reflectiongaussian3.c

Transmission/reflection of a wave propagating across an interface between two fluids
- References

Transmission/reflection of a wave propagating across an interface between two fluids

In this test proposed by Denner et al. 2018 a linear wave propagating in an ideal gas is partially transmitted to another ideal gas with a different acoustic impedance.

#include "grid/multigrid1D.h"
#include "compressible/two-phase.h"
#include "compressible/Mie-Gruneisen.h"

Parameters of the problem.

double tend = 0.1;
double uper = 0.0001;
double freq = 4000.;
double uref = 347.8;
double p0, rho20, rho10;

int main()
{  
  
  CFLac = 0.25;

The EOS for an adiabatic perfect gas is defined by its polytropic coefficient \Gamma = \gamma = 1.4.

  gamma1 = 1.4;
  gamma2 = 1.667;
  rho20  = 0.164/1.157;
  rho10 = 1;;

  p0 = 1./gamma1;
  freq *= sqrt(gamma2/gamma1/rho20)/uref;

We perform a convergence study.

  for (N = 256; N <= 1024; N *= 2)
    run();
}

event init (i = 0)
{
  foreach() {
    double perturb = uper*exp(- sq((x - 0.4)*freq));
    f[] = (x > 0.5);
    p[] = p0 + perturb;
    frho1[] = f[]*rho10*(1. + perturb);
    frho2[] = (1. - f[])*rho20;
    q.x[] = (frho1[] + frho2[])*perturb*sqrt(gamma2*p[]/rho20);
    fE1[] = f[]*p[]/(gamma1 - 1.) + 0.5*sq(q.x[]/(frho1[] + frho2[]))*frho1[];
    fE2[] = (1. - f[])*p[]/(gamma2 - 1.) + 0.5*sq(q.x[]/(frho1[] + frho2[]))*frho2[];
  }
}

event endprint (t = tend) 
{
  foreach()
    fprintf (stderr, "%i %f %f\n", N, x, (p[] - p0)/1e-4);
}

ZR = 1.
ZL = 0.164/1.157*sqrt(1.667*1.157/0.164/1.4)
set ylabel '{/Symbol D}p/{/Symbol D}p_0'
set xlabel 'x'
set cblabel '{/Symbol s}/{/Symbol D}x'
set xrange[0.2:0.4]
p "log" u 2:3:(0.1*$1) not w l palette, (ZR-ZL)/(ZR+ZL) t 'theory' w l lc 0

Reflected wave (script)

set xrange[0.5:0.65]
p "log" u 2:3:(0.1*$1) not w l palette, 1./(ZR+ZL)*2*ZR t 'theory' w l lc 0

Transmitted wave (script)

References

[denner2018]

Fabian Denner, Cheng-Nian Xiao, and Berend G.M. van Wachem. Pressure-based algorithm for compressible interfacial flows with acoustically-conservative interface discretisation. Journal of Computational Physics, 367:192–234, 2018. [ DOI | http ]