# src/vof.h

# Volume-Of-Fluid advection

We want to approximate the solution of the advection equations $${\partial}_{t}{c}_{i}+{\mathbf{\text{u}}}_{f}\cdot \nabla {c}_{i}=0$$ where ${c}_{i}$ are volume fraction fields describing sharp interfaces.

This can be done using a conservative, non-diffusive geometric VOF scheme.

We also add the option to transport diffusive tracers confined to one side of the interface i.e. solve the equations $${\partial}_{t}{t}_{i,j}+\nabla \cdot ({\mathbf{\text{u}}}_{f}{t}_{i,j})=0$$ with ${t}_{i,j}={c}_{i}{f}_{j}$ (or ${t}_{i,j}=(1-{c}_{i}){f}_{j}$) and ${f}_{j}$ is a volumetric tracer concentration.

The list of tracers associated with the volume fraction is stored in the *tracers* attribute. For each tracer, the “side” of the interface (i.e. either $c$ or $1-c$) is controlled by the *inverse* attribute).

```
attribute {
scalar * tracers;
bool inverse;
}
```

We will need basic functions for volume fraction computations.

```
#include "fractions.h"
```

The list of volume fraction fields `interfaces`

, will be provided by the user.

The face velocity field `uf`

will be defined by a solver as well as the timestep.

```
extern scalar * interfaces;
extern face vector uf;
extern double dt;
```

On trees, we need to setup the appropriate prolongation and refinement functions for the volume fraction fields.

```
event defaults (i = 0)
{
#if TREE
for (scalar c in interfaces)
c.refine = c.prolongation = fraction_refine;
#endif
}
```

We need to make sure that the CFL is smaller than 0.5 to ensure stability of the VOF scheme.

```
event stability (i++) {
if (CFL > 0.5)
CFL = 0.5;
}
```

## One-dimensional advection

The simplest way to implement a multi-dimensional VOF advection scheme is to use dimension-splitting i.e. advect the field along each dimension successively using a one-dimensional scheme.

We implement the one-dimensional scheme along the x-dimension and use the foreach_dimension() operator to automatically derive the corresponding functions along the other dimensions.

```
foreach_dimension()
static void sweep_x (scalar c, scalar cc)
{
vector n[];
scalar α[], flux[];
double cfl = 0.;
```

If we are also transporting tracers associated with $c$, we need to compute their gradient i.e. ${\partial}_{x}{f}_{j}={\partial}_{x}({t}_{j}/c)$ or ${\partial}_{x}{f}_{j}={\partial}_{x}({t}_{j}/(1-c))$ (for higher-order upwinding) and we need to store the computed fluxes. We first allocate the corresponding lists.

```
scalar * tracers = c.tracers, * gfl = NULL, * tfluxl = NULL;
if (tracers) {
for (scalar t in tracers) {
scalar gf = new scalar, flux = new scalar;
gfl = list_append (gfl, gf);
tfluxl = list_append (tfluxl, flux);
}
```

The gradient is computed using a standard three-point scheme if we are far enough from the interface (as controlled by *cmin*), otherwise a two-point scheme biased away from the interface is used.

```
foreach() {
scalar t, gf;
for (t,gf in tracers,gfl) {
double cl = c[-1], cc = c[], cr = c[1];
if (t.inverse)
cl = 1. - cl, cc = 1. - cc, cr = 1. - cr;
gf[] = 0.;
static const double cmin = 0.5;
if (cc >= cmin) {
if (cr >= cmin) {
if (cl >= cmin) {
if (t.gradient)
gf[] = t.gradient (t[-1]/cl, t[]/cc, t[1]/cr)/Δ;
else
gf[] = (t[1]/cr - t[-1]/cl)/(2.*Δ);
}
else
gf[] = (t[1]/cr - t[]/cc)/Δ;
}
else if (cl >= cmin)
gf[] = (t[]/cc - t[-1]/cl)/Δ;
}
}
}
boundary (gfl);
}
```

We reconstruct the interface normal $\mathbf{\text{n}}$ and the intercept $\alpha $ for each cell. Then we go through each (vertical) face of the grid.

```
reconstruction (c, n, α);
foreach_face(x, reduction (max:cfl)) {
```

To compute the volume fraction flux, we check the sign of the velocity component normal to the face and compute the index `i`

of the corresponding *upwind* cell (either 0 or -1).

```
double un = uf.x[]*dt/(Δ*fm.x[]), s = sign(un);
int i = -(s + 1.)/2.;
```

We also check that we are not violating the CFL condition.

```
if (un*fm.x[]*s/cm[] > cfl)
cfl = un*fm.x[]*s/cm[];
```

If we assume that `un`

is negative i.e. `s`

is -1 and `i`

is 0, the volume fraction flux through the face of the cell is given by the dark area in the figure below. The corresponding volume fraction can be computed using the `rectangle_fraction()`

function.

