The Euler equations

The behaviour of an ideal compressible gas is described by the Euler equations:

$${\partial \rho \over \partial t} + {\bf\nabla}\cdot\rho{\bf v}=0$$ (1)

$${\partial \rho{\bf v}\over \partial t} + {\bf\nabla}\cdot\rho{\bf vv} + {\bf\nabla}p = 0$$ (2)

$${\partial e \over \partial t} + {\bf\nabla}\cdot(e+p){\bf v}=0\;,$$ (3)

with

$$e={1\over 2}\rho v^2+e_{\rm int}\;,$$ (4)

the internal energy of the gas.

Reducing these to one spatial dimension and using cartesian coordinates one can write them as

$${\partial {\bf W}\over\partial t}+{\partial {\bf F}\over\partial x}=0\;,$$ (5)

with ${\bf W}$ the so called state vector ($\rho$, $\rho v$, $e$)$^{\rm T}$, and ${\bf F}$ the flux vector ($\rho v$, $\rho v^2 + p$, $(e+p)v$$^{\rm T}$). The elements of the state vector are also called the conserved quantities since the Euler equations basically say that mass, momentum and energy are conserved. The variables $\rho$, $v$, and $p$ are often referred to as the primitive variables.

The Euler equations are not complete without an equation of state. We choose an ideal gas for which

$$e_{\rm int}=p/(\gamma -1)\;,$$ (6)

where $\gamma=5/3$ for a monatomic gas, and

$$p={\rho k T \over \mu m_{\rm H}}\;,$$ (7)

with $\mu$ the mean molecular weight.

A more general form of the Euler equations is

$${\partial {\bf W}\over\partial t}+{\partial {\bf F}\over\partial \xi}={\bf S}\;,$$ (8)

in which $\xi$ now is an arbitrary spatial coordinate and ${\bf S}$ is the source vector. ${\bf S}$ can be split into two parts: geometrical source terms which arise in the case of non-cartesian coordinates, and physical source terms such as radiative heating and cooling, gravitation, etc.

We will be using the spherical coordinate $r$. For this the momentum equation is

$${\partial \rho v \over \partial t}+{1\over r^2}{\partial \over \partial r} (r^2\rho v^2) =-{\partial p \over \partial r}\;,$$ (9)

and similar expressions can be written down for the continuity and energy equations. To write this in the form of Eq. (8) we take

$${\bf W} = r^2(\rho,\rho v,e)^{\rm T}$$ (10)

$${\bf F} = r^2(\rho v,\rho v^2+p,(e+p)v )^{\rm T}$$ (11)

$${\bf S} = (0,2rp,0)=r^2(0,2p/r,0)^{\rm T}\;.$$ (12)

${\bf W}$ is still the vector of conserved quantities vector since it is total mass, momentum and energy which are conserved, not their densities. In spherical coordinates a volume element is proportional to $r^2$ (really $r^2\sin{\theta}$ but we are not considering the $\theta$ coordinate here), so by multiplying the densities by the position dependent volume factor $r^2$, one obtains the conserved quantities.

One sees that choosing a non-cartesian coordinate leads to the appearance of a geometrical source term. There are actually several ways in which to split off the geometrical source terms, but here we choose the one shown in Eqs. (10)-(12).