Incompressible Navier-Stokes in Stream Function - Vorticity Formulation
I am trying to solve the incompressible, steady state Navier-Stokes in stream function - vorticity formulation, for a given domain (attached pictures).
for the boundary conditions of the vorticity I use Thompson's condition (1930).
I am solving it using an iterative method I developed from the differential equations. I use central finite differences schemes (attached as well).
I solve the vorticity equation - than the stream function equation - than calculate velocities - check convergence - if not, do it again.
I use successive under relaxation for the vorticity for without it the program diverges after a few iterations.
My problem is that as I increase the Reynolds number my solution becomes extremely unstable and it diverges. let say for Re=100 I need only a relaxation factor of 0.5 for the vorticityand the program converges - I get physical results and all is well. But for Re=700 It doesn't converge even for 0.01...
I tried formulating my equation with upwind-schemes but it seemed to only worsen my stability.
Does anyone had this experience with this problem? how do I overcome it?
It seems my problem comes from the convection terms in the equation (the ones Re multiplies)
I am attaching my code that is written in matlab to who ever want to take a look.
use main.m to start it. change Re number if you need.
relax1 - relaxation factor for the vorticity
relax2 - -||- stream function
relax3 - -|| - velocities
it is set now on Re=100. and the mesh is pretty coarse but still gets nice results. sorry for not writing titles on the figures.. u can check the code to see what's in them.
in fact .... why do you use FDM? im using another meshless method. maybe FVM will be better than FDM in your case.
The reason is that we didn't cover FVM in our course. Only FDM :)
but thanks though.
If I'm here I'll update that I solved the problem using very strong relaxation (0.05). It's a banging your head against the wall solution but it worked and I was out of time so it had to do.
I guess the problem is in your scheme for the vorticity function. I saw your schemes it seems to me you are going to solve a parabolic equation (since you assumed burger viscid equation for the vorticity function) which means it should be a time dependent equation but I can't see any term that contains dt), and also in the same step I think you took a centeral difference in space for the diffusion term. If I was right, there would be w(i,j).
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