[TurbJet]

Greetings,

As far as I know, there are some in-house codes (e.g., the code I am using) for solving high-speed compressible flows based on collocated grid (with finite difference of course), and they can generate good results without further efforts such as applying Rhie-Chow interpolation.

In contrast, in incompressible flow, collocated mesh will lead to some high-frequency oscillations in the solution due to the velocity-pressure decoupling (if I remember correctly), and either have to apply Rhie-Chow 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!

[FMDenaro]

Originally Posted by TurbJet

Greetings,

As far as I know, there are some in-house codes (e.g., the code I am using) for solving high-speed compressible flows based on collocated grid (with finite difference of course), and they can generate good results without further efforts such as applying Rhie-Chow interpolation.

In contrast, in incompressible flow, collocated mesh will lead to some high-frequency oscillations in the solution due to the velocity-pressure decoupling (if I remember correctly), and either have to apply Rhie-Chow interpolation to fix it or just switch to staggered mesh.

I am confused : why collocated mesh does work for compressible flow, and yet it doesn't 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!

The decopuling appearing in the incompressible flow solvers is due to the elliptic equation for the pressure that must enforce the divergence-free 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.

[TurbJet]

Originally Posted by FMDenaro

The decopuling appearing in the incompressible flow solvers is due to the elliptic equation for the pressure that must enforce the divergence-free 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.

Under what circumstances would the compressible flow have oscillations?

Besides, do you have any references recommend on this topic?

[sbaffini]

It also depends from the specific compressible solver in use and the convective scheme. The preconditioned Roe flux difference splitting scheme has a Rhie-Chow 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.

[TurbJet]

Originally Posted by sbaffini

It also depends from the specific compressible solver in use and the convective scheme. The preconditioned Roe flux difference splitting scheme has a Rhie-Chow 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.

That sounds about right; I've read a paper by S.K.Lele says that if shock waves is involved in the flow, collocated mesh will no longer work, so I guess the R-C interpolation has to be applied.

But I am just wondering, for low-speed 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?

[sbaffini]

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.

Rhie-Chow, in a sense, just does that, adds dissipation to the mass flux in the continuity equation.

[TurbJet]

Originally Posted by sbaffini

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.

Rhie-Chow, in a sense, just does that, adds dissipation to the mass flux in the continuity equation.

Actually, for subsonic-to-transonic flow, the code I use can directly use 4th-order DRP scheme (which is an optimized central difference). But it's true that sometimes extra artificial filtering (dissipation) needs to be added to keep the solution stable.

[agd]

If you are using a dispersion-preserving 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 high-order 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.

[TurbJet]

Originally Posted by agd

If you are using a dispersion-preserving 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 high-order 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.

I am using DRP scheme developed by C.Bogey & C.Bailly. I do use filtering. What I meant in the previous thread is just that there are some cases I can live without it.

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?

[agd]

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.

[FMDenaro]

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.

[TurbJet]

Originally Posted by agd

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.

Now it makes me wondering if anyone adopt such approach to incompressible (introducing the filtering and without the Rhie-Chow interpolation) to eliminate the oscillations from collocated mesh?

Also, assuming I am using staggered mesh, is the artificial viscosity still needed?

[TurbJet]

Originally Posted by FMDenaro

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.

So you mean the filtering kills all possible oscillations, no matter where they stems from?

[FMDenaro]

Originally Posted by TurbJet

So you mean the filtering kills all possible oscillations, no matter where they stems from?

Well, it depends on the type of filtering, some filter can totally remove high wavenumbers components other types can smooth the content...