src/test/dispersion_gravitocap.c
Dispersion relation of gravito-capillary waves
This test is an adaptation of the dispersion.c test to gravito-capillary waves. The dependence of the oscillation frequency of a gravito-capillary wave of wavenumber k on the water depth h is measured. The corresponding phase velocity c=\omega/k is then compared to the exact linear dispersion relation c_e = \sqrt{\left(\frac{g}{k} + \frac{\gamma k}{\rho}\right)\tanh(kh)}
We use a 1D grid and the implicit multi-layer solver with surface tension.
#include "grid/multigrid1D.h"
#include "layered/hydro-tension.h"
#include "layered/remap.h"
#include "layered/nh.h"
double emax = 1.;
double h0 = 10.;
double k = 1.;
double L = 1;
double c0 = 1;We initialise a wave of small relative amplitude (10^{-3}) to be in the linear regime.
The code below locates the time of zero “up-crossing” of the amplitude in the middle of the wave and computes the corresponding averaged wave period.
double Tm = 0., Es = 1., Ee = 1.;
int nm = 0;
event logfile (i++) {
double pe = 0., ke = 0.;
foreach() {
double H = 0;
foreach_layer(){
ke += h[]*Delta*(sq(u.x[]) + sq(w[]))/2.;
H += h[];
}
pe += sigma[]*sq(eta[1]-eta[-1])/(8.*Delta);
pe += Delta*G*sq(H - h0)/2.;
}
double dh = (interpolate (eta, L0/2., linear = false) - h0)/h0;
static double told = 0., hold = 0., eold = 0., tsold = -1;
if (i == 0) {
Tm = 0., nm = 0;
Es = 0.;
told = 0., hold = 0., tsold = -1;
}
else {
if (i > 1 && hold < 0. && dh > 0.) {
// this is an (upward) zero-crossing at time ts
double ts = told - hold*(t - told)/(dh - hold);
Ee = eold - hold*(ke + pe - eold)/(dh - hold);
if (Es == 0.)
Es = Ee;
if (tsold > 0. && nm < 4) {
// we average the periods
Tm += ts - tsold;
nm++;
}
tsold = ts;
}
told = t;
hold = dh;
eold = ke + pe;
}
}After ten (exact) wave periods, we stop and dump the solution.
event dumps (t = 10.*L/c0) // ten wave periods
{
FILE * fp = fopen ("prof", "w");
fprintf (fp, "x");
for (scalar s in all)
fprintf (fp, " %s", s.name);
fprintf (fp, "\n");
foreach() {
fprintf (fp, "%g", x);
for (scalar s in all)
fprintf (fp, " %g", s[]);
fprintf (fp, "\n");
}
fprintf (fp, "\n");
fclose (fp);
}We output the wave period and relative energy variation and compute
the relative error emax.
event end (t = end)
{
emax = (L/c0)/(Tm/nm);
fprintf (stderr, "%g %g %g %g %g %g %d %d %d\n",
k, h0, c0, L/(Tm/nm), emax, (Ee - Es)/Es, mgp.i, mgp.nrelax, i);
fflush (stderr);
}We run for several numbers of layers and many water depths, stopping when the error becomes too large.
int main()
{
periodic (right);
DT = HUGE [0]; // dimensionless time
N = 128;
NITERMAX = 1000;
linearised = true;
TOLERANCE = 1e-8;
CFL_H = 0.5;
G = 1.;
for (nl = 1; nl <= 4; nl*=2){
char name[80];
sprintf (name, "log-%d", nl);
freopen (name, "w", stderr);
emax = 1.;
for (k = 0.01; k <= 10. && emax > 0.96; k *= 1.3) {
c0 = sqrt(tanh(k*h0)*(1.*k + G/k));
L = 2.*pi/k;
size (L);
run();
}
}
}The discrete dispersion relations can be computed using this Maxima script.
g = 1.
k = 1.
