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I'm looking for analytic solutions to time dependent fluid flow problems (can be compressible or incompressible, Euler or Navier-Stokes equations). The main thing though is that I'd like there to be 'interesting' physics going on and for this 'interesting' physics to develop as time proceeds. I'm fine with having source terms to force the solution to be 'interesting'.

To give an idea of what I mean by interesting -- I'm thinking along the lines of something that appears like transitioning to turbulence, developing shocks and/or vortex shedding.

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  • $\begingroup$ Are these analytic solutions for testing? $\endgroup$ – Geoff Oxberry Nov 26 '13 at 19:55
  • $\begingroup$ What I'm trying to do is some in-situ analysis to see if I can properly capture this 'interesting' behaviour automatically. So yes, this is for testing but not for checking the accuracy of a CFD simulation code but of checking the ability of a general feature extraction algorithm. It's also meant to be a mini-app (code.google.com/p/hpc-proto-apps) so it's important that it isn't expensive to compute (i.e. a true CFD solution). $\endgroup$ – andybauer Nov 26 '13 at 20:03
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Try looking into the solutions available within MASA. MASA is a library of manufactured solutions containing many steady and transient problems for the flow physics you describe. These don't tend to be "interesting" in the sense that one stares at the results because they describe canonical engineering flows. However, they're "interesting" in the sense that many or all of the terms in the governing equations are active. For example, the library contains a transient manufactured solution for the compressible Navier--Stokes equations with a power law viscosity that I helped to develop.

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Frankly, what you want doesn't exist. If we could generate arbitrary, interesting solutions with simple techniques, we wouldn't have to write sophisticated Navier-Stokes solvers to find them.

You can manufacture arbitrary solutions as Rhys notes, but they will likely be arbitrarily tuned to your feature extractor. As such, they won't tell you much. Your best bet will be to develop a catalog of solutions to real fluids problems using a CFD solver and save them. Loading these from disk should be orders of magnitude faster than regenerating them for most interesting flows.

You might also try finding some CFD communities to solicit some interesting solutions from.

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For the time-dependent 1D compressible Euler equations, Sod's shock tube is a standard test case with an exact solution from method of characteristics. It has all the interesting elements of 1D Euler equations: shocks, expansions, and thermodynamic discontinuities. For 2D Euler, the mach 3 flow over a step problem is popular in CFD and presents some extremely complex features (but no exact solution I'm aware of).

Stokes' problems for 2D unsteady incompressible low-Re viscous flow admit exact solutions.

Vortex shedding is not something you're going to find an exact analytical solution to, although there is active research in proper orthogonal decomposition for simple geometries. Turbulent transition is also not something you'll find an analytical solution for. There is some well-developed theory for homogeneous isotropic turbulence.

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Aside from those mentioned above, there are a couple more well-known hydrodynamics problems that are convenient for code testing: (i) strong explosion in atmosphere (spherical shock wave) - described well in Landau,Livshits "Fluid dynamics"; and (ii) Rayleigh-Taylor instability - described well in "Hydrodynamic and Hydromagnetic Stability" by Chandrasekhar.

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