src/momentum.h

    Momentum-conserving formulation for two-phase interfacial flows

    The interface between the fluids is tracked with a Volume-Of-Fluid method. The volume fraction in fluid 1 is f=1 and f=0 in fluid 2. The densities and dynamic viscosities for fluid 1 and 2 are rho1, mu1, rho2, mu2, respectively.

    #include "all-mach.h"
#include "vof.h"

scalar f[], * interfaces = {f};
double rho1 = 1., mu1 = 0., rho2 = 1., mu2 = 0.;

    Auxilliary fields are necessary to define the (variable) specific volume \alpha=1/\rho and average viscosity \mu (on faces) as well as the cell-centered density.

    face vector alphav[], muv[];
scalar rhov[];

event defaults (i = 0) {
  alpha = alphav;
  rho = rhov;
  mu = muv;

    We use (strict) minmod slope limiting for all components.

      theta = 1.;
  foreach_dimension()
    q.x.gradient = minmod2;
}

    The density and viscosity are defined using arithmetic averages by default. The user can overload these definitions to use other types of averages (i.e. harmonic).

    #ifndef rho
# define rho(f) (clamp(f,0,1)*(rho1 - rho2) + rho2)
#endif
#ifndef mu
# define mu(f)  (clamp(f,0,1)*(mu1 - mu2) + mu2)
#endif

event properties (i++) {
  foreach()
    rhov[] = rho(f[])*cm[];
  foreach_face () {
    double ff = (f[] + f[-1])/2.;
    alphav.x[] = fm.x[]/rho(ff);
    muv.x[] = fm.x[]*mu(ff);
  }
}

    We overload the vof() event to transport consistently the volume fraction and the momentum of each phase.

    static scalar * interfaces1 = NULL;

event vof (i++) {

    We split the total momentum q into its two components q1 and q2 associated with f and 1 - f respectively.

      vector q1 = q, q2[];
  foreach()
    foreach_dimension() {
      double u = q.x[]/rho(f[]);
      q1.x[] = f[]*rho1*u;
      q2.x[] = (1. - f[])*rho2*u;
    }

    Momentum q2 is associated with 1 - f, so we set the inverse attribute to true. We use the same limiting for q1 and q2.

      foreach_dimension() {
    q2.x.inverse = true;
    q2.x.gradient = q1.x.gradient;
  }

#if TREE

    The refinement function is modified by vof_advection(). To be able to restore it, we store its value.

      void (* refine) (Point, scalar) = q1.x.refine;
#endif

    We associate the transport of q1 and q2 with f and transport all fields consistently using the VOF scheme.

      scalar * tracers = f.tracers;
  f.tracers = list_concat (tracers, (scalar *){q1, q2});
  vof_advection ({f}, i);
  free (f.tracers);
  f.tracers = tracers;

    We recover the total momentum.

      foreach()
    foreach_dimension()
      q.x[] = q1.x[] + q2.x[];

#if TREE

    We restore the refinement function for the total momentum.

      for (scalar s in {q}) {
    s.refine = s.prolongation = refine;
    s.dirty = true;
  }
#endif // TREE

    We set the list of interfaces to NULL so that the default vof() event does nothing (otherwise we would transport f twice).

      interfaces1 = interfaces, interfaces = NULL;
}

    We set the list of interfaces back to its default value.

    event tracer_advection (i++) {
  interfaces = interfaces1;
}

    See also

    Usage

    Tests