Source of a river

    In this example we impose a variable flow rate for a “source” located inside the computation domain.

    #include "saint-venant.h"
    #include "discharge.h"

    The domain is 10 metres squared, centered on the origin. Time is in seconds.

    #define LEVEL 8
    int main()
      size (10.);
      origin (- L0/2., - L0/2.);
      G = 9.81;
      N = 1 << LEVEL;

    We start with a dry riverbed, so that the problem does not have a natural timescale the Saint-Venant solver can use. We set a maximum timestep to set this timescale.

      DT = 1e-2;

    Boundary conditions

    We create a new boundary for the source, with a Neumann condition for the normal velocity (i.e. an inflow).

    bid source;
    u.n[source] = neumann(0);

    The flow rate varies in time and is set by computing the the elevation \eta_s of the water surface necessary to match this flow rate.

    double etas;
    event inflow (i++) {
      etas = eta_b (0.1*(1.1 - cos(4.*pi*t)), source);
      h[source] = max (etas - zb[], 0.);
      eta[source] = max (etas - zb[], 0.) + zb[];

    Initial conditions

    The river bed is a single valley. The source is created by masking and is a narrow slot located at the head of the valley.

    event init (i = 0)
      mask (fabs(x) < 0.5 && fabs(y - 3.5) < Delta/2. ? source : none);
        zb[] = (- cos(x) + y)/2.;


    We compute the time-derivative of the total water volume (i.e. the net flow rate), and make a GIF movie.

    event logfile (i++; t <= 2.) {
      static double volo = 0., to = 0.;
      double vol = statsf(h).sum;
      if (i > 0)
        fprintf (stderr, "%g %.6f %g %g\n", t, (vol - volo)/(t - to), vol, etas);
      volo = vol, to = t;
    event output (i += 5) {
      output_ppm (h, min = 0, max = 0.05, file = "source.gif");


    set key top left
    set xlabel 'Time'
    set ylabel 'Flow rate'
    plot './log' u 1:2 w l t 'obtained', 0.1*(1.1-cos(4.*pi*x)) w l t 'imposed'
    Evolution of the flow rate (script)

    Evolution of the flow rate (script)

    Evolution of the water level

    Evolution of the water level

    See also