Solving the Poisson equation using Python

This  post is part of the CFDPython series that shows how to solve the Navier Stokes equations with finite difference method by use of Python. This is the last step to the small solver we want to create. The Poisson equation is actually the Laplace equation to which we add a source term to the right hand side:

∂2p∂x2+∂2p∂y2=b

2D Poisson equation solve for 4 source terms

To better understand how this appears, we must return at the Navier Stokes equations; for a incompressible fluid we must resolve the continuity and the momentum equations; the continuity equation creates a kinematic constraint which implies that div(U) should be zero everywhere in the domain and is not coupled with the pressure; that is why we cannot use the continuity directly in the developing a solving procedure.

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Solving the pressure pulse traveling in a tube with sonicFoam

This is the shock tube problem solved 1D using OpenFOAM.
Problem description: a cylindrical tube is separated at the middle with a non-deformable membrane; in right hand side chamber the pressure is 10 000 Pa and temperature is 248 K; in the left hand side chamber the pressure is 100 000 Pa and temperature 402 K. These are the initial conditions.
The membrane disappears suddenly and the gas starts to move; compute the distributions for pressure, temperature and velocity 7 milliseconds after the membrane is removed.

Velocity field after 7 milliseconds

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Creating grids with blockMesh

OpenFOAM, like any other CFD package that uses the finite volume method, needs a mesh on which to solve the NS equations.

There are a lot of tools to make meshes these days, free or commercial.

Today I will focus on one of the tools provided with OpenFOAM: the blockMesh; this is mainly a dictionary in which the user types in vertices, edges, blocks and settings for the mesh and afterwards defines the boundary conditions.

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