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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.

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    $\begingroup$ 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. $\endgroup$ – Geoff Oxberry Oct 14 '14 at 8:15
  • $\begingroup$ 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. $\endgroup$ – Jan Oct 14 '14 at 8:24
  • $\begingroup$ Taylor-Green vortex can also be used for verification and convergence rate checks for 3D codes, its not exclusive only for 2D codes. $\endgroup$ – Johntra Volta Oct 14 '14 at 18:52
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    $\begingroup$ 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.... $\endgroup$ – Rhys Ulerich Oct 14 '14 at 22:32
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    $\begingroup$ Acutally, there was already a large list here: physics.stackexchange.com/questions/60476/… $\endgroup$ – Dr_Sam Oct 16 '14 at 13:33
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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.

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    $\begingroup$ 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. $\endgroup$ – Jan Oct 16 '14 at 12:20
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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.

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  • $\begingroup$ 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. $\endgroup$ – Jan Oct 14 '14 at 12:06
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    $\begingroup$ 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. $\endgroup$ – Dr_Sam Oct 14 '14 at 12:14
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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.

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