Hi, I am trying to use the pressure-correction technique(projection method) to solve 3D Incompressible shear flow. I have zero derivatives at the boundaries for the velocities. Since I am enforcing continuity through the Poission Equation for Pressure,I dont need boundary conditions for pressure. Initially I am trying to simulate a plug flow.Time stepping is Adam-Basforth and I am using central difference for spatial derivaties.My velocities are blowing up by orders of magnitude after the initial few time steps. I had been trying to resolve this problem for some months now and now it had turn out to be a nerve-wrecking experience for me.I don't know where I am going wrong.Did anybody tried to simulate a 3D Incompressible Flow on similar lines . I would be glad if you share your experiences. Thanks Saurav

How are you treating the flux derivatives? If you are centrally differnecing those then your scheme will be neutrally stable at best and this could be your problem.

i had similar problem, my solver used to work perfectly fine with structured meshes and when i tried to use it with unstructured meshes, the converging solution would blow up same as you mentioned. and yes it took me three months to find the solution. originally i was thinking the bug is in the orthogonal correction and with unstructured meshes non orthogonolity creating trouble, so i implemented all the possible treatments of these terms but no avail. finally it turns out that the problem was the way i was calculating gradients. i was also using central differences for calculating gradients.

anyway you can drop me email to talk more about this issue. i think i will be help.

Why don't you use linear interpolation boundary value for the boundaries for the velocities.

Are you using a staggered grid arrangement? If not, you will have unstable solutions (Read more in textbook by Patankar). To remedy this you will have to add a fourth order dissipation term to your poisson equation or use rhie-chow interpolation.

Have you looked at your solution immediately after the first time step? If solution looks bad near boundaries, it could be a boundary issue.

Good Luck.

It is a staggered grid formulation. Thanks Saurav

I also had similar problem....however, rather than meshing up in finding out the subtle wrongs and likewise stuffs, I decided to solve pressure-poisson equation to obtain the pressure field (staggered grid, cylindrical co-ordinate, central difference, MAC) and I got a stable solution immediately. I implemented what was suggested in the original MAC paper by Horlow and Welch.

You mentioned that you do not need a boundary condition for pressure. Why is that? For a second order pressure poisson equation, you will need two boundary conditions in each direction. Normally a dirichlet or a Neumann condition is used.

Without explcitily imposing boundary conditions for pressure the matrix for the pressure equation is singular. Since you only want pressure gradients and not pressure itself this is not a problem and quite normal. However, in order to get a solution for the pressure field (to within an arbitrary constant) the RHS must integrate exactly to zero. If not, every iteration of the solver the average level of the pressure will be pushed up or down because there are no boundary conditions to prevent this.

Simply add a loop to add up the mass errors and check that it is a roundoff number. If not, scale your outflow velocities so they exactly match your inflow velocities. For incompressible flow you cannot pump in more/less mass than flows out so the ajustment is required on physical grounds as well.