# Film Boiling

The film boiling configuration consists in a superheated solid wall, covered by a thin vapor layer, which is perturbed by the phase change phenomena releasing bubbles. The simulation setup used here was inspired by Welch and Wilson, 2000. The domain is initialized with a uniform constant temperature equal to the saturation value. The bottom wall is maintained at a specific superheating temperature, causing the vapor layer to expand and leading to the departure of bubbles, which rise the liquid column unil the breakup with the free surface.

Evolution of the temperature field and the gas-liquid interface

## Phase change setup

We do not use the Stefan flow shifting procedure. The volume fraction source term is divided by the density of the gas phase. We compute the interface gradients just in gas-phase, and we use the same velocity used for the volume fraction also for the tracers in phase 2: TG. Defining the variable WELCH you can use the setup described in the paper.

#define BOILING_SETUP
#define SOLVE_GASONLY

## Simulation setup

We use the centered solver with the evaporation source term. The velocity potential method is adopted to obtain a divergece-free velocity extension for the VOF advection. The temperature-gradient phase change model is employed, since just the temperature field has to be solved.

#if POTENTIAL
# include "navier-stokes/centered-evaporation.h"
# include "navier-stokes/velocity-potential.h"
#else
# include "navier-stokes/velocity-jump.h"
#endif
#include "two-phase.h"
#include "tension.h"
#include "reduced.h"
#include "evaporation.h"
#include "tag.h"
#include "view.h"

The characteristic length of the problem is the most dangerous Taylor wavelength, defined as: \displaystyle \lambda_0 = 2\pi \left( \dfrac{3\sigma}{(\rho_l - \rho_g)g} \right)^{1/2}

Then, we declare variables necessary for the temperature gradient model, the maximum level of refinement, and the temperature of the solid wall.

double wavelength = 0.0786844;
int maxlevel = 8;

double lambda1, lambda2, cp1, cp2, dhev, TIntVal;
double TL0, TG0, Twall, Tsat, gas_vol0, width;

We set the boundary conditions for velocity, pressure and temperature at the bottom, which is considered as as superheated solid wall. The velocity potential ps boundary conditions are set in order to be coherent with the pressure boundary conditions.

#if POTENTIAL
u.n[bottom] = dirichlet (0.);
u.t[bottom] = dirichlet (0.);
p[bottom] = neumann (0.);
ps[bottom] = neumann (0.);

u.n[top] = neumann (0.);
u.t[top] = neumann (0.);
p[top] = dirichlet (0.);
ps[top] = dirichlet (0.);
#else
u1.n[bottom] = dirichlet (0.);
u1.t[bottom] = dirichlet (0.);
u2.n[bottom] = dirichlet (0.);
u2.t[bottom] = dirichlet (0.);
p[bottom] = neumann (0.);
ps[bottom] = neumann (0.);
pg[bottom] = neumann (0.);

u1.n[top] = neumann (0.);
u1.t[top] = neumann (0.);
u2.n[top] = neumann (0.);
u2.t[top] = neumann (0.);
p[top] = dirichlet (0.);
ps[top] = dirichlet (0.);
pg[top] = dirichlet (0.);
#endif

f[bottom] = dirichlet (0.);
TG[bottom] = dirichlet (Twall);
TL[bottom] = dirichlet (Twall);

int main (void) {

We set the values of the material properties of the fluids.

  rho1 = 200., rho2 = 5.;
#ifdef WELCH
mu1 = 0.1, mu2 = 0.005;
#else
mu1 = 0.01, mu2 = 0.005;
#endif
lambda1 = 40., lambda2 = 1.;
cp1 = 400., cp2 = 200.;
#ifdef WELCH
dhev = 1.e4;
#else
dhev = 1.e5;
#endif

We set the initial and wall temperatures.

