src/examples/breaking.c

    3D breaking Stokes wave (multilayer solver)

    A steep, 3D, third-order Stokes wave is unstable and breaks. This is the 3D equivalent of this test case. The bathymetry is given by \displaystyle z_b(y) = - 0.5 + \sin(\pi y)/4 i.e. it is shallower toward the back of the domain which causes the wave to break earlier there.

    Animation of the free-surface. The surface is coloured according to the x-component of the surface velocity.

    The solution is obtained using the layered model and demonstrates its robustness and a degree of realism even for this complex case.

    #include "layered/hydro.h"
#include "layered/nh.h"
#include "layered/remap.h"
#include "layered/perfs.h"
#include "view.h"

    The initial conditions are given by the wave steepness ak and the Reynolds number Re=c\lambda/\nu with c the phase speed of the gravity wave and \lambda its wavelength.

    double ak = 0.33;
double RE = 40000.;
double k_ = 2.*pi, h_ = 0.5, g_ = 1.;

#define T0  (k_*L0/sqrt(g_*k_))

    The domain is periodic in x and resolved using 256^2 points and 30 layers.

    int main()
{
  origin (-L0/2., -L0/2.);
  periodic (right);
  N = 256;
  nl = 30;
  G = g_;
  nu = 1./RE;

    Some implicit damping is necessary to damp fast modes. This may be related to the slow/fast decoupling of the \theta-scheme mentioned by Durran & Blossey, 2012.

      theta_H = 0.51;

  run();
}

    The initial conditions for the free-surface and velocity are given by the third-order Stokes solution.

    #include "../test/stokes.h"

event init (i = 0)
{

    We can use a larger CFL, in particular because we are not dealing with shallow-water/wetting/drying.

      CFL = 0.8;

    The layer thicknesses follow a geometric progression, starting from a top layer with a thickness of 1/3 that of the regular distribution.

      geometric_beta (1./3., true);

  double a = 0.25;
  foreach() {
    zb[] = - 0.5 + a*sin(k_/2.*y);
    double H = wave(x, 0) - zb[];
    double z = zb[];
    foreach_layer() {
      h[] = H*beta[point.l];
      z += h[]/2.;
      u.x[] = u_x(x, z);
      w[] = u_y(x, z);
      z += h[]/2.;
    }
  }
}

    We add (an approximation of) horizontal viscosity.

    event viscous_term (i++)
  horizontal_diffusion ((scalar *){u}, nu, dt);

    We log the evolution of the kinetic and potential energies.

    set xlabel 't/T0'
plot [0:6]'log' u 1:2 w l t 'kinetic', '' u 1:3 w l t 'potential', \
     '' u 1:(($2+$3)/2.) w l t 'total/2'
    Evolution of the kinetic, potential and total energy (script)

    Evolution of the kinetic, potential and total energy (script)

    event logfile (i++; t <= 8.*T0)
{
  double ke = 0., gpe = 0.;
  foreach (reduction(+:ke) reduction(+:gpe)) {
    foreach_layer() {
      double norm2 = sq(w[]);
      foreach_dimension()
	norm2 += sq(u.x[]);
      ke += norm2*h[]*dv();
    }
    gpe += sq(eta[])*dv();
  }
  fprintf (stderr, "%g %g %g\n", t/T0, ke/2., g_*gpe/2.);
}

    And generate the movie of the free surface (this is quite expensive). The movie is 45 seconds at 25 frames/second.

    event movie (t += 8.*T0/(45*25))
{
  view (fov = 17.3106, quat = {0.475152,0.161235,0.235565,0.832313},
	tx = -0.0221727, ty = -0.0140227, width = 1200, height = 768);
  char s[80];
  sprintf (s, "t = %.2f T0", t/T0);
  draw_string (s, size = 80);
  for (double x = -1; x <= 1; x++)
    translate (x) {
      squares ("u29.x", linear = true, z = "eta", min = -0.15, max = 0.6);
    }
  save ("movie.mp4");
}

    Parallel run

    The simulation was run in parallel on the Occigen machine on 64 cores, using this script

    local% qcc -source -D_MPI=1 breaking.c
local% scp _breaking.c user@occigen.cines.fr:
    #!/bin/bash
#SBATCH -J breaking
#SBATCH --constraint=HSW24
#SBATCH --ntasks=64
#SBATCH --threads-per-core=1
#SBATCH --time=1:00:00
#SBATCH --exclusive
#SBATCH --mail-type=BEGIN,END,FAIL
#SBATCH --mail-user=popinet@basilisk.fr

module purge
module load openmpi
module load intel gcc

NAME=breaking
mpicc -Wall -std=c99 -D_XOPEN_SOURCE=700 -O2 _$NAME.c -o $NAME \
    -L/home/popinet/local/gl -L/home/popinet/local/lib \
    -lglutils -lfb_tiny -lppr -lgfortran -lm

export LD_LIBRARY_PATH=/home/popinet/local/lib:$LD_LIBRARY_PATH
export PATH=$PATH:/home/popinet/local/bin
rm -f *.ppm
srun --mpi=pmi2 -K1 --resv-ports -n $SLURM_NTASKS $NAME 2> log > out

    The number of timesteps was 4159 and the runtime was 44 minutes with movie generation.

    References

    [durran2012]

    Dale R Durran and Peter N Blossey. Implicit–explicit multistep methods for fast-wave–slow-wave problems. Monthly Weather Review, 140(4):1307–1325, 2012. [ DOI ]