:

CD = F x

(1/2)U2D, CL =

F y(1/2)U2D

, (33)

whereF =

[pI+ (U + U)] ndl. nis the normal vector of the cylinder sur-

face. The vortex shedding frequency f, which is expressed by the Strouhal number

ofS t= f D/U, can be obtained by measuring the final period of the lift coefficient.

Time

DragCoefficient

0 50 100 150 2000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

(a) Whole history

Time

DragCoefficient

150 160 170 180 190 200

1.25

1.3

1.35

1.4

1.45

1.5

(b) Close-up

Fig. 6. Time evolution of drag coefficients for different Reynolds numbers ( Re = 50;Re = 100; Re = 150).


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Time

LiftCoefficient

0 50 100 150 200-1

-0

.75

-0

.5

-0

.25

0

0.25

0.5

0.75

1

(a) Whole history

Time

LiftCoefficient

150 160 170 180 190 200-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

(b) Close-up

Fig. 7. Time evolution of lift coefficients for different Reynolds numbers ( Re = 50;Re = 100; Re = 150).

Figures 6 and 7 show the time evolution of the drag and lift coefficients for

different Reynolds numbers. The initially irregular variations of these coefficients

can be attributed to the initial disturbance. After a certain time, they gradually

evolve to periodic oscillations, which become stronger with increase of the Reynolds

number. The lift coefficient has a stronger oscillation than the drag coefficient. From

the close-up of these two figures, one can observe that the variation of the drag

coefficient is two times faster than that of lift coefficient. The reason may be due to

the fact that the drag coefficient is mainly affected by the vortex shedding processes

from both sides of the cylinder. This observation has been justified by the study of

Braza et al.19

Table 1. Comparisons of average and oscillatory drag and lift

coefficients and Strouhal numbers with previous studies.

Reynolds number 50 100 150

Williamson14 0.123 0.164 0.183St He et al.8 0.121 0.161 0.179

Present 0.123 0.164 0.176

Tritton20 1.450 1.350 1.330CD He et al.

8 1.394 1.250 1.261Present 1.461 1.3668 1.314

Braza et al.19 0.03

CD He et al.8

0.002 0.018 0.048Present 0.002 0.0242 0.5

Braza et al.19 0.60 CL He et al.

8 0.11 0.64 0.98Present 0.18 0.75 1.08


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Table 1 lists the detailed comparison of the average values and oscillations (peak

to peak) of the drag and lift coefficients, and the calculated Strouhal number St

with those of the previous studies.8,14,19,20 In Table 1,CD is the mean value of thedrag coefficient, while CD and CL are oscillations of drag and lift coefficients

from peak to peak. The results of He and Doolen8 were obtained by the LBM with

interpolation technique. The results of Tritton20 were experimental data. Other

data shown in Table 1 were the results of NavierStokes equations. Obviously, the

present results agree very well with those published previously. As compared to

the experimental data of Tritton,20 it seems that the present results have a better

accuracy than those of He and Doolen.8

Since the flow patterns are similar for all the simulated Reynolds numbers, only

the flow patterns at Re = 100 are presented in the paper. Figure 8 shows the se-quence of vortex shedding over a complete cycle using a sequence of instantaneous

streamlines and vorticity contours separated by intervals of T /8, where T is the

period of the shedding cycle. From the evolution of the streamlines (left of Fig. 8),

one can observe that a large vortex sheds from the top while a small vortex appears

at the bottom of the cylinder initially. In the meantime, fluid below the cylinder is

drawn down into the large recirculation region [Fig. 8(a)]. The small vortex form-

ing at the bottom of the cylinder grows and the large vortex travels to downstream

(a)t= T /8

(b) t= T /4

(c) t= 3T /8

Fig. 8. Streamlines (left) and vorticity contours (right) near the wake separated by an intervalofT /8.


