Poisson Equation in CFD
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> |
Re: Poisson Equation in CFD
You could use lapl P(x,y) = -divergence(u dot grad(u))
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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...
Maciek |
Re: Poisson Equation in CFD
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 |
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. |
Re: Poisson Equation in CFD
thank you, that is something what I've been looking for!
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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 |
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?
Maciej |
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, Tom. |
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.... Regards, ZubenUbi |
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