integral.h
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Integral formulation for surface tension
See Al Saud et al., 2018 and Popinet & Zaleski, 1999 for details.
The surface tension field $\sigma$ will be associated to each levelset tracer. This is done easily by adding the following field attributes.
extern scalar * tracers;
attribute {
scalar sigmaf;
}Surface tension is a source term in the right-hand-side of the evolution equation for the velocity of the centered Navier--Stokes solver i.e. it is an acceleration. If necessary, we allocate a new vector field to store it.
event defaults (i = 0) {
if (is_constant(a.x)) {
a = new face vector;
foreach_face() {
a.x[] = 0.;
dimensional (a.x[] == Delta/sq(DT));
}
}
}Stability condition
The surface tension scheme is time-explicit so the maximum timestep is the oscillation period of the smallest capillary wave.
with $\rho_m=(\rho_1+\rho_2)/2.$ and $\rho_1$, $\rho_2$ the densities on either side of the interface.
event stability (i++)
{
double sigma = 0.;
for (scalar d in tracers)
if (is_constant (d.sigmaf))
sigma += constant (d.sigmaf);
double sigmamax = sigma;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)
reduction(max:sigmamax))
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;The maximum timestep is set using the sum of surface tension coefficients.
double sigmai = sigma;
for (scalar d in tracers)
if (!is_constant (d.sigmaf) && fabs(d[]) < 2.*Delta) {
scalar sigmaf = d.sigmaf;
sigmai += sigmaf[];
}
if (sigmai > sigmamax)
sigmamax = sigmai;
}
double rhom = (1./amin + 1./amax)/2.;
if (sigmamax) {
double dt = sqrt (rhom*cube(dmin)/(pi*sigmamax));
if (dt < dtmax)
dtmax = dt;
}
}Curvature
This function computes the curvature from the levelset function *d* using:
#define CURVATURE 1 // set to 1 (resp. 2) to use curvature (resp. linear) interpolation of curvature
#if CURVATURE
static inline double distance_curvature (Point point, scalar d)
{
double dx = (d[1] - d[-1])/2.;
double dy = (d[0,1] - d[0,-1])/2.;
double dxx = d[1] - 2.*d[] + d[-1];
double dyy = d[0,1] - 2.*d[] + d[0,-1];
double dxy = (d[1,1] - d[-1,1] - d[1,-1] + d[-1,-1])/4.;
double dn = sqrt(sq(dx) + sq(dy)) + 1e-30;
return (sq(dx)*dyy - 2.*dx*dy*dxy + sq(dy)*dxx)/cube(dn)/Delta;
}
#endif // CURVATURESurface tension term
The calculation of the acceleration is done by this event, overloaded from its definition in the centered Navier--Stokes solver.
#if AXI
# include "fractions.h"
#endif
event acceleration (i++)
{We check whether surface tension is associated with any levelset function *d*.
for (scalar d in tracers)
if (d.sigmaf.i) {
(const) scalar sigma = d.sigmaf;
#if CURVATURE == 2We first compute the curvature.
scalar kappa[];
foreach()
kappa[] = distance_curvature (point, d);
#endif // CURVATURE == 2We allocate the surface tension stress tensor "manually", because the symmetries of the default tensor allocation in Basilisk are not what we want (this is messy).
scalar Sxx[], Sxy[], Syy[], Syx[];
tensor S; S.x.x = Sxx, S.x.y = Sxy, S.y.y = Syy, S.y.x = Syx;We compute the diagonal components of the tensor.
foreach()
foreach_dimension() {
S.y.y[] = 0.;
for (int i = -1; i <= 1; i += 2)
if (d[]*(d[] + d[i]) < 0.) {
double xi = d[]/(d[] - d[i]);
double nx = ((d[1] - d[-1])/2. +
xi*i*(d[-1] - 2.*d[] + d[1]))/Delta;
double sigmai = sigma[] + xi*(sigma[i] - sigma[]);
#if CURVATURE
#if CURVATURE == 2 // does not make much difference
double ki = kappa[] + xi*(kappa[i] - kappa[]);
#else
double ki = distance_curvature (point, d);
#endif
S.y.y[] += sigmai*(fabs(nx)/Delta - sign(d[])*ki*(0.5 - xi));
#else // CURVATURE == 0Here we use the pressure jump instead of the curvature. See Appendix A of Al Saud et al., 2018. The noise induced by pressure jumps can be problematic for some cases however, for example for Marangoni translation at small capillary numbers Ca.
S.y.y[] += sigmai*fabs(nx)/Delta + (p[] - p[i])*(0.5 - xi);
#endif // CURVATURE == 0
}
}We compute the off-diagonal components of the tensor.
foreach_vertex()
foreach_dimension()
if ((d[] + d[0,-1])*(d[-1] + d[-1,-1]) > 0.)
S.x.y[] = 0.;
else {
double xi = (d[-1] + d[-1,-1])/(d[-1] + d[-1,-1] - d[] - d[0,-1]);
double ny = (xi*(d[] - d[-1] + d[-1,-1] - d[0,-1]) +
d[-1] - d[-1,-1])/Delta;
double sigmai = (sigma[-1] + sigma[-1,-1] +
xi*(sigma[] + sigma[0,-1] - sigma[-1] - sigma[-1,-1]))/2.;
S.x.y[] = - sigmai*sign(d[] + d[0,-1])*ny/Delta;
}Finally, we add the divergence of the surface tension tensor to the acceleration and the axisymmetric term if necessary.
face vector av = a;
foreach_face() {
av.x[] += alpha.x[]/(fm.x[] + SEPS)*(S.x.x[] - S.x.x[-1] + S.x.y[0,1] - S.x.y[])/Delta;Axisymmetric term
The axisymmetric acceleration is computed using the volumetric surface tension formulation as
with the principal axisymmetric curvature given by
and using the approximation
with $f$ the volume fraction.
#if AXI
coord n = {
(d[] - d[-1])/Delta,
(d[0,1] + d[-1,1] - d[0,-1] - d[-1,-1])/(4.*Delta)
};
extern scalar f;
av.x[] -= alpha.x[]/sq(fm.x[] + SEPS)*(sigma[] + sigma[-1])/2.*n.y*(f[] - f[-1])/Delta;
#endif // AXI
}
}
}References
~~~bib @hal{alsaud2018, hal-01706565}
@article{popinet1999, title={A front-tracking algorithm for accurate representation of surface tension}, author={S. Popinet and S. Zaleski}, journal={International Journal for Numerical Methods in Fluids}, volume={30}, number={6}, pages={775--793}, year={1999}, publisher={Wiley Online Library} } ~~~