wave.c

Breaking wave

Breaking wave

We solve the two-phase Navier–Stokes equations with surface tension and using a momentum-conserving transport of each phase. Gravity is taken into account using the “reduced gravity approach” and the results are visualised using Basilisk view.

#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "navier-stokes/conserving.h"
#include "tension.h"
#include "reduced.h"
#include "view.h"
#include "tag.h"

We log some profiling information.

#include "navier-stokes/perfs.h"
#include "profiling.h"

The primary parameters are the wave steepness ak, the Bond and Reynolds numbers.

double ak = 0.55;
double BO = 1000.;
double RE = 40000.;

The default maximum level of refinement depends on the dimension.

int LEVEL = dimension == 2 ? 9 : 6;

The error on the components of the velocity field used for adaptive refinement.

double uemax = 0.005;

The density and viscosity ratios are those of air and water.

#define RATIO (1./850.)
#define MURATIO (17.4e-6/8.9e-4)

Define if we want to use a Dirac viscous layer initialization.

int DIRAC = 0;

The wave number, fluid depth and acceleration of gravity are set to these values.

#define k_  (2.*pi)
#define h_   0.5
#define g_   1.

The program takes optional arguments which are the level of refinement, steepness, Bond and Reynolds numbers, and optional Dirac initialisation.

int main (int argc, char * argv[])
{
  if (argc > 1)
    LEVEL = atoi (argv[1]);
  if (argc > 2)
    ak = atof(argv[2]);
  if (argc > 3)
    BO = atof(argv[3]);
  if (argc > 4)
    RE = atof(argv[4]);    
  if (argc > 5)
    DIRAC = atof(argv[5]);

The domain is a cubic box centered on the origin and of length L0=1, periodic in the x- and z-directions.

  origin (-L0/2, -L0/2, -L0/2);
  periodic (right);
#if dimension > 2
  periodic (front);
#endif

Here we set the densities and viscosities corresponding to the parameters above.

  rho1 = 1.;
  rho2 = RATIO;
  mu1 = 1.0/RE; //using wavelength as length scale
  mu2 = 1.0/RE*MURATIO;
  f.sigma = 1./(BO*sq(k_));
  G.y = -g_;

When we use adaptive refinement, we start with a coarse mesh which will be refined as required when initialising the wave.

#if TREE  
  N = 32;
#else
  N = 1 << LEVEL;
#endif
  run();
}

Initial conditions

These functions return the shape of a third-order Stokes wave with the wavenumber and steepness given by the parameters above (ak and _k_).

double wave (double x, double y) {
  double a_ = ak/k_;
  double eta1 = a_*cos(k_*x);
  double alpa = 1./tanh(k_*h_);
  double eta2 = 1./4.*alpa*(3.*sq(alpa) - 1.)*sq(a_)*k_*cos(2.*k_*x);
  double eta3 = -3./8.*(cube(alpa)*alpa - 
			3.*sq(alpa) + 3.)*cube(a_)*sq(k_)*cos(k_*x) + 
    3./64.*(8.*cube(alpa)*cube(alpa) + 
	    (sq(alpa) - 1.)*(sq(alpa) - 1.))*cube(a_)*sq(k_)*cos(3.*k_*x);
  return eta1 + ak*eta2 + sq(ak)*eta3 - y;
}

double eta (double x, double y) {
  double a_ = ak/k_;
  double eta1 = a_*cos(k_*x);
  double alpa = 1./tanh(k_*h_);
  double eta2 = 1./4.*alpa*(3.*sq(alpa) - 1.)*sq(a_)*k_*cos(2.*k_*x);
  double eta3 = -3./8.*(cube(alpa)*alpa - 
			3.*sq(alpa) + 3.)*cube(a_)*sq(k_)*cos(k_*x) + 
    3./64.*(8.*cube(alpa)*cube(alpa) + 
	    (sq(alpa) - 1.)*(sq(alpa) - 1.))*cube(a_)*sq(k_)*cos(3.*k_*x);
  return eta1 + ak*eta2 + sq(ak)*eta3;
}

We also calculate an approximation to a Dirac distribution on the wave surface. This allows us to calculate a vortex sheet on the surface to provide a boundary layer in the air above the water surface.

