void
NaClFluidProperties::e_dpT(Real pressure, Real temperature, Real & e, Real & de_dp, Real & de_dT) const
{
Real h, dh_dp, dh_dT;
h_dpT(pressure, temperature, h, dh_dp, dh_dT);
Real rho, drho_dp, drho_dT;
rho_dpT(pressure, temperature, rho, drho_dp, drho_dT);
e = h - pressure / rho;
de_dp = dh_dp + pressure * drho_dp / rho / rho - 1.0 / rho;
de_dT = dh_dT + pressure * drho_dT / rho / rho;
}
void
NaClFluidProperties::rho_e_dpT(Real pressure,
Real temperature,
Real & rho,
Real & drho_dp,
Real & drho_dT,
Real & e,
Real & de_dp,
Real & de_dT) const
{
rho_dpT(pressure, temperature, rho, drho_dp, drho_dT);
e_dpT(pressure, temperature, e, de_dp, de_dT);
}
Real
StiffenedGasFluidProperties::beta(Real pressure, Real temperature) const
{
// The volumetric thermal expansion coefficient is defined as
// 1/v dv/dT)_p
// It is the fractional change rate of volume with respect to temperature change
// at constant pressure. Here it is coded as
// - 1/rho drho/dT)_p
// using chain rule with v = v(rho)
Real rho, drho_dp, drho_dT;
rho_dpT(pressure, temperature, rho, drho_dp, drho_dT);
return -drho_dT / rho;
}
void
StiffenedGasFluidProperties::rho_e_dps(Real pressure,
Real entropy,
Real & rho,
Real & drho_dp,
Real & drho_ds,
Real & e,
Real & de_dp,
Real & de_ds) const
{
// compute rho(p, T(p,s)) and e(p, rho(p, T(p,s)))
this->rho_e_ps(pressure, entropy, rho, e);
// compute temperature
const Real aux =
(entropy - _q_prime + _cv * std::log(std::pow(pressure + _p_inf, _gamma - 1))) / _cv;
const Real T = std::pow(std::exp(aux), 1 / _gamma);
// dT/dp
const Real dT_dp = 1.0 / _gamma * std::pow(std::exp(aux), 1.0 / _gamma - 1.0) * std::exp(aux) /
std::pow(pressure + _p_inf, _gamma - 1.0) * (_gamma - 1.0) *
std::pow(pressure + _p_inf, _gamma - 2.0);
// dT/ds
const Real dT_ds =
1.0 / _gamma * std::pow(std::exp(aux), 1.0 / _gamma - 1.0) * std::exp(aux) / _cv;
// Drho/Dp = d/dp[rho(p, T(p,s))] = drho/dp + drho/dT * dT/dp
Real drho_dp_partial, drho_dT;
rho_dpT(pressure, T, rho, drho_dp_partial, drho_dT);
drho_dp = drho_dp_partial + drho_dT * dT_dp;
// Drho/Ds = d/ds[rho(p, T(p,s))] = drho/dT * dT/ds
drho_ds = drho_dT * dT_ds;
// De/Dp = d/dp[e(p, rho(p, T(p,s)))] = de/dp + de/drho * Drho/Dp
const Real de_dp_partial = 1.0 / ((_gamma - 1.0) * rho);
const Real de_drho = -(pressure + _gamma * _p_inf) / ((_gamma - 1.0) * rho * rho);
de_dp = de_dp_partial + de_drho * drho_dp;
// De/Ds = d/ds[e(p, rho(p, T(p,s)))] = de/drho * Drho/Ds
de_ds = de_drho * drho_ds;
}
void
IdealGasFluidPropertiesPT::rho_e_dpT(Real pressure,
Real temperature,
Real & rho,
Real & drho_dp,
Real & drho_dT,
Real & e,
Real & de_dp,
Real & de_dT) const
{
Real density, ddensity_dp, ddensity_dT;
rho_dpT(pressure, temperature, density, ddensity_dp, ddensity_dT);
rho = density;
drho_dp = ddensity_dp;
drho_dT = ddensity_dT;
Real energy, denergy_dp, denergy_dT;
e_dpT(pressure, temperature, energy, denergy_dp, denergy_dT);
e = energy;
de_dp = denergy_dp;
de_dT = denergy_dT;
}