[optimux]

Hi All,

I'm searching for spectral method used in a two-phase flow. Unfortunately, only a few papers that have something to do with this topic. Let's take the attached figure as an example: a box filled with hot liquid alloy, cooled from right and bottom walls while left and top walls are insulated. As it cools, liquid near right and bottom walls are going to crystallize, some crystals will attach to the walls while others will move along convection cells.

If boldly we assume all solid will attach to the walls, then this system can be described by the equations attached. The subscript L and S are liquid and solid. Darcy's law is also included for partially solidified region but we can neglect it here. The bar means time averaged value but for simplicity here we assume it's a constant. Latent heat denoted by Hs could be determined in physical space I guess.

So the problem is: is there a way to apply any kind of spectral method to this system? Most spectral method are working in a single phase. This problem is a two-phase flow and the solid region grows with time. All physical variables are multiplied by a solid or liquid fraction (fS or fL), I cannot get rid of them in a vorticity-streamfunction way.

Any ideas?


Best,
Mingming