# src/iforce.h

# Interfacial forces

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

The interfacial force potential $\varphi $ 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;
foreach_face()
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 $\varphi $ 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.

```
foreach()
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);
#endif
```

Finally, for each interface for which $\varphi $ is allocated, we compute the interfacial force acceleration $$\varphi \mathbf{\text{n}}{\delta}_{s}/\rho \approx \alpha \varphi \nabla f$$

```
face vector ia = a;
foreach_face()
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] :
0.;
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);
#endif
```

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);
}
```

## References

See Section 3, pages 8-9 of:

**popinet2018**- S. Popinet, "Numerical models of surface tension",
*Annual Review of Fluid Mechanics*, 2018, article