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 Petrov June 8, 2011 22:25

Projection Methods

I've got a incompressible Navier-Stokes code that works reasonably well (compared to the taylor-green vortex) that uses a pressure-correction time integration method. I want to implement a projection method now, both as a learning experience and to compare the accuracies.

I'm just trying to get a gist of how it works and how it compares to the pressure correction method. From what I understand, I get an intermediate velocity (probably from the old pressure) and then I project the velocity onto divergence-free vector field (by projecting the vector onto the null-space of another matrix?!), then I solve for pressure, and then I make it divergence-free again.

Is this correct? If not, could you describe projection methods to me?
Are there any good (and free?!) articles you could point me to? I have a strict \$0 budget for learning CFD.

EDIT:
I also want the method to be accurate at the no-slip boundary conditions so that I can calculate the drag/lift forces. I was reading that projection methods may have a problem with this. Should I look at other options?

 valgrinda June 10, 2011 03:32

pjm

Hi,

you got the projection almost right, except that the intermediate velocity is calculated with the pressure not included. This is the major difference to SIMPLE type methods: There you include the pressure gradient in the Navier-Stokes equation, so you only need to solve the poisson equation for the pressure correction.
The procedure for the projection method is (with explicit time treatment for the velocities):
- calculate intermediate velocity u* without the pressure gradient
- calculate the divergence of the velocity field, this term will be the right hand side of the poisson equation for the pressure
- discretize the poisson equation for the pressure (trivial for constant density)
- solve the poisson equation iteratively
- correct the velocities (explicitly) with the pressure gradient
- go to next timestep

Here is a link to a paper, where the projection method is used and explained with some detail:
http://wissrech.ins.uni-bonn.de/rese.../croce/lsm.pdf

Here is book, where the discretization of the projection method is described with even more detail: