# Test Case with Known Solution for 3D Navier Stokes Equations

Is there a test case for 3D incompressible Navier Stokes Equations like the Taylor vortex in two dimensions?

I know, I can easily construct 3D manufactured solutions but I would like to have something more physical.

• Stokes flow is probably your best bet, since you can use Green's functions (Stokeslets) to construct solutions analytically for simple geometries (alternatively, streamfunctions). The classic examples tend to be 2-D, and other classic physical flows, such as Poiseuille flow, Couette flow, and stagnation point flows tend to be 1-D, or maybe 2-D at best. These could, of course, be rigged up in 3-D, but that doesn't sound like what you want. Commented Oct 14, 2014 at 8:15
• Didn't know about Stokeslets. I will have a look into it. Maybe I can add some convective elements to get manufactured solutions with some physical meaning.
– Jan
Commented Oct 14, 2014 at 8:24
• Taylor-Green vortex can also be used for verification and convergence rate checks for 3D codes, its not exclusive only for 2D codes. Commented Oct 14, 2014 at 18:52
• Why do you care if they are physical for a benchmark problem? Only seems to matter if you want to show your numeric better resolve some feature than competitor X at resolution Y for compute time Z on vendor A's new chip B at frequency C.... Commented Oct 14, 2014 at 22:32
• Acutally, there was already a large list here: physics.stackexchange.com/questions/60476/… Commented Oct 16, 2014 at 13:33

Chapter 7 of the (free in PDF form) book "I do like CFD, Vol. 1" collects many different known solutions for multidimensional flows that can be used for benchmarking.

• This looks like a pretty helpful resource. The example for the 3D incompressible Navier-Stokes equations is the Ethier-Steinman solution mentioned by @dr_sam in his answer.
– Jan
Commented Oct 16, 2014 at 12:20

There are several benchmarks, like the flow around a cylinder, which is described in details at the FEATFLOW web page here. It is a well defined configuration of a flow passing by a cylinder obstacle and values such as the drag and lift can be compared with values obtained with different softwares (a file containing the data is provided).

Otherwise there are "less physical" flows, like the Ethier-Steinman problem described here, which however has a known exact solution.

• Good point. Benchmark cases are also useful to verify code. However, I don't think that the comparison of derived quantities like drag or lift can replace the convergence analysis in the actual variables.
– Jan
Commented Oct 14, 2014 at 12:06
• Indeed, it cannot replace the convergence analysis. However, I doubt that you will be able to find a problem for which you have an exact solution and that has some nice physical features. Commented Oct 14, 2014 at 12:14

Let me add, the cavity flow in three dimensions. A very good reference is R. Glowinski, Handbook of Numerical Analysis, Numerical Methods for Fluids, Chapter 9, Section 44.3. Here the test is very well detailed.