Lid Driven Cavity (SIMPLE)
 

Hello, I am trying to code the lid driven cavity problem using SIMPLE method and FVM discretization with staggered grid and ghost cells.

Steps followed
1. Found the velocity coefficient matrix for u,v (incorporated the BC in the coefficient matrix for corner and boundary cell)
2. Found the updated velocities using momentum equation
3. Found the coefficient matrix for pressure correction (incorporated the BC in the coefficient matrix for corner and boundary cell)
4. updated p,u,v

Central differencing scheme used to discretize.

I am facing some issues in the results for centerline velocity.

Thanks in advance

What is the problem? Why do you think to get exactly the same results obtained by Fluent?

I am not looking for the exact results but I'm concerned about the curvature it achieves after reaching the minima. That curvature increases as I decrease the grid spacing and hence the error.

I suggest to compare to the well known results in literature (for example Ghia & Ghia) using the same Re and a similar grid.
Check both velocity profiles in the centerline and the recirculation region.
Check if the divergence-free constraint is satisfied.

Yes, the divergence free condition is satisfied. I've posted the comparison for Ghia et al. Probably this would give you a better picture

What about the Re number? What about the comparison to the other velocity component?
Plot the streamlines and check that your solution achieved a convergent result.

The Re is 100. V velocity is actually worse

It seems like you have not reached a fully convergence, how do you stop the run? What about a refining of the grid?

If the successive change in velocity is less than 1e-6, it stop the solution. The grid size is 31x31 which is quite decent. The other results from similar mesh seem to have got a pretty accurate result.

That makes no sense in terms of convergence, you have to monitor the residuals

What do you mean it doesn't make sense ? velocity residuals are calculated in terms of the difference in old and new values (https://www.researchgate.net/publica..._driven_cavity).

Also, keep in mind that am monitoring residuals from continuity equation.

Not at all, the differences between old and new values are not the residuals of the equations.


A.x^k-q=r^k is the k-th residual

I tried the approach you mentioned. It doesn't seem to work. Do you mind looking at my code ? Please share your email id if so.

I cannot debug your code, tell us what is the problem, did you compute the residuals and they diverge?

No it does not diverge. To me it seems that either something is wrong with the link coefficients for u,v in momentum equation as I incorporate the boundary conditions. The interior nodes look good to me. Or there could be something wrong in the way I am implementing relaxation factors.

what do you mean? the norm on the u and v residual become constant?