axi_rising_bubble.c

Soluble gas diffusing from a rising bubble
- References

Soluble gas diffusing from a rising bubble

This is the example discussed in section 3.3.2 of Farsoiya et al., 2021.

The four cases illustrated below are considered. The left-half of each figure is the vertical velocity component and the right-half the tracer concentration when close to the stationary regime.


Case 1	Case 2	Case 3	Case 4

The Sherwood number characterises the gas diffusion from the bubble to the liquid. The computed Sherwood number is compared to the theory of Levich, 1962.

alpha_c = 1.0/30
r0 = 1.

sherwood(D) = (ci = $2/$5,  \
               co = $3/$6, \
               area = $7,   \
               dtr = (tr - $2)/0.01, \
               tr = $2,              \
               Sh = dtr/area/(ci*alpha_c - co)*2.*r0/D)

# Levich formula to calculate transfer rate using terminal velocity
levich(ut, D) = 2.*sqrt(D*ut/2./r0/pi)*2.*r0/D

# get the rise velocity averaged for the last steps
array ut[4]
array et[4] = [ 140, 250, 250, 150 ]

do for [i in "1 2 3 4"] {
  stats 'log' index 'case '.i u 1:4 every ::et[1] nooutput
  ut[i] = STATS_mean_y
}
array D[4] = [ 0.4472, 0.5029, 0.17416, 0.08944 ]

tr = 0.
set key center right
set xlabel 't U/d_0'
set ylabel 'Sh'
set xrange [0.01:4]
set yrange [0:14]
plot for [i = 1:4]					       \
     'log' index 'case '.i u ($1*ut[i]/2./r0):(sherwood(D[i])) \
     w l t 'Axi Case '.i lt i,				       \
     for [i = 1:4] levich(ut[i], D[i]) w l t '' lt i dt 2

Sherwood number vs time (script)

References

[farsoiya2021]	Palas Kumar Farsoiya, Stéphane Popinet, and Luc Deike. Bubble-mediated transfer of dilute gas in turbulence. Journal of Fluid Mechanics, 920(A34), June 2021. [ DOI \| http \| .pdf ]
[levich1962]	Veniamin Grigorʹevich Levich. Physicochemical hydrodynamics. Prentice-Hall Inc., 1962.

#include "axi.h"
#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "navier-stokes/conserving.h"
#include "tension.h"
#include "reduced.h"
#include "henry.h"
#include "view.h"

scalar c[], * stracers = {c};
double bubble_radius = 1.;
double box_size = 20.;
double conc_liq1 = 0, conc_gas1 = 1.;
double end_time[4] = {7.5,3,3,2};	
	
int MAXLEVEL, dcase = 1;

int main (int argc, char **argv)
{
  size (box_size);
	
  MAXLEVEL = 9;
#if !TREE
  N = 1 << MAXLEVEL;
#endif

  rho1 = 1.;
  rho2 = 0.01;
  c.alpha = 1./30.;
  TOLERANCE = 1e-4 [*];
	
  for (dcase = 1; dcase <= 4; dcase++) {
    switch (dcase) {
    case 1:
      c.D1 = 0.4472;
      f.sigma = 10.0;	
      G.x = - 2.5;	
      break;
    case 2:
      c.D1 = 0.5029;
      f.sigma = 10.0;	
      G.x = - 7.8125;	
      break;
    case 3:
      c.D1 = 0.17416;
      f.sigma = 1.0;	
      G.x = - 10.0;
      break;
    case 4:
      c.D1 = 0.08944;
      f.sigma = 10.0;	
      G.x = - 7.8125;
      break;
    default:
      fprintf (stderr, "Error: must specify case\n");
      exit (1);
    }

    mu1 = c.D1;
    mu2 = c.D1/20.;
    c.D2 = c.D1*100.;
    
    fprintf (stderr, "\n\n# case %d\n", dcase);
    run();
  }
}

event init (t = 0)
{
#if TREE  
  refine (sq(2.*bubble_radius) - sq(x - box_size*0.2) - sq(y) > 0 &&
	  level < MAXLEVEL);
#endif
  
  fraction (f, - (sq(bubble_radius) - sq(x - box_size*0.2) - sq(y)));

  foreach()
    c[] = conc_liq1*f[] + conc_gas1*(1. - f[]);
}

#if TREE
event adapt (i++) {
  adapt_wavelet ({f, c, u}, (double[]){0.01,0.01,0.01,0.01,0.01},
		 maxlevel = MAXLEVEL); 
}
#endif
	
event extract (t = 0; t += 0.01; t <= end_time[dcase-1])
{	
  double yb = 0., vb = 0., vbx = 0., area = 0., ci = 0., co = 0.;
  foreach (reduction(+:yb) reduction(+:vb) reduction(+:vbx)
	   reduction(+:ci) reduction(+:co)
	   reduction(+:area)) {
    double dvb = (1. - f[])*dv();
    vb += dvb;          // volume of the bubble
    yb += x*dvb;	// location of the bubble
    vbx += u.x[]*dvb;	// bubble velocity

    ci += c[]*(1. - f[])/(f[]*c.alpha + (1. - f[]))*dv();
    co += c[]*f[]*c.alpha/(f[]*c.alpha + (1. - f[]))*dv();
    
    if (f[] > 1e-6 && f[] < 1. - 1e-6) {
      coord n = interface_normal (point, f), p;      
      double alpha = plane_alpha (f[], n);
      // area of the bubble interface
      area += y*pow(Delta, dimension - 1)*plane_area_center (n, alpha, &p);      
    }
  }
	
  if (i == 0)
    fprintf (stderr, "t ci co vbx vb vbo area dt\n");
  fprintf (stderr,"%g %g %g %g %g %g %g %g\n",
	   t,
	   ci*2.*pi, co*2.*pi,
	   vbx/vb, 2.*pi*vb, 2.*pi*statsf(f).sum, 2.*pi*area,
	   dt);
}

event pictures (t = end)
{
  char name[80];
#if 0  
  sprintf (name, "dump-%d", dcase);
  dump (name);
#endif
  
  double ty[] = { - 0.6, - 0.5, - 0.5, - 0.5};
  view (fov = 9, quat = {0.707,0.707,0,0},
	ty = ty[dcase - 1],
	width = 400, height = 800);  
  squares ("c", spread = -1, linear = true, map = cool_warm);
  draw_vof ("f");
  mirror ({0,1}) {
    squares ("u.x", spread = -1, linear = true, map = cool_warm);
    draw_vof ("f");
  }
  sprintf (name, "final-%d.png", dcase);
  save (name);
}