gravity and boundary condition

Tom

I'm not sure what fi is physically (I'm not familiar with the level set method) howeve you do not have enough boundary conditions to define your problem!

The flow is inviscid so that the wall condition is simply n.u = 0 where n is the normal to the wall ( in your case your wall appears to be vertical so n.u = U =0 - if your wall is horizontal you should have n.u = V=0). You cannot apply no-slip!

On your interface you need to prescibe the pressure in order to drive the flow ( along with suitable "inlet/outlet" bcs - your system is elliptic and so needs conditions on all boundaries)

I think you need to thoroughly check the mathematical definition of your problem to ensure it is completely self consistent,

Tom.