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Poisson Equation in CFD

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Old   November 8, 2004, 16:25
Default Poisson Equation in CFD
  #1
Maciej Matyka
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Hello,

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="http://panoramix.ift.uni.wroc.pl/~maq/eng/">http://panoramix.ift.uni.wroc.pl/~maq/eng/</a>
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Old   November 8, 2004, 22:54
Default Re: Poisson Equation in CFD
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agg
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You could use lapl P(x,y) = -divergence(u dot grad(u))
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Old   November 9, 2004, 02:19
Default Re: Poisson Equation in CFD
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Maciej Matyka
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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...

Maciek
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Old   November 9, 2004, 04:03
Default Re: Poisson Equation in CFD
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Mikael Ersson
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Hey,

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
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Old   November 9, 2004, 05:42
Default Re: Poisson Equation in CFD
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Tom
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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.
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Old   November 9, 2004, 06:04
Default Re: Poisson Equation in CFD
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Maciej Matyka
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thank you, that is something what I've been looking for!
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Old   November 9, 2004, 23:02
Default Re: Poisson Equation in CFD
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Adrin Gharakhani
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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
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Old   November 10, 2004, 02:50
Default Re: Poisson Equation in CFD
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Maciej Matyka
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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?

Maciej
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Old   November 10, 2004, 05:30
Default Re: Poisson Equation in CFD
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Tom
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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,

Tom.
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Old   November 10, 2004, 12:30
Default Re: Poisson Equation in CFD
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ZubenUbi
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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....

Regards,

ZubenUbi
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