# Capillary wave

This is the classical test case first proposed in Popinet & Zaleski, 1999.

We use a constant-resolution grid, the Navierâ€“Stokes solver with VOF interface tracking (optionally coupled with levelset) and surface tension (optionally using the integral formulation).

#include "grid/multigrid.h"
#include "navier-stokes/centered.h"
#if CLSVOF
# include "two-phase-clsvof.h"
# include "integral.h"
# include "curvature.h"
#else
# include "vof.h"
# include "tension.h"

The interface is represented by the volume fraction field c.

scalar f[], * interfaces = {f};
#endif
#include "prosperetti.h"

We make sure that the boundary conditions for the face-centered velocity field are consistent with the centered velocity field (this affects the advection term).

uf.n[left]   = 0.;
uf.n[right]  = 0.;
uf.n[top]    = 0.;
uf.n[bottom] = 0.;

We will store the accumulated error in se and the number of samples in ne.

double se = 0; int ne = 0;

int main() {

The domain is 2x2 to minimise finite-size effects. The surface tension is one and the viscosity is constant.

  size (2. [1]);
Y0 = -L0/2.;
#if CLSVOF
const scalar sigma[] = 1.;
d.sigmaf = sigma;
#else
f.sigma = 1.;
#endif
TOLERANCE = 1e-6 [*];
const face vector muc[] = {0.0182571749236, 0.0182571749236};
mu = muc;

We vary the resolution to check for convergence.

  for (N = 16; N <= 128; N *= 2) {
se = 0, ne = 0;
run();
}
}

The initial condition is a small amplitude plane wave of wavelength unity.

event init (t = 0) {
double k = 2., a = 0.01;
#if CLSVOF
foreach()
d[] = y - a*cos (k*pi*x);
#else
fraction (f, y - a*cos (k*pi*x));
#endif
}

By default tracers are defined at t-\Delta t/2. We use the first keyword to move VOF advection before the amplitude output i.e.Â at t+\Delta/2. This improves the results.

event vof (i++, first);

We output the amplitude at times matching exactly those in the reference file.

event amplitude (t += 3.04290519077e-3; t <= 2.2426211256) {

To get an accurate amplitude, we reconstruct interface position (using height functions) and take the corresponding maximum.

  scalar pos[];
position (f, pos, {0,1 [0]});
double max = statsf(pos).max;

We output the corresponding evolution in a file indexed with the number of grid points N.

  char name[80];
sprintf (name, "wave-%d", N);
static FILE * fp = fopen (name, "w");
fprintf (fp, "%g %g\n", t*11.1366559937, max);
fflush (fp);

To compute the RMS error, we get data from the reference file prosperetti.h and add the difference to the accumulated error.

  se += sq(max - prosperetti[ne][1]); ne++;
}

At the end of the simulation, we output on standard error the resolution (number of grid points per wavelength) and the relative RMS error.

event error (t = end)
fprintf (stderr, "%g %g\n", N/L0, sqrt(se/ne)/0.01);

#if 0
event gfsview (i += 1) {
static FILE * fp = popen ("gfsview2D -s ../capwave.gfv", "w");
output_gfs (fp);
}
#endif

## Results

set xlabel 'tau'
set ylabel 'Relative amplitude'
plot '../prosperetti.h' u 2:4 w l t "Prosperetti", \
'wave-128' every 10 w p t "Basilisk", \
'../capwave-clsvof/wave-128' every 10 w p t "Basilisk (CLSVOF)"
set xlabel 'Number of grid points'
set ylabel 'Relative RMS error'
set logscale y
set logscale x 2
set grid
plot [5:200][1e-4:1]\
'log' t "Basilisk" w lp, 2./x**2 t "Second order", \
'../capwave-clsvof/log' t "Basilisk (CLSVOF)" w lp, \
2./x**2 t "Second order"