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Why collocated grid can be used for solving compressible flow? 

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January 18, 2019, 18:49 
Why collocated grid can be used for solving compressible flow?

#1 
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Greetings,
As far as I know, there are some inhouse codes (e.g., the code I am using) for solving highspeed compressible flows based on collocated grid (with finite difference of course), and they can generate good results without further efforts such as applying RhieChow interpolation. In contrast, in incompressible flow, collocated mesh will lead to some highfrequency oscillations in the solution due to the velocitypressure decoupling (if I remember correctly), and either have to apply RhieChow interpolation to fix it or just switch to staggered mesh. I am confused : why collocated mesh can work for compressible flow, and yet it doesn't work for incompressible flow? I read some explanations saying that it relates to energy conservation. But I have no clue about the details. Could anything provide some insights? Or maybe some papers I can refer to? Appreciate it! Last edited by TurbJet; January 19, 2019 at 19:59. 

January 19, 2019, 03:33 

#2  
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Filippo Maria Denaro
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Quote:
The decopuling appearing in the incompressible flow solvers is due to the elliptic equation for the pressure that must enforce the divergencefree velocity constraint (have a look to the Peric & Ferziger textbook). Conversely, in compressible flow solvers this equation does not exist as the pressure is thermodinamic. However, some oscillations can appear also in this case. 

January 19, 2019, 11:22 

#3  
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Besides, do you have any references recommend on this topic? Last edited by TurbJet; January 19, 2019 at 20:00. 

January 19, 2019, 16:46 

#4 
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It also depends from the specific compressible solver in use and the convective scheme. The preconditioned Roe flux difference splitting scheme has a RhieChow like term for the pressure in the incompressible limit. Too drunk now to remember if it is in place also in general and/or for the non preconditioned case.


January 19, 2019, 17:27 

#5  
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Quote:
But I am just wondering, for lowspeed compressible flow (e.g., subsonic, no shock waves), why the collocated mesh can work? Is it just because no Poisson equation as in incompressible flow? 

January 21, 2019, 05:26 

#6 
Senior Member

The continuity equation, no matter in which kind of solver, always include the divergence of the mass flux. If this is just evaluated centrally, I think you are going to have problems as well, no matter what.
I think compressible codes don't have such problems because they actually use upwinding/matrix dissipation for the mass flux in the continuity equation. RhieChow, in a sense, just does that, adds dissipation to the mass flux in the continuity equation. 

January 21, 2019, 11:17 

#7  
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January 21, 2019, 13:24 

#8 
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If you are using a dispersionpreserving scheme such as that developed by C. Tam, I would be surprised if there is not some artificial damping added into the code to maintain stability. It may be in the form of explicit damping or a highorder filter to remove spurious oscillations, but it will be there. All central difference schemes require some form of damping/upwinding/filtering as the Mach number increases to account for the hyperbolic nature of the equations of motion.


January 21, 2019, 15:24 

#9  
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Um, so I guess it is the filtering eliminates those oscillations so that they do not show up? Namely, the collocated mesh still introduce oscillations (same as in incompressible flow) but just killed by filtering? Or it is the different oscillations you are talking about? 

January 21, 2019, 15:39 

#10 
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No, that's it. The filtering removes the oscillatory behavior that would tend to develop. It serves the same function as artificial viscosity, or added damping, or upwinding. It is just a different approach to the same end state.


January 21, 2019, 15:50 

#11 
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Filippo Maria Denaro
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Generally, one analyses the number of zero in the Fourier symbol of the operator. If more than the fundamental mode is found (more than one dimension), the zeros are at the Nyquist frequencies implying a possible oscillations in the solution. The adoption of filtering aims to eliminate such modes.


January 21, 2019, 15:57 

#12  
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Quote:
Also, assuming I am using staggered mesh, is the artificial viscosity still needed? Last edited by TurbJet; January 21, 2019 at 17:47. 

January 21, 2019, 15:59 

#13  
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January 21, 2019, 16:04 

#14 
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Filippo Maria Denaro
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Tags 
collocated grid, compressible flows, incompressible flows 
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