# Rising bubble

A two-dimensional bubble is released in a rectangular box and raises under the influence of buoyancy. This test case was proposed by Hysing et al, 2009 (see also the FEATFLOW page).

We solve the incompressible, variable-density, Navier–Stokes equations with interfaces and surface tension. We can solve either the axisymmetric or planar version. We can used standard or “reduced” gravity.

#if AXIS
# include "axi.h" // fixme: does not run with -catch
#endif
#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "tension.h"
#if REDUCED
# include "reduced.h"
#endif

#ifndef LEVEL
# define LEVEL 8
#endif

The boundary conditions are slip lateral walls (the default) and no-slip on the right and left walls.

u.t[right] = dirichlet(0);
u.t[left]  = dirichlet(0);

We make sure there is no flow through the top and bottom boundary, otherwise the compatibility condition for the Poisson equation can be violated.

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

int main() {

The domain will span [0:2]\times[0:0.5] and will be resolved with 256\times 64 grid points.

  size (2);
init_grid (1 << LEVEL);

Hysing et al. consider two cases (1 and 2), with the densities, dynamic viscosities and surface tension of fluid 1 and 2 given below.

  rho1 = 1000., mu1 = 10.;
#if CASE2
rho2 = 1., mu2 = 0.1, f.sigma = 1.96;
#else
rho2 = 100., mu2 = 1., f.sigma = 24.5;
#endif

We reduce the tolerance on the Poisson and viscous solvers to improve the accuracy.

  TOLERANCE = 1e-4;
#if REDUCED
G.x = -0.98;
Z.x = 1.;
#endif
run();
}

event init (t = 0) {

The domain is a rectangle. We only simulate half the bubble.

  mask (y > 0.5 ? top : none);

The bubble is centered on (0.5,0) and has a radius of 0.25.

  fraction (f, sq(x - 0.5) + sq(y) - sq(0.25));
}

We add the acceleration of gravity.

#if !REDUCED
event acceleration (i++) {
face vector av = a;
foreach_face(x)
av.x[] -= 0.98;
}
#endif

A utility function to check the convergence of the multigrid solvers.

void mg_print (mgstats mg)
{
if (mg.i > 0 && mg.resa > 0.)
printf ("%d %g %g %g %d ", mg.i, mg.resb, mg.resa,
mg.resb > 0 ? exp (log (mg.resb/mg.resa)/mg.i) : 0.,
mg.nrelax);
}

We log the position of the center of mass of the bubble, its velocity and volume as well as convergence statistics for the multigrid solvers.

event logfile (i++) {
double xb = 0., vb = 0., sb = 0.;
foreach(reduction(+:xb) reduction(+:vb) reduction(+:sb)) {
double dv = (1. - f[])*dv();
xb += x*dv;
vb += u.x[]*dv;
sb += dv;
}
printf ("%g %g %g %g %g %g %g %g ",
t, sb, -1., xb/sb, vb/sb, dt, perf.t, perf.speed);
mg_print (mgp);
mg_print (mgpf);
mg_print (mgu);
putchar ('\n');
fflush (stdout);
}

At t=3 we output the shape of the bubble.

event interface (t = 3.) {
output_facets (f, stderr);
}

If gfsview is installed on the system, we can also visualise the simulation as it proceeds.

#if 0
event gfsview (i += 10) {
static FILE * fp = popen("gfsview2D rising.gfv", "w");
scalar vort[];
vorticity (u, vort);
output_gfs (fp);
}
#endif

}
#endif

## Results

The final shape of the bubble is compared to that obtained with the MooNMD Lagrangian solver (see the FEATFLOW page) at the highest resolution. We also display the shape of the axisymmetric version of the test. The axisymmetric bubble moves much faster.

set term push
set term @SVG size 640,320
set size ratio -1
set grid
plot [][0:0.4]'../c1g3l4s.txt' u 2:($1-0.5) w l t 'MooNMD', \ 'log' u 1:2 w l t 'Basilisk', \ '../rising-axi/log' u 1:2 w l t 'Basilisk (axisymmetric)' Bubble shapes at the final time (t=3) for test case 1. (script) For test case 2, the mesh in Basilisk is too coarse to accurately resolve the skirt. plot [][0:0.4]'../c2g3l4s.txt' u 2:($1-0.5) w l t 'MooNMD', \
'../rising2/log' u 1:2 w l t 'Basilisk' Bubble shapes at the final time (t=3) for test case 2. (script)

The agreement for the bubble rise velocity with time is also good.

set term pop
reset
set grid
set xlabel 'Time'
set key bottom right
plot [0:3][0:]'../c1g3l4.txt' u 1:5 w l t 'MooNMD', \
'out' u 1:5 w l t 'Basilisk', \
'../rising-axi/out' u 1:5 w l t 'Basilisk (axisymmetric)' Rise velocity as a function of time for test case 1. (script)

reset
set grid
set xlabel 'Time'
set key bottom right
plot [0:3][0:]'../c2g3l4.txt' u 1:5 w l t 'MooNMD', \
'../rising2/out' u 1:5 w l t 'Basilisk' Rise velocity as a function of time for test case 2. (script)