Euler Equations, Sod shock tube & conservation

[Antigravity324]

Is momentum conserved?
I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be conserved wrt time. Density and total energy are indeed conserved, however, momentum is not.
The Euler equations are given by
$$\begin{bmatrix} \rho \\ \rho u \\ E \end{bmatrix}_t + \begin{bmatrix} \rho u \\ \rho u^2 + p \\ u(E +p) \end{bmatrix}_x = 0$$
where pressure is related to the conserved quantities by
$$p = (\gamma - 1)(E - 0.5 \rho u^2)$$
The Sod shock problem splits the domain into two regions separated by a density and pressure discontinuity with initial velocity zero. That is, $$\rho_L = 1, \rho_R = .125; p_L=1, p_R =.1; u_L=u_R =0$$ These initial conditions imply that ρu=0, therefore, momentum should be zero throughout the simulation.

The solution profiles are well known and can be found here or here. We see that ρ >0 and u > 0, therefore, there is no way for momentum ρu = 0 (which it should be from the intial conditions). As a result, I do not even see why it is reasonable to expect that momentum would be conserved. A paper by Sod himself which surveys some methods for solution, on page 20, list a table which shows momentum is not conserved, but no reason why. I generally do not work in this area, so maybe I am missing something basic. Can anyone shed some light on this? Thanks!

 

[Nate810]

No, momentum is not conserved. This is because the Euler equations are only valid for ideal fluids, which do not take into account viscous or dissipative forces that act to reduce the momentum of a system. As the fluid flows and encounters obstacles or other fluid elements, some of its momentum is dissipated and converted into heat. Over time, this can lead to a decrease in the total momentum of the system.