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(d)t= T /2

(e) t= 5T /8

(f) t= 3T /4

(g) t= 7T /8

(h) t= T

Fig. 8. (Continued)

while its strength is reduced gradually [Figs. 8(b)8(d)]. When the strength of the

small vortex grows to its maximum, it breaks off. After that, another shedding pro-

cess from the bottom of the cylinder is repeated [Figs. 8(e)8(h)]. The mechanismof the vortex shedding is reflected also on the instantaneous vorticity contours. As

shown in Fig. 8 (right), the alternative change of the strength of a pair of asymmet-

ric secondary vortices is just the initiator of the vortex shedding from the cylinder.

The flow patterns in Fig. 8 match well with the numerical simulations by Eaton21


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and experimental observation by Perry et al.22 The centers (points surrounded by

closed streamlines), saddles (points where a streamline crosses itself), separatrices

(streamlines containing a saddle point) and instant alleyways between two sep-aratrices described by them are observed also in our simulation. However, we did

not observe the coexistence of centers and saddles of two shed vortices suggested

by Perry et al.22 Our simulation result is consistent with the simulation results of

He and Doolen8 and Eaton.21

3.2. Oscillating flow in a lid driven cavity atRe = 400

Another test case is the oscillating flow in a square cavity. A periodic velocity

waveform is imposed on the cavity lid and the time evolution observed in the flowis compared with the numerical results of Soh and Goodrich.16 The periodic lid

velocity is given by a sinusoidal waveform:

u(t) =Ucos(t) , (34)

where U is the maximum lid velocity during the cycle, is the frequency of the

oscillation andt is the time. The period of oscillation,T, is related to the frequency

byT = 2/.

Our simulation is performed with Re = 400 (Re = UL/, where L is the char-

acteristic length set as the value of top wall), frequency of = 1 and U = 0.15.

Initially, a periodic velocity given by Eq. (34) is imposed. A nonuniform grid of

97 97 with more densely distribution near the boundaries is used, and as a con-

sequence, the time step is set as, in the unit ofL/U, 4.5 104. The oscillatory

flow reaches the stable periodic state when the velocity components u and v of each

point in the domain at two consequent flow cycles are with a small tolerance of

= 105.

(a) Initial evolution (b) Later stage of evolution

Fig. 9. Time evolution of drag on the cavity lid.


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It was found that the flow reaches the stable periodic state after seven cycles.

Figure 9 shows the time evolution of the viscous drag on the lid, which is estimated

using the same formula as used by Soh and Goodrich.16 It can be observed thatthe drag settles down to be periodic very quickly, much quicker than the entire flow

field. The value of maximum drag obtained by the present method agrees very well

with that predicted by Soh and Goodrich.16

The instantaneous streamlines obtained by the present method match very well

with those of Soh and Goodrich,16 which are shown in Figs. 10 and 11 at t =

(5 +)T where = 0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.7, 0.8, 0.85, 0.9, 0.95 and 1.0,

respectively. Figure 10 shows the streamlines in the first half of the cycle while

Fig. 11 displays the streamlines in the second half of the cycle. As time advances,

the direction of the lid movement and the center of the vortex change. The lidvelocity passes through zero at t = T /4, after which it reverses its direction. As a

result, at t = 0.3T, a counter rotating vorticity is formed in the flow field at the

left corner of the cavity. As the magnitude of the velocity increases in the negative

x direction, the size of the second vortex created in the upper left corner of the

cavity also increases. At the same time, the primary vortex continuously shrinks

until t = T /2. At this point, the velocity reaches its maximum in the negative x

direction and the second vortex, which has formed in the left corner of the cavity,

(a) t= 0.2T (b) t= 0.3T (c) t= 0.35T

(d) t= 0.4T (e) t= 0.45T (f) t= 0.5T

Fig. 10. Instantaneous streamlines in the first half of a cycle obtained by the present method.


