src/test/fall.c

Impact of a viscoelastic drop on a solid

We solve the axisymmetric, incompressible, variable-density, Navier–Stokes equations with two phases and use the log-conformation method to include viscoelastic stresses. The curvature module is used to compute interface properties.

#include "axi.h"
#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "log-conform.h"
#include "curvature.h"

The density and viscosity ratios are 1000. The Reynolds number based on the droplet diameter, velocity and viscosity is 5 and the Froude number is 2.26.

#define RHO_r 0.001
#define MU_r 0.001
#define RE 5.
#define FR 2.26
#define LEVEL 7

The dimensionless viscoelastic properties used are the ratio of the solvent to the total viscoelastic viscosity (polymeric plus solvent) and the Weissenberg number.

#define BETA 0.1
#define WI 1.0

scalar lambdav[], mupv[];

The drop comes from the right. We allow the fluid to get through that boundary.

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

The wall is at the left side. We apply a no-slip boundary condition and a non-wetting condition for the VOF tracer.

u.t[left] = dirichlet(0);
tau_qq[left] = dirichlet(0);
f[left]   = 0.;

int main() {

The domain spans [0:2.6]\times[0:2.6].

  size (2.6);
  init_grid (1 << LEVEL);

The densities and viscosities are defined by the parameters above.

  rho1 = 1.;
  rho2 = RHO_r;
  mu1 = BETA/RE;
  mu2 = MU_r/RE;

The viscoelastic fields will be set below.

  mup = mupv;
  lambda = lambdav;

We set a maximum timestep. This is necessary for proper temporal resolution of the viscoelastic stresses.

  DT = 2e-3;
  run();
}

event init (t = 0) {

At a wall of normal \mathbf{n} the component of the viscoelastic stress tensor tau_p_{nn} is zero. Since the left boundary is a wall, we set tau_p_{xx} equal to zero at that boundary.

  scalar s = tau_p.x.x;
  s[left] = dirichlet(0.);

The drop is centered on (2,0) and has a radius of 0.5.

  fraction (f, - sq(x - 2.) - sq(y) + sq(0.5));

The initial velocity of the droplet is -1.

  foreach()
    u.x[] = - f[];
}

We add the acceleration of gravity.

event acceleration (i++) {
  face vector av = a;
  foreach_face(x)
    av.x[] -= 1./sq(FR);
}

We update the viscoelastic properties. Only the droplet is viscoelastic.

event properties (i++) {
  foreach() {
    mupv[] = (1. - BETA)*clamp(f[],0,1)/RE;
    lambdav[] = WI*clamp(f[],0,1);
  }
}

We adapt the solution at every timestep based on the interface and velocity errors.

#if TREE
event adapt (i++) {
  adapt_wavelet ({f, u.x, u.y}, (double[]){1e-2, 5e-3, 5e-3},
		 maxlevel = LEVEL, minlevel = LEVEL - 2);
}
#endif

We track the spreading diameter of the droplet.

event logfile (i += 20; t <= 5) {
  scalar pos[];
  position (f, pos, {0,1});
  fprintf (stderr, "%g %g\n", t, 2.*statsf(pos).max);
}

We generate a movie of the interface shape.

#include "view.h"

event viewing (i += 10) {
  view (width = 400, height = 400, fov = 20, ty = -0.5,
	quat = {0, 0, -0.707, 0.707});

  clear();
  draw_vof ("f", lw = 2);
  squares ("u.x", linear = true);
  box (notics = true);
  mirror ({0,1}) {
    draw_vof ("f", lw = 2);
    squares ("u.y", linear = true);
    box (notics = true);
  }
  save ("movie.mp4");

We can optionally visualise the results while we run.

#if 0
  static FILE * fp = popen ("bppm","w");
  save (fp = fp);
#endif
}

Results

The time evolution of the maximum diameter is close to that reported by Figueiredo et al. (2016).

reset
set ylabel 'Maximum diameter'
set xlabel 't'
plot 'fall.figueiredo' lt 3 pt 5 t 'Figueiredo et al. (2016)', \
     'log' w l lw 2 t 'Basilisk'

Time evolution of the maximum diameter (script)

References