Basilisk source code (http://basilisk.fr/src/)

root / src / vof.h.page

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/**
# Volume-Of-Fluid advection

We want to approximate the solution of the advection equations
$$
\partial_tc_i + \mathbf{u}_f\cdot\nabla c_i = 0
$$
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
$$
\partial_tt_{i,j} + \nabla\cdot(\mathbf{u}_ft_{i,j}) = 0
$$
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()](/Basilisk C#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[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)/Delta;
	      else
		gf[] = (t[1]/cr - t[-1]/cl)/(2.*Delta);
	    }
	    else
	       gf[] = (t[1]/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[]), 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.
    
    ![Volume fraction flux](figures/flux.svg)
    
    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[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
  $$
  \partial_tc = -\nabla_x\cdot(\mathbf{u}_f c) + c\nabla_x\cdot\mathbf{u}_f
  $$
  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
  $$
  \partial_tt_j = -\nabla_x\cdot(\mathbf{u}_f t_j)
  $$
  */

  foreach() {
    c[] += dt*(flux[] - flux[1] + cc[]*(uf.x[1] - uf.x[]))/(cm[]*Delta);
    scalar t, tflux;
    for (t, tflux in tracers, tfluxl)
      t[] += dt*(tflux[] - tflux[1])/(cm[]*Delta);
  }
  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](/src/references.bib#weymouth2010) 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);