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SIMPLE pressure correction on unstructured mesh

Hi, could anyone provide some details or references on the derivation of this equation using the finite volume method for discretization?

Thanks in advance.

So I can get as far as this,

The continuity equation is defined as

$\int\limits_{CV} \nabla \cdot \left(\rho \vec{u} \right) dV = 0$

and using Gauss' Theorem we can say that

$\sum_i \int\limits_{A_i} \rho \vec{n}_i \cdot \vec{u} dA_i = 0$

Approximating this integral for each control volume surface we get

$\sum_i \rho \vec{n}_i \cdot \vec{u} \Delta A_i = 0$

and the SIMPLE approximations for the velocity correction (collocated grid) is

$\vec{u} = - \vec{d}_i \nabla p$

where

$\vec{d}_i = A_i / a_P$ and p is the pressure.

Normally (on a structured mesh) we would be able to approximate the pressure gradient term to get an equation with coefficients on each neighboring point. The problem is how is this done for an unstructured mesh? I suspect the thing I am missing is a pressure gradient approximation that would allow be to construct the pressure correction equation. Any ideas?

Thanks again in advance.

I am also having trouble getting the latex equations to show up in my post.. sorry

Dear abcdef,

Well, I maynot be able to directly help you out in writing down the pressure correction equation for SIMPLE, but I can point out how it is done on unstructured meshes. Please note that I am not writing down equations here either, so kindly excuse me. The basic steps are as follows.

1. The momentum equation gives you an auxiliary velocity, call it u*
2. Thsi auxiliary velocity doesnot satisfy continuity and this leads us to a pressure correction equation.
3. The PCE is of the form div u* = div(grad(dp))=Lap(dp)

The idea then is to integrate this equation over a control volume and apply the divergence theorem on both sides.

The LHS is merely \sum(u*_f s_f) where u*_f is the auxiliary velocity at the face (the velocities are at the face centroids in a staggered approach; if a collocated approach is used these are interpolated from the centroidal velocities with a suitable correction to avoid pressure-velocity decoupling). This term acts as the source because the auxiliary velocity is known by solving the momentum equation.

The RHS leads to the normal pressure gradient, which can be approximated in different ways. The way that I use, and would recommend (although there exist other approaches) is to use an orthogonal correction to the normal pressure gradient; which consists of two parts: one which is obtained using a finite difference of pressures in cells sharing the face and the other which depends on the gradients of pressures in these cells. For simplicity, in a collocated approach, as a first try, you can neglect the second term.

The result is an implicit sytem of equations for the pressures that can be solved using any suitable linear solver.

For more details, you could refer to the following paper (and references therein) or also google for related sources.

A numerical method for large-eddy simulation in complex geometries, Mahesh et.al., JCP 2005.

Hope this helps.

Regards,

Ganesh