# Scriven Problem

Expansion of a bubble in a superheated environment. A spherical bubble is initialized on the bottom of an AXI domain. The bubble is at saturation temperature while the liquid environment is superheated. The phase change pheomena, promoted by the temperature gradient between the surface of the bubble and the liquid environment leads to the expansion of the bubble. The simulation setup used here was inspired by Tanguy et al., 2014, and is characterized by Ja=3. The analytic solution to this problem was proposed by Scriven, 1959, and it describes the dynamic of the bubble radius in time: \displaystyle R(t) = 2\beta\sqrt{\alpha_l t} where \alpha_g is the thermal diffusivity, defined as \alpha_l = \lambda_l/\rho_l/cp_l, while \beta is the growth constant, which is obtained as a function of the physical properties of the simulation: \displaystyle \dfrac{\rho_l cp_l (T_{bulk}-T_{sat})}{\rho_g (\Delta h_{ev} + (cp_l - cp_g)(T_{bulk} - T_{sat}))} = 2\beta^2 \int_{0}^1 \exp \left(-\beta^2((1-x)^{-2} -2(1-\dfrac{\rho_l}{\rho_g})x - 1)\right)dx

The animation shows the map of the temperature field, which is initialized with the analytic solution at the beginning of the simulation. We let the bubble expand until reaching a radius which is twice the initial radius. The Stefan flow contribution must be considered because it provides an additional transport of the temperature field in liquid phase. The analytic solution takes into account this phenomena.

Evolution of the interface and temperature field

## Phase Change Setup

We move the interface using the velocity uf, with the expansion term shifted toward the gas-phase. In this way uf is divergence-free at the interface. The double pressure velocity couping is used to obtain an extended velocity, used to transport the gas phase tracers.

#define INT_USE_UF
#define CONSISTENTPHASE2
#define SHIFT_TO_GAS
#define INIT_TEMP

## Simulation Setup

We use the centered solver with the divergence source term, and the extended velocity is obtained using the centered-doubled procedure. The evaporation model is used in combination with the temperature-gradient mechanism, which manages the solution of the temperature field.

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

We declare the variables required by the temperature-gradient model.

double lambda1, lambda2, cp1, cp2, dhev;
double TL0, TG0, TIntVal, Tsat, Tbulk;

### Boundary conditions

Outflow boundary conditions for velocity and pressure are imposed on the top, left, and right walls. The temperature on these boundaries is imposed to the bulk value.

u.n[top] = neumann (0.);
u.t[top] = neumann (0.);
p[top] = dirichlet (0.);
uext.n[top] = neumann (0.);
uext.t[top] = neumann (0.);
pext[top] = dirichlet (0.);

u.n[left] = neumann (0.);
u.t[left] = neumann (0.);
p[left] = dirichlet (0.);
uext.n[left] = neumann (0.);
uext.t[left] = neumann (0.);
pext[left] = dirichlet (0.);

u.n[right] = neumann (0.);
u.t[right] = neumann (0.);
p[right] = dirichlet (0.);
uext.n[right] = neumann (0.);
uext.t[right] = neumann (0.);
pext[right] = dirichlet (0.);

TL[top] = dirichlet (Tbulk);
TL[right] = dirichlet (Tbulk);
TL[left] = dirichlet (Tbulk);

### Problem Data

We declare the maximum and minimum levels of refinement, the \lambda parameter, the initial radius of the droplet, the growth constant, and additional post-processing variables.

int maxlevel, minlevel = 5;
double R0 = 1.e-3;
double XC, YC;
double V0, tshift;
double betaGrowth = 3.32615013;

### Initial Temperature

We use gsl to compute the integral required to obtain the analytic temperature profile: \displaystyle T(r,t) = T_{bulk} - 2\beta^2 \left( \dfrac{\rho_g (\Delta h_{ev} + (cp_l-cp_g)(T_{bulk} - T_{sat}))}{\rho_l cp_l} \right) \int_{1-R/r}^1 \exp \left(-\beta^2((1-x)^{-2} -2(1-\dfrac{\rho_l}{\rho_g})x - 1)\right)dx

