Evaporation of a Pure Droplet in Buoyancy-Driven Flows

    This module reports the simulation setup for a generic pure droplet at different gravity conditions. This configuration has been used in several experimental works, by suspending the droplet in a static configuration by means of a solid fiber. This module mimick the experimental setup by combining a droplet suspension strategy with the interface-resolved phase change model and the variable properties formulation, which allows buoyancy-driven flows to be resolved directly. These phenomena promote non-radial gas-phase convective fluxes, and liquid phase internal recirculation, which break the spherical symmetry of the evaporation process in microgravity conditions. The simulation setup can be used with any gravity value, not only in normal gravity. This feature allows small residual gravity values to be simulated, making this simulation setup useful to drive the experimental investigation also for droplets in microgravity.

    This simulation includes variable thermodynamic and transport properties, and the interface radiation contribution. We neglect the heat transfer from the suspending solid fiber, which may affect experimental data if the fiber is large. The simulation is performed by simulating a droplet at the edge of a square domain, exploting the axial-symmetry, which is acceptable for small Reynolds number.

    Default Simulation Data

    The simulation setup is independent from the liquid under investigation, and we may want to perform the same simulation studying the sensitivity of the numerical results to the operative conditions (temperature, pressure, ecc…) and to the liquid fuel or the ambient conditions. To avoid code duplication, we set a bunch of default compiler variables, which are overwritten in the Makefile to create different cases with different operative conditions.

    The default properties are:

    • initial ambient temperature: TEMPERATURE = 773 K
    • initial droplet temperature: TEMPERATURE_DROPLET = 300 K
    • constant thermodynamic pressure: PRESSURE = 10 atm
    • initial droplet diameter: DIAMETER = 1 mm
    • emissivity of the liquid fuel: RADIATION_INTERFACE = 0.93
    • name of the liquid fuel: FUEL = n-heptane
    • name of the inert/ambient species: INERT = nitrogen
    • path of the kinetics folder: KINFOLDER = evaporation/n-heptane-in-nitrogen
    • path of the liquid properties folder: LIQFOLDER = LiquidProperties
    • gravitational acceleration: GRAVITY = -9.81 m/s^2
    • ratio between solid fiber diameter and droplet diameter: FIBER = 0.1

    these properties describe the evaporation of a n-heptane droplet in nitrogen in normal gravity conditions, including the interface radiation. These properties can be easily changed by overwriting the compilation variables, for example by setting -DTEMPERATURE=473 -DPRESSURE=1 to change the ambient temperature and pressure. The interface radiation is suppressed by setting a null value of emissivity: -DRADIATION_INTERFACE=0.

    Eventually, one may consider the possibility of compiling a code which reads the properties directly from an input file (using libconfig for example).

    #define STRINGIFY(x) #x
    #define TOSTRING(x) STRINGIFY(x)
    #ifndef TEMPERATURE
    # define TEMPERATURE 773.
    # define TEMPERATURE_DROPLET 300.
    #ifndef PRESSURE
    # define PRESSURE 10.
    #ifndef DIAMETER
    # define DIAMETER 1.e-3
    #ifndef FUEL
    # define FUEL NC7H16
    #ifndef INERT
    # define INERT N2
    #ifndef KINFOLDER
    # define KINFOLDER evaporation/n-heptane-in-nitrogen
    #ifndef LIQFOLDER
    # define LIQFOLDER LiquidProperties
    # define RADIATION_INTERFACE 0.93
    #ifndef GRAVITY
    # define GRAVITY -9.81
    #ifndef FIBER
    # define FIBER 0.1

    Phase Change Setup

    We define the names of the gas and the liquid species, and their intial composition (in mass fractions).

    #define NGS 2
    #define NLS 1
    char* gas_species[NGS] = {TOSTRING(FUEL), TOSTRING(INERT)};
    char* liq_species[NLS] = {TOSTRING(FUEL)};
    char* inert_species[1] = {TOSTRING(INERT)};
    double gas_start[NGS] = {0., 1.};
    double liq_start[NLS] = {1.};

    We declare all the properties necessary for the multicomponent phase change model. The properties are set to null values because they are overwritten by the variable-properties formulation, which computes all the physical properties as a function of the thermodynamic state of the mixture.

    double inDmix1[NLS] = {0.};
    double inDmix2[NGS] = {0.};
    double inKeq[NLS] = {0.};
    double lambda1 = 0.;
    double lambda2 = 0.;
    double dhev = 0.;
    double cp1 = 0.;
    double cp2 = 0.;

    We set the initial temperature of the liquid and of the gas phase.

    double TG0 = TEMPERATURE;

    We solve both mass fractions and temperature fields, also in the interface jump condition. The GSL library is used for root finding operations, whose tolerance is controlled by the variable FSOLVE_ABSTOL. The thermodynamic equlibrium is computed from Antoine’s law, and we solve the momentum equation in non-conservative form by setting the variable NO_ADVECTION_DIV to 1.

