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The comments on the accepted answer of my previous question here have left me with a more general question about accurately capturing shockwaves in fluid calculations.

For the sake of having an example, let's say we have a standing shockwave with a pressure or density ratio of 10:1 measured at two points one unit length apart. Let's also assume that I am attempting to capture it with 1D fluid calculations.

In this case, is the shockwave a true singularity, where infinite mesh refinement at the location of the shockwave always leading to one cell having 10 times the flow variable value of an adjacent cell? Or is it the case that the gradients of flow variables may be several orders of magnitude larger within the area of the shockwave than anywhere else, but are still continuous? Does the answer to the above question depend on the source of the shockwave?

And as a tangent question, assuming that the second case is true, over the one unit length thickness of the shockwave, what would be a good rule of thumb for an initial mesh density? 100 cell? 1000?

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  • $\begingroup$ Please write down the equations you are solving -- they are essential to the answer to this question. If the system is hyperbolic, shocks will be singularities. If it has dissipative terms, typically shocks will not be singularities. $\endgroup$
    – David Ketcheson
    Commented Jul 17, 2012 at 7:56
  • $\begingroup$ Let's say its the full Navier-Stokes Equations. I am currently not solving them, but will likely be building a 1D solver to handle such a case in the near future. $\endgroup$
    – Godric Seer
    Commented Jul 17, 2012 at 16:37

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This depends on the equations in case. If your equations are first-order hyperbolic equations such as the Euler equations of gas dynamics, then the equations allow solutions that have truly discontinuous shocks -- in that case, there is no amount of mesh refinement that can resolve the shock.

On the other hand, if your equations are transport dominated but still have a diffusion term (like, I believe, what you call the Navier-Stokes equations, by which I believe you mean the compressible equations that still contain a certain amount of viscosity with a term that contains second order derivatives) then in general the solutions may have large gradients but are still continuous. In that case, at least conceptually, shocks are continuous and have a finite, positive width that can be resolved using mesh refinement -- although oftentimes the length scale of the shock is so small that you won't be able to resolve it with any practically usable mesh.

So, in essence, it all boils down to which equations exactly you are considering, and if it has a diffusion/viscosity term.

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  • $\begingroup$ That all makes sense. I am working with the compressible viscous equations. Now I just have to hope that mine is a case where a reasonable mesh capable of capturing the shockwave is possible. $\endgroup$
    – Godric Seer
    Commented Jul 17, 2012 at 20:25
  • $\begingroup$ Anisotropic mesh refinement in 1D should allow you to resolve the shock without blowing up the number of grid points/cells. $\endgroup$
    – Bill Barth
    Commented Jul 17, 2012 at 21:32

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