Hi, All, It is known that, FE/FV discretization for NS based on node centered linear (bilinear) functions (both for pressure and velocity) is unstable.

Does somebody know if the cell centered, colocated discretization for NS is stable? I mean any references on this topic, or somebody's computattional expereince.

STAR-CD, FLUENT AND CFX all use cell-centered colocated FV arrangement and there is no such problem with stability if one does it properly.

Hi there,

it's not unstable if done properly. Bunch a FV collocated codes are available. some people have introduced some artificial dissipation by using QUICK or UPWIND schemes, but this is really not necessary. As long as your code is mass, momentuum, and preferably energy conservative, then it'll be stable. Check the following paper where the conservatives properties are discussed for several different grid layouts:

Morinishi, Y., Lund, T., Vasilyev and Moin. "Fully conservative Higher order finite difference schemes for incompressible flow". Journal of computational Physics, Vol 143, pp 90-124, 1998.

Sincerely,

Frederic Felten.

If you introduced numerical dissipation there will be no stability problem. But, will the cell-centered approach, yaou have to be aware of orthogonality problems (i.e. evaluation of the flux between two cells) when your mesh is skewed.

If you refer to the occurence of spurious pressure modes when equal order interpolation for velocity and pressure is used, then co-located (equal order) FVM are not an exception and "pressure stabilisation" is (traditionally) accomplished by adding a fourth order pressure difference in the continutity eqn. This is most often referenced as the Rhie and Chow correction, from their paper AIAA J, 21(11):1525-1532 (1983). The technique somewhat resembles what is used in (pressure) stabilised FEM, e.g. PSPG.