Hello,

For purposes of my MSc Thesis I am now doing a set of test benchmarks (test cases) for code validation.

I found really nice test case with an analytical solution as presented above:

It has been taken from:

Amara, "Vorticity–velocity–pressure formulation for Navier–Stokes equations", Comput Visual Sci 6: 47–52 (2004)

Unfortunately author of this paper did not provide enough information about this test case. They call it as "Academic Test" - and I am not sure about how to specify boundary conditions for it. It seems like a driven cavity type flow with different velocities on all the edges, but I am not sure.

Anybody know that test case and is able to help me a little bit to find other reference article for it, or give an idea about how to specify conditions of the flow here? (I am working with primitive variable SIMPLE solver, no stream function and vorticity).

Best Regards Maciej Matyka <A HREF="http://panoramix.ift.uni.wroc.pl/~maq/eng">http://panoramix.ift.uni.wroc.pl/~maq/eng</a>

ps. If you have any other idea about interesting test cases other than Driven Cavity and Couette Flow (those two I have done so far), please give me a hint. I am working now with code: in primitive variables, for incompressible flow, without free surface, without temperature convection terms.

You can specify the exact solution on the boundary which is known.

it is so simple and obvoius that it is a shame I did not find that way...

thank you! Maciej

It's Taylor's array of vortices. General time-dependent solution is as follows:

$$u(x,y,t)=-\cos(nx) \sin (ny) \exp (-2n^2\nu t)$$

$$v(x,y,t)=-\sin(nx) \cos (ny) \exp (-2n^2\nu t)$$

$$p(x,y,t)=-{1/4} (\cos(2nx) + \cos (2ny)) \exp (-4n^2\nu t)$$

Your equations are for $t \to \infty$. Of course, these are nice analytic solutions, but fundamentally, they are too simple. These might be just a starting point for validations of your code. The flow has no shear whatsoever, i.e.:

$$\sigma_{12}=\sigma_{21}=0$$.

This is one of the reasons you need more thorough test cases. Good luck. MMG