src/compressible/NASG.h
The Noble-Abel Stiffened-Gas (NASG) Equation Of State
This EOS is typically used in combination with the two-phase compressible solver with thermal effects.
The general form of the NASG EOS (Le Métayer & Saurel, 2016) is \displaystyle \rho_i e_i = \frac{p_i + \Gamma_i \Pi_i}{\Gamma_i - 1}(1 - \rho_i b_i) + \rho_i q_i with \rho_i, e_i and p_i the densities, internal energies and pressures of each phase.
These are the coefficients of the NASG EOS for each phase.
double gamma1 = 1.4 [0], gamma2 = 1.4 [0], PI1 = 0., PI2 = 0.;
double b1 = 0., b2 = 0.;
double q1 = 0., q2 = 0.;
double cv1 = 0., cv2 = 0.;
Sound speed
In mixture cells, this function returns the maximum between the speeds in both phases.
double sound_speed (Point point)
{
double fc = clamp (f[],0.,1.);
double c2speed1 = 0., c2speed2 = 0.;
double Ek = 0.;
foreach_dimension()
Ek += sq(q.x[]);
Ek /= 2.*(frho1[] + frho2[]);
if (fc > 0.00001) {
double fe1 = fE1[] - fc*Ek;
double p = fe1/fc*(gamma1 - 1.) - gamma1*PI1;
c2speed1 = fc*gamma1*(p + PI1)/frho1[];
}
if (fc < 0.99999) {
double fe2 = fE2[] - (1. - fc)*Ek;
double p = fe2/(1. - fc)*(gamma2 - 1.) - gamma2*PI2;
c2speed2 = (1. - fc)*gamma2*(p + PI2)/frho2[];
}
return sqrt (max (c2speed1, c2speed2));
}
Average pressure
#define PIGAMMA double invgammaavg = (fc - frho1[]*b1)/(gamma1 - 1.) + \
(1. - fc - frho2[]*b2)/(gamma2 - 1.), \
PIGAMMAavg = (fc - frho1[]*b1)*PI1*gamma1/(gamma1 - 1.) + frho1[]*q1 + \
(1. - fc - frho2[]*b2)*PI2*gamma2/(gamma2 - 1.) + frho2[]*q2
double average_pressure (Point point)
{
double fc = clamp (f[],0.,1.);
PIGAMMA;
double Ek = 0.;
foreach_dimension()
Ek += sq(q.x[]);
Ek /= 2.*(frho1[] + frho2[]);
return (fE1[] + fE2[] - Ek - PIGAMMAavg)/invgammaavg;
}
Bulk compressibility of the mixture
i.e. \rho c^2.
double bulk_compressibility (Point point)
{
double fc = clamp (f[],0.,1.);
// Arithmetic mean of the mixture compressibility
double rhoc2v1 = fc ? gamma1*(p[] + PI1)/(1. - frho1[]*b1/fc) : 1.;
double rhoc2v2 = (1. - fc) ? gamma2*(p[] + PI2)/(1. - frho2[]*b2/(1. - fc)) : 1.;
return fc*rhoc2v1 + (1. - fc)*rhoc2v2;
}
Internal energy
double internal_energy (Point point, double fc)
{
PIGAMMA;
return p[]*invgammaavg + PIGAMMAavg;
}
Average temperature
double average_temperature (Point point, double p)
{
double fc = clamp (f[],0.,1.);
double rhocpmcvavg = (cp1 - cv1)*frho1[] + (cp2 - cv2)*frho2[];
double const1 = (fc - frho1[]*b1) + (1. - fc - frho2[]*b2);
double const2 = (fc - frho1[]*b1)*PI1 + (1. - fc - frho2[]*b2)*PI2;
return (const1*p + const2)/rhocpmcvavg;
}
Thermal expansion coefficient
double thermal_expansion (Point point)
{
double fc = clamp (f[],0.,1.);
return (1. - fc)*(gamma2 - 1.)*cv2/((gamma2 - 1.)*cv2*Ts[] + b2*(ps[] + PI2));
}
See also
References
[metayer2016] |
Olivier Le Métayer and Richard Saurel. The Noble–Abel stiffened-gas equation of state. Physics of Fluids, 28(4), 2016. |