src/test/oscillation.c

    Shape oscillation of an inviscid droplet

    This test case is discussed in Popinet, 2009.

    A two-dimensional elliptical droplet (density ratio 1/1000) is released in a large domain. Under the effect of surface-tension the shape of the droplet oscillates around its (circular) equilibrium shape. The fluids inside and outside the droplet are inviscid so ideally no damping of the oscillations should occur. As illustrated on the figures some damping occurs in the simulation due to numerical dissipation.

    This simulation is also a stringent test case of the accuracy of the surface tension representation as no explicit viscosity can damp eventual parasitic currents.

    We use either the momentum-conserving or standard Navier–Stokes solver with VOF interface tracking and surface tension.

    #if MOMENTUM

# include "momentum.h"
# include "tension.h"
# define cf

#elif COMPRESSIBLE

# include "compressible/two-phase.h"
# include "compressible/Mie-Gruneisen.h"
# include "tension.h"
# include "compressible/tension.h"

double rho1, rho2;

#else // standard centered Navier--Stokes solver

# include "navier-stokes/centered.h"
# define FILTERED 1
# include "two-phase.h"
# include "tension.h"

#endif // standard centered Navier--Stokes solver

    The diameter of the droplet is 0.2.

    #define D 0.2

    We will vary the level of refinement to study convergence.

    FILE * fp = NULL;
int LEVEL;

int main()
{

    The density is variable.

      rho1 = 1, rho2 = 1e-3;

#if COMPRESSIBLE

    The parameters of the EOS for the liquid and gas.

      PI1 = 300.;
  gamma1 = 7.14;
  gamma2 = 1.4;
#if 0 // does not work
  PI2 = PI1;
  gamma2 = gamma1;
#endif
#endif

#if 0
  double nu1 = 1e-4, nu2 = 1e-4;
  mu1 = nu1*rho1;
  mu2 = nu2*rho2;
#endif

    The surface tension is unity. Decreasing the tolerance on the Poisson solve improves the results. We cleanup existing files and vary the level of refinement.

      f.sigma = 1.;
  L0 = 0.5 [0];
  TOLERANCE = 1e-4 [*];
  remove ("error");
  remove ("laplace");
  for (LEVEL = 4; LEVEL <= 7; LEVEL++) {
    N = 1 << LEVEL;

    We open a file indexed by the level to store the time evolution of the kinetic energy.

        char name[80];
    sprintf (name, "k-%d", LEVEL);
    fp = fopen (name, "w");
    run();
    fclose (fp);
  }

    We use grep to filter the lines generated by gnuplot containing the results of the fits (see below).

      system ("grep ^fit out >> log");
}

event init (i = 0) {

    We initialise the shape of the interface, a slightly elliptic droplet.

      fraction (f, D/2.*(1. + 0.05*cos(2.*atan2(y,x))) - sqrt(sq(x) + sq(y)));

#if COMPRESSIBLE

    For compressible flows we initialize the densities, pressure and energies.

      // fixme: The initial condition for pressure/energy does not respect
  // the Laplace pressure because the curvature is not constant
  double p0L = 1, p0 = p0L + f.sigma/D*2;
  foreach() {
    frho1[] = rho1*f[];
    frho2[] = rho2*(1. - f[]);
    p[]   = p0*f[] + p0L*(1. - f[]);
    fE1[] = f[]*(p0/(gamma1 - 1.) + PI1*gamma1/(gamma1 - 1.));
    fE2[] = (1. - f[])*p0L/(gamma2 - 1.);
  }
#endif // COMPRESSIBLE
}

    At each timestep we output the kinetic energy.

    event logfile (i++; t <= 1) {
  double ke = 0.;
  foreach (reduction(+:ke))
#if MOMENTUM || COMPRESSIBLE
    ke += dv()*(sq(q.x[]) + sq(q.y[]))/rho[];
#else
    ke += dv()*(sq(u.x[]) + sq(u.y[]))*rho(sf[]);
#endif
  fprintf (fp, "%g %g %d\n", t, ke, mgp.i);
  fflush (fp);
}

    At the end of the simulation, we use gnuplot to fit a function of the form \displaystyle k(t) = ae^{-bt}(1-\cos(ct)) to the kinetic energy. This gives estimates of the oscillation pulsation c and of the damping b.

