I'm trying to benchmark a new numerical method for the incompressible Navier-Stokes equations. I'd like to use the most widely-accepted test cases possible. In particular, I'm looking for a reference setup for the 3-dimensional Rayleigh-Taylor instability in a periodic domain. It would be especially helpful if:

  • The initial conditions are described precisely
  • The diagnostics (measures of correctness) are described precisely
  • Code and data are available

If necessary, I will settle for a reference for which code is not available or the data is only available in figures.

I'd prefer single-mode RTI, but a benchmark setup for multi-mode RTI that is widely accepted would be okay.


2 Answers 2


It's not quite the problem for which you're looking for solutions, but if you're willing to consider something slightly out of the box, there are numerous benchmarks for the R-T instability in the case of the Stokes equation without time derivative -- i.e., the Boussinesq approximation. Rather than list a bunch of papers, I'm simply going to reference the ASPECT manual that reproduces many of them, with appropriate references:


(Go to Info > Manual.)

  • $\begingroup$ I had run across the van Keken paper, but not the variants in your manual. This is potentially helpful, but I'm actually focused on convincing the turbulent CFD community that my method works for the flows they're interested in, and I think this is too different. $\endgroup$ Nov 16, 2016 at 12:02
  • $\begingroup$ Yes, I knew it was a tangent. Either way, there are many papers that test the RTI for the Boussinesq approximation. I don't know the N-S field, though. $\endgroup$ Nov 16, 2016 at 16:35


I had forgotten about this, but a while ago a friend of mine did his whole thesis work on the Rayleigh-Taylor instability in the context of statistical analysis of turbulence. You can find it enclosed here : http://www.theses.fr/2011DENS0035

The thesis is in French, but the work is extremely thorough concerning the RT instability. You might find the quantitative information (growth rate, turbulent fluctuations, etc.) that you are interested in as well as the problem setup. If you need help with some parts, I can help with the translation.


On another note, if you wish to study the order of convergence of your method (if you wish to study stability, ignore what I am about to say) the best approach to test your convergence is to design test cases using the method of manufactured solution.


Reference book : https://www.amazon.ca/Verification-Validation-Scientific-Computing-Oberkampf/dp/0521113601

Example of a recent article using this (this one is by me, but there are many others) :http://www.sciencedirect.com/science/article/pii/S0045793015000675


Concretly, the method of manufactured solution allows you to create analytical solutions to the incompressible Navier-Stokes equations that are infinitely continuous, that stimulate every term in the Navier-Stokes equation equally and that may even possibly be unsteady. This allows you to fully assess the order of convergence (or the run time, etc.) on fully complex test cases that are representative of the real applications of your CFD code. Furthermore, they can be combined with any type of boundary condition (Dirichlet, Periodic, Neumann, Robin).

That is a good way to benchmark a numerical method and may be even used to evaluate RANS models (with or without wall models). For the latter, see the work of Pr. Pelletier's group.

If this is not what you are looking for, then the Rayleigh-Taylor flow can be good example of an unsteady test case. I will try to look-up a reference.

  • $\begingroup$ A reference would be much appreciated. I've found many, but none quite fit the description in my question. $\endgroup$ Nov 16, 2016 at 12:02
  • $\begingroup$ Please see the edit I made. This is a full thesis dedicated to the Rayleight-Taylor instability in 3D. $\endgroup$
    – BlaB
    Nov 16, 2016 at 13:41
  • $\begingroup$ Thanks! A 2011 thesis available only in French presumably isn't the most widely referenced setup for this problem. But perhaps your friend can answer my question? $\endgroup$ Nov 16, 2016 at 18:18
  • $\begingroup$ Sure, I could ask him if I can get in touch with him. At least, I know is literature review is very thorough, so if you look in the reference therein you should get a very wide idea of what has been done on that exact topic. $\endgroup$
    – BlaB
    Nov 16, 2016 at 18:21

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