Interfacial forces

We assume that the interfacial acceleration can be expressed as \displaystyle \phi\mathbf{n}\delta_s/\rho with \mathbf{n} the interface normal, \delta_s the interface Dirac function, \rho the density and \phi a generic scalar field. Using a CSF/Peskin-like approximation, this can be expressed as \displaystyle \phi\nabla f/\rho with f the volume fraction field describing the interface.

The interfacial force potential \phi is associated to each VOF tracer. This is done easily by adding the following field attributes.

attribute {
scalar phi;
}

Interfacial forces are a source term in the right-hand-side of the evolution equation for the velocity of the centered Navier–Stokes solver i.e. it is an acceleration. If necessary, we allocate a new vector field to store it.

event defaults (i = 0) {
if (is_constant(a.x)) {
a = new face vector;
foreach_face()
a.x[] = 0.;
}
}

The calculation of the acceleration is done by this event, overloaded from its definition in the centered Navier–Stokes solver.

event acceleration (i++)
{

We check for all VOF interfaces for which \phi is allocated. The corresponding volume fraction fields will be stored in list.

  scalar * list = NULL;
for (scalar f in interfaces)
if (f.phi.i) {
list = list_add (list, f);

To avoid undeterminations due to round-off errors, we remove values of the volume fraction larger than one or smaller than zero.

      foreach()
f[] = clamp (f[], 0., 1.);
}

On trees we need to make sure that the volume fraction gradient is computed exactly like the pressure gradient. This is necessary to ensure well-balancing of the pressure gradient and interfacial force term. To do so, we apply the same prolongation to the volume fraction field as applied to the pressure field.

#if TREE
for (scalar f in list) {
f.prolongation = p.prolongation;
f.dirty = true; // boundary conditions need to be updated
}
#endif

Finally, for each interface for which \phi is allocated, we compute the interfacial force acceleration \displaystyle \phi\mathbf{n}\delta_s/\rho \approx \alpha\phi\nabla f

  face vector ia = a;
foreach_face()
for (scalar f in list)
if (f[] != f[-1] && fm.x[] > 0.) {

We need to compute the potential phif on the face, using its values at the center of the cell. If both potentials are defined, we take the average, otherwise we take a single value. If all fails we set the potential to zero: this should happen only because of very pathological cases e.g. weird boundary conditions for the volume fraction.

	scalar phi = f.phi;
double phif =
(phi[] < nodata && phi[-1] < nodata) ?
(phi[] + phi[-1])/2. :
phi[] < nodata ? phi[] :
phi[-1] < nodata ? phi[-1] :
0.;

ia.x[] += alpha.x[]/fm.x[]*phif*(f[] - f[-1])/Delta;
}

On trees, we need to restore the prolongation values for the volume fraction field.

#if TREE
for (scalar f in list) {
f.prolongation = fraction_refine;
f.dirty = true; // boundary conditions need to be updated
}
#endif

Finally we free the potential fields and the list of volume fractions.

  for (scalar f in list) {
scalar phi = f.phi;
delete ({phi});
f.phi.i = 0;
}
free (list);
}

References

See Section 3, pages 8-9 of:

 [popinet2018] Stéphane Popinet. Numerical Models of Surface Tension. Annual Review of Fluid Mechanics, 50:49 – 75, January 2018. [ DOI | http | .pdf ]