# Advection of two fluids at two different pressures

This problem proposed by Johnsen and Colonius (JCP, 2006) tests the solver for the advection of an interface between two different ideal gases at uniform velocity and pressure quantifying the amplitude of spurious pressure and velocity oscillations induced by the method when advecting an interface with different material properties.

\displaystyle (\rho,u,p,\gamma)^T_L = (1, 0.5, 1/1.4, 1.2)^T \displaystyle (\rho,u,p,\gamma)^T_R = (10, 0.5, 1/1.4, 1.4)^T

By construction the method should keep the pressure and velocity uniform. During the advection step the energy is advected avoiding any diffusion and therefore both the provisional pressure and velocity are uniform and do not need to be corrected during the projection step.

#include "grid/multigrid.h"
#include "two-phase-compressible.h"

We set the problem parameters, size domain and boundary conditions

double rhoL = 1., rhoR = 10.;
double pL, pR;
double tend = 8;

int main() {

L0 = 2.;
X0 = Y0 = -L0/2.;

TOLERANCE = 1.e-6;

periodic (right);

pL = pR = 1./1.4;
gamma1 = 1.2;
gamma2 = 1.4;

We use 1st order upwind method for advection

  f.gradient = zero;

N = 128;
run();

}

The initial conditions are:

event init (i = 0) {

double u0 = 0.5;
foreach() {
f[] = (x < 0.);
p[] = f[]*pL + (1. - f[])*pR;
frho1[] = f[]*rhoL;
frho2[] = (1. - f[])*rhoR;
fE1[] = f[]*pL/(gamma1 - 1.) +  0.5*frho1[]*pow(u0,2);
fE2[] = (1.-f[])*pR/(gamma2 - 1.) +  0.5*frho2[]*pow(u0,2);
q.x[] = (frho1[] + frho2[])*u0;
q.y[] = 0.;
}

foreach_face(x)
uf.x[] = u0;

foreach_face(y)
uf.x[] = 0.;

boundary ((scalar *){q,frho1,frho2,p,fE1,fE2});

}

We output the field variables at the end of the simulation

event outputdata ( t = tend )
{

scalar perr[], uerr[];

foreach () {

perr[] = fabs(p[] - 1./1.4);
uerr[] = fabs(q.x[]/rho[] - 0.5);

double Ek = 0.;
foreach_dimension ()
Ek += pow(q.x[],2);

printf("%g %g %g %g %g %g %g \n", x, t, p[], rho[], q.x[]/(rho[]),f[],
(fE1[] + fE2[] - 0.5*Ek/rhov[])/(f[]/(gamma1-1.) + (1. - f[])/(gamma2 - 1.)));

}

assert ( (statsf(perr).sum/pow(L0,2) < 1.e-3) && (statsf(uerr).sum/pow(L0,2) < 1.e-3) );
}
set xlabel 'x'
set ylabel 'p'
set cblabel 't'
p "out" u 1:3 t 'Sim' w lp pt 6 Pressure profile (script)

reset
set xlabel 'x'
set ylabel '{/Symbol r}'
set cblabel 't'
p "out" u 1:4 t 'Sim' w lp pt 6 Density profile (script)

reset
set xlabel 'x'
set ylabel 'u'
set cblabel 't'
p "out" u 1:5 t 'Sim' w lp pt 6 Velocity profile (script)

reset
set xlabel 'x'
set ylabel 'VOF function'
set cblabel 't'
p "out" u 1:6 t 'f' w l VOF comp (script)