#include <gsl/gsl_integration.h>

double intfun (double x, void * params) {
double beta = *(double *) params;
return exp(-sq(beta)*(pow(1. - x, -2.) - 2.*(1. - rho2/rho1)*x - 1 ));
}

double tempsol (double r, double R) {
gsl_integration_workspace * w
= gsl_integration_workspace_alloc (1000);
double result, error;
double beta = betaGrowth;
gsl_function F;
F.function = &intfun;
F.params = &beta;
gsl_integration_qags (&F, 1.-R/r, 1., 1.e-9, 1.e-5, 1000,
w, &result, &error);
gsl_integration_workspace_free (w);
return Tbulk - 2.*sq(beta)*(rho2*(dhev + (cp1 - cp2)*(Tbulk - Tsat))/rho1/cp1)*result;
}

int main (void) {

We set the material properties of the fluids.

  rho1 = 958.; rho2 = 0.59;
mu1 = 2.82e-4; mu2 = 1.23e-6;
lambda1 = 0.6, lambda2 = 0.026;
cp1 = 4216., cp2 = 2034.;
dhev = 2.257e+6;

The initial bubble temperature and the interface temperature are set to the saturation value.

  Tbulk = 373.989, Tsat = 373., TIntVal = 373.;
TL0 = Tbulk, TG0 = TIntVal;

We change the dimension of the domain, the surface tension coefficient, and the coordinates of the center of the bubble.

  L0 = 12.e-3;
XC = 0.5*L0, YC = 0.;
f.sigma = 0.001;

We reduce the tolerance of the Poisson equation solver.

  TOLERANCE = 1.e-6;

We compute the time shifting factor (post-processing), since the bubble Radius at simulation time t=0 is not zero.

  double alpha = lambda1/rho1/cp1;
tshift = sq(R0/2./betaGrowth)/alpha;

We define a list with the maximum time steps and the maximum levels of refinement.

  double dtlist[] = {1.e-4, 5.e-5, 5.e-5, 5.e-5};
int mllist[] = {6, 7, 8, 9};

We run the simulation for different levels of refinement.

  for (int sim=0; sim<4; sim++) {
DT = dtlist[sim];
maxlevel = mllist[sim];
init_grid (1 << maxlevel);
run();
}
}

#define circle(x, y, R) (sq(R) - sq(x - XC) - sq(y - YC))

event init (i = 0) {
fraction (f, circle(x,y,R0));

foreach()
f[] = 1. - f[];

We initialize the temperature field. The liquid phase temperature is set to the analytic value.

  foreach() {
double r = sqrt( sq(x - XC) + sq(y - YC) );
#ifdef INIT_TEMP
TL[] = f[]*tempsol (r, R0);
#else
TL[] = f[]*TL0;
#endif
TG[] = (1. - f[])*TG0;
T[]  = TL[] + TG[];
}
}

We refine the domain according to the interface and the temperature field.

#if TREE
double uemax = 1e-2;
(double[]){1e-3,uemax,uemax,uemax,1e-3}, maxlevel, minlevel, 1);
}
#endif

## Post-Processing

The following lines of code are for post-processing purposes.

### Exact Solution

We write a function that computes the exact solution to the thickness of the vapor layer.

double exact (double time) {
return 2.*betaGrowth*sqrt(lambda1/rho1/cp1*time);
}

### Output Files

We write the bubble radius and the analytic solution on a file.

event output (t += 0.005) {
scalar fg[];
foreach()
fg[] = 1. - f[];

double rsol = exact (t+tshift);
double relerr = (rsol > 0.) ? fabs (rsol - effective_radius) / rsol : 0.;

char name[80];
sprintf (name, "OutputData-%d", maxlevel);

static FILE * fp = fopen (name, "w");
fprintf (fp, "%g %g %g %g\n",
fflush (fp);
}

### Temperature Profile

We write on a file the temperature profile at the final time step.

event profiles (t = end) {
char name[80];
sprintf (name, "Temperature-%d", maxlevel);

We create an array with the temperature profile for each processor.

