Roe flux scheme, why Roe's averages important
Dear colleagues,
Could anyone please explain to me or point me to some reference on why Roe's averages are so important in Roe flux scheme. A simple question I have is what happens if I simply take the arithmetic average or geometric average of the left and right state, instead of strictly follow the Roe's average formulation. In fact, I remember I read in Fluent's manual that the Roe's scheme implemented therein uses simple arithmetic average, don't know why. Couldn't find the link now, so Fluent folks, forgive me if this is not true. I read Roe's original paper but constantly find myself lost, mainly due to my poor command of mathematics. Regards, Shenren 
Found two references online from people that I think know what they are talking about. Hope this can be of help to someone asking the same question.
1) From him (http://www.ita.uniheidelberg.de/~dullemond/index.shtml) https://docs.google.com/viewer?a=v&q...Ec3Sn1lZcrYI1g 2) From him (http://www.hiroakinishikawa.com/) http://www.cfdnotes.com/cfdnotes_roe...d_density.html Quote:

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In a few words, if one replaces the nonlinear problem dq/dt + df(q)/dx with a linearized Riemann problem at each interface, dq/dt + A*dq/dx, then a problem arises in defining the Jacobian A. A natural choice could be A = f'(qavg), where f' = df/dq One could simply use qavg = 0.5(qr + ql), averaging right and left states (as Fluent does). For some problems, however (e.g. Euler equations, shallow water equations, ...), a special average value exists for which a number of "nice" properties are satisfied  and that's why Roe's average is important. In some way, the idea is more or less the same of using the mean value theorem between right and left states  although I'm not sure about the mathematical formalism for this. 
Ciao Francesco,
Could you please elaborate a little bit more on the 'nice' properties of Roe's averages and why those properties are important to a good simulation? You see, my question is exactly why the Roe's averages are superior than using the arithmetic average, which as you confirmed, is what Fluent does. I'll read the book you recommended. Thank you! Shenren Quote:

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unfortunately a comprehensive answer to your question needs a detailed mathematical formulation, which you can find in the abovementioned book. In simple words, use of arithmetic average would result in a spurious behaviour near shocks, which on the contrary would be wellresolved using Roe's average  see Figure 15.2 from Leveque's book. This is a consequence of some mathematical constraints which are violated using a simple average. Unfortunately, for practical problems a Roe linearization is not available and many solvers (e.g. Fluent) use arithmetic average (which, however, works well in regions where the flow is smooth). Hope this helps, Francesco 
Hi Francesco,
1) I'm reading the book you recommended and as you explained, the superiority of Roe averages indeed is much more evident when there's large gradient in the flow, such as shocks. No wonder Roe scheme is known for shock capturing. 2) I thought Fluent uses a simple average only because of its simplicity. In what circumstance would Roe's linearisation be not available? Could you give one or two simple examples? In that case, what shall we do for shockcapturing in those cases then? Regards, Shenren Quote:

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As a general consideration, however, I think you should ask yourself about the real importance of numerical accuracy in your activity: if you are doing "engineering" work, and you are concerned with global properties of the flow (let's say, drag coefficient, heat flux, etc.), then I wouldn't be too worried about the details of the flux scheme  just try to achieve a solution which is grid and schemeindependent and very good confidence would be placed in your results. On the other hand, a very different situation arises if you are doing scientific work, then numerical accuracy becomes a primary issue... but that is just my two cents :D Regards, Francesco 
Keep in mind that Roe averaging dramatically slower than simple linear averaging

Hi SergeAS,
I admit that Roe averaging is more costly then linear averaging. But is it really dramatically slower? I thought the most costly part is the wave amplitude \times the wave speed, i.e., A\cdot \delta_w. But I may be wrong, it's just that your argument is not apparent to me. Regards, Shenren Quote:

Hi
Is rhoPimpleFoam in OpenFOAM appropriate to capturing of shock wave? does Roe term is div(phi,U)? 
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