I'm a graduate student.Now I work on the topic of numerical combustion.I should solve the N-S equations on unstructured mesh.I have a question about convective flux and need your help. I compute the convective flux of N-S equations by using Roe's approximate Rieman solver.With the evolution,it will give a density which is less than zero.(the value \rho continue to decrease) What's the reason of diverge?Is Roe's approximate Rieman solver not suitable to solve the field of reactive flows(\gama=Cp/Cv changes greatly in the computation domain,1.25~1.34) I need your suggestion!Thanks!

Roe's scheme has been extended to deal with real gas effects (i.e. changing values of gamma). Seems like there could a problem with your mass conservation. If you are using a FV scheme, you should be able to check this.

Try to check for cell-wise (and global) mass conservation at every step from the start of the simulation (much before the density goes -ve). If mass is conserved in the first few steps and then the problem arises, you could be voilating a stability criterion (most likely the CFL condition).

Real-gas extensions might not help. Roe's scheme (for real or ideal gas) can give negative densities. If your problem has strong expansions and low densitites, you don't want to use Roe's scheme.

dear kalyan and jess davies,

Thank you very much for your kind explanation!

it seems that roe's scheme is not a good choice.Then how about osher's approximate riemann solver,or kinetic boltzmann method?Do they have the same problem too?

If a correct algorithm is employed to avoid non-physical solution(keep \rho>0),can it keep the solution to relax to a steady-state?

Two thoughts on your Roe scheme questions:

(1) One way around the negative density problem some researchers have used is to add dissipation to the different waves. One of the more common is due to Harten and Yee (I don't have the exact reference).

(2) Another possibility that is less costly than the exact Riemann solvers is to use a low dissipation flux vector scheme like AUSM, AUSM+ or LDFSS. You should be able to find papers relative to these schemes in the AIAA Journal. Check especially for papers by Meng-Seng Liou of NASA and J. R. Edwards from N. C. State University.

I have used Roe scheme for computing the reating flows and I solved shock tube cases with strong discontinuities. I did not have such problems. If you put in the entropy fix I think it should help. Otherwise, you can try using HLLE, HLLEM which has more dissipation than Roe scheme and are found to be better for some cases.

But extension of OSher scheme to real-gas flows is a real tough problem . You can refer to Sureh and Liou's paper in Int J. for Numerical methods in fluids, 14:219-232, 1992

Just because Roe's scheme works for some problems is no guarantee it will always work. Roe (and Solomon/Osher) are known to die under fairly mild expansions. Discontinuities can be as strong as you like - this is not what causes these schemes to die.

It is an over-simplification to say that methods with extra dissipation (entropy fixes) will cure this. (For one thing, it may not !) The flaw in Roe's scheme is non-physical (eg. rho<0) states. An exact solution will not have this problem, but this doesn't mean Roe's approximate solver is less dissipative/more accurate than an exact Riemann solver.

You could also get negative densities because of coding errors. The only way you'll ever know for sure is by using a method which doesn't allow negative densities.

(1). well, when you have negative densities, you are automatically in the research domain, that is, it is problem dependent. (2). And the implementation of the method is also critical. Even with a method which allows only positive densities, the difference in thinking between the east and the west, will also determine whether it will be implemented successfully. (3). The west tends to explore the new way of doing things, which can lead to a new method of dealing with difficult situations. On the other hand, the east tends to follow the old method and try to push it to the extreme to overcome the difficulty. This is not always a good approach and can lead to fatal accident. (4). My suggestion is: define the key issue of the problem first, eliminate the unnecessary variables, and select the simple method to deal with the key issues in the problem. Don't try to put the chemical reactions, compressible flows, and shock waves all in one problem, unless you are already an expert.

There can be many reasons which make the solution divergent. Assuming entropy inequality is a problem...

