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Finite Element approximation of the vorticity-stream

function equations for incompressible 2-dimensional

Navier-Stokes flows.

Ga Hyung Jo 1 and Hi Jun Choe 1

1) Department of Mathematics Yonsei University, Seoul 120-749, Korea

ABSTRACT

We compute numerical solution of incompressible two-dimensional Navier Stokes equation

with finite element method. The Navier Stokes equation has terms of velocity and pressure,

they establish relations among the rates of change, for example, in inviscid case, the rate of

velocity is proportional to the gradient of pressure, there is difficulty to use same finite element

mesh for velocity and pressure.

We can avoid pressure term by considering vorticity and stream function for Navier Stokes

equation and then our system is divided into smaller systems which are uncoupled, so our prob-

lem is less serious. Moreover we can use same mesh for vorticity and stream. However instead

of getting some advantage, we meet a difficulty what is boundary condition for vorticity is un-

known. Thus we need some technique to find its boundary condition.

We think way to find boundary condition for vorticity from relation stream function and some

physical assumption on the boundary and then we solve two-dimensional Naiver Stokes equa-tion in term of vorticity and stream function numerically.

1 INTRODUCTION

We have the incompressible Navier-Stokes Equation and conservation law of mass,

u

t+ (u )u +

1

p 2u = f (1)

t +u

= 0(2)

where u : velocity, : density, : viscosity and f body force per unit mass. We have theincompressible condition,

u = 0 (3)

In two dimensional case, we write that u = (u, v), we define vorticity wz = ( u)z = =yu+xv and stream function byu = y, v = x. By using basic identity, incompressiblecondition and assumption there is no external force, we get that

Vorticity equation : t + ux + vy =

2, in ;

= , on . (4)

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Stream function :

2 = , in ;

= g or nn = q. on .(5)

Finally we get uncoupled two smaller system so that our problem is less serious than the pri-

mary. But we dont know boundary value of vorticity, . Hence we need way to find boundarycondition for vorticity.

2 DISCRETIZATION OF BOUNDARY CONDITION FOR VORTICITY

In conventional dicretization of boundary condition for vorticity by using Finite Difference

Method(FDM) is B = 3 (N B vsy) /y N/2. The conventional method is ad-missible when only rectangle mesh. In physically we can assume that viscosity is zero near the

boundary, we have

2 = 0 (6)

near the boundary. From equation (6), vorticity and stream function relation (1) and by some

rearrangement we get that

BB =

nB +

B

IB (7)

where B and I are test function based on the nodes on boundary and near respectively. Tosolve the Naiver Stokes equation, apply Crank-Nicolson to time,

n+1

n

t +un+

1

2

12n+1 + n

= 2

2n+1 + 2n

(8)

3 CONCLUSION

The incompressible two-dimensional Navier-Stokes equations have been solved by the fi-

nite element method using a new vorticity-stream function formulation and this method can beapplied resolve not only regular domains but also complicate domains.

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REFERENCES

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Naiver Stokes equations, , Int. J. Numer. Methods Fluids., Vol. 7, 1987, pp. 17-27.

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3. Camprion-Renson, A. and Crochet, On the stream function-voricity finite element

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1818.

4. Tezduyar, T. E., Glowinski, R., and Liou, J., Petro-Galerkin methods on multiply

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