wind_pool.c

Constant wind over constant depth pool
Adaptation
Adaptivity

Constant wind over constant depth pool

Example of a constant wind over a square constant depth pool

A constant wind of 1 m/s blows in the x direction over a square pool of 1000 m side length with constant depth 2 m.

We assume quadratic bottom friction with coefficient 2.5e-3. Wind stress \tau is defined by the formula \frac{\tau}{\rho} = C_{10} U_{10}^2, where C_{10} is taken from Wu (1982) JGRC C_{10}=(0.8 + 0.065 U_{10}) \times 10^{-3}. So for a U_{10}=10 m/s we have a wind stress of \frac{\tau}{\rho} = 0.145.

//#include "grid/cartesian1D.h"
#include "sandbox/Emily/storm_surge.h"

#define MAXLEVEL 8
#define MINLEVEL 4
#define ETAE 1e-8

double t1=7200.;

Here we can set standard parameters in Basilisk

int main()
{
#if QUADTREE
  // 32^2 grid points to start with
  N = 1 << MINLEVEL;
#else // Cartesian
  // 1024^2 grid points
  N = 1 << MAXLEVEL;
#endif

Here we setup the domain geometry. For the moment Basilisk only supports square domains. This case uses metres east and north. We set the size of the box L0 and the coordinates of the lower-left corner (X0,Y0). In this case we are assuming a square ‘pool’ of length 1000 m.

  // the domain is
  L0 = 1000.;
  X0 = -L0/2.;
  Y0 = -L0/2.;

G is the acceleration of gravity required by the Saint-Venant solver. This is the only dimensional parameter..

  G = 9.81;
  run();
}

Adaptation

Here we define an auxilliary function which we will use several times in what follows. Again we have two #if...#else branches selecting whether the simulation is being run on an (adaptive) quadtree or a (static) Cartesian grid.

We want to adapt according to two criteria: an estimate of the error on the free surface position – to track the wave in time – and an estimate of the error on the maximum wave height hmax – to make sure that the final maximum wave height field is properly resolved.

We first define a temporary field (in the automatic variable η) which we set to h+z_b but only for “wet” cells. If we used h+z_b everywhere (i.e. the default \eta provided by the Saint-Venant solver) we would also refine the dry topography, which is not useful.

int adapt() {
#if QUADTREE
  scalar eta[];
  foreach()
    eta[] = h[] > dry ? h[] + zb[] : 0;
  boundary ({eta});

  astats s = adapt_wavelet ({eta}, (double[]){ETAE},
			    MAXLEVEL, MINLEVEL);
  fprintf (stderr, "# refined %d cells, coarsened %d cells\n", s.nf, s.nc);
  return s.nf;
#else // Cartesian
  return 0;
#endif
}

event initiate(i=0)
{

The initial still water surface is at z=0 so that the water depth h is…

  foreach() {
    zb[] = -2;
    h[] = max(0., - zb[]);
    ts.x[] = 0;
    ts.y[] = 0;
  }
  boundary ({h,zb}); 
}

We want the simulation to stop when we are close to steady state. To do this we store the h field of the previous timestep in an auxilliary variable hn.

scalar hn[];

event init_hn (i = 0) {
  foreach() {
    hn[] = h[];
  }
}

Every timestep we check whether h has changed by more than 10^-8. If it has not, the event returns 1 which stops the simulation. We also output running statistics to the standard error.

event logfile (i++; i <= 100000) {
  double dh = change (h, hn);
  if ( (t> t1 && dh < 1e-8) || i==100000) {
  foreach()
    fprintf (stderr, "%g %g\n", x, h[]);
    return 1; /* stop */
  }
  stats s = statsf (h);
  norm n = normf (u.x);
  if (i == 0)
    fprintf (stderr, "t i h.min h.max h.sum u.x.rms u.x.max dh  dt\n");
  fprintf (stderr, "%12.8g %d %12.8g %12.8g %12.8g %12.8g %12.8g %12.8g %12.8g\n", t, i,
	    s.min, s.max, s.sum, n.rms, n.max, dh, dt);
}

We also use a simple implicit scheme to implement quadratic bottom friction i.e. \frac{d\mathbf{u}}{dt} = - C_f|\mathbf{u}|\frac{\mathbf{u}}{h} with C_f=2.5 \times 10^{-3}.

Also assume that we have a constant wind blowing in the x direction

event source (i++) {
  double ramp = t < t1 ? t/t1 : 1.;
  foreach() {
    ts.x[] = ramp*1.7e-4;
    ts.y[] = 0;
    double a_inv = (h[] < dry ? 0. : h[]/(h[] + 2.5e-3*dt*norm(u)));
    //double a = h[] < dry ? HUGE : 1. + 2.5e-3*dt*norm(u)/h[];
    foreach_dimension()
      u.x[] = u.x[] * a_inv;
  }
  boundary ({h,u});
}

Every 5 minutes, the h, z_b and hmax fields are interpolated bilinearly onto a n x n regular grid and written on standard output.

event snapshots (t += 300) {
  scalar ux[], uy[];
  foreach(){
    ux[]=u.x[];
    uy[]=u.y[];
  }
  printf ("%% file: t-%g\n", t);
  output_field ({h, eta, zb, ux, uy}, stdout, n = 1 << MAXLEVEL, linear = true);
}

event movies (t+=60) {
  static FILE * fp = NULL;
  if (!fp) fp = popen ("ppm2mpeg > eta.mpg", "w");
  scalar m[], etam[];
  foreach() {
    etam[] = eta[]*(h[] > dry);
    m[] = etam[] - zb[];
  }
  boundary ({m, etam});
  output_ppm (etam, fp, min = -0.005, max = 0.005 , n = 512, linear = true);
  //  output_ppm (etam, fp, n = 512, linear = true);
#if 0
  static FILE * fp1 = NULL;
  if (!fp1) fp1 = popen ("ppm2mpeg > level.mpg", "w");
  scalar l = etam;
  foreach()
    l[] = level;
  output_ppm (l, fp1, min = MINLEVEL, max = MAXLEVEL, n = 512);
#endif
}

Adaptivity

We apply our adapt() function at every timestep.

event do_adapt (i++) adapt();