    Additional compiler variables are used to activate terms in the governing equations. In particular: FICK_CORRECTED is used to force the diffusive fluxes to close to zero, even if the diffusivity of every chemical species is different; MOLAR_DIFFUSION is used to correct Fick’s law considering that the diffusivity values are mole fractions-based; MASS_DIFFUSION_ENTHALPY includes the species diffusion contribution in the temperature eqation. All of them should be used.

    #define USE_GSL 0
    #define FSOLVE_ABSTOL 1.e-3
    #define FICK_CORRECTED
    #define NO_ADVECTION_DIV 1

    Simulation Setup

    In this simulation, the Navier-Stokes equations can be solved both using the centered solver with divergence source term, or using the centered solver with velocity jump. The latter should be preferred since it is able to limit oscillations in the velocity field. The two-phase.h solver is extended to variable physical properties by including a policy for the calculation of such properties. In this case, we use correlations implemented in OpenSMOKE++ libraries. The solution of the jump conditions and of the temperature and mass fraction fields is performed by the multicomponent phase change model. We include the recoil pressure contribution, and the non-reduced gravity contribution. Using gravity.h simplifies the boundary condition for pressure, analogously to the reduced.h approach but considering variable liquid and gas phase densities.

    #include "axi.h"
    #if JUMP
    # include "navier-stokes/velocity-jump.h"
    # include "navier-stokes/centered-evaporation.h"
    # include "navier-stokes/centered-doubled.h"
    #include "opensmoke-properties.h"
    #include "pinning.h"
    #include "two-phase.h"
    #include "tension.h"
    #include "gravity.h"
    #include "recoil.h"
    #include "evaporation.h"
    #include "multicomponent-varprop.h"
    #include "view.h"

    Boundary conditions

    We initialize the droplet at the lower edge of the domain, and we exploit axial symmetry. We set no-slip boundary conditions on the side in contact with the droplet (left), and outflow boundary conditions on the other sides of the domain.

    #if JUMP
    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.);
    u1.n[left] = neumann (0.);
    u1.t[left] = neumann (0.);
    u2.n[left] = neumann (0.);
    u2.t[left] = neumann (0.);
    p[left] = dirichlet (0.);
    ps[left] = dirichlet (0.);
    pg[left] = dirichlet (0.);
    u1.n[right] = neumann (0.);
    u1.t[right] = neumann (0.);
    u2.n[right] = neumann (0.);
    u2.t[right] = neumann (0.);
    p[right] = dirichlet (0.);
    ps[right] = dirichlet (0.);
    pg[right] = dirichlet (0.);
    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.);
    uf1.n[bottom] = 0.;
    uf1.t[bottom] = 0.;
    uf2.n[bottom] = 0.;
    uf2.t[bottom] = 0.;
    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.);
    u.n[bottom] = dirichlet (0.);
    u.t[bottom] = dirichlet (0.);
    p[bottom] = neumann (0.);
    uext.n[bottom] = dirichlet (0.);
    uext.t[bottom] = dirichlet (0.);
    pext[bottom] = neumann (0.);
    uf.n[bottom] = 0.;
    uf.t[bottom] = 0.;
    ufext.n[bottom] = 0.;
    ufext.t[bottom] = 0.;

    Simulation Data

    We declare the maximum and minimum levels of refinement, the initial droplet diameter, and additional data for post-processing. We also transport a scalar tracer tr which can be useful to visualize liquid internal recirculation.

    int maxlevel, minlevel = 2;
    double D0 = DIAMETER, effective_radius0;
    double effective_radius = 0.5*DIAMETER, d_over_d02 = 1., tad = 0.;
    double volumecorr = 0., trmin = 0., trmax = 0.;
    scalar tr[];
    int main (void) {