  Tsat = 500., Twall = 506.;
TL0 = Tsat, TG0 = Tsat, TIntVal = Tsat;

We set the surface tension coefficient, and we apply the gravity force using the reduced.h approach.

  f.sigma = 0.1;
G.y = -9.81;

The dimensions of the problems are a function of the wavelength. In principle, the vapor bubbles, should be positioned in a squared pattern separated by a distance equivalent to \lambda_0.

  size (3.*wavelength);
init_grid (1 << maxlevel);
run();
}

We initialize the volume fraction field and the temperature field accordingly. The amount of gas in the domain at the beginning of the simulation is stored.

#define sin(x,y)(y-wavelength/128*(4+cos(2*pi*x/wavelength)))

event init (i = 0) {
width = 0.5*wavelength;
mask (x > width ? right : none);
fraction (f, sin(x,y));
#ifndef WELCH
foreach()
f[] = (y > 2./3.*L0) ? 0. : f[];
#endif
foreach() {
TL[] = TL0*f[];
TG[] = TG0*(1. - f[]);
T[]  = TL[] + TG[];
}

foreach (reduction(+:gas_vol0))
gas_vol0 += (1. - f[])*dv();
}

We write in the stdout the simulation time, the amount of gas in the domain, the Nusselt number from this simulation: \displaystyle Nu = \dfrac{\lambda_0}{w \left(T_{wall}-T_{sat}\right)} \int_{0}^{w} \left.\dfrac{\partial T}{\partial y}\right\vert_{y=0} dw and the Nusselt number from the Berenson correlation: \displaystyle Nu_B = 0.425 \left( \dfrac{\rho_g(\rho_l-\rho_g)g\Delta h_{ev}} {\lambda_g \mu_g (T_{wall}-T_{sat})} \right)^{1/4} (\lambda_0)^{3/4}

event logfile (i++) {
double gas_vol = 0.;
foreach (reduction(+:gas_vol))
gas_vol += (1. - f[])*dv();

foreach()
T[] = TL[] + TG[];

double Nu = 0.;
foreach_boundary (bottom, reduction(+:Nu)) {
T[0,-1] = 2.*Twall - T[];
Nu += (T[] - T[0,-1]);
}
Nu *= -wavelength/(Twall - Tsat)/width;

double NuB = 0.425*pow((rho2*(rho1 - rho2)*9.81*dhev)
/(lambda2*mu2*(Twall - Tsat)), 1./4.)*pow(wavelength, 3./4.);

fprintf (stdout, "%f %f %f %f\n", t, gas_vol / gas_vol0, Nu, NuB);
fflush (stdout);
}

We remove small bubbles formed during the breakups that are not important for the process under investigation, but that give some problems during the solution of the diffusion part of the temperature equation.

event remove_droplets (i++) {
remove_droplets (f, threshold=F_ERR, bubbles=true);
}

We output a movie with the evolution of the temperature field and the volume fraction facets.

event movie (t += 0.01; t <= 6) {
clear();
view (ty = -0.5);
draw_vof ("f", lw = 1.5);
squares ("T", min = Tsat, max = Twall,
map=blue_white_red, linear = true);
mirror ({1,0}) {
draw_vof ("f", lw = 1.5);
squares ("T", min = Tsat, max = Twall,
map=blue_white_red, linear = true);
}
save ("movie.mp4");
}

## Results

We compare the evolution of the Nusselt number obtained from the simulation with the Berenson correlation. A displacement is expected since the correlation describes a 3D system, while the simulation is 2D.

set yr[0:60]

set xlabel "t [s]"
set ylabel "Nu [-]"

p "out" u 1:3 w l t "Results", "out" u 1:4 w l t "Berenson"

The amount of gas phase volume fraction increases according to the phase change. The discontinuities correspond to the bubbles release.

reset
set xlabel "t [s]"
set ylabel "gas volume fraction [-]"

p "out" u 1:2 w l t "volume fraction"

## References

 [welch2000volume] Samuel WJ Welch and John Wilson. A volume of fluid based method for fluid flows with phase change. Journal of computational physics, 160(2):662–682, 2000.