shocks in a 1d spherically symmetric model
Hi all,
I am a complete novice in the field of CFD and I am having trouble capturing shocks in the following situation. I would greatly appreciate any help or advise you can give me. I am modeling a spherically symmetric collapse of a gas bubble in liquid. The bubble wall's motion is a prescribed function of time. The equations inside are compressible NavierStokes + thermal conduction in Lagrange coordinate formulation (lagrange coordinates are very convenient here since I have a boundary that moves with the gas). I am using 2nd order accurate explicit finite difference numerical scheme and it works just fine for relatively slow collapses that do not develop shocks. What is the best way to capture shocks in this situation? Also, the shock seems to be forming in a region within the bubble where density is low, and thus grid points are sparse (thanks to the Lagrange coordinates). Thanks! 
Re: shocks in a 1d spherically symmetric model
A great deal is known about shock capturing and flux limiting in Eulerian coordinates. A lot of this should translate to Lagrangian coordinates, though I am unfamiliar with the literature for the latter. Check the wiki here at cfdonline (click on the link for the free CFD reference at top right of this webpage) or check Wikipedia.
Regarding the grid sparsity, should this not be addressable by suitably clustering gridpoints in your initial condition? You could examine your final solution or your solution before it goes haywire, and prescribe what would be a good gridpoint distribution for shockcapturing at that time. Then map this better grid back to the initial condition using the Lagrange coordinates. Now use this better initial grid to recompute the bubble collapse. 
Re: shocks in a 1d spherically symmetric model
I wonder how to treat the interface between the bubble and liquid(?water). The nonphysical error can be made from incorrect interface treatment. If the flow is compressible, you could insert any flux limiter or slope limiter.

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