[FluidApprentice]

I am simulating a two-phase flow with an Eulerian-Eulerian model, and the concept of "closure laws" and "closing a set of equations" keeps coming up. I loosely understand that it has something to do with making the different phases interact properly and something to do with the interfacial forces, but could someone explain what it physically does to eg. the continuity equations?

[LuckyTran]

A closure problem means there are more unknowns than equations. Hence additional closure laws or closure equations are needed to make the problem solvable. Generally speaking these additional closure relations are not derivable from one of the physical equations being solved and are pulled out of a magic hat. For example in RANS, you have these Reynolds Stresses that appear in the momentum equation but you have no more conservation laws to apply. Thus you need a turbulence model to make the problem solvable and that's your closure model. A closure problem is easiest to recognize as a new or different variable appearing in your equation of interest, i.e. reynolds stresses in the mean momentum equation. Reynolds stresses are not the velocity variables that you want to solve for. Material properties also appear, but you usually have equations of state for each of these (because they are material properties) and so you do not consider it a closure problem. It is a bit semantic what is and what is not a closure problem.

[FluidApprentice]

Originally Posted by LuckyTran

A closure problem means there are more unknowns than equations. Hence additional closure laws or closure equations are needed to make the problem solvable. Generally speaking these additional closure relations are not derivable from one of the physical equations being solved and are pulled out of a magic hat. For example in RANS, you have these Reynolds Stresses that appear in the momentum equation but you have no more conservation laws to apply. Thus you need a turbulence model to make the problem solvable and that's your closure model. A closure problem is easiest to recognize as a new or different variable appearing in your equation of interest, i.e. reynolds stresses in the mean momentum equation. Reynolds stresses are not the velocity variables that you want to solve for. Material properties also appear, but you usually have equations of state for each of these (because they are material properties) and so you do not consider it a closure problem. It is a bit semantic what is and what is not a closure problem.

Wow, great answer! From an equation point of view in the context of multiphase flows, the interfacial forces Drag, Lift, Turbulent Dispersion and Virtual Mass Force are often mentioned in the context of closure laws. They are included in the momentum balance equation, in its interphase transfer term, but what unknown variables (in which equation) are they addressing?

[FluidApprentice]

Originally Posted by FluidApprentice

Wow, great answer! From an equation point of view in the context of multiphase flows, the interfacial forces Drag, Lift, Turbulent Dispersion and Virtual Mass Force are often mentioned in the context of closure laws. They are included in the momentum balance equation, in its interphase transfer term, but what unknown variables (in which equation) are they addressing?

I figured it out. It's the interphase momentum transfer term in the momentum balance.