Multiphase VOF Simulation

Why do we need multiphase simulation?

When a ship hull cuts through water, a surf fin pierces the surface, or a propeller ventilates in rough sea conditions, we are not dealing with single-phase flow. There are two fluids — water and air — with densities differing by three orders of magnitude (rho_water ~ 1000 kg/m3, rho_air ~ 1.2 kg/m3). The Volume of Fluid (VOF) method, introduced by Hirt and Nichols in 1981, is the standard tool for capturing the interface between immiscible fluids in CFD simulations.

Mathematical foundations of VOF

The VOF method solves a single momentum equation for the mixture, sharing velocity and pressure between phases, and a transport equation for the volume fraction alpha:

da/dt + nabla·(U a) + nabla·[Uc a(1 - a)] = 0

Where alpha = 1 in cells fully occupied by water, alpha = 0 in cells with air, and 0 < alpha < 1 at the interface. The additional term with U_c (compression velocity) counteracts numerical diffusion and keeps the interface sharp — it is the key to OpenFOAM's interFoam solver.

The effective properties in each cell are calculated as a weighted average:

rho = a rho_water + (1 - a) rho_air
mu = a mu_water + (1 - a) mu_air

Surface tension is incorporated as a volumetric force using Brackbill's Continuum Surface Force (CSF) model:

F_sigma = sigma kappa nabla a

Where sigma is the surface tension coefficient (0.072 N/m for water-air at 20C) and kappa is the interface curvature calculated from the gradient of alpha.

Practical case: surf fin water entry

Simulation setup

We simulate a surf fin entering water at 8 m/s with a 12-degree angle of attack. The domain is a 4 x 2 x 2 meter box with the bottom half filled with water (alpha = 1) and the top half with air (alpha = 0). We use interFoam with the k-omega SST turbulence model and adaptive mesh refinement at the interface.

/* setFieldsDict - Phase initialization */
defaultFieldValues
(
    volScalarFieldValue alpha.water 0
);

regions
(
    boxToCell
    {
        box (-2 -1 -2) (2 0 2);
        fieldValues
        (
            volScalarFieldValue alpha.water 1
        );
    }
);

/* dynamicMeshDict - Adaptive refinement at the interface */
dynamicFvMesh dynamicRefineFvMesh;

dynamicRefineFvMeshCoeffs
{
    refineInterval 5;
    field alpha.water;
    lowerRefineLevel 0.001;
    upperRefineLevel 0.999;
    unrefineLevel 10;
    nBufferLayers 2;
    maxRefinement 3;
    maxCells 4000000;
}

Results

The simulation captures several complex physical phenomena:

• Spray formation: small droplets detach from the fin's leading edge upon impact with the free surface
• Ventilation: at high angles of attack (>15), an air pocket adheres to the suction side, collapsing lift abruptly
• Surface wave: the depression and elevation of the free surface generates a wave pattern that modifies the effective angle of attack

The calculated lift coefficient (C_l ~ 0.62) agrees within 8% with experimental measurements in a test flume, validating the approach.

VOF limitations and alternatives

VOF is not perfect:

• The interface blurs on coarse meshes; requires adaptive refinement
• Does not model phase change (evaporation, cavitation) without additional coupling
• Computational cost scales with interfacial area, not volume

Alternatives include Level-Set (sharper interface but poorer mass conservation) and Front Tracking (more accurate but more complex to implement in 3D). CLSVOF (Coupled Level-Set VOF) hybrids combine the best of both worlds.

Conclusions

VOF is the go-to tool for multiphase flows with well-defined interfaces. In the context of surface hydrodynamics — fins, propellers, hulls — it captures phenomena that single-phase models simply ignore, such as ventilation and spray, which can be critical to performance.