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Maciej Matyka November 8, 2004 16:25

Poisson Equation in CFD

I am writing a thesis about solution to incompressible NS equations. I have a question regarding Poisson equation, since I need to create a simple example of solution to Poisson equation (I am making comparison of convergence of different Poisson equation solvers as a part of this thesis).

The problem I have is to find a physically meaning of seperated poisson equation:

lapl P(x,y) = -rho(x,y)

I've used an example from electrostatic (p- is a potential and rho is a charge density) but it does not suit the subject of thesis and I am looking for an example from CFD field.

Ok, so question - do you have an idea what can be described (I am looking for simple, simple case) by seperated Poisson equation and will have connection to CFD?

Best Regards, Maciej Matyka <A HREF=""></a>

agg November 8, 2004 22:54

Re: Poisson Equation in CFD
You could use lapl P(x,y) = -divergence(u dot grad(u))

Maciej Matyka November 9, 2004 02:19

Re: Poisson Equation in CFD
Yes, I know that - but I am looking for the most simple poisson equation. That one you've mentioned introduce convective term on right hand, so example from electrostatic is more easy to understand...


Mikael Ersson November 9, 2004 04:03

Re: Poisson Equation in CFD

I don't have a direct answer to your question, but I have a vague memory of this subject being discussed maybe some 6 months ago in this forum. A search might help.

Also a good book I can recommend is Fundamentals of CFD by Roache. He covers the subject quite well.

Good Luck

Tom November 9, 2004 05:42

Re: Poisson Equation in CFD
If you mean you want an example of the solution of the Poisson equation which is directly applicable to fluid flow try

grad^2 \phi = -w

with w constant on patches; e.g. w = -1 for x^2 + y^2 <1 and zero otherwise.

In this case psi is the streamfunction and w is the vorticity.

Maciej Matyka November 9, 2004 06:04

Re: Poisson Equation in CFD
thank you, that is something what I've been looking for!

Adrin Gharakhani November 9, 2004 23:02

Re: Poisson Equation in CFD
If you use patches of vorticity (or even the simpler case would be to use point singularities), you'll be able to use the Biot-Savart law to evaluate the streamfunction and the velocity fields "exactly" (analytically per patch), which you can then use as your benchmark ...

Adrin Gharakhani

Maciej Matyka November 10, 2004 02:50

Re: Poisson Equation in CFD
Is the Biot-Savart law applicable in CFD field? As far as I know Biot-Savart law is something in Electricity/Magnetism. So is that only a mathematical law?


Tom November 10, 2004 05:30

Re: Poisson Equation in CFD
Yes - the Biot-Savart law is applicable. Basically the origin of the concept of vorticity lies in the fact that if you introduce a quantity w=curl(u) then if div(u)=0 the equations linking u and w are those for an electro-magnetic field - div(u)=0 is not referred to as the solenoidal condition for nothing.

See the introduction to the book "vortex dynamics" by P.G. Saffman,


ZubenUbi November 10, 2004 12:30

Re: Poisson Equation in CFD
With a Dirichlet boundary condition at solid walls and a Neumann boundary condition at inlet and outlet, P could be regard as a "pseudo" distance from the wall (taking into account all walls instead of the nearest. This is of interest for some turbulence model in channel and in vicinity of edges. I used to solve smth like that.

Moreover, the damping function in turbulence model, which usually depends on distance from the wall, modelize, in fact, the non-local effect of pressure fluctuation....



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