Euler equations in primitive form for a real gas For an ideal gas, it is relatively easy to express the Euler equations in primitive form (variables $\rho$, $u$, $p$), starting from their expression in conservative variables ($\rho$, $\rho u$, $E$).
I did not find any example of such derivation for a general real gas, governed by any equation of state. Is it possible to express the Euler equations in primitive form for any (unknown) real gas (involving the speed of sound somewhere)?
 A: Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then
$$u_t=\rho^{-1}((\rho u)_t-u\rho_t)$$
and$$i_t=\rho^{-1}[(\rho i+\rho u^2/2)_t-(i+u^2/2)\rho_t-\rho uu_t]$$
where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.
A: In general, you can always convert between any two variable sets using the Jacobian that connects them. For example, if we define our conservative variable vector as:
$$ \mathbf{W} = \lbrace \rho, \rho u, \rho E \rbrace^T $$
and our primitive variable vector as:
$$ \mathbf{Q} = \lbrace \rho, u, p \rbrace^T $$
then we want to find the mapping that converts $\mathbf{W}$ into $\mathbf{Q}$. Abusing math notation in ways that make mathematicians cringe, we can say that:
$$ \partial \mathbf{W} = \frac{\partial\mathbf{W}}{\partial\mathbf{Q}} \partial \mathbf{Q} $$
where $\partial \mathbf{W} / \partial \mathbf{Q}$ is the Jacobian matrix. This results in a matrix, which for our system here is:
$$ 
\frac{\partial \mathbf{W}}{\partial \mathbf{Q}} = 
\begin{bmatrix} 
1 & 0 & \frac{\partial \rho}{\partial p} \\
u & \rho & \frac{\partial \rho}{\partial p} u \\
E+\rho \frac{\partial E}{\partial \rho} & \frac{\partial E}{\partial u} & \frac{\partial \rho}{\partial p} E + \rho \frac{\partial E}{\partial p}
\end{bmatrix}
$$
where the derivatives of one thermodynamic variable with respect to another, $\partial \rho /\partial p, \partial E/\partial \rho, \partial E/\partial p$, come from your equation of state and can be computed for any equation of state that is analytical.
You can then plug this into the Euler equations:
$$ 
\begin{aligned}
\frac{\partial \mathbf{W}}{\partial t} &+ \frac{\partial \mathbf{F}\left(\mathbf{W}\right)}{\partial x} = 0 \\
\rightarrow \frac{\partial \mathbf{W}}{\partial \mathbf{Q}} \frac{\partial \mathbf{Q}}{\partial t} &+ \frac{\partial \mathbf{F}\left(\mathbf{W}\right)}{\partial x} = 0
\end{aligned}$$
where the fluxes are still expressed in terms of the conservative variables. If you wanted to derive governing equations for the primitives entirely, there are some additional steps to do to convert the fluxes to primitive variables through the flux Jacobian, which is another matrix constructed using derivatives of the flux vector with respect to the conservative variable vector. I'll leave that as an exercise for the reader.
You can find the Jacobians for a real gas system in pressure-temperature primitive variables in the appendix of this paper, including the flux Jacobians.