  Array * arrtemp = array_new();
for (double x = 0.5*L0; x < L0; x += 0.5*L0/(1 << maxlevel)) {
double val = interpolate (T, x, 0.);
val = (val == nodata) ? 0. : val;
array_append (arrtemp, &val, sizeof(double));
}
double * temps = (double *)arrtemp->p;

We sum each element of the arrays in every processor.

  @if _MPI
int size = arrtemp->len/sizeof(double);
MPI_Allreduce (MPI_IN_PLACE, temps, size, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
@endif

The master node writes the temperature profile.

  if (pid() == 0) {
FILE * fpp = fopen (name, "w");
int count = 0;
for (double x = 0.5*L0; x < L0; x += 0.5*L0/(1 << maxlevel)) {
double r = x - 0.5*L0;
double R = exact (t+tshift);
double tempexact = (r >= R) ? tempsol (r, R) : Tsat;
fprintf (fpp, "%g %g %g\n", x, temps[count], tempexact);
count++;
}
fflush (fpp);
fclose (fpp);
}
array_free (arrtemp);
}

### Movie

We write the animation with the evolution of the temperature field and the gas-liquid interface.

event movie (t += 0.01; t <= 0.5) {
clear();
view (tx = -0.5, ty = -0.5);
draw_vof ("f", lw = 1.5);
squares ("T", min = Tsat, max = Tbulk);
save ("movie.mp4");
}

## Results

reset
set xlabel "t [s]"
set key top left
set size square
set grid

plot "OutputData-6" every 8 u 1:3 w p ps 2 t "Analytic", \
"OutputData-6" u 1:2 w l lw 2 t "LEVEL 6", \
"OutputData-7" u 1:2 w l lw 2 t "LEVEL 7", \
"OutputData-8" u 1:2 w l lw 2 t "LEVEL 8", \
"OutputData-9" u 1:2 w l lw 2 t "LEVEL 9"
reset

stats "OutputData-6" using (last6=$4) nooutput stats "OutputData-7" using (last7=$4) nooutput
stats "OutputData-8" using (last8=$4) nooutput stats "OutputData-9" using (last9=$4) nooutput

#stats "OutputData-6" using 4 nooutput name "LEVEL6"
#stats "OutputData-7" using 4 nooutput name "LEVEL7"
#stats "OutputData-8" using 4 nooutput name "LEVEL8"
#stats "OutputData-9" using 4 nooutput name "LEVEL9"

set print "Errors.csv"

#print sprintf ("%d %.12f", 2**6, LEVEL6_mean)
#print sprintf ("%d %.12f", 2**7, LEVEL7_mean)
#print sprintf ("%d %.12f", 2**8, LEVEL8_mean)
#print sprintf ("%d %.12f", 2**9, LEVEL9_mean)

print sprintf ("%d %.12f", 2**6, last6)
print sprintf ("%d %.12f", 2**7, last7)
print sprintf ("%d %.12f", 2**8, last8)
print sprintf ("%d %.12f", 2**9, last9)

unset print

reset
set xlabel "N"
set ylabel "Relative Error"

set logscale x 2
set logscale y

set xr[2**5:2**10]
set yr[1e-4:10]

set size square
set grid

plot "Errors.csv" w p pt 8 ps 2 title "Results", \
50*x**(-1) lw 2 title "1^{st} order", \
1000*x**(-2) lw 2 title "2^{nd} order"
reset
set ylabel "Temperature [K]"

set yr[372.9:374.1]

set key bottom right
set size square
set grid

plot "Temperature-6" u 1:3 w p ps 2 t "Analytic", \
"Temperature-6" u 1:2 w l lw 2 t "LEVEL 6", \
"Temperature-7" u 1:2 w l lw 2 t "LEVEL 7", \
"Temperature-8" u 1:2 w l lw 2 t "LEVEL 8", \
"Temperature-9" u 1:2 w l lw 2 t "LEVEL 9"

## References

 [tanguy2014benchmarks] Sébastien Tanguy, Michaël Sagan, Benjamin Lalanne, Frédéric Couderc, and Catherine Colin. Benchmarks and numerical methods for the simulation of boiling flows. Journal of Computational Physics, 264:1–22, 2014. [scriven1959dynamics] LE Scriven. On the dynamics of phase growth. Chemical engineering science, 10(1-2):1–13, 1959.