sandbox/hasansh/fast-capwave/fast-capwave-air-water.c

Capillary wave for fluids with different densities using new poisson solver

We use a constant-resolution grid, the Navier–Stokes solver with VOF interface tracking and surface tension. Since we’re using the new fast poisson solver we include fast-navierstokes.h instead of navier-stokes/centered.h

#include "grid/multigrid.h"
#include "fast-navierstokes.h"
#include "two-phase.h"
#include "tension.h"
#include "prosperetti-air-water.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.

  L0 = 2.;
  Y0 = -L0/2.;
  f.sigma = 1;

We reduce TOLERANCE and CFL numbers to get better results

  TOLERANCE = 1e-14;
  CFL=1e-4;

We are using viscosity and density ratios corresponding to an air-water interface.

  rho1=1;
  rho2=0.0012;
  mu1= 0.0182571749236;
  mu2= 0.0182571749236*(1.8e-5/1.003e-3);

\rho_0 is defined as minimum of \rho_1 and \rho_2.

  rho0 = min(rho1,rho2);

We change method and resolution to compare convergnece of FastPn, FastP*, and Basilisk methods.

  for(int iter= unsplit; iter<=fastpstar;iter++){      
     for (N = 16; N <= 128; N *= 2) {
     se = ne = 0;
     run();
     }
  poisson_solver=iter+1;
  } 
}

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

event init (t = 0) {
  fraction (f, y - 0.01*cos (2.*pi*x));
}

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. 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 += 0.00198785108553814829; t <= 1.58928694288774963184) {

To get an accurate amplitude, we reconstruct the height function field and take the corresponding maximum.

  vector h[];
  heights (f, h);
  double max = - HUGE;;
  foreach() 
    if (f[] > 0 && f[] < 1) {
      double yi = y + height(h.y[])*Delta;
      if (yi > max)
	max = yi;
    }

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

   char name[80];
  sprintf (name, "wave-%i-%d",poisson_solver ,N);
  static FILE * fp = fopen (name, "w");
  fprintf (fp, "%g %g\n", t*15.7402, max);
  fflush (fp);

To compute the RMS error, we get data from the reference file prosperetti-air-water.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 a file, indexed with the method, the resolution (number of grid points per wavelength) and the relative RMS error.

event error (t = end){
  char name[80];
  sprintf (name, "convergence-%i",poisson_solver); 
  FILE * fp = fopen (name, "a");
  if (fp!=NULL){
  	fprintf (fp, "%g %g\n", N/L0, sqrt(se/ne)/0.01);
  	fflush (fp);    
  }
  else{
        FILE * fp = fopen (name, "w");
  	fprintf (fp, "%g %g\n", N/L0, sqrt(se/ne)/0.01);
  	fflush (fp);    
  }

}

Results

set output 'amplitude.png'
set xlabel 'tau'
set ylabel 'Relative amplitude'
plot '../prosperetti-air-water.h' u 2:4 w l t "Prosperetti", \
     'wave-0-128' every 10 w p t "Basilisk",\
     'wave-1-128' every 10 w p t "FastPn",\
     'wave-2-128' every 10 w p t "FastP*"

set output 'convergence.png'
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] 2./x**2 t "Second order",'convergence-0' t "Basilisk" w lp,'convergence-1' t "FastPn" w lp, 'convergence-2' t "FastP*" w lp

See also

• Same test with Gerris