staggered vs non staggered
 

does any body know that in compressible flow, we need staggered grid?

Staggering has only one typical field of application in incompressible flow: ensuring the exact projection in the subspace of divergence-free functions with central second order discretization (actually, the recent methods ensure that also on non-staggered grids). Neverthless, if you like it and want complicate your CFD life you can use also in any other case. ;)

thanks Filippo

you mean that I do not need to do that in compressible case or I should use specific interpolation such as rhi-chow or something like that?
my case is compressible, not incompressible!

generally, for compressible flows the problem is the undesired presence of "wiggles" caused by decoupled flux reconstructions. The literature in this field is full of methods working on non-staggered grids that help in that.
Thus, there is no real reasons to complicate a code (immagine a 3D Navier-Stokes code) using staggering.
Neverthless, you can find people working on that, just for example see

https://www.cs.ubc.ca/~rbridson/docs/mbonner_msc.pdf

The problem solved by staggered meshes (and approximately solved via Rhie-Chow and similar methods) are present in incompressible and compressible flow...as well as in computational electromagnetics. The problem arises because of the significant effect of a grad(phi) operator in one or more transport equation. When the grad term doesn't offer a rationale to use upwinding as with convection terms, you will find that the gradient formula can involve ONLY the neighbor values and not the cell value itself. This, by itself, does not guarantee that even-odd decoupling will occur, but it opens the door for it. This has physical ramifications. For example, the pressure in a given cell will not drive the mass fluxes, so a cell with a little too much mass (relative to its neighbor cells) will have a higher pressure than its neighbors and yet, on a collocated mesh with centered gradient formulation, there may be NO OUTFLOW from the cell due to that higher pressure/density.

Just to be clear, this is caused byf the discretization, not the particular solution algorithm. It is calamitous in pressure-based solvers because it gives you a pressure matrix that may not have unique solutions. In density-based solvers, it can just lead to artifacts that are not easy to damp out and can lead to solution drift until you get a negative density or something similarly unphysical.

If the pressure terms are only weakly driving your flow, you will likely be OK. If your compressible scheme uses Riemann solvers to construct the face fluxes, you may be OK. If you use Rhie-Chow interpolation, you should always be OK.

So, avoid the complexity of using staggered grids, but if you start seeing even-odd decoupling, you will likely need to use one of the above options or those noted by Prof Denaro,

Note that when using the integral form (often used for problem with non regular solution) the pressure gradient disappears. You get the integral of the pressure over the surface of the finite volume and the key (to work on non staggered grids) is in the proper evaluation of the pressure flux (for example like in the AUSM).