As a beginner to the free surface flows, I would like to ask your ideas. Currently I am studying two-layered natural convection. The materals are heavy metalic melts such as ferrite and allumina. The density ratio is more than two, so they separate by the gravitational force. As you can imagine, it is a kind of free surface flows. There are some codes to solve free surface flows like sola-vof and Ripple. Some say light material is neglected if density ratio is very high between two materials such as water and air. But it is possible to solve two materials at the same time no matter hwo the density ratio is.

In my case I should solve two materials for heat and momenta, but the interface between the two materials is assumed not changed. I am using my unstructured-mesh solver which is based on SIMPLE, Co-located(nonstaggered), Rhie-Chow interpolation.

The problem is how to treat gravitational body force. The book written by Peric said hydrostatic head is added to pressure therm. And it makes the computation more stable. I agree on that.

But if there are two different materials, I think it is impossible.

At first I used the gravitational source term as rho*g = rho_ref*[1- beta*(T-T_ref)]*g --(1) ,where hydrostatic head is not subtracted. When the eq (1) is used, I modified the boundary pressure extrapolation like p_b = p_o + rho*g*( x_b - x_o ). Except the problem with garvitational force boundary face pressure is same as boundary cell pressure.(Zero order extrapolation)

I didn't use free surface capturing schem, in other words I didn't solve VOF equation. I just fixed the interface. When the density ratio is near one I could get the solution to the two-layered natural convection. The materials are naturally stratified. But with the density ratio very larger than 1 it failed. So I simplified the problem by not solving energy equation. This is just hydrostatic problem, and exact solution is stationary flow with hydrostatic pressure field( p = rho*g*h ). The cells adjacent to interface have infinitesimal source of the difference between dp/dy and rho*g_y. This discrepancy is amplified and the code is diverged.

---------Oh. I am sorry, it is too long to explain my trouble. I would like to simplify my questions.

1) How to treat the gravitational body force and pressure gradient terms.

2) Is there a special way to integrate pressure gradient term in momentum equations(In case of non-staggered algorithm)

3) If you know some papers related two-layered free surface problem with non-staggered method, could you tell me them?

4) What is the initial pressure field for that? Some solved air, water and bubble interactions. I think it is very difficult to initialize the hydrostatic pressure field.

Currently I am studying the basics of the numerical methods for free surface. If you give me your idea, I can reduce the time to dig the problem. In advence I want to thank you for your forever kindness.

(1). You have a natural convection problem, where the density of the liquid is not uniform. (2). If you ignore the shape of the free surface and the interface between two liquids, then you basically have two problems in two domains. (3). The interaction between the two problem is through the fixed interface surface. (4). What I would suggest is, solve it as coupled problem, each with uniform density (material). that is, a natural convection problem-A with heavy material on the bottom, and a given interface condition. And another natural convection problem with a light material on the top. (5). With this approach, the material will be uniform (not the density), and the boundary conditions are specified on the fixed given locations for each problem and domain. (6). You can used zero interface velocity boundary condition to simulate a real wall, or you can relax it to simulate the real interface condition.