# Volume-Of-Fluid advection

We want to approximate the solution of the advection equations \displaystyle \partial_tc_i + \mathbf{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 \displaystyle \partial_tt_{i,j} + \nabla\cdot(\mathbf{u}_ft_{i,j}) = 0 with t_{i,j} = c_if_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 alpha[], flux[];
double cfl = 0.;

If we are also transporting tracers associated with c, we need to compute their gradient i.e. \partial_xf_j = \partial_x(t_j/c) or \partial_xf_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;
if (t.inverse)
cl = 1. - cl, cc = 1. - cc, cr = 1. - cr;
gf[] = 0.;
static const double cmin = 0.5;
if (cc >= cmin && t.gradient != zero) {
if (cr >= cmin) {
if (cl >= cmin) {
if (t.gradient)
gf[] = t.gradient (t[-1]/cl, t[]/cc, t/cr)/Delta;
else
gf[] = (t/cr - t[-1]/cl)/(2.*Delta);
}
else
gf[] = (t/cr - t[]/cc)/Delta;
}
else if (cl >= cmin)
gf[] = (t[]/cc - t[-1]/cl)/Delta;
}
}
}
boundary (gfl);
}

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

  reconstruction (c, n, alpha);

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/(Delta*fm.x[] + SEPS), 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[] + SEPS) > cfl)
cfl = un*fm.x[]*s/(cm[] + SEPS);

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. Volume fraction flux

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]}, alpha[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]*Delta/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 = 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 = (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 = (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 \displaystyle \partial_tc = -\nabla_x\cdot(\mathbf{u}_f c) + c\nabla_x\cdot\mathbf{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 \displaystyle \partial_tt_j = -\nabla_x\cdot(\mathbf{u}_f t_j)

  foreach() {
c[] += dt*(flux[] - flux + cc[]*(uf.x - uf.x[]))/(cm[]*Delta + SEPS);
scalar t, tflux;
for (t, tflux in tracers, tfluxl)
t[] += dt*(tflux[] - tflux)/(cm[]*Delta + SEPS);
}
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;
for (d = 0; d < dimension; d++)
sweep[(i + d) % dimension] (c, cc);
}
}

event vof (i++)
vof_advection (interfaces, i);