# What numerical methods are recommendable for simulating two phase immiscible fluid flow through a pipe with high capillary pressure?

I'm simulating two phase immiscible drainage (air displacing water) in a rectangular domain of size .6mm x 2.4mm (2 dimensions) using Ansys FLUENT software. I am using an implicit Volume of Fluid formulation with a PRESTO pressure solver and PISO pressure-velocity coupling, with 2nd order upwind schemes for momentum and volume fraction. The problem is that even with a highly refined mesh, my scheme diverges very quickly (within the first few timesteps, using an algebraic multigrid solver). When I try taking smaller timesteps (about $10^{-6}$ s), the solution takes a lot longer to diverge, but still diverges nonetheless.

I'm not sure if the problem is with the choice of solver itself or if the boundary conditions were not set correctly. I chose to use a constant inlet velocity of .0001m/s, and zero pressure conditions at the outlet. I used no-slip boundary conditions at the other two walls, with wall adhesion and a constant air-to-water contact angle of 135 degrees. Any help with this would be greatly appreciated! :)

It sounds like you are using Ansys Fluent. As I recall from some VOF articles most of them were using projection methods with explicit time stepping. If you are using Fluent you could try finding Non-iterative time advancement (NITE) option for time stepping.

I would also suggest you trying to write your own solver!

There are really good algorithms that you could try in that case. One of those is Moment of fluid (MOF) interface reconstruction developed by researchers from LANL. Here are some links that might be helpful:

• Actually, yes. I am using FLUENT. I have tried using the non-iterative time stepping mechanism too, but I run into the same problem. I have noticed that when I use a larger domain (let's say about on the order of 1cm or larger), everything works perfectly. But when I scale the problem down to millimeters or smaller, it just doesn't work the same... Do you have any suggestions? – Paul Mar 10 '12 at 1:53
• In that case next thing that I would try is to increase the number of sub-iterations in timestep. – Johntra Volta Mar 10 '12 at 7:56
• I browsed the web a little bit and what I found is similar setup like in your case on page 8 of this Fluent tutorial, also an interesting paper which benchmarks several commercial CFD codes. About BC's in all references they say they used velocity inlet for inlet, pressure outlet (0 Pa) for outlet, and no-slip wall. – Johntra Volta Mar 10 '12 at 8:09
• Yes. This is very similar to the kind of problem that I'm trying to setup. There is an interface which is clearly defined and should not be diffusing as it advances through the medium. I just wish the tutorial had a step by step procedure on how to set it up and solve it. – Paul Mar 10 '12 at 15:38

Here's our (open source) entry to the field: http://www.dealii.org/developer/doxygen/deal.II/step_43.html

• This is a nice example, but flow in porous media is a very different problem. – Jed Brown Mar 10 '12 at 18:31
• Err, no -- flow in porous media is Darcy flow, which is exactly one half of what this program solves... – Wolfgang Bangerth Mar 12 '12 at 16:59
• And the two phase flow problem (water in air) that Paul's question here is about does not involve porous media at all. – Jed Brown Mar 12 '12 at 19:54
• You are completely correct. My brain simply equated two-phase flow with porous media flow. Thanks for pointing it out. – Wolfgang Bangerth Mar 14 '12 at 20:44