vof.h

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/** @file vof.h
 */
/**
# Volume-Of-Fluid advection
 
We want to approximate the solution of the advection equations
\f[
\partial_tc_i + \mathbf{u}_f\cdot\nabla c_i = 0
\f]
where \f$c_i\f$ 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 (aka "VOF
concentrations") confined to one side of the interface i.e. solve the
equations
\f[
\partial_tt_{i,j} + \nabla\cdot(\mathbf{u}_ft_{i,j}) = 0
\f]
with \f$t_{i,j} = c_if_j\f$ (or \f$t_{i,j} = (1 - c_i)f_j\f$) and \f$f_j\f$ is a
volumetric tracer concentration (see [Lopez-Herrera et al, 2015](#lopez2015)).
 
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 \f$c\f$ or \f$1 - c\f$) is controlled by the *inverse*
attribute). */
 
attribute {
  scalar * tracers, c;
  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 vector uf;
extern double dt;
 
/**
The gradient of a VOF-concentration `t` 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. */
 
for (int _d = 0; _d < dimension; _d++)
static double vof_concentration_gradient_x (Point point, scalar c, scalar t)
{
  static const double cmin = 0.5;
  double cl = c[-1], cc = c[], cr = c[1];
  if (t.inverse)
    cl = 1. - cl, cc = 1. - cc, cr = 1. - cr;
  if (cc >= cmin && t.gradient != zero) {
    if (cr >= cmin) {
      if (cl >= cmin) {
  if (t.gradient)
    return t.gradient (t[-1]/cl, t[]/cc, t[1]/cr)/Delta;
  else
    return (t[1]/cr - t[-1]/cl)/(2.*Delta);
      }
      else
  return (t[1]/cr - t[]/cc)/Delta;
    }
    else if (cl >= cmin)
      return (t[]/cc - t[-1]/cl)/Delta;
  }
  return 0.;
}
 
/**
On trees, VOF concentrations need to be refined properly i.e. using
volume-fraction-weighted linear interpolation of the concentration. */
 
#if TREE
static void vof_concentration_refine (Point point, scalar s)
{
  scalar f = s.c;
  if (cm[] == 0. || (!s.inverse && f[] <= 0.) || (s.inverse && f[] >= 1.))
    for (int _c = 0; _c < 4; _c++) /* foreach_child */
      s[] = 0.;
  else {
    coord g;
    for (int _d = 0; _d < dimension; _d++)
      g.x = Delta*vof_concentration_gradient_x (point, f, s);
    double sc = s.inverse ? s[]/(1. - f[]) : s[]/f[], cmc = 4.*cm[];
    for (int _c = 0; _c < 4; _c++) /* foreach_child */ {
      s[] = sc;
      for (int _d = 0; _d < dimension; _d++)
  s[] += child.x*g.x*cm[-child.x]/cmc;
      s[] *= s.inverse ? 1. - f[] : f[];
    }
  }
}
 
/**
On trees, we need to setup the appropriate prolongation and
refinement functions for the volume fraction fields. */
 
/** @brief Event: defaults (i = 0) */
void event_defaults (void) 
{
  for (int _c = 0; _c < 1; _c++) /* scalar in interfaces */ {
    c.refine = fraction_refine;
    set_prolongation (c, fraction_refine);
    scalar * tracers = c.tracers;
    for (int _t = 0; _t < 1; _t++) /* scalar in tracers */ {
      set_restriction (t, restriction_volume_average);
      t.refine = vof_concentration_refine;
      set_prolongation (t, vof_concentration_refine);
      t.c = c;
    }
  }
}
#endif // TREE
 
/**
Boundary conditions for VOF-advected tracers usually depend on
boundary conditions for the VOF field. */
 
/** @brief Event: defaults (i = 0) */
void event_defaults (void) 
{
  for (int _c = 0; _c < 1; _c++) /* scalar in interfaces */ {
    scalar * tracers = c.tracers;
    for (int _t = 0; _t < 1; _t++) /* scalar in tracers */
      t.depends = list_add (t.depends, c);
  }
}
 
/**
We need to make sure that the CFL is smaller than 0.5 to ensure
stability of the VOF scheme. */
 
/** @brief Event: stability (i++) */
void event_stability (void) {
  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 [for (int _d = 0; _d < dimension; _d++)](/Basilisk C#foreach_dimension) operator to
automatically derive the corresponding functions along the other
dimensions. */
 
for (int _d = 0; _d < dimension; _d++)
static void sweep_x (scalar c, scalar cc, scalar * tcl)
{
  vector n[];
  scalar alpha[], flux[];
  double cfl = 0.;
 
  /**
  If we are also transporting tracers associated with \f$c\f$, we need to
  compute their gradient i.e. \f$\partial_xf_j = \partial_x(t_j/c)\f$ or
  \f$\partial_xf_j = \partial_x(t_j/(1 - c))\f$ (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 (int _t = 0; _t < 1; _t++) /* scalar in tracers */ {
      scalar gf = {0} /* new scalar */, flux = {0} /* new scalar */;
      gfl = list_append (gfl, gf);
      tfluxl = list_append (tfluxl, flux);
    }
 
    /**
    The gradient is computed using the "interface-biased" scheme above. */
 
    for (int _i = 0; _i < _N; _i++) /* foreach */ {
      scalar t, gf;
      for (t,gf in tracers,gfl)
  gf[] = vof_concentration_gradient_x (point, c, t);
    }
  }
  
