Hi guys, I'm having a bash at the Murman & Cole method for solving transonic flow. It's relatively straightforward and I think I've got it working correctly. I've read that the reason their scheme works is that the transition from central to upwind differencing (as the flow accelerates from sub to supersonic) introduces an artificial viscosity - which allows the shock to be captured. I understand why this happens - by expanding the finite difference operators using taylor series.

What I don't understand is why this artificial viscosity is important? Is artificial viscosity needed to correctly capture a shock or does it have some effect on the stability of the scheme?

I'd really appreciate any advice you guys could offer. D

It stabilizes the scheme, and keeps it from blowing up at the shock.

why numerical schemes blowing up at the shock?

because of the thing called Gibbs Phenomena. it occurs as a consequence of using fixed stencils (second or higher-order) across a discontinuity. while fixed stencil tries to reconstruct the polynomial it fails to obtain a non-oscillatory solution. these oscillations do not decay in magnitude even if the mesh is refined around that region. applying artificial viscosity or limiters, or eno-weno type candidate stencil based methods can reduce or eliminate these oscillations.

Am getting this even for first order upwind scheme. Is it something related to the physical background of the model equations - incompressible nonviscous model?

are you getting oscillations for Euler equations? perhaps it would be good to implement the scheme for 1D or 2D scalar equations, like advection or Burger's equations. by this way capturing the flaws of impelementations could be easier. in addition, incompressible algorithm for shock capturing seems not the right case.

No it is not for Euler equations, these are fine for me.

What is the best incompressible algorithm for shock capturing then.

Why are you trying to capture shocks in an incompressible flow?

Because of sharp discontinuities in my liquid-liquid two-phase flow model. In fact it is a stationary discontinuity any ideas please!

Shock capturing is not the same thing as handling interface discontinuities between phases. You need to go back and take a look at multi-phase algorithms such as VOF, front-tracking, level-set, etc. coupled to a good pressure-based incompressible flow solver.

What about if I have a point source terms then?

shock capturing is the only choice for me as I can not use other methods.

If you don't have shocks (which you won't in an incompressible fluid) then shock capturing can't be your only choice because it isn't a choice to begin with. Why can you not use other methods that are actually designed to work with the problem you have described?

I have a stationary discontinuity i.e. one contact wave corresponding to one eigenvalue which is a linearly degenerate, and two injected discontinuities these are just point source terms. So sharp discontinuities will appear. The numerical solutions are mesh (grid) dependent. Problems starts with fine mesh.

Is it because of the linearly degenerate? Or the mathematical model or the point source terms effect the hyperbolicity of the model?

Your views very much appreciated.

Are you dealing with an incompressible flow?

yes it is incompressible flow

Then how are you handling the infinite eigenvalues that correspond to the acoustic waves? Are you using a density-based algorithm or a pressure-based, segregated algorithm?

I have one repeated eigenvalue. This eigenvalue corresponds to one phase velocity i.e. u_{dispersed phase} and it is not like the gas dynamics eigenvalues, no speed of sound. This eigenvalue is a contact wave â€" as I said before. Am trying to use finite volume methods based on the Riemann problem. What do you think then?

What are these methods density-based algorithm or a pressure-based, segregated algorithm? How can I start learning them...?

Do you know anything about carbuncle phenomena or carbuncle waves if it is correct?

What equation set are you solving? Are you using the Riemann solver for the fluid equations? Are you treating the dispersed phase in a Lagrangian or Eulerian fashion?

The equations are in Eulerian fashion- the usual equations, no energey. It is not possible to use Riemann solvers since the system is linearly degenerate just one repeated wave.

At this point I'm not sure what to suggest. The problem you are trying to solve (interface tracking between two phases) would be best handled by tracking the interface in a Lagrangian fashion, and I'm not sure that the eigenvalue you are left with in an eigenvalue analysis really corresponds to the motion of the interface (could be wrong, but I've never seen any other work to indicate that). Shock capturing is a different animal than interface tracking, and you need to review the approaches used for interface tracking. I don't think that artificial viscosity is going to help you, since the most common forms are driven by strong pressure gradients, which don't enter into the interface problem, since the pressure gradients are not typically large.