double gaus (double y, double yc, double T){
  double deltaw = sqrt(2.0/RE)/k_;
  double deltaa = sqrt(2.0/RE*MURATIO/RATIO)/k_;
  double r = y - yc;
  return 2.0/(sqrt(2.0*pi*sq(deltaa)) + sqrt(2.0*pi*sq(deltaw))) *
    (T*exp(-sq(r)/(2.0*sq(deltaw))) + (1.0 - T)*exp(-sq(r)/(2.0*sq(deltaa))));
}

We either restart (if a “restart” file exists), or initialise the wave using the third-order Stokes wave solution.

event init (i = 0)
{
  if (!restore ("restart")) {
    do {
      fraction (f, wave(x,y));

To initialise the velocity field, we first define the potential.

      scalar Phi[];
      foreach() {
	double alpa = 1./tanh(k_*h_);
	double a_ = ak/k_;
	double sgma = sqrt(g_*k_*tanh(k_*h_)*
			   (1. + k_*k_*a_*a_*(9./8.*(sq(alpa) - 1.)*
					      (sq(alpa) - 1.) + sq(alpa))));
	double A_ = a_*g_/sgma;
	double phi1 = A_*cosh(k_*(y + h_))/cosh(k_*h_)*sin(k_*x);
	double phi2 = 3.*ak*A_/(8.*alpa)*(sq(alpa) - 1.)*(sq(alpa) - 1.)*
	  cosh(2.0*k_*(y + h_))*sin(2.0*k_*x)/cosh(2.0*k_*h_);
	double phi3 = 1./64.*(sq(alpa) - 1.)*(sq(alpa) + 3.)*
	  (9.*sq(alpa) - 13.)*
	  cosh(3.*k_*(y + h_))/cosh(3.*k_*h_)*a_*a_*k_*k_*A_*sin(3.*k_*x);
	Phi[] = phi1 + ak*phi2 + ak*ak*phi3;
      }
      boundary ({Phi});
      
      if (DIRAC) {

We calculate the vorticity in the Dirac layer. We need a separate foreach here because we need the derivative of the potential phi.

	scalar vort2[];
	scalar psi[];
	foreach() {
	  vort2[] = -2.0*gaus(y,wave(x,y)+y,f[])*(Phi[1,0]-Phi[-1,0])/(2.*Delta);
	  psi[] = 0.0;
	}
	boundary ({vort2,psi});
	psi[top] = dirichlet(0.);
	psi[bottom] = dirichlet(0.);

Solve the Poisson problem for the streamfunction psi given the vorticity field.

	poisson (psi, vort2);

And then define the velocity field using centered-differencing of the streamfunction.

	foreach() {
	  u.x[] = (psi[0,1] - psi[0,-1])/(2.*Delta);
	  u.y[] = -(psi[1] - psi[-1])/(2.*Delta);
	}
      }
      else {

If we choose not to use the Dirac layer, instead initialize in the water only according to the potential already calculated.

	foreach()
	  foreach_dimension()
	    u.x[] = (Phi[1] - Phi[-1])/(2.0*Delta) * f[];
      }
      boundary ((scalar *){u});
    }

On trees, we repeat this initialisation until mesh adaptation does not refine the mesh anymore.

#if TREE  
    while (adapt_wavelet ({f,u},
			  (double[]){0.01,uemax,uemax,uemax}, LEVEL, 5).nf);
#else
    while (0);
#endif
  }
}

Outputs

We are interested in the viscous dissipation rate in both water and air.

