sandbox/wind_pool.c

    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 C_f=2.5 \times 10^{-3}. Wind stress \tau is defined by the formula $ = C_{10} U_{10}^2, $ where C_{10} is taken from Wu (1982) JGRC $C_{10}=(0.8 + 0.065 U_{10}) 10^{-3}. $ So for a U_{10}=10 m/s we have a wind stress of \frac{\tau}{\rho_{air}} = 0.145 m^2/s^2 and thus \frac{\tau}{\rho_{water}} = 1.7 \times 10^{-4} m^2/s^2.

    #include "grid/cartesian1D.h"
#include "saint-venant_ss.h"

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

    Here we can set standard parameters in Basilisk

    int main()
{
#if QUADTREE
  // 32^2 grid points to start with
  init_grid(1 << MINLEVEL);
#else // Cartesian
  // 1024^2 grid points
  init_grid(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 is 1000 m and the coordinates of the lower-left corner (X0,Y0) are (-500,-500). In this case we are assuming a square ‘pool’ of length 1000 m

      // the domain is
  size (1000.);
  origin(-L0/2.,-L0/2.);
     `G` is the acceleration of gravity required by the Saint-Venant
 solver. This is the only dimensional parameter..
      // acceleration of gravity in m/s^2
  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});

    We can now use wavelet adaptation on the list of scalars {η,hmax} with thresholds {ETAE,HMAXE}. The compiler is not clever enough yet and needs to be told explicitly that this is a list of doubles, hence the (double[]) (type casting).

    The function then returns the number of cells refined.

      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
}

    Initiation

    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}); 
}

    Stopping condition

    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 10 timesteps we check whether u 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+= 10; i <= 10000000) {
  double dh = change (h, hn);
  if ( (i > 100 && dh < 1e-8) || i==10000000) {
  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);
}

    Bottom Friction and Wind Forcing

    We also use a simple implicit scheme to implement quadratic bottom friction i.e. \displaystyle \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 of

    event source (i++) {
  double ramp = t < 12000. ? t/12000. : 1.;
  struct { double x, y; } ts = {1.7e-4,0};

  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});
}

    Output

    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) {
  printf ("# file: t-%g\n", t);
  //  output_field ({h, zb}, stdout, n = 1 << MAXLEVEL, linear = true);
  foreach()
    printf ("%10.4g %10.4g %10.8g %10.8g\n", x, y, h[], zb[]);
  printf ("\n");
}

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();