Contact angles

    This file is used to impose contact angles on boundaries for interfaces described using a VOF tracer and height functions.

    We first overload the default function used to compute the normal, defined in fractions.h.

    coord interface_normal (Point point, scalar c);
    #define interface_normal(point, c) interface_normal (point, c)
    #include "fractions.h"
    #include "curvature.h"

    We will compute the normal using height-functions instead. If this is not possible (typically at low resolutions) we revert back to the Mixed-Youngs-Centered approximation.

    coord interface_normal (Point point, scalar c)
      coord n;
      if (!c.height.x.i || (n = height_normal (point, c, c.height)).x == nodata)
        n = mycs (point, c);
      return n;

    The height functions are stored in the vector field associated with each VOF tracer. They need to be updated every time the VOF field changes. For the centered Navier-Stokes solver, this means after initialisation and after VOF advection.

    Note that strictly speaking this should be done for each sweep of the direction-split VOF advection, which we do not do here i.e. we use the normal at the beginning of the timestep and assume it is constant during each sweep. This seems to work fine.

    extern scalar * interfaces;
    event init (i = 0) {
      for (scalar c in interfaces)
        if (c.height.x.i)
          heights (c, c.height);
    event vof (i++) {
      for (scalar c in interfaces)
        if (c.height.x.i)
          heights (c, c.height);

    The macro below can be used to impose a contact angle on a boundary by setting the corresponding tangential component of the height function.

    Note that the equivalent function for the normal component of the height function is not defined yet. This limits the range of accessible contact angles, since values of the normal component of the height function will be required to compute curvature at shallow angles.

    #if dimension == 2
    #define contact_angle(theta)					\
      (val(_s) == nodata ? nodata : val(_s) +			\
       (orientation(val(_s)) ? -1. : 1.)/tan(theta))

    Three-dimensional implementation

    While the 2D implementation is trivial, in 3D one must take into account the projection onto the boundary of the normal to the interface (see Afkhami & Bussmann, 2009 for details). This leads to the code below, where the only complication comes from taking into account the relative orientations of the boundary and height-function components.

    From a user point-of-view, using the contact_angle() macro is as simple as in 2D.

    #else // dimension == 3
    #define contact_angle(theta) contact_angle_ (point, neighbor, _s, theta)
    static double contact_z (Point point, scalar h, double theta)
      if (h.i == h.v.z.i) {
        fprintf (stderr,
    	     "contact_angle() cannot be used for '%s' which is the normal\n"
    	     "  component of the height vector\n",
        exit (1);
      if (h[] == nodata)
        return nodata;
        if (h.i == h.v.x.i)
          foreach_dimension(2) {
    	coord n = normal2_x (point, h.v);
    	if (n.x != nodata && n.y != nodata)
    	  return h[] + 1./(tan(theta)*n.x/sqrt(sq(n.x) + sq(n.y)));
      return h[]; // 90 degree contact angle if the normal is not defined
    double contact_angle_ (Point point, Point neighbor, scalar h, double theta)
      if (neighbor.i != point.i)
        return contact_x (point, h, theta);
      if (neighbor.j != point.j)
        return contact_y (point, h, theta);
      if (neighbor.k != point.k)
        return contact_z (point, h, theta);
      assert (false); // not reached
      return 0.;
    #endif // dimension == 3



    S Afkhami and M Bussmann. Height functions for applying contact angles to 3d vof simulations. International Journal for Numerical Methods in Fluids, 61(8):827–847, 2009. [ .pdf ]