Hello Jai,

             Can you state the equations which you have used? Or is it the Orr-Sommerfeld equations? It might give an insight as to where the problem could be.

Thank you sir for taking interest.

I am solving simple 2D incompressible Navier Stokes equations. And the algorithm can be found in the work of Dalal et al. ( Dalal, A., Eswaran, V., and Biswas, G., 2008, “A Finite Volume Method for Navier-Stokes Equations on Unstructured Meshes”, Numerical Heat Transfer: Part B, Volume 54, Issue 3, Pages 238-259. )

Thankyou sir (Xavier Dechamps) for taking time to explain the condition in such an elaborate manner. I am grateful to you.

I will try the test case as you mentioned. The solution is diverging after few time steps only, as the velocities are increasing at the cells adjacent to periodic boundary.

I just have one more question. Will the effect of periodic boundary condition be reflected in matrix A ( Ax=b, A is a coefficient matrix, x unknowns and b is the source term vector)? I mean, the boundary condition should be applied after each time step or within the iterations performed to solve Ax=b?

Thanks for immediate response sir (Xavier Dechamps ).  And thanks to all, for the kind support.

Finally I got the answer. When I updated the periodic boundary condition at inner iteration loop ( i.e loop while solving Ax=b, so that both the periodic cell should be treated like neighbors), it converged and the results are attached. I have considered the pressure drop per unit length B=0.01228 for a Umean of 1 m/s. The flow is in between two parallel plates (2D in nature), so the Umax should be 1.5 times Umean. I got Umax=1.484, I am sure if I will use finner mesh, it would be even better.

Thanks for the forum and special  thanks to the active members for helping new learners like me.

Thanks and Regards

Jai

Dear Jai!

I'm working with navier-stokes with periodic boundary conditons using finite difference method thod. And i have problem with periodic boundary conditions when i restructions Laplace operations for periodic boundary conditions (ex. Lx=b). Thus, can you help me to build Laplace matrix for periodic boundary conditions?!

Thanks and best regards

Le

Dear Le

Sorry for the late reply, as I was on leave from last 1 month.

Now regarding your problem, I could not understood your query properly. I worked in finite volume, but I have little knowledge about finite difference method as well. For the sake of clarity in periodic boundary condition, I would like to mention that, it is just a boundary condition that has to be updated every single iteration and a little change in the matrix entries.

Like in 1D problem if u have n discrete points, the first and last being the boundaries, then the boundary conditions are:

phin = phi2

and

phi1 = phi(n-1)

accordingly your matrix entries will change for node 2 and node n-1.

hope it will help.

Best regards

Jai

Thank Jai for your answer! I 

Best regards

Le

i often use a repeted replacement procedure for the periodic problem.

we can copy the out let value to the inlet at every iteration.

Dear Ztdep Zt

If you are  copying the outlet value to inlet, what boundary condition you are using initially in the outlet face?

https://diglib.tugraz.at/download.php?id=581303c7c91f9&location=browse

You can read chapter 7 in this thesis. I think there are a lot of ways to implement the periodic BC. Maybe the simplest one is the explicit way which is Dirichlet-Dirichlet method and you need iterations to make it converged.

Implicit coupled way is more favorable for its lower degree of freedom to solve. I think you need to transform the boundary and adjust the coefficients in the matrix.