omega1_keller(h_0)=sqrt((4*h_0*g*k**2)/(h_0**2*k**2+4))
omega2_keller(h_0,h_1)=sqrt(((4*h_0*h_1**2+4*h_0**2*h_1)*g*k**4+(16*h_1+16*h_0)*g*k**2)/(h_0**2*h_1**2*k**4+(4*h_1**2+16*h_0*h_1+4*h_0**2)*k**2+16))
omega3_keller(h_0,h_1,h_2)=sqrt((((4*h_0*h_1**2+4*h_0**2*h_1)*h_2**2+4*h_0**2*h_1**2*h_2)*g*k**6+((16*h_1+16*h_0)*h_2**2+(16*h_1**2+64*h_0*h_1+16*h_0**2)*h_2+16*h_0*h_1**2+16*h_0**2*h_1)*g*k**4+(64*h_2+64*h_1+64*h_0)*g*k**2)/(h_0**2*h_1**2*h_2**2*k**6+((4*h_1**2+16*h_0*h_1+4*h_0**2)*h_2**2+(16*h_0*h_1**2+16*h_0**2*h_1)*h_2+4*h_0**2*h_1**2)*k**4+(16*h_2**2+(64*h_1+64*h_0)*h_2+16*h_1**2+64*h_0*h_1+16*h_0**2)*k**2+64))
omega4_keller(h_0,h_1,h_2,h_3)=sqrt(((((4*h_0*h_1**2+4*h_0**2*h_1)*h_2**2+4*h_0**2*h_1**2*h_2)*h_3**2+4*h_0**2*h_1**2*h_2**2*h_3)*g*k**8+(((16*h_1+16*h_0)*h_2**2+(16*h_1**2+64*h_0*h_1+16*h_0**2)*h_2+16*h_0*h_1**2+16*h_0**2*h_1)*h_3**2+((16*h_1**2+64*h_0*h_1+16*h_0**2)*h_2**2+(64*h_0*h_1**2+64*h_0**2*h_1)*h_2+16*h_0**2*h_1**2)*h_3+(16*h_0*h_1**2+16*h_0**2*h_1)*h_2**2+16*h_0**2*h_1**2*h_2)*g*k**6+((64*h_2+64*h_1+64*h_0)*h_3**2+(64*h_2**2+(256*h_1+256*h_0)*h_2+64*h_1**2+256*h_0*h_1+64*h_0**2)*h_3+(64*h_1+64*h_0)*h_2**2+(64*h_1**2+256*h_0*h_1+64*h_0**2)*h_2+64*h_0*h_1**2+64*h_0**2*h_1)*g*k**4+(256*h_3+256*h_2+256*h_1+256*h_0)*g*k**2)/(h_0**2*h_1**2*h_2**2*h_3**2*k**8+(((4*h_1**2+16*h_0*h_1+4*h_0**2)*h_2**2+(16*h_0*h_1**2+16*h_0**2*h_1)*h_2+4*h_0**2*h_1**2)*h_3**2+((16*h_0*h_1**2+16*h_0**2*h_1)*h_2**2+16*h_0**2*h_1**2*h_2)*h_3+4*h_0**2*h_1**2*h_2**2)*k**6+((16*h_2**2+(64*h_1+64*h_0)*h_2+16*h_1**2+64*h_0*h_1+16*h_0**2)*h_3**2+((64*h_1+64*h_0)*h_2**2+(64*h_1**2+256*h_0*h_1+64*h_0**2)*h_2+64*h_0*h_1**2+64*h_0**2*h_1)*h_3+(16*h_1**2+64*h_0*h_1+16*h_0**2)*h_2**2+(64*h_0*h_1**2+64*h_0**2*h_1)*h_2+16*h_0**2*h_1**2)*k**4+(64*h_3**2+(256*h_2+256*h_1+256*h_0)*h_3+64*h_2**2+(256*h_1+256*h_0)*h_2+64*h_1**2+256*h_0*h_1+64*h_0**2)*k**2+256))
set xlabel 'kH'
set ylabel 'c/c_e'
set key bottom left
set xr [0.01:10]
set yr [0.96:1.04]
set logscale x
set grid
plot omega1_keller(10*x)/sqrt(tanh(10*x)) t '1 layer', \
'log-1' u 1:5 t '1 layers' pt 1 lt 2, \
omega2_keller(10*x/2.,10*x/2.)/sqrt(tanh(10*x)) t '2 layers', \
'log-2' u 1:5 t '2 layers' pt 2 lt 4, \
omega4_keller(10*x/4.,10*x/4.,10*x/4.,10*x/4.)/sqrt(tanh(10*x)) t '4 layers', \
'log-4' u 1:5 t '4 layers' pt 3 lt 6, \
'log-8' u 1:5 t '8 layers' pt 4 lt 8, \
'log-16' u 1:5 t '16 layers' pt 6 lt 10
set ylabel 'c'
set xlabel 'kH'
set logscale x
set xr [*:*]
set yr [*:*]
plot 'log-1' u 1:4 t '1 layers' pt 1 lt 2, \
'log-2' u 1:4 t '2 layers' pt 2 lt 4, \
'log-4' u 1:4 t '4 layers' pt 3 lt 6, \
'log-8' u 1:4 t '8 layers' pt 4 lt 8, \
sqrt(tanh(10.*x)*(x + 1./x)) t 'dispersion relation'
set ylabel 'energy loss'
set xlabel 'kH'
set xr [*:*]
set yr [-0.1:0.1]
plot 'log-1' u ($1):($6*10.) t '1 layers' pt 1 lt 2, \
'log-2' u ($1):($6*10.) t '2 layers' pt 2 lt 4, \
'log-4' u ($1):($6*10.) t '4 layers' pt 3 lt 6, \
'log-8' u ($1):($6*10.) t '8 layers' pt 4 lt 8
