# sandbox/ecipriano/run/staticbi.c

# Isothermal Evaporation of a Static Binary Droplet

A binary liquid droplet is placed on the lower-left edge of the domain. The two chemical species in liquid phase have the same physical properties, but different volatility. The relative volatility between the heavy and the light species is equal to 0.5. Therefore, we expect the light component to start to evaporate first, increasing its mass fractions in gas phase and decreasing its concentration in the liquid phase. The heavy component accumulates in the liquid phase in response to the evaporation of the light species. The gas phase is initially full of an inert compound, important in combustion simulations, which always remains in gas phase.

## Phase Change Setup

We let the default settings of the evaporation model: the Stefan flow is shifted toward the liquid phase, and the consistent phase is the liquid, which is advected using the extended velocity. The multicomponent model requires the number of gas and liquid species to be set as compiler variables. We don’t need to solve the temperature field because the vapor pressure is set to a constant value, different for each chemical species. Using GSL at level 1 we can activate the coupled solution of the interface jump condition.

```
#define NGS 3
#define NLS 2
```

## Simulation Setup

We use the centered solver with the evaporation source term in the projection step. The extended velocity is obtained from the doubled pressure-velocity coupling. We use the evaporation model together with the multiomponent phase change mechanism.

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

### Data for multicomponent model

We define the lists with the names of the chemical species in gas and in liquid phase. The initial mass fractions are defined for each component, as well as the diffusion coefficients and the thermodynamic equilibrium constant. Since we set the vapor pressure to a constant value, we don’t need to solve the temperature field.

```
char* gas_species[NGS] = {"A", "B", "C"};
char* liq_species[NLS] = {"A", "B"};
char* inert_species[1] = {"C"};
double gas_start[NGS] = {0.0, 0.0, 1.0};
double liq_start[NLS] = {0.5, 0.5};
double inDmix1[NLS] = {4.e-6, 4.e-6};
double inDmix2[NGS] = {8.e-5, 8.e-5, 8.e-5};
double inKeq[NLS] = {0.8, 0.4};
```

### Boundary conditions

Outflow boundary conditions are set at the top and right sides of the domain.

```
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[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.);
```

### Simulation Data

We declare the maximum and minimum levels of refinement, the initial radius and diameter, and the radius from the numerical simulation.

```
int maxlevel, minlevel = 5;
double D0 = 0.4e-3, effective_radius0;
int main (void) {
```

We set the material properties of the fluids.

```
rho1 = 10.; rho2 = 1.;
mu1 = 1.e-4; mu2 = 1.e-5;
```

We change the dimension of the domain as a function of the initial diameter of the droplet.

` D0 = 0.4e-3; L0 = 4.*D0;`

We change the surface tension coefficient.

` f.sigma = 0.03;`

We run the simulation at three different levels of refinement.

```
for (maxlevel = 7; maxlevel <= 7; maxlevel++) {
//CFL = 0.1;
init_grid (1 << maxlevel);
run ();
}
}
#define circle(x, y, R) (sq(R) - sq(x) - sq(y))
```

We initialize the volume fraction field and we compute the initial radius of the droplet. We don’t use the value D0 because for small errors of initialization the squared diameter decay would not start from 1.

```
event init (i = 0) {
fraction (f, circle (x, y, 0.5*D0));
effective_radius0 = sqrt (4./pi*statsf(f).sum);
#ifdef BALANCES
mb.liq_species = liq_species;
mb.gas_species = gas_species;
mb.YLList = YLList;
mb.YGList = YGList;
mb.mEvapList = mEvapList;
mb.liq_start = liq_start;
mb.gas_start = gas_start;
mb.rho1 = rho1;
mb.rho2 = rho2;
mb.inDmix1 = inDmix1;
mb.inDmix2 = inDmix2;
mb.maxlevel = maxlevel;
mb.boundaries = true;
#endif
}
```