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(a)t= 0.7T (b) t= 0.8T (c)t= 0.85T

(d) t= 0.9T (e) t= 0.95T (f) t= T

Fig. 11. Instantaneous streamlines in the second half of a cycle obtained by the present method.

attains its maximum size and occupies the entire domain. After this point, the

streamlines at each time step are the mirror images of the streamlines at the time

t T /2. This finding can be obviously observed by comparing the results of Fig. 10

with those of Fig. 11. The same finding has been shown in the results of Soh

and Goodrich.16

3.3. Effect of grid size

To show that the present results are accurate, we performed numerical simulations

for the vortex shedding flow at Re = 100 and the oscillating flow in a lid driven

cavity by using two different mesh sizes. For the vortex shedding case, two mesh

sizes of 201 201 and 241 241 are used and the time steps are correspondingly

set to be, in units ofD/(2U), 0.0045 and 0.00375. For the oscillating flow in the

lid driven cavity, the mesh sizes of 81 81 and 97 97 with the same time step

are used. The computed Strouhal number St, average value of drag coefficient,

oscillations (peak to peek) of drag and lift coefficients for the vortex shedding case,and the oscillation of drag and phase shift for the oscillating flow case are listed in

Table 2. It was found from Table 2 that all discrepancies caused by different mesh

sizes are less than 5%. This implies that the mesh sizes of 241 241 for the vortex

shedding flow and 97 97 for the oscillating flow used in this work are fine enough

to provide accurate numerical results.


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Table 2. Effect of grid size.

Vortex shedding behind the cylinder at Re = 100

Grid St CD CD CL

201 201 0.156 1.3363 0.024 0.71241 241 0.164 1.3668 0.0242 0.75

Oscillating flow in the lid driven cavity at Re = 400

Grid Oscillation of drag Phase shift

81 81 58.968 16.0897 97 59.713 15.96

4. Conclusions

An explicit Taylor series expansion- and least square-based lattice Boltzmann

method is presented, and applied to simulate the two-dimensional unsteady vor-

tex shedding flow behind a circular cylinder and the oscillating flow in the lid

driven cavity. The dynamic parameters and flow patterns obtained by the present

method are in good agreement with those of previous numerical and experimental

studies. The success of this study shows that the present method can be used as

a versatile CFD tool to simulate real unsteady flows. The major advantage of the

present scheme is that it still keeps the local and explicit features of the standardlattice Boltzmann method. Since the method has a meshless feature and no coor-

dinate transformation is involved, it is very convenient for the present method to

be applied to the flow problems with complex geometry.

References

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4. Y. H. Qian, D. dHumieres, and P. Lallemand, Europhys. Lett. 17, 479 (1992).5. S. Chen and G. D. Doolen, Ann. Rev. Fluid Mech. 30, 329 (1998).6. Q. Zou and X. He, Phys. Fluids9, 1591 (1997).7. X. He and G. D. Doolen,J. Comp. Phys. 134, 306 (1997).8. X. He and G. D. Doolen,Phys. Rev. E56, 434 (1997).9. X. He, L.-S. Luo, and M. Dembo,J. Comp. Phys. 129, 357 (1996).

10. R. Mei and W. Shyy, J. Comp. Phys. 143, 426 (1998).11. H. Chen,Phys. Rev. E58, 3955 (1998).12. G. Peng, H. Xi, C. Duncan, and S. H. Chou,Phys. Rev. E59, 4675 (1999).13. E. Berger and R. Wille, Ann. Rev. Fluid Mech. 4, 313 (1972).

14. C. H. K. Williamson,Phys. Fluids31

, 2742 (1988).15. M. Hammache and M. Gharib,Phys. Fluids A 1, 1611 (1989).16. W. Y. Soh and J. W. Goodrich, J. Comp. Phys. 79, 113 (1988).17. A. L. Gaitinde, Int. J. Numer. Meth. Eng. 41, 1153 (1998).18. C. H. K. Williamson, Ann. Rev. Fluid Mech. 23, 477 (1996).19. M. Braza, P. Chassaing, and M. H. Ha, J. Fluid Mech. 165, 79 (1986).


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20. D. J. Tritton,J. Fluid Mech. 6, 547 (1959).21. B. E. Eaton,J. Fluid Mech. 6, 547 (1959).

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chew shu niu (ijmpc) 2002

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