    We also compute the relative error on the pulsation, using the theoretical value \omega_0 as reference.

    event fit (t = end) {
  FILE * fp = popen ("gnuplot 2>&1", "w");
  fprintf (fp, 
           "k(t)=a*exp(-b*t)*(1.-cos(c*t))\n"
           "a = 3e-4\n"
           "b = 1.5\n"
	   "\n"
           "D = %g\n"
           "n = 2.\n"
           "sigma = 1.\n"
           "rhol = 1.\n"
           "rhog = 1./1000.\n"
           "r0 = D/2.\n"
           "omega0 = sqrt((n**3-n)*sigma/((rhol+rhog)*r0**3))\n"
	   "\n"
           "c = 2.*omega0\n"
           "fit k(x) 'k-%d' via a,b,c\n"
	   "level = %d\n"
	   "res = D/%g*2.**level\n"
	   "print sprintf (\"fit %%g %%.6f %%.2f %%.0f\\n\", res, a, b, c, D)\n"
	   "\n"
	   "set table 'fit-%d'\n"
	   "plot [0:1] 2.*a*exp(-b*x)\n"
	   "unset table\n"
	   "\n"
	   "set print 'error' append\n"
	   "print res, c/2./omega0-1., D\n"
	   "\n"
	   "set print 'laplace' append\n"
	   "empirical_constant = 30.\n"
	   "print res, (1./(b**2.*D**3.))*empirical_constant**2, D\n"
	   "\n",
	   D, LEVEL, LEVEL, L0, LEVEL);
  pclose (fp);
}

#if TREE
event adapt (i++) {
#if MOMENTUM || COMPRESSIBLE
  vector u[];
  foreach()
    foreach_dimension()
      u.x[] = q.x[]/rho[];
#endif
  adapt_wavelet ({f,u}, (double[]){5e-3,1e-3,1e-3}, LEVEL);
}
#endif

    Results

    set xlabel 'Time'
set ylabel 'Kinetic energy'
set logscale y
plot [0:1][8e-5:]'k-8' t "51.2" w l, 'k-7' t "25.6" w l,               \
  'k-6' t "12.8" w l, 'k-5' t "6.4" w l,			       \
  'fit-8' t "" w l lt 7, 'fit-7' t "" w l lt 7, 'fit-6' t "" w l lt 7, \
  'fit-5' t "" w l lt 7
    Evolution of the kinetic energy as a function of time for the spatial resolutions (number of grid points per diameter) indicated in the legend. The black lines are fitted decreasing exponential functions. (script)

    Evolution of the kinetic energy as a function of time for the spatial resolutions (number of grid points per diameter) indicated in the legend. The black lines are fitted decreasing exponential functions. (script)

    set xlabel 'Diameter (grid points)'
set ylabel 'Frequency error (%)'
set logscale x 2
unset grid
set xzeroaxis
set key spacing 1.5 top right
ftitle(a,b,c) = sprintf("%.0f/x^{%4.2f} (%s)", exp(a), -b, c)
f(x)=a+b*x
fit f(x) 'error' u (log($1)):(log(abs($2)*100.)) via a,b
f1(x)=a1+b1*x
fit f1(x) '../oscillation-momentum/error' u (log($1)):(log(abs($2)*100.)) via a1,b1
f2(x)=a2+b2*x
fit f2(x) '../oscillation-compressible/error' u (log($1)):(log(abs($2)*100.)) via a2,b2
plot 'error' u ($1):(abs($2)*100.) t "" w p pt 5 ps 1, \
     '../oscillation-momentum/error' u ($1):(abs($2)*100.) t "" w p pt 7 ps 1, \
     '../oscillation-compressible/error' u ($1):(abs($2)*100.) t "" w p pt 9 ps 1, \
      exp(f(log(x))) t ftitle(a,b,"standard"), \
      exp(f1(log(x))) t ftitle(a1,b1,"momentum"), \
      exp(f2(log(x))) t ftitle(a2,b2,"compressible")
    Relative error in the oscillation frequency as a function of resolution. (script)

    Relative error in the oscillation frequency as a function of resolution. (script)

    The amount of numerical damping can be estimated by computing an equivalent viscosity. With viscosity, kinetic energy is expected to decrease as: \displaystyle \exp(-C\nu/D^2t) where C is a constant, \nu the viscosity and D the droplet diameter. Using curve fitting the damping coefficient b=C\nu/D^2 can be estimated (black curves on Figure ). An equivalent Laplace number can then be computed as: \displaystyle La=\frac{\sigma D}{\rho\nu^2}=\frac{\sigma C^2}{\rho b^2 D^3} The equivalent Laplace number depends on spatial resolution as illustrated below.

    set xlabel 'Diameter (grid points)'
set ylabel 'Equivalent Laplace number'
set grid
set key bottom right
plot 'laplace' t "standard" w p pt 5 ps 1, \
     '../oscillation-momentum/laplace' t "momentum" w p pt 7 ps 1, \
     '../oscillation-compressible/laplace' t "compressible" w p pt 9 ps 1
    Equivalent Laplace number estimated from the numerical damping of kinetic energy. (script)

    Equivalent Laplace number estimated from the numerical damping of kinetic energy. (script)

    See also