Simulation of 2D Flow around TwoParallel Circular Cylinders in Curvilinear Coordinates

Guangcai Sun Wenli Wei

Weinan Teachers CollegeXi’an University of Technology, Xi’an, China, 710048

Wei_wenli@ 126.com

Abstract: This paper presents a numerical method to simulatethe 2D flow around two parallel circular cylinders incurvilinear coordinates. In order to overcome thecomputational difficulties in complicated boundary figures,curvilinear boundary-fitted grid is used, the irregular domainin physical plane is transformed into a rectangular domain intransformed plane, and the depth averaged momentumequations and mass equation are rewritten and discretizedbased on the alternating direction implicit finite differencescheme in curvilinear coordinates. Practical application of themethod is illustrated by an example for the flow around twoparallel circular cylinders.Keywords curvilinear coordinates; numerical simulation; 2Dflow around two parallel circular cylinders

I. INTRODUCTIONIn the following, a 2D numerical model based on the

alternating direction implicit finite difference scheme for flowunder the curvilinear coordinate is suggested, the irregulardomain in physical plane is transformed into a rectangulardomain in computational plane, and the depth-averagedmomentum equations and mass equation are rewritten anddiscretized in curvilinear coordinates. A practical application ofthe model is illustrated by the example for the flow around twoparallel circular cylinders.

II. MATHEMATICAL MODEL

A. Generation of Orthogonal Curvilinear GridsGenerating a grid in an arbitrary physical domain involves

a coordinate transformation from the physical plane (x, y) tothe computational plane ( , ). This is done here by solvinga system of Poisson equation [1].

12 pxxax

22 pyyay

where

2222 ,, yxyyxxyxa

and

., 22

21 QypyJpQxpxJp

In the above equations, J is the Jacobian ( yxyx ) oftransformation; and P and Q are control functions which can bechosen to provide a denser distribution of points in certainregions.

In Eq. (1), determining the function of P and Q is oftendifficult problem. Wei Wen-Li [2,3] has proposed a new methodto determine them more efficiently. By using the method, adesired boundary-fitted curvilinear coordinate grid can beautomatically generated.

B. Transformed governing equationSince the numerical computations are performed on a

orthogonal mesh, it is necessary to convert Cartesian formulasinto transformed equations written in terms of the boundary-fitted orthogonal curvilinear coordinates and . Thetransformed shallow water governing equations are[3]

011HVg

ggHUg

ggtz 3a

Bg

Ag

zgg

fVHCVUU

gg

ggV

g

ggUVU

gVU

gU

tU

11

2

222

3b

)11(

2

222

Bg

Ag

zgg

fUHCVUV

gg

ggU

g

ggUVV

gVV

gU

tV

c

Corresponding author: weiwenli@xaut.edu.cn

2850978-1-4577-1415-3/12/$26.00 ©2012 IEEE

Sgg

HDggg

gHD

gg

HVggg

HUgggt

H

)(1)(1

)(1)(1)(

(3d)

where

;/)]()([

;/)]()([

ggUgVgB

ggVgUgA

.

;22

22

yxg

yxg

n

t is time; z is water surface elevation; H is water depth; g isgravity acceleration; C is Chezy roughness coefficient; isdepth eddy viscosity; f is Coriolis parameter; U and V aredepth-averaged velocity components in the -direction and -direction respectively; is the pollutant concentration;

DD , are mixing coefficients in the and direction,

respectively; S is the source term. The velocities u (in the x-direction) and v (in the y-direction) are related to U and V by

vgxugyU )()( a

vgxugyV )()( b

C. Numerical schemeThe transformed governing equations (3a)~(3c) are

discretized on a staggered ( , ) grid and solved usingalternating direction implicit finite difference scheme. Thevelocity variables are fully staggered, and the water levelmodes are located at the center of the continuity flow cell asillustrated in Fig. 2.

Fig.1 Grid arrangement ——U +——z, ——V

and are defined as the distances in the transformeddomain between the velocity vector positions. Since the rangeof the coordinates and in the computational plane is

completely arbitrary, the mesh increments and arespecified, for convenience, as unity.

A further index, n, is now introduced to denote the timelevel of the discretized hydrodynamic variables. Thealternating direction algorithm splits each time step into twointervals. The procedure can be demonstrated with respect tothe transformed equations as follows.

Step 1, in the first half time step:

Continuity equation (3a) becomes

0)(1)(1 121

gVHgg

gUHggt

z nnnn

n

(5a)

-momentum equation (3b) becomes

)11(

)()()( 21

2

222

111

nn

n

nnnnn

nnnnnnn

Bg

Ag

zgg

fVHCVUU

ggV

gggVUU

gVU

gU

tU

(5b)

Step2, in the second half-time step:

Continuity equation (3a) becomes

0)(1)(1 121

1211

gVHgg

gUHggt

z nnnnn

(6a)

-momentum equation (3c) becomes

)11(

)()()( 11

21

2

22121

11111

nn

nn

n

nnnn

nnnnnnn

Bg

Ag

zgg

fUHC

VUVg

ggU

g

ggVUV

gVV

gU

tV

(6b)

By adding the above equations for the two half-time steps,it can be shown that the combined effect steps 1 and 2 resultsare the final results ( 111 , nnn andVzU ) in one time step (fromt=n t to t=(n+1) t).

D. Numerical discretizationWhen we solve the equations (3a)~(3c), the viscosity of

water is neglected. For brevity, only the -direction ADIdiscretization of the continuity equation and -momentum

(3d)

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equation will be presented. A more detailed discussion ofcomputational scheme can be found in Ref. [2].

