# A bootstrapping bubble

In the example of a bouncing droplet, some strange behaviour was observed regarding the advection of a droplet that is subject to surface tension. It appeared that for specific values of the dimensionless ratios, the kinetic energy of the system increases as the droplet advects trough the domain. This is unexpected as there was only supposed to be a viscous drainage of energy due to the finite Reynoldsnumber.

## Set-up

We follow the set-up of the aforementioned bouncing droplet example, except that on this page we are not interested in the bounce event itself and hence use periodic boundaries instead of the (hydrophobic) no-slip bottom.

The dimensionless parameters are:

\displaystyle Re = \frac{\rho_a UR}{\mu_a} \approx 30,

\displaystyle We = \frac{\rho_w U^2R}{\sigma} \approx 0.2,

\displaystyle \Pi_1 = \frac{\mu_w }{\mu_a} \approx 100,

\displaystyle \Pi_1 = \frac{\rho_w }{\rho_a} \approx 1000 \rightarrow 100,

#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "tension.h"
#include "view.h"

int maxlevel = 8; //The presented results are converged-ish with
//respect to those obtained when using 9 levels of
//refinement (not shown)
int j;

int main(){
mu2 = 1./30.;
mu1 = 100./30.;
rho2 = 1.;
rho1 = 100;
f.sigma = 500.;
foreach_dimension()
periodic(left);
init_grid (1<<7);
L0 = 10;
X0 = Y0 = -L0/2;

## Three cases

For our analysis of the issue we run the set-up for three cases: First a (naive) default run is performed, second refinement is enfored arround the droplet’s interface, and finally we re-run de default case with a decreased value for the surface tension parameter (\sigma). Run 1 to 3 are tagged with value j=0 to 2, respectively.

  j = 0;
run();
j = 1;
run();
j = 2;
f.sigma=250;
run();
}

event init (i = 0){
refine (sq(x) + sq(y) < sq(1. + 0.5) && sq(x) + sq(y) > sq(1.-0.25) && level < maxlevel);
fraction(f, sq(1.) - sq(x) - sq(y));
foreach()
u.y[] = -f[];
}

## Force refinement on the interface

It is not well known (?) that the default (smart) adaptation rules tend to apply a coarser resolution when an interface is alligned with the grid (see here). For a near circular bubble, I would guess that this makes no sense when evaluating the curvature (?). Therefore, during the second run (i.e. j=1), the algorithm is forced to adapt all interfacial cells to the maximum level of refinement, by using a dummy field (ff) that uses the default bilinear prolongation technique.

event adapt (i++){
if (j == 1){
scalar ff[];
foreach()
ff[] = f[];
boundary({ff});
adapt_wavelet((scalar *){u,f,ff}, (double []){0.02, 0.02, 0.001, 0.001}, maxlevel);
}  else {
adapt_wavelet((scalar *){u,f}, (double []){0.02, 0.02, 0.001}, maxlevel);
}
}

## Output

We monitor the evolution of the kinetic energy and write the result to seperate files for the different runs.

event diagnose (i += 25){
double e = 0;
double e1 = 0;
double e2 = 0;
foreach(reduction(+:e) reduction(+:e1) reduction(+:e2)){
e +=  0.5*sq(Delta)*(sq(u.x[]) + sq(u.y[])) * ((rho1*f[]) + (rho2*(1-f[])));
e1 += 0.5*sq(Delta)*(sq(u.x[]) + sq(u.y[])) * ((rho1*f[]));
e2 += 0.5*sq(Delta)*(sq(u.x[]) + sq(u.y[])) * (rho2*(1. - f[]));
}
char fname[100];
sprintf(fname,"dataj=%d", j);
static FILE * fp = fopen (fname,"w");
fprintf (fp, "%g\t%g\t%g\t%g\n", t, e, e1, e2);
fflush (fp);
}

For the all-important visual reference, some movies are rendered that display the bubble and the used computational mesh.

event bviewer (t += 0.1; t <= 5){
clear();
view (fov = 20, width = 720, height = 720);
squares ("f", min = 0, max = 1);
cells();
draw_vof ("f", lc = {1,0,1}, lw = 3.);
char fname[99];
sprintf (fname, "%d.mp4", j);
save (fname);
}

## Results

One may view the difference in the gridding between the default case (top) and the radially refined case (bottom).

Reference case

The used strategy of enforcing refinment seems to have worked as intented. A plot of the evolution of the kinetic energy as was diagnosed for the three different runs is presented below:

   set xlabel 'time'
set ylabel 'Energy'
set key box top left
set size square
plot 'dataj=0' u 1:2 w l lw 3 t 'Reference (default)' ,	\
'dataj=1' u 1:2 w l lw 3 t 'Radially refined',		\
'dataj=2' u 1:2 w l lw 3 t 'Reduced surface tension'

From run 1 and 3 it appears that the surface-tension-force term is responsible for a spurrious source of kinetic energy. The ansantz is that this is due to improper reconstruction of the interface at the locations of the coarser level interfacial cells. These are not balanced arround the droplet because it does not have an exact circular shape and also the flow arround the droplet warrants an non-radially symmetric refinement. Fortunately this ansantz can be tested using the results from the second run. It appears from the plot that when refinement is enforced at the droplets interface, the spurrious source of energy is atleast much less prominent.

I am guessing that this issue arrises from the fact that the wavelet-based error estimation for a volume-fraction field as defined in two-phase.h/vof.h is designed to only concern the errors for advection of an interface. However, when the additional module tension.h is included, the estimation of the curvature gives rise to an addional source of numerical errors. Without being burdened with any knowedge of the details, I expect that the latter procedure is different enough from evaluating the advection scheme such that the default volume-of-fluid-fiield wavelet-estimated error is not able to identify the challinging regions for curvature reconstruction correctly.

Some addional analysis is presented here.