15.2 Relativistic hydrodynamics (RHD)

15.2.1 Overview

FLASH provides support for solving the equations of special relativistic hydrodynamics (RHD) in one, two and three spatial dimensions.

Relativistic fluids are characterized by at least one of the following two behaviors: (i) bulk velocities close to the speed of light (kinematically relativistic regime), (ii) internal energy greater than or comparable to the rest mass density (thermodynamically relativistic regime). As can be seen from the equations in Sec:RHD_equations, the two effects become coupled by the presence of the Lorentz factor; as a consequence, transverse velocities do not obey simple advection equations. Under these circumstances, Newtonian hydrodynamics is not adequate and a correct description of the flow must take relativistic effects into account.

15.2.2 Equations

The motion of an ideal fluid in special relativity is described by the system of conservation laws

$$\frac{\partial }{\partial t}\left(\begin{array}{c} D \boldsymb... ... \boldsymbol{v} + p\boldsymbol{I} \boldsymbol{m} \end{array} \right) = 0 ,$$ (15.14)

where $D$, $\boldsymbol{m}$, $E$, $\boldsymbol{v}$ and $p$ define, respectively, the fluid density, momentum density, total energy density, three-velocity and pressure of the fluid. (15.14) is written in units of $c = 1$, where $c$ is the speed of light. The same convention is used throughout this section, and is also adopted in the FLASH code.

At present, only Cartesian (1, 2 and 3-D), 2-D cylindrical ($x=r$, $y=z$) and 1-D spherical (1-D, $x=r$) geometries are supported by FLASH. Gravity is not included, although it can be easily added with minor modifications.

An equation of state (Eos) provides an additional relation between thermodynamic quantities and closes the system of conservation laws ((15.14)). The current version of FLASH supports only the ideal equation of state, for which the specific enthalpy $h$ may be expressed as

$$h = 1 + \frac{\Gamma}{\Gamma - 1}\frac{p}{\rho}$$ (15.15)

where $\Gamma$ (constant) is the specific heat ratio and $\rho$ is the proper rest mass density. Causality ($c_s < c$) is preserved for $\Gamma < 2$. The specific heat ratio is specified as a runtime parameter ("gamma").

As in classical hydrodynamics, relativistic fluids may be described in terms of a state vector of conservative, $\boldsymbol{U} = (D, \boldsymbol{m}, E)$, or primitive, $\boldsymbol{V} = (\rho, \boldsymbol{v}, p)$, variables. The connection between the two sets is given by

$$\displaystyle D = \gamma \rho , \quad \displaystyle \boldsymbo... ...oldsymbol{v} , \quad \displaystyle E = \displaystyle \rho h \gamma^2 - p ,$$ (15.16)

where $\gamma = \big(1 - \boldsymbol{v}^2\big)^{-1/2}$ is the Lorentz factor. Notice that the total energy density includes the rest mass contribution. The inverse relation, giving $\boldsymbol{V}$ in terms of $\boldsymbol{U}$, is

$$\rho = \frac{D}{\gamma} , \quad \boldsymbol{v} = \frac{\boldsymbol{m}}{E + p} , \quad p = Dh\gamma - E .$$ (15.17)

This inverse map is not trivial due to the non-linearity introduced by the Lorentz factor $\gamma$; it can be shown, in fact, that (15.17) can be combined together to obtain the following implicit expression for $p$:

$$p = Dh\big(p,\tau(p)\big)\gamma(p) - E .$$ (15.18)

(15.18) has to be solved at least once per time step in order to recover the pressure from a set of conservative variables $\boldsymbol{U}$. Notice that $\tau = \tau(p)$ depends on the pressure $p$ through $\tau = \gamma(p)/D$ and that the specific enthalpy $h$ is, in general, a function of both $p$ and $\tau$, $h = h(p,\tau(p))$. The conversion routines are implemented in the rhd_conserveToPrimitive.F90 and rhd_primitiveToConserve.F90 source files.

15.2.3 Relativistic Equation of State

A variant version of the ideal gamma law Eos_wrapped.F90 routine is required by the RHD unit and is provided in Eos/EosMain/Gamma/RHD. In order to do so the unit requires a typical Config file which should look like this:

REQUIRES physics/Eos/EosMain/Gamma/RHD

For this specific purpose, the current RHD implementation supports MODE_DENS_EI (a default mode) and MODE_DENS_PRES only (but not MODE_DENS_TEMP) in making a Eos_wrapped call.

15.2.4 Additional Runtime Parameter

One additional runtime parameter used with the RHD unit is

Table 15.6: Additional parameters in the RHD unit.

Variable	Type	Default	Description
rhd_reconType	integer	1	Order of reconstruction scheme: 1 for piecewise liner; 2 for piecewise parabolic