  /**
  We reconstruct the interface normal \f$\mathbf{n}\f$ and the intercept
  \f$\alpha\f$ for each cell. Then we go through each (vertical) face of
  the grid. */
 
  reconstruction (c, n, alpha);
  for (int _i = 0; _i < _N; _i++) /* foreach_face */) {
 
    /**
    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 EMBED
    if (cs[] >= 1.)
#endif
    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](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);
  
  /**
  We warn the user if the CFL condition has been violated. */
 
  if (cfl > 0.5 + 1e-6)
    fprintf (ferr, 
       "src/vof.h:%d: warning: CFL must be <= 0.5 for VOF (cfl - 0.5 = %g)\n", 
       LINENO, 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
  \f[
  \partial_tc = -\nabla_x\cdot(\mathbf{u}_f c) + c\nabla_x\cdot\mathbf{u}_f
  \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 corresponding one-dimensional update is
  \f[
  \partial_tt_j = -\nabla_x\cdot(\mathbf{u}_f t_j) + t_j\nabla_x\cdot\mathbf{u}_f
  \f]
  The compressive second-term can be removed by defining the
  NO_1D_COMPRESSION macro. */
 
#if !EMBED
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
    c[] += dt*(flux[] - flux[1] + cc[]*(uf.x[1] - uf.x[]))/(cm[]*Delta);
#if NO_1D_COMPRESSION
    scalar t, tflux;
    for (t, tflux in tracers, tfluxl)
      t[] += dt*(tflux[] - tflux[1])/(cm[]*Delta);
#else // !NO_1D_COMPRESSION
    scalar t, tc, tflux;
    for (t, tc, tflux in tracers, tcl, tfluxl)
      t[] += dt*(tflux[] - tflux[1] + tc[]*(uf.x[1] - uf.x[]))/(cm[]*Delta);
#endif // !NO_1D_COMPRESSION
  }
#else // EMBED
  /**
  When dealing with embedded boundaries, we simply ignore the fraction
  occupied by the solid. This is a simple approximation which has the
  advantage of ensuring boundedness of the volume fraction and
  conservation of the total tracer mass (if it is computed also
  ignoring the volume occupied by the solid in partial cells). */
  
  for (int _i = 0; _i < _N; _i++) /* foreach */
    if (cs[] > 0.) {
      c[] += dt*cs[]*(flux[] - flux[1] + cc[]*(uf.x[1] - uf.x[]))/(cm[]*Delta);
#if NO_1D_COMPRESSION
      for (t, tflux in tracers, tfluxl)
  t[] += dt*cs[]*(tflux[] - tflux[1])/(cm[]*Delta);
#else // !NO_1D_COMPRESSION
      scalar t, tc, tflux;
      for (t, tc, tflux in tracers, tcl, tfluxl)
  t[] += dt*cs[]*(tflux[] - tflux[1] + tc[]*(uf.x[1] - uf.x[]))/(cm[]*Delta);
#endif // !NO_1D_COMPRESSION
    }
#endif // EMBED
 
  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 (int _c = 0; _c < 1; _c++) /* scalar 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[], * tcl = NULL, * tracers = c.tracers;    
    for (int _t = 0; _t < 1; _t++) /* scalar in tracers */ {
#if !NO_1D_COMPRESSION
      scalar tc = {0} /* new scalar */;
      tcl = list_append (tcl, tc);
#endif // !NO_1D_COMPRESSION
#if TREE
      if (t.refine != vof_concentration_refine) {
  t.refine = vof_concentration_refine;
        set_prolongation (t, vof_concentration_refine);
  set_restriction (t, restriction_volume_average);
  t.c = c;
      }
#endif // TREE
    }
    for (int _i = 0; _i < _N; _i++) /* foreach */ {
      cc[] = (c[] > 0.5);
#if !NO_1D_COMPRESSION
      scalar t, tc;
      for (t, tc in tracers, tcl) {
  if (t.inverse)
    tc[] = c[] < 0.5 ? t[]/(1. - c[]) : 0.;
  else
    tc[] = c[] > 0.5 ? t[]/c[] : 0.;
      }
#endif // !NO_1D_COMPRESSION
    }
 
    /**
    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, scalar *);
    int d = 0;
    for (int _d = 0; _d < dimension; _d++)
      sweep[d++] = sweep_x;
    for (d = 0; d < dimension; d++)
      sweep[(i + d) % dimension] (c, cc, tcl);
    delete (tcl), free (tcl);
  }
}
 
/** @brief Event: vof (i++) */
void event_vof (void) 
  vof_advection (interfaces, i);
 
/**
## References
 
~~~bib
@Article{lopez2015,
  title = {A VOF numerical study on the electrokinetic effects in the 
           breakup of electrified jets},
  author = {J. M. Lopez-Herrera and A. M. Ganan-Calvo and S. Popinet and
            M. A. Herrada},
  journal = {International Journal of Multiphase Flows},
  pages = {14-22},
  volume = {71},
  year = {2015},
  doi = {doi.org/10.1016/j.ijmultiphaseflow.2014.12.005},
  url = {https://gerris.dalembert.upmc.fr/papers/lopez2015.pdf}
}
~~~
*/