# Surface tension

We will need to compute the curvature of the interface, using its Volume-Of-Fluid description.

``#include "curvature.h"``

The surface tension $\sigma$ and interface curvature $\kappa$ will be associated to each VOF tracer. This is done easily by adding the following field attributes.

``````attribute {
double σ;
scalar κ;
}

event defaults (i = 0) {``````

Surface tension is 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.

``````  if (is_constant(a.x)) {
a = new face vector;
foreach_face()
a.x[] = 0.;
boundary ((scalar *){a});
}``````

Each interface for which $\sigma$ is not zero needs a new field to store the curvature.

``````  for (scalar c in interfaces)
if (c.σ && !c.κ.i) {
scalar κ = new_scalar ("kappa");
foreach()
κ[] = 0.;
boundary ({κ});
c.κ = κ;
}
}``````

## Stability condition

The surface tension scheme is time-explicit so the maximum timestep is the oscillation period of the smallest capillary wave. $T=\sqrt{\frac{{\rho }_{m}{\Delta }_{min}^{3}}{\pi \sigma }}$ with ${\rho }_{m}=\left({\rho }_{1}+{\rho }_{2}\right)/2.$ and ${\rho }_{1}$, ${\rho }_{2}$ the densities on either side of the interface.

``event stability (i++) {``

We first compute the minimum and maximum values of $\alpha /{f}_{m}=1/\rho$, as well as ${\Delta }_{min}$.

``````  double amin = HUGE, amax = -HUGE, dmin = HUGE;
foreach_face (reduction(min:amin) reduction(max:amax) reduction(min:dmin)) {
if (α.x[]/fm.x[] > amax) amax = α.x[]/fm.x[];
if (α.x[]/fm.x[] < amin) amin = α.x[]/fm.x[];
if (Δ < dmin) dmin = Δ;
}
double rhom = (1./amin + 1./amax)/2.;``````

We then consider each VOF interface with an associated value of $\sigma$ different from zero and set the maximum timestep.

``````  for (scalar c in interfaces)
if (c.σ) {
double dt = sqrt (rhom*cube(dmin)/(π*c.σ));
if (dt < dtmax)
dtmax = dt;
}
}``````

## Surface tension term

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 $\sigma$ is non-zero. The corresponding volume fraction fields will be stored in list.

``````  scalar * list = NULL;
for (scalar c in interfaces)
if (c.σ) {

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

``````      foreach()
c[] = clamp (c[], 0, 1);
boundary ({c});``````

We update the values of $\kappa$ using the height-function curvature calculation.

``````      assert (c.κ.i);
curvature (c, c.κ);
}``````

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 surface tension term. To do so, we apply the same prolongation to the volume fraction field as applied to the pressure field.

``````#if TREE
for (scalar c in list)
c.prolongation = p.prolongation;
boundary (list);
#endif``````

Finally, for each interface for which $\sigma$ is non-zero, we compute the surface tension acceleration $\sigma \kappa \mathbf{\text{n}}{\delta }_{s}/\rho \approx \alpha \sigma \kappa \nabla c$

``````  face vector st = a;
foreach_face()
for (scalar c in list)
if (c[] != c[-1]) {
scalar κ = c.κ;``````

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

``````	double kf =
(κ[] < nodata && κ[-1] < nodata) ?
(κ[] + κ[-1])/2. :
κ[] < nodata ? κ[] :
κ[-1] < nodata ? κ[-1] :
0.;

st.x[] += α.x[]/fm.x[]*c.σ*kf*(c[] - c[-1])/Δ;
}``````

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

``````#if TREE
for (scalar c in list)
c.prolongation = fraction_refine;
boundary (list);
#endif``````

Finally we free the list of interfacial volume fractions.

``````  free (list);
}``````