In discretizing the equations (5), the average and finitedifference of the function F (such as U, V and z, etc.) may berepresented as

)(21

,21,

21,

jijiji FFF , )(

21

21,

21,

,jiji

ji FFF

)(41

21,

21

21,

21

21,

21

21,

21,

jijijijiji FFFFF

)(1,21

,21

,jiji

ji

FFF , )(1

21,

21

,,

jiji

ji

FFF

Thus the transformed continuity equation for the first half-time step may be discretized and rearranged to give

11,211

2/1,1

1,211 DUCzBUA n

jinji

nji (7a)

where

][12

;1;0.2;1

2/1,2/1,,2/1,,2/1,2/1,,2/1

,

,1

,2/1,2/1

,11,2/1,2/1

,1

jinjijjijji

njiji

ji

nji

jiji

ji

jiji

ji

gVHgVHJt

zD

gHJ

Ct

BgHJ

A

nn

nn

And the transformed -momentum equation for the firsthalf-time step may be discretized and rearranged to give

22/1

,121,2/12

2/1,2 DzCUBzA n

jin

jinji 7b

where

][][

])()[(

)()(

;

;)(

10.2;

2/1,2/1,,2/1,2/1

,2/1

,2/12

2/12,2/1

2,2/1,2/1

,,1,2/1

2,2/1

,2/1

,2/1,2/12

,2/12

,2/1

2/1,2/12/1,2/1,2/1

,2/12

,2/12

nji

nji

nji

nji

n

ji

ji

n

jin

jin

ji

jiji

ji

n

ji

n

ji

n

jin

ji

ji

ji

jiji

n

ji

n

jiji

BBg

AAg

VfHC

VUUg

ggJV

UgV

t

UD

gg

C

J

ggV

Ugt

Bgg

A

During the above -direction release the -momentumcross-convective terms< >are differenced using a two-orderupwind scheme and discretized as

0212

23

0212

23

,211,

21,,

,211,

21,,

n

ji

nji

n

ji

nji

n

ji

nji

n

ji

nji

n

VifUUU

VifUUUU

and

0212

23

0212

23

,21,1,

21,

,21,1,

21,

n

ji

nji

n

ji

nji

n

ji

nji

n

ji

nji

n

UifUUU

UifUUUU

Using the ADI method to solve the tri-diagonal matrixequations about U and z, we can obtain the values of Un+1 andzn+0.5. These are the results for the first half-time step; and theresults are prepared for the initial values of the second step.And then, using the similar method, we can obtain the valuesof Vn+1 and zn+1. Up to now, final results (Un+1 Vn+1 and zn+1)are obtained for one time step. When the values of U, V and Zare obtained, we use upwind difference scheme to discretizethe convective term and central difference scheme todiscretize the diffusion term in equation (3d), and can obtainthe value of at each time step.

E. Boundary ConditionsThe boundary conditions for flow are that the upstream

and downstream water level is specified and the flux throughthe solid boundaries is zero; and the initial conditions are thatwater level at any flow element is the averaged value ofupstream and downstream water level and velocity is set equalto zero.

The boundary conditions for water quality are that thepollutant concentration through the solid boundaries is zero,and at the inlet boundary and at the outlet boundary it doesn’tchange in the flowing direction and at the discharge place isspecified. The initial condition is that is set to be zero.

F. Technique of moving boundary [3 4]

In the numerical solution of unsteady flow, the riverbedmay be exposed to the water surface. In order to deal with thechangeable computational region, the technique of movingboundary (or the method of condensation) is used in thecomputation. The basic idea of this method is to make theroughness of the computational finite cell very large when itsriverbed is exposed to the water surface. This enables thevelocity of the fluid in the computational finite cell to be zero,just looking like the fluid being condensed to solid.

III. RESULTS OF COMPUTATIONThe sketch shown in Fig. 1 illustrates the complete flow simulation

area used in this investigation. Due to the complex geometry of thecircular cylinder walls; curvilinear coordinates are utilized to refine

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the wall modeling. The discretization of the flow domain uses controlvolumes in a body-fitted coordinate system.The simulation area is 10m long, 6m wide and 2 m high and the

diameter of the circular cylinders is 4m.The distance between the twocircular cylinders is1m.The physical domain is shown in Fig.1; The grid of the

computational domain on horizontal plan is shown in Fig.2 Thecomputed velocity vector on the plane is shown in Fig.3; Thecomputed streamline distribution is shown Fig.4.

Fig.1 physical domain

Fig.2 Grid on horizontal plan

Fig.3 Computed velocity vector

Fig.4 Computed streamline distribution

IV. CONCLUSIONSThe advanced technique of computational fluids dynamics,

the technique of curvilinear grid generation, presented aboveis convenient and effective in dealing with the complicatedboundary of physical region. The method of alternatingdirection implicit finite difference scheme used to solve thepartial differential equations has good stability, convergence,and accuracy.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China(Grant No.51178391), Shanxi Provinceeducation department funds 2009JK649, 2010JK761 andthe Shanxi Province key subject construction funds.

REFERENCES[1] J.F.Thompson, “Numerical Solution of Flow Problems Using Body-

Fitted Coordinate Systems”, Computational Fluid Dynamics, 1980, pp.1-98

[2] Wei Wenli, Liu Yuling, Li Jianzhong, “The Application of OrthogonalBoundary-Fitted Coordinate System and Fractional Step Method to theCalculation of 2D Flow Field in River Course”, Journal of Xi’anUniversity of Technology, 1996 (in Chinese)

[3] Wei wenli, “Numerical Solution for Unsteady 2-D Flow Using theTransformed Shallow Water Equations”, Journal of Hydrodynamics, Ser.B,1993(3), pp.65-71.

[4] Wei WenLi, “Study on Curvilinear Grid Generation”, 9th nationalHydrodynamics Conference, Nanjing, China, 1995, pp.162 166 (inChinese)

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