Surface tension

    Surface tension can be expressed as the interfacial force density \displaystyle \phi\nabla f with f the volume fraction describing the interface and the potential \displaystyle \phi = \sigma\kappa with \sigma the (constant) surface tension coefficient and \kappa the interface mean curvature.

    #include "iforce.h"
    #include "curvature.h"

    The surface tension coefficient is associated to each VOF tracer.

    attribute {
      double sigma;

    Stability condition

    The surface tension scheme is time-explicit so the maximum timestep is the oscillation period of the smallest capillary wave. \displaystyle T = \sqrt{\frac{\rho_{m}\Delta_{min}^3}{\pi\sigma}} with \rho_m=(\rho_1+\rho_2)/2. and \rho_1, \rho_2 the densities on either side of the interface.

    We first compute the minimum and maximum values of \alpha/f_m = 1/\rho, as well as \Delta_{min}.

      double amin = HUGE, amax = -HUGE, dmin = HUGE;
      foreach_face (reduction(min:amin) reduction(max:amax) reduction(min:dmin))
        if (fm.x[] > 0.) {
          if (alpha.x[]/fm.x[] > amax) amax = alpha.x[]/fm.x[];
          if (alpha.x[]/fm.x[] < amin) amin = alpha.x[]/fm.x[];
          if (Delta < dmin) dmin = Delta;
      double rhom = (1./amin + 1./amax)/2.;

    The maximum timestep is set using the sum of surface tension coefficients.

      double sigma = 0.;
      for (scalar c in interfaces)
        sigma += c.sigma;
      if (sigma) {
        double dt = sqrt (rhom*cube(dmin)/(pi*sigma));
        if (dt < dtmax)
          dtmax = dt;

    Definition of the potential

    We overload the acceleration event to define the potential \phi=\sigma\kappa.

    We check for all VOF interfaces for which \sigma is non-zero.

      for (scalar f in interfaces)
        if (f.sigma) {

    If \phi is already allocated, we add \sigma\kappa, otherwise we allocate a new field and set it to \sigma\kappa.

          scalar phi = f.phi;
          if (phi.i)
    	curvature (f, phi, f.sigma, add = true);
          else {
    	phi = new scalar;
    	curvature (f, phi, f.sigma, add = false);
    	f.phi = phi;