Calculating pressure in Navier-Stokes finite difference solver

[jojodi]

I've been working for the last month trying to figure this out.
I am trying to produce a computational solver for fluid dynamics based on the Navier-Stokes Equations. I have derived the finite difference form of all the terms and worked out the operators, but I have been unable in finding any way to claculate the pressure at a given point in the next frame. If someone could explain or possibly show me some pseudo-code to the procedure, I would be appreciative.

Thanks in Advance,
Brandon

 

[jojodi]

hmmm.. bump?

 

[Integral]

I'll move this over to math perhaps someone there can help out.

Meanwhile, if you posted what you have for the discretized equations it may spur some conversation.

 

[jojodi]

The equations for Navier-Stokes that I have been working on also have some problems as well now that I look at them. I assume I am using an incorrect order of operatins.

$$\frac{\partial \bold{u}}{\partial t} + \bold{u} \cdot \nabla{\bold{u}} = -\frac{\nabla{P}}{\rho} + \nu\nabla^2\bold{u}$$

Where $$\bold{u}$$ is the fluid parcel velocity, $$\rho$$ is the parcel density, and $$P$$ is the pressure at that point.

I have discretized this (I'm sure with some mistake(s) ) to...

$$\frac{\bold{u}_{t+\delta}-2 \bold{u}_{t}+\bold{u}_{t-\delta}}{\delta t^2} + \bold{u} \cdot<\frac{\partial \bold{u}}{\partial x},\frac{\partial \bold{u}}{\partial y}>= -\frac{<\frac{\partial P}{\partial x},\frac{\partial P}{\partial y}>}{\rho} + \nu({\frac{\partial^2 \bold{u}}{\partial x^2}+\frac{\partial^2 \bold{u}}{\partial y^2}}) \bold{u}$$

then translates to (for the new frame) :

$$\bold{u}_{t+\delta}= \delta^2 ( -\frac{<\frac{\partial P}{\partial x},\frac{\partial P}{\partial y}>}{\rho} + \nu({\frac{\partial^2 \bold{u}}{\partial x^2}+\frac{\partial^2 \bold{u}}{\partial y^2}} - <\bold{u}\cdot\frac{\partial \bold{u}}{\partial x},\frac{\partial \bold{u}}{\partial y}>) \bold{u} ) + 2\bold{u}_{t}-\bold{u}_{t-\delta}$$

For each dimension:
(under construction )
$$\bold{u}_{x,t+\delta}= \delta^2 ( -\frac{\frac{\partial P}{\partial x}{\rho} + \nu(\frac{\partial^2 \bold{u}}{\partial x^2} + \frac{\partial^2 \bold{u}}{\partial y^2} - \bold{u}\cdot\frac{\partial \bold{u}}{\partial x}) \bold{u} ) + 2\bold{u}_{t}-\bold{u}_{t-\delta}$$