Interfacial forces

We assume that the interfacial acceleration can be expressed as ϕnδs/ρ with n the interface normal, δs the interface Dirac function, ρ the density and ϕ a generic scalar field. Using a CSF/Peskin-like approximation, this can be expressed as ϕf/ρ with f the volume fraction field describing the interface.

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

attribute {
  scalar φ;

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;
      a.x[] = 0.;
    boundary ((scalar *){a});

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 ϕ is allocated. The corresponding volume fraction fields will be stored in list.

  scalar * list = NULL;
  for (scalar f in interfaces)
    if (f.φ.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.

	f[] = clamp (f[], 0, 1);
      boundary ({f});

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;
  boundary (list);

Finally, for each interface for which ϕ is allocated, we compute the interfacial force acceleration ϕnδs/ραϕf

  face vector ia = a;
    for (scalar f in list)
      if (f[] != f[-1]) {

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 φ = f.φ;
	double phif =
	  (φ[] < nodata && φ[-1] < nodata) ?
	  (φ[] + φ[-1])/2. :
	  φ[] < nodata ? φ[] :
	  φ[-1] < nodata ? φ[-1] :
	ia.x[] += α.x[]/fm.x[]*phif*(f[] - f[-1])/Δ;

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;
  boundary (list);

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

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


See Section 3, pages 8-9 of:

S. Popinet, "Numerical models of surface tension", Annual Review of Fluid Mechanics, 2018, article