# Incompressible Navier-Stokes in Stream Function - Vorticity Formulation

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September 15, 2012, 10:20
Incompressible Navier-Stokes in Stream Function - Vorticity Formulation
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Join Date: Aug 2012
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Hello,

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.

Thanks,

Daniel
Attached Images
 pic.jpg (50.6 KB, 59 views) figure1.jpg (22.5 KB, 48 views)
Attached Files
 N-S_incomp_steady.zip (9.2 KB, 52 views)

 October 9, 2012, 16:22 #2 New Member   Thyago de Lellys Join Date: May 2011 Posts: 1 Rep Power: 0 Hi Daniel, in fact .... why do you use FDM? im using another meshless method. maybe FVM will be better than FDM in your case.

 October 9, 2012, 16:38 #3 New Member   Join Date: Aug 2012 Posts: 9 Rep Power: 7 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. Thanks, Daniel

 May 29, 2016, 01:28 #4 New Member   ahmed Join Date: Jan 2016 Posts: 1 Rep Power: 0 Hello Deco! 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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