int dissipation_rate (double* rates)
{
  double rateWater = 0.0;
  double rateAir = 0.0;
  foreach (reduction (+:rateWater) reduction (+:rateAir)) {
    double dudx = (u.x[1]     - u.x[-1]    )/(2.*Delta);
    double dudy = (u.x[0,1]   - u.x[0,-1]  )/(2.*Delta);
    double dudz = (u.x[0,0,1] - u.x[0,0,-1])/(2.*Delta);
    double dvdx = (u.y[1]     - u.y[-1]    )/(2.*Delta);
    double dvdy = (u.y[0,1]   - u.y[0,-1]  )/(2.*Delta);
    double dvdz = (u.y[0,0,1] - u.y[0,0,-1])/(2.*Delta);
    double dwdx = (u.z[1]     - u.z[-1]    )/(2.*Delta);
    double dwdy = (u.z[0,1]   - u.z[0,-1]  )/(2.*Delta);
    double dwdz = (u.z[0,0,1] - u.z[0,0,-1])/(2.*Delta);
    double SDeformxx = dudx;
    double SDeformxy = 0.5*(dudy + dvdx);
    double SDeformxz = 0.5*(dudz + dwdx);
    double SDeformyx = SDeformxy;
    double SDeformyy = dvdy;
    double SDeformyz = 0.5*(dvdz + dwdy);
    double SDeformzx = SDeformxz;
    double SDeformzy = SDeformyz;
    double SDeformzz = dwdz; 
    double sqterm = 2.*dv()*(sq(SDeformxx) + sq(SDeformxy) + sq(SDeformxz) +
			     sq(SDeformyx) + sq(SDeformyy) + sq(SDeformyz) +
			     sq(SDeformzx) + sq(SDeformzy) + sq(SDeformzz)) ;
    rateWater += mu1/rho[]*f[]*sqterm; //water
    rateAir   += mu2/rho[]*(1. - f[])*sqterm; //air
  }
  rates[0] = rateWater;
  rates[1] = rateAir;
  return 0;
}

We also want to count the drops and bubbles in the flow.

event countDropsBubble(i++)
{
  scalar m1[]; //droplets
  scalar m2[]; //bubbles
  foreach(){
    m1[] = f[] > 0.5; //i.e. set m true if f[] is close to unity (droplets)
    m2[] = f[] < 0.5; //m true if f[] close to zero (bubbles)
  }
  int n1 = tag(m1);
  int n2 = tag(m2);

Having counted the bubbles, now we find their size. This example is similar to the jet atomization problem. We are interested in the volumes and positions of each droplet/bubble.

  double v1[n1]; //droplet
  coord b1[n1];  //droplet
  double v2[n2]; //bubble
  coord b2[n2];  //bubble

We initialize:

  for (int j=0; j<n1; j++)
      v1[j] = b1[j].x = b1[j].y = b1[j].z = 0.0;
  for (int j=0; j<n2; j++)
      v2[j] = b2[j].x = b2[j].y = b2[j].z = 0.0;

We proceed with calculation.

  foreach_leaf() {
    // droplets
    if (m1[] > 0) {
      int j = m1[] - 1;
      v1[j] += dv()*f[]; //increment the volume of the droplet
      coord p = {x,y,z};
      foreach_dimension()
	b1[j].x += dv()*f[]*p.x;
    }
    // bubbles
    if (m2[] > 0) {
      int j = m2[] - 1;
      v2[j] += dv()*(1.0-f[]);
      coord p = {x,y,z};
      foreach_dimension()
	b2[j].x += dv()*(1.0-f[])*p.x;
    }
  }

Reduce for MPI.

#if _MPI
  MPI_Allreduce (MPI_IN_PLACE, v1, n1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
  MPI_Allreduce (MPI_IN_PLACE, b1, 3*n1, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
  MPI_Allreduce (MPI_IN_PLACE, v2, n2, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
  MPI_Allreduce (MPI_IN_PLACE, b2, 3*n2, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
#endif

Output the volume and position of each droplet to file.

  static FILE * fdrop = fopen("droplets.dat","w");
  static FILE * fbubb = fopen("bubbles.dat","w");
  for (int j=0; j<n1; j++)
    fprintf (fdrop, "%d %g %d %g %g %g\n", i, t,
	     j, v1[j], b1[j].x/v1[j], b1[j].y/v1[j]);
  for (int j=0; j<n2; j++)
    fprintf (fbubb, "%d %g %d %g %g %g\n", i, t,
	     j, v2[j], b2[j].x/v2[j], b2[j].y/v2[j]);
}

We log the evolution of the kinetic and potential energies and dissipation rate as functions of the non-dimensional time.