When the upwind cell is entirely full or empty we can avoid this computation.

```
double cf = (c[i] <= 0. || c[i] >= 1.) ? c[i] :
rectangle_fraction ((coord){-s*n.x[i], n.y[i], n.z[i]}, α[i],
(coord){-0.5, -0.5, -0.5},
(coord){s*un - 0.5, 0.5, 0.5});
```

Once we have the upwind volume fraction *cf*, the volume fraction flux through the face is simply:

` flux[] = cf*uf.x[];`

If we are transporting tracers, we compute their flux using the upwind volume fraction *cf* and a tracer value upwinded using the Bell–Collela–Glaz scheme and the gradient computed above.

```
scalar t, gf, tflux;
for (t,gf,tflux in tracers,gfl,tfluxl) {
double cf1 = cf, ci = c[i];
if (t.inverse)
cf1 = 1. - cf1, ci = 1. - ci;
if (ci > 1e-10) {
double ff = t[i]/ci + s*min(1., 1. - s*un)*gf[i]*Δ/2.;
tflux[] = ff*cf1*uf.x[];
}
else
tflux[] = 0.;
}
}
delete (gfl); free (gfl);
```

On tree grids, we need to make sure that the fluxes match at fine/coarse cell boundaries i.e. we need to *restrict* the fluxes from fine cells to coarse cells. This is what is usually done, for all dimensions, by the `boundary_flux()`

function. Here, we only need to do it for a single dimension (x).

```
#if TREE
scalar * fluxl = list_concat (NULL, tfluxl);
fluxl = list_append (fluxl, flux);
for (int l = depth() - 1; l >= 0; l--)
foreach_halo (prolongation, l) {
#if dimension == 1
if (is_refined (neighbor(-1)))
for (scalar fl in fluxl)
fl[] = fine(fl);
if (is_refined (neighbor(1)))
for (scalar fl in fluxl)
fl[1] = fine(fl,2);
#elif dimension == 2
if (is_refined (neighbor(-1)))
for (scalar fl in fluxl)
fl[] = (fine(fl,0,0) + fine(fl,0,1))/2.;
if (is_refined (neighbor(1)))
for (scalar fl in fluxl)
fl[1] = (fine(fl,2,0) + fine(fl,2,1))/2.;
#else // dimension == 3
if (is_refined (neighbor(-1)))
for (scalar fl in fluxl)
fl[] = (fine(fl,0,0,0) + fine(fl,0,1,0) +
fine(fl,0,0,1) + fine(fl,0,1,1))/4.;
if (is_refined (neighbor(1)))
for (scalar fl in fluxl)
fl[1] = (fine(fl,2,0,0) + fine(fl,2,1,0) +
fine(fl,2,0,1) + fine(fl,2,1,1))/4.;
#endif
}
free (fluxl);
#endif
```

We warn the user if the CFL condition has been violated.

```
if (cfl > 0.5 + 1e-6)
fprintf (ferr,
"WARNING: CFL must be <= 0.5 for VOF (cfl - 0.5 = %g)\n",
cfl - 0.5), fflush (ferr);
```

Once we have computed the fluxes on all faces, we can update the volume fraction field according to the one-dimensional advection equation $${\partial}_{t}c=-{\nabla}_{x}\cdot ({\mathbf{\text{u}}}_{f}c)+c{\nabla}_{x}\cdot {\mathbf{\text{u}}}_{f}$$ The first term is computed using the fluxes. The second term – which is non-zero for the one-dimensional velocity field – is approximated using a centered volume fraction field `cc`

which will be defined below.

For tracers, the one-dimensional update is simply $${\partial}_{t}{t}_{j}=-{\nabla}_{x}\cdot ({\mathbf{\text{u}}}_{f}{t}_{j})$$

```
foreach() {
c[] += dt*(flux[] - flux[1] + cc[]*(uf.x[1] - uf.x[]))/(cm[]*Δ);
scalar t, tflux;
for (t, tflux in tracers, tfluxl)
t[] += dt*(tflux[] - tflux[1])/(cm[]*Δ);
}
boundary ({c});
boundary (tracers);
delete (tfluxl); free (tfluxl);
}
```

## Multi-dimensional advection

The multi-dimensional advection is performed by the event below.

```
void vof_advection (scalar * interfaces, int i)
{
for (scalar c in interfaces) {
```

We first define the volume fraction field used to compute the divergent term in the one-dimensional advection equation above. We follow Weymouth & Yue, 2010 and use a step function which guarantees exact mass conservation for the multi-dimensional advection scheme (provided the advection velocity field is exactly non-divergent).

```
scalar cc[];
foreach()
cc[] = (c[] > 0.5);
```

We then apply the one-dimensional advection scheme along each dimension. To try to minimise phase errors, we alternate dimensions according to the parity of the iteration index `i`

.

```
void (* sweep[dimension]) (scalar, scalar);
int d = 0;
foreach_dimension()
sweep[d++] = sweep_x;
boundary ({c});
for (d = 0; d < dimension; d++)
sweep[(i + d) % dimension] (c, cc);
}
}
event vof (i++)
vof_advection (interfaces, i);
```