    We set the kinetics folder, which defines the species of the simulation, and it is used by OpenSMOKE++ for the calculation of the thermodynamic and transport properties. The path is relative to OpenSMOKEppInterface/kinetics. We do the same for the folder gathering the properties of the liquid species.

      kinfolder = TOSTRING(KINFOLDER);
      liqfolder = TOSTRING(LIQFOLDER);

    We set additional simulation properties. Be careful, these are “dummy” properties which are overwritten by the variable-properties formulation already at the first iteration. To start the simulation without any problem of division by zero, the viscosity value should be non-null.

      rho1 = 1.; rho2 = 1.;
      mu1 = 1.; mu2 = 1.;

    We set the thermodynamic pressure in SI units (Pa).

      Pref = PRESSURE*101325.;

    We change the dimensions of the domain as a function of the initial diameter of the droplet. The domain is large with respect to the droplet diameter, in order to avoid the influence of the domain boundaries on the droplet evaporation dynamics.

      double RR = 7.986462e+01;
      L0 = 0.5*RR*D0;

    We set the gravity contribution, and we shift the origin along the y coordinate, according to the radius of the solid fiber.

      G.x = GRAVITY;
      double df = FIBER*D0;
      X0 = -0.5*L0, Y0 = 0.5*df;

    We use a constant surface tension value, and we transport tr as a VOF tracer.

      f.sigma = 0.03;
      f.tracers = {tr};

    We run the simulation at different maximum levels of refinement.

      for (maxlevel = 10; maxlevel <= 10; maxlevel++) {

    The pinning point ap is adjusted according to the value of Y0. The pinning center ac is also adjusted depending on the grid.

        pinning.ap = sqrt (sq (0.5*D0) - sq (Y0)); = pinning.ap - 2.*L0/(1 << maxlevel);
        init_grid (1 << 9);
    #define circle(x,y,R)(sq(R) - sq(x) - sq(y))

    We initialize the volume fraction field according to D0, after refining the region around the droplet to the maximum level of refinement.

    event init (i = 0) {
      refine (circle (x, y, 4.*D0) > 0. && level < maxlevel);
      fraction (f, circle (x, y, 0.5*D0));

    We compute initial variables useful for post-processing.

      //volumecorr = 2.*pi*statsf(f).sum - (4./3.*pi*pow (0.5*D0, 3.));
      volumecorr = 0.;
      effective_radius0 = pow(3./4./pi*(2.*pi*statsf(f).sum - volumecorr), 1./3.);
      effective_radius = effective_radius0;
      foreach (reduction(+:mLiq0))
        mLiq0 += rho1v[]*f[]*dv();

    We set the molecular weights of the chemial species involved in the simulation (by default inMW=1).

      foreach_elem (YGList, jj)
        inMW[jj] = OpenSMOKE_MW (jj);

    The scalar tracer is initialized with the value of the horizontal coordinate.

        tr[] = x*f[];
      trmin = statsf(tr).min;
      trmax = statsf(tr).max;

    On the edge in contact with the liquid droplet we set Neumann boundary conditions for the scalar fields, while we set Dirichlet boundary conditions for the other sides.

    event bcs (i = 0) {
      scalar fuel  = YGList[OpenSMOKE_IndexOfSpecies (TOSTRING(FUEL))];
      scalar inert = YGList[OpenSMOKE_IndexOfSpecies (TOSTRING(INERT))];
      fuel[top] = dirichlet (0.);
      fuel[left] = dirichlet (0.);
      fuel[right] = dirichlet (0.);
      inert[top] = dirichlet (1.);
      inert[left] = dirichlet (1.);
      inert[right] = dirichlet (1.);
      TG[top] = dirichlet (TG0);
      TG[left] = dirichlet (TG0);
      TG[right] = dirichlet (TG0);

    We adapt the grid according to the fuel mass fraction, the temperature, and the velocity fields.