A well-known drawback of Roe scheme is that it may resolve non-physical solutions, which is called the "carbuncle phenomenon" by the researchers at NASA. The reason of occuring it is not known clearly yet. Flux vector splitting does not suffer from such a problem, but it is too dissipative and gives thick boundary layer in viscous flow calculations. To cure carbuncle phenomenon, ie. to break expansion shocks, corrected eigenvalues are used in calculation. Some known formulae for entropy correction are Harten's formula, Roe's formula, Yee's formula and van Leer's formula, etc. The function of these formulae is to add dissipation near the sonic points. If you want to avoid those annoying jobs, use HLLE solver. Once you programmed Roe scheme, it is very easy to add entropy correction formula to your code. The original formula is for 1D or stuructured 2D grids. However, it is straightforward to extend to multidimensional unstructured grids. Try to test it.

I think there is some reason that Roe scheme is widespread now. Do not judge Roe scheme is a bad choice because of a carbuncle phenomenon. It can be cured by entropy correction. Many numerical experiments showed the abilitiy of Roe scheme. There are many Riemann solvers and they have their own advantages and disadvantages.

References

Harten & Hyman, Self adjusting grid methods for one dimensional hyperbolic conservation laws. J. Comp. Phys, 50(1983),235-269

Jeremie Gressier & Jean-Marc Moschetta, Robustness versus accuracy in shock-wave computations, Int'l J. for Numer. Meth. in Fluids, 33(2000), 313-332.

Hong-Chia Lin, Dissiaption additions to flux-difference splitting, J. Comp. Phys, 117(1995), 20-27.

P.L. Roe, Some Contributions to the modelling of discontinuous flow, In Lectures in Applied Mathematics 22, AMS, 163-193.

Francois Dubois & Guillaume Mehlman, A non-parameterized entropy correction for Roe's approximate Riemann solver, Numerische Mathematik, Springler-Verlag, 73(1996), 169-208.

Bram van Leer et. al, Sonic point capturing, AIAA-89-1945.

Good Luck~

Graduate Student, Department of Civil Engineering, Hanyang University, Seoul, Korea.

> I think there is some reason that Roe scheme is widespread now.

Yes. Historically, it was one of the first approximate Riemann solvers.

> Do not judge Roe scheme is a bad choice because of a carbuncle phenomenon.

No, I judge it a bad choice because in this application we want to keep our densities positive. The caurbuncle phenomenon is a separate, unrelated problem.

> Many numerical experiments showed the abilitiy of Roe scheme.

Many numerical experiments also showed the ability of central differencing. That does not mean central-differencing is suitable for all applications.

Thanks for your comments.

>Yes. Historically, it was one of the first approximate Riemann solvers.

-> I think the popularity of Roe scheme is more than that. Osher's solver is unpopular now. And HLL solver is more disspative than Roe's. In addition, I did not say Roe scheme is applicable for all problems.

>Many numerical experiments also showed the ability of central differencing. That does not mean central-differencing is suitable for all applications.

-> Yes it is right. As you probably know, central differencing is the last choice for shock problem. Bear in mind our topic is "shock". Your statement is out of point.

> Bear in mind our topic is "shock". Your statement is out of point.

I think most people will understand my analogy.

If I suggest we should not follow Roe like lemmings, I don't intend this to be taken literally. I am not really talking about rope-less bungee jumping.

Hi Hai-Wen Ge,

you can look at http://www.geocities.com/andrei_chernousov/freecfd.htm or simply download some examples from there(dealing with fluxes in flow with combustion), namely:

http://www.geocities.com/andrei_cher.../normal.tar.gz,

and

http://www.geocities.com/andrei_cher...lcs2mix.tar.gz,

and

http://www.geocities.com/andrei_cher...S/solver.ps.gz.

Hope it helps! Best wishes,

Andrei

dear Andrei Chernousov,

thank you very much!I will download them and try them.

dear Mr. Andrei Chernousov,

I fail to connect to www.geocities.com .Could you kindly send the code by e-mail?(mailto:heaven@mail.ustc.edu.cn)

Thank you very much