Staggered mesh
 

I am coding a cfd code using FV method, and I wonder if I have to use staggered mesh for simple algorithm. The problem is lamina 2d back step for incomprehensible flow.

For incompressible flow, it's highly possibly you have to use staggered mesh; otherwise you have to use some methods to eliminate the oscillations in the pressure field, e.g., Rhie-Chow interpolation.

As far as I know, and correct me if I am wrong, the whole point of staggered mesh for incompressible is to preserve energy. If not, spurious energy will be generated, and it's reflected in your pressure field ('cauz pressure is directly relates to energy).

Anyway, in a word, yes, it might be the easiest way for you to use staggered mesh for incompressible simulation.

The main reason to use the staggered mesh (that I know of) is to avoid false uniform pressure that will lead to zero velocity as illustrated in Versteeg Book as in the attached picturehttps://drive.google.com/file/d/1aAI...ew?usp=sharing.
I think this will lead to destroying energy (kinematic) energy and the solution may never converge.

But what I was wondering about is, even for simple flow with good initial values, will checker-board condition (That what I think you meant by pressure oscillation) happen?

Since your case is a back-step, it will probably lead to large-scale separation, which is always challenging for CFD simulation; so probably you want to choose your numerical method conservatively. Also, in the image you posted, the checker-board is there, it's the nature of the collocated mesh; it has nothing to do with your initial conditions. So I would personally suggest using staggered mesh anyway.

BTW, thanks for the clarification on the pressure thing.

You are welcome and thanks for your help.

Is it necessary to use a staggered grid? The answer is no, you can use a non-staggered grid provided that proper care in the formulation is in act


Is it useful to use a staggered grid? the answer is that can be simple and useful only for second order central discretization.


The problem with the non-staggered grid is in the possible decoupling of spurious solutions in the pressure field when a large stencil is used for the pressure. But to see this in terms of kinetic energy production you need to see the divergence-free constraint. Indeed, p*(Div v) is the production term in the balance equation for the kinetic energy.

so if I am planning to use upwind discretization I may use the collocated mesh??
Could you please clarify what kind of formulation you mean for usage of non-staggered mesh?
Thanks for correcting the energy generation point

Upwind introduce a lot of numerical viscosity in the discretization of the convective term but is not used in the pressure equation that is elliptic.

Hello guys, some points were already arised during your discussion. Just to give you my point of view, staggered mesh will solve the checkboarding problem but they are not necessarily easier to implement since the staggering can create some confusion in your head. Moreover they are unsuitable for unstrucured domains or non-rectangular domain. Since you are in the very beginning consider to do a collocated arrangement which nowdays is the common practice. Rhie-Chow interpolation in easy to perform and well documented. Regarding your last question, yes you need rhie-chow even with upwind. The point is that you have to interpolate your momentum equation at cell faces in collocated arrangement (you don't in staggered) and this may cause spurious oscillations in the pressure field.

Pressures equation ?! pressure correction equation in simple algorithm?
And I know about the false diffusion in the upwind scheme however I am currently not so focused on accuracy however I am targeting roboust simple implementation.

Yes, RC interpolation is easy to implement and produce stable and unique pressure solutions. It is suitable to start using it. However, such artificial dissipation terms can affect the quality of the solution in some specific formulations like LES.

I developed a more sophisticated formulation that does not introduce dissipation. However, it is introduced in the framework of th Exact projection methods and requires some effort in the implementation and has some computational cost.



https://www.researchgate.net/publica...y-driven_flows


https://www.researchgate.net/publica...taggered_grids

Yes, pressure-correction. At some point you have to compute the coeffs as rho*vol/aC and interpolate them at faces.

Upwind is a point to start. You have a lot of diffusion (espescially if the flow is tilted), but it is the most stable option you have. To get the 2nd order go for a linear-upwind, still robust (more than central difference)

But don't I need to use convection schemes also at velocity control volume faces at staggered mesh, so momentum interpolation is done in both meshes so differencing scheme also, right?

My flow is mainly in same direction of grid except for the separation after the step so, based on the recommendations in this thread I will go for up wind simple with collocated grid then I may try RC also with up wind or central collocated grid