    #if TREE
    event adapt (i++) {
      scalar fuel = YList[OpenSMOKE_IndexOfSpecies (TOSTRING(FUEL))];
      adapt_wavelet_leave_interface ({fuel,T,u.x,u.y}, {f},
          (double[]){1.e-1,1.e0,1.e-1,1.e-1}, maxlevel, minlevel, 1);


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

    Grashof Number

    We compute the Grashof number to quantify the intensity of the natural convective fluxes.

    struct Grashof {
      double rhob, rhos;
      double r, g, nu;
      double value;
    struct Grashof Gr;
    event grashof (i++) {
      if (i == 0) {
        ThermoState tsg;
        tsg.T = TG0;
        tsg.P = Pref;
        tsg.x = gas_start;
        Gr.rhob = tp2.rhov (&tsg); = tp2.muv (&tsg)/tp2.rhov (&tsg);
        effective_radius = effective_radius0;
      Gr.r = effective_radius;
      Gr.g = fabs (GRAVITY);
      scalar YGIntFuel = YGIntList[0];
      scalar YGIntInert = YGIntList[1];
      double TIntAvg = avg_interface (TInt, f, tol=0.1);
      double YIntAvgFuel = avg_interface (YGIntFuel, f, tol=0.1);
      double YIntAvgInert = avg_interface (YGIntInert, f, tol=0.1);
      double YIntAvg[] = {YIntAvgFuel, YIntAvgInert};
      double XIntAvg[NGS];
      correctfrac (YIntAvg, NGS);
      mass2molefrac (XIntAvg, YIntAvg, inMW, NGS);
      ThermoState tsg;
      tsg.P = Pref;
      if (i == 0) {
        tsg.T = TL0;
        tsg.x = gas_start;
      else {
        tsg.T = TIntAvg;
        tsg.x = XIntAvg;
      Gr.rhos = tp2.rhov (&tsg);
      Gr.value = (Gr.rhos - Gr.rhob)*pow (Gr.r, 3.)*Gr.g/(Gr.rhob*sq(;

    Output Files

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

    event output_data (i += 50) {
      char name[80];
      sprintf (name, "OutputData-%d", maxlevel);
      static FILE * fp = fopen (name, "w");
      effective_radius = pow(3./4./pi*(2.*pi*statsf(f).sum - volumecorr), 1./3.);
      double d_over_d02_old = d_over_d02;
      double tad_old = tad;
      d_over_d02 = sq (effective_radius / effective_radius0);
      tad = t/sq(D0*1e3);

    The vaporization rate is computed according to the formula in Liu & Avedisian, 2011, pag. 777 bottom.

      double kv = 0.;
      if (i > 1)
        kv = fabs ((d_over_d02 - d_over_d02_old)/(tad - tad_old));
      double mLiq = 0.;
        mLiq += rho1v[]*f[]*dv();

    We compute and print useful average quantities such as the average interface temperature and mass fractions, and the average droplet temperature.

      scalar YGIntFuel = YGIntList[0];
      double TIntAvg = avg_interface (TInt, f, tol=0.1);
      double YIntAvg = avg_interface (YGIntFuel, f, tol=0.1);
      int counter = 0;
      double TDropAvg = 0.;
      foreach(reduction(+:TDropAvg) reduction(+:counter)) {
        if (f[] > 1.-F_ERR) {
          TDropAvg += TL[];
      TDropAvg = (counter > 0.) ? TDropAvg/counter : 0.;
      fprintf (fp, "%g %g %g %g %g %g %g %g %g %g\n", t, tad, effective_radius,
          d_over_d02, mLiq/mLiq0, kv, Gr.value, TIntAvg, YIntAvg, TDropAvg);
      fflush (fp);


    We write the animation with the evolution of the fuel mass fraction, the interface position and the temperature field.

    #if MOVIE
    event movie (t += 0.01) {
      view (tx = 0.025, fov = 3.5, samples = 2);
      draw_vof ("f", lw = 1.5);
      squares ("T", min = TL0, max = TG0, linear = true);
      save ("temperature.mp4");
      view (tx = 0.025, fov = 3.5, samples = 2);
      draw_vof ("f", lw = 1.5);
      squares (TOSTRING(FUEL), min = 0., max = 1., linear = true);
      save ("fuel.mp4");


    Output dump files for restore or post-processing.

    #if DUMP
    event snapshots (t += 0.1) {
      if (i > 1) {
        char name[80];
        sprintf (name, "snapshots-%g", t);
        dump (name);

    Stopping Condition

    We stop the simulation when the droplet is almost fully consumed.

    event stop (i++) {
      if (d_over_d02 <= 0.05)
        return 1;

    We run the simulation for long time, which is not reached because the stopping condition on the droplet diameter is reached first.

    event end (t = 50.);