sandbox/M1EMN/Exemples/higuera.c

    Viscous hydraulic Jump

    We want to reproduce the hydraulic jump of Higuera (1994) using Audusse et al (2011) multilayer algorithm.

    #include "grid/cartesian1D.h"
#include "saint-venant.h"
double Cf;

int main() {
  X0 = 0.14;
  L0 = 1. - X0;
  G  = 1.;
  N  = 128*2;

    one layer Poiseuille

      nl = 1;
  Cf = 3;  
  run();

    one layer Watson

      Cf = 2.2799;
  run();

    multi layer Higuera

      nl = 25;
  nu = 1.;
  run();
}

    We impose boundary condition for h and \eta.

    h[left] = dirichlet (.2);
eta[left] = dirichlet (.2);

h[right] = dirichlet (0.0);
eta[right] = dirichlet (0.0);

    Initialization

    We define a scalar field hc to check for convergence on h.

    scalar hc[];

event init (i = 0) {

    We set a constant velocity at the inlet and a free outlet.

      for (vector u in ul) {
    u.n[left] = 5.90;
    u.n[right] = neumann(0.);
  }

    We initialize h and hc.

      foreach()
    hc[] = h[] = 0.2;
}

    friction, only in the case of one layer

    event frotte (i++) {
  if(nl==1){
   foreach()
  {    
    double ff = h[] < dry ? HUGE : (1. + Cf*dt/h[]/h[]);
    u.x[] /= ff;
    }
  }
}

    We check for convergence.

    event logfile (t += 0.1; i <= 50000) {
  double dh = change (h, hc);
  printf ("%g %g\n", t, dh);
  if (i > 0 && dh < 1e-5)
    return 1;
}

    Output

    We print the elevation and the stress, C_f u/h or \partial u/\partial z|_0.

    event output (t = end) {
  vector u0 = ul[0];
  double tau ;
 
  foreach(){
     if(nl==1){tau = Cf*(u.x[]/h[]);}else{tau = 2.000*u0.x[]/(h[]/nl);}
    fprintf (stderr, "%g %g %g\n", x, eta[], tau);
  }
  fprintf (stderr, "\n\n");  
}

    Run

    compile and run:

     
qcc higuera.c -o higuera -lm
./higuera 2> log

make higuera.tst  
make higuera/plots
make higuera.c.html

    Results

     
plot[][:2]  'log' u 1:2 t 'h' w l,'' u 1:3 t 'tau' w l,0 not w l linec -1
    raw plot. (script) 
    set output 'f.png'
X0 = 169
X1 = 604
Y0 = 222.24
Y1 = 528
unset tics
plot [0:][0:605] '../Img/higueragraph.png' binary filetype=png with rgbimage not, \
  'log' i 2 u (X0+$1*(X1-X0)):($2/2*(Y1-Y0)+Y0) t 'h' w l,\
  '' i 2 u (X0+$1*(X1-X0)):($3/15*(Y1-Y0)+Y0) t 'tau' w l
    Comparison with Figure 2 of Higuera (1994). (script)

    We compare several solutions, the jump appears sooner with Poiseuille than with Watson.

    reset
set xlabel "x"
set ylabel "h"
set y2label "{/Symbol t}_b"
set y2range [-1:30]
set y2tics border out 
H= .0

  p [0:][-.1:3] 0 not w l linec -1,1.813*(x<.15? x:NaN) not linec -1,\
            (x>.35? (H+(12*(1-x))**.25):NaN)  not linec -1,\
              'log' i 0 u 1:2 t'P' w l ls 1,'' i 0 u 1:($3) not w l ls 1 axes x1y2,\
                   '' i 1 u 1:2 t'W' w l ls 2,'' i 1 u 1:($3) not w l ls 2 axes x1y2,\
                 '' i 2 u 1:2 t'ML' w l ls 3,'' i 2 u 1:($3) not w l ls 3 axes x1y2
    Comparison with one layer (two closures, Red Poiseuille, Green Watson) and multilayer (blue) (script)

    The same in two graphs

    reset
set xlabel "x"
set ylabel "h" 
H= .0
 set multiplot layout 2,1 title "Jump"
  p [0:][-.1:3.5] 0 not w l linec -1,1.813*(x<.15? x:NaN) not linec -1,\
              'log' i 0 u 1:2 t'P' w l ls 1,\
                   '' i 1 u 1:2 t'W' w l ls 2,\
                 '' i 2 u 1:2 t'ML' w l ls 3
 set ylabel "{/Symbol t}_b"                
   p [0:][-1:25]0 not w l linec -1,\
              'log'   i 0 u 1:($3) t'P' w l ls 1,\
                   '' i 1 u 1:($3) t'W' w l ls 2  ,\
                 ''   i 2 u 1:($3) t'ML' w l ls 3  
  
  unset multiplot
    Comparison with one layer (two closures, Red Poiseuille, Green Watson) and multilayer (blue) (script)

    Bibliography