# src/test/rising.c

# 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.

```
#if AXIS
# include "axi.h" // fixme: does not run with -catch
#endif
#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "tension.h"
#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.σ = 1.96;
#else
rho2 = 100., mu2 = 1., f.σ = 24.5;
#endif
```

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

```
TOLERANCE = 1e-4;
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.

```
event acceleration (i++) {
face vector av = a;
foreach_face(x)
av.x[] -= 0.98;
}
```

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
#if ADAPT
event adapt (i++) {
adapt_wavelet ({f,u}, (double[]){5e-4,1e-3,1e-3}, LEVEL);
}
#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.

For test case 2, the mesh in Basilisk is too coarse to accurately resolve the skirt.

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