Conservative energy flux of a polytropic ideal gas ($E=\frac{1}{2}\rho u^2+\frac{p}{\gamma -1},F_E=u_x(E+p)$)

For an unmagnetized ideal gas without gravity, the pressure equation gives internal-energy balance ${\partial }_{t}e+{\partial }_x(u_{x}e)=-p{\partial }_x{u}_x$, where $e=p/(\gamma -1)$. The momentum equation gives kinetic-energy balance with source $-u_x{\partial }_{x}p$. Their sum is conservative with the displayed total-energy flux. Tangential velocities contribute to kinetic energy even when all derivatives are in the normal direction. Smooth polytropic flow advects its entropy parameter; a physical shock need not preserve that parameter across the discontinuity.