sorry I guess my post was rather vague. Let's try again
The Navier Stokes equations for incompressible fluid flow are the following:
\frac{\partial u} {\partial t} = -(u. \nabla )u - \frac {1} {\rho} \nabla p + v \nabla ^2 u + f
\nabla .u = 0
The second equation (the incompressibility equation) is the one I'm curious about.
I've read that this can be resolved through the pressure equation, or through the helmholtz-hodge decomposition.
One paper suggested this was done through substituting a pressure update formula into the divergence formula.
the pressure update formula is
u^{n+1} _{i+1/2,j} = u_{i+1/2,j} - \delta t \frac{1}{\rho} \frac{p_{i+1,j} - pi,j}{\delta x}
u^{n+1} _{i,j+1/2} = u_{i,j+1/2} - \delta t \frac{1}{\rho} \frac{p_{i,j+1} - pi,j}{\delta x}
substituting into the divergence formula gives
\frac{\delta t}{\rho} (\frac{4p_{i,j}-p_{i,j+1}-p_{i+1,j}-p_{i-1,j}-p_{i,j-1}}{\Delta x^2}) = -\frac{u_{i+1/2,j}-u_{i-1/2,j}}{\Delta x} + \frac{v_{i,j+1/2} - v_{i,j-1/2}}{\Delta x}
The helmholtz-hodge decomposition is
<br />
<br />
\xi = \nabla u + \nabla .v +h<br />
where u is a scalar potential field (note that \nabla * (\nabla u) = 0
where v is a vector potential field (note that \nabla .(\nabla * v) = 0
where h is the harmonic vector field (note that \nabla .h = 0
One of the papers I've read states that it uses a Poisson equation to derive a height field, which is subtracted from \xi to yield a divergence free flow (it should be noted that this paper altogether ignores the harmonic vector field). The paper can be found here:
http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf
in a few papers I've read the authors have implied a relationship between the pressure and helmholtz-hodge solutions. However I don't see how they're related; in fact in that paper I just mentioned the author doesn't even acknowledge the pressure equation in his statement of the NS equations as pertaining to his solver, so I'm finding difficulty in tieing these things together.
I hope this has made my question more clear, and look forward to any answers.