We overwrite the boundary conditions for the chemical species mass fractions in the *bcs* event.

```
event bcs (i = 0) {
scalar YLA = YLList[0], YLB = YLList[1];
scalar YGA = YGList[0], YGB = YGList[1], YGC = YGList[2];
YLA[top] = dirichlet (0.);
YLB[top] = dirichlet (0.);
YLA[right] = dirichlet (0.);
YLB[right] = dirichlet (0.);
YGA[top] = dirichlet (0.);
YGB[top] = dirichlet (0.);
YGC[top] = dirichlet (1.);
YGA[right] = dirichlet (0.);
YGB[right] = dirichlet (0.);
YGC[right] = dirichlet (1.);
}
```

We adapt the grid according to the mass fractions of the species A and B, the velocity and the interface position.

```
#if TREE
event adapt (i++) {
scalar YA = YList[0], YB = YList[1];
adapt_wavelet_leave_interface ({YA,YB,u.x,u.y}, {f},
(double[]){1.e-4,1.e-4,1.e-3,1.e-3}, maxlevel, minlevel, 1);
}
#endif
```

## Post-Processing

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

### Output Files

We write on a file the squared diameter decay and the dimensionless time.

```
event output_data (i++) {
char name[80];
sprintf (name, "OutputData-%d", maxlevel);
static FILE * fp = fopen (name, "w");
double effective_radius = sqrt (4./pi*statsf(f).sum);
double tad = t*inDmix2[0]/sq (2.*effective_radius0);
double d_over_d0 = effective_radius / effective_radius0;
double d_over_d02 = sq (d_over_d0);
fprintf (fp, "%g %g %g %g\n", t, tad, d_over_d0, d_over_d02);
fflush (fp);
}
```

### Movie

We write the animation with the evolution of the chemical species, the interface position and the grid refinement.

```
event movie (t += 2.e-5; t <= 0.005) {
clear();
draw_vof ("f", lw = 1.5);
squares ("B", linear = true, min = 0., max = 0.56);
mirror ({0,1}) {
draw_vof ("f", lw = 1.5);
squares ("C", linear = true, min = 0., max = 1.);
}
mirror ({1,0}) {
draw_vof ("f", lw = 1.5);
squares ("A", linear = true, min = 0., max = liq_start[0]);
}
mirror ({1,1}) {
cells ();
draw_vof ("f", lw = 1.5);
}
save ("movie.mp4");
}
#if DUMP
event snapshot (t += 2.e-4) {
char name[80];
sprintf (name, "snapshot-%g", t);
dump (name);
}
#endif
```

## Results

```
reset
set xlabel "t [s]"
set ylabel "(D/D_0)^2"
set key top right
set size square
set grid
plot "OutputData-7" u 2:4 w l lw 2 t "LEVEL 7"
```

The conservation tests compare the mass of the chemical species in liquid phase with the total amount of the same species that evaporates. If the global conservation is considered, the volume fraction is used instead of the mass fraction field. See balances.h for details.

```
reset
set xlabel "t [s]"
set ylabel "(m_L - m_L^0) [kg]"
set key top right
set size square
set grid
plot "balances-7" every 500 u 1:10 w p ps 1.2 lc 1 title "Evaporated Mass Species A", \
"balances-7" every 500 u 1:11 w p ps 1.2 lc 2 title "Evaporated Mass Species B", \
"balances-7" every 500 u 1:4 w p ps 1.2 lc 3 title "Evaporated Mass Total", \
"balances-7" u 1:(-$5) w l lw 2 lc 1 title "Variation Mass Species A", \
"balances-7" u 1:(-$6) w l lw 2 lc 2 title "Variation Mass Species B", \
"balances-7" u 1:(-$2) w l lw 2 lc 3 title "Variation Mass Total"
```

```
reset
set xlabel "t [s]"
set ylabel "(m_G - m_G^0) [kg]"
set key top left
set size square
set grid
plot "balances-7" every 500 u 1:(-$10) w p ps 1.2 lc 1 title "Evaporated Mass Species A", \
"balances-7" every 500 u 1:(-$11) w p ps 1.2 lc 2 title "Evaporated Mass Species B", \
"balances-7" every 500 u 1:(-$4) w p ps 1.2 lc 3 title "Evaporated Mass Total", \
"balances-7" u 1:7 w l lw 2 lc 1 title "Variation Mass Species A", \
"balances-7" u 1:8 w l lw 2 lc 2 title "Variation Mass Species B", \
"balances-7" u 1:3 w l lw 2 lc 3 title "Variation Mass Total"
```