event graphs (i++) {
  static FILE * fpwater = fopen("budgetWater.dat", "w");
  static FILE * fpair = fopen("budgetAir.dat", "w");
  double ke = 0., gpe = 0.;
  double keAir = 0., gpeAir = 0.;
  foreach(reduction(+:ke) reduction(+:gpe) 
	  reduction(+:keAir) reduction(+:gpeAir)) {
    double norm2 = 0.;
    foreach_dimension()
      norm2 += sq(u.x[]);
    ke += rho[]*norm2*f[]*dv();
    keAir += rho[]*norm2*(1.0-f[])*dv();
    gpe += rho1*g_*y*f[]*dv();
    gpeAir += rho2*g_*y*(1.0-f[])*dv();
  }
  double rates[2];
  dissipation_rate(rates);
  double dissWater = rates[0];
  double dissAir   = rates[1];
    if (i == 0) {
    fprintf (fpwater, "t ke gpe dissipation\n");
    fprintf (fpair, "t ke gpe dissipation\n");
    }
  fprintf (fpwater, "%g %g %g %g\n",
	   t/(k_/sqrt(g_*k_)), ke/2., gpe + 0.125, dissWater);
  fprintf (fpair, "%g %g %g %g\n",
	   t/(k_/sqrt(g_*k_)), keAir/2., gpeAir + 0.125, dissAir);
  fprintf (ferr, "%g %g %g %g\n",
	   t/(k_/sqrt(g_*k_)), ke/2., gpe + 0.125, dissWater);
}

set key auto col
plot 'log' u 1:2 w l, '' u 1:3 w l, '' u 1:(($2+$3)/2.) t 'total/2' w l, \
'' u 1:4 w l

Evolution of kinetic, potential and total energies and dissipation rate. (script)

Visualisation

We use Basilisk view (and output_ppm()) to display animations of the results.

event movies (t += 0.01) {

We first do simple movies of the volume fraction, level of refinement fields. In 3D, these are in a z=0 cross-section.

  {
    static FILE * fp = popen ("ppm2mp4 f.mp4", "w");
    output_ppm (f, fp, min = 0, max = 1, n = 512);
  }

#if TREE
  {
    scalar l[];
    foreach()
    l[] = level;
    static FILE * fp = popen ("ppm2mp4 level.mp4", "w");
    output_ppm (l, fp, min = 5, max = LEVEL, n = 512);
  }
#endif

Wave breaking. Animation of the level of refinement.

We use Basilisk view differently in 2D and 3D.

  scalar omega[];
  vorticity (u, omega);
#if dimension == 2
  view (width = 800, height = 600, fov = 18.8);
  clear();

We repeat the drawing periodically in the x-direction.

  for (double x = -L0; x <= L0; x += L0)
    translate (x) {
      draw_vof ("f");
      squares ("omega", linear = true);
    }

This gives the following movie.

#else // dimension == 3

In 3D, we generate a first movie seen from below.

  view (width = 1600, height = 1200, theta = pi/4, phi = -pi/6, fov = 20);
  clear();
  for (double x = -2*L0; x <= L0; x += L0)
    translate (x) {
      squares ("omega", linear = true, n = {0,0,1}, alpha = -L0/2);
      for (double z = -3*L0; z <= L0; z += L0)
	translate (z = z)
	  draw_vof ("f");
    }
  save ("below.mp4");

And a second movie, seen from above.

  view (width = 1600, height = 1200, theta = pi/4, phi = pi/6, fov = 20);
  clear();

In 3D, we are doubly-periodic (along x and z).

  for (double x = -2*L0; x <= L0; x += L0)
    translate (x) {
      squares ("omega", linear = true, n = {0,0,1}, alpha = -L0/2);
      for (double z = -3*L0; z <= L0; z += L0)
	translate (z = z)
	  draw_vof ("f");
    }
#endif // dimension == 3
  save ("movie.mp4");
}

Dump/restore

To be able to restart, we dump the entire simulation at regular intervals.

event snapshot (i += 200) {
  dump();
}

End

The wave period is k_/sqrt(g_*k_). We want to run up to 2 (alternatively 4) periods.

event end (t = 2.*k_/sqrt(g_*k_)) {
  fprintf (fout, "i = %d t = %g\n", i, t);
  dump ("end");
}

Mesh adaptation

On trees, we adapt the mesh according to the error on volume fraction and velocity.

#if TREE
event adapt (i++) {
  adapt_wavelet ({f,u}, (double[]){0.01,uemax,uemax,uemax}, LEVEL, 5);
}
#endif

Running in parallel

This file will work in 2D or 3D, either with parallel multigrid (without adaptivity), using for example:

qcc -source -D_MPI=1 -grid=multigrid3D wave.c
scp _wave.c occigen.cines.fr:

and then following a recipe similar to that of the atomisation example.

To use adaptivity, just do something like:

qcc -source -D_MPI=1 -grid=octree wave.c
scp _wave.c occigen.cines.fr: