# Sound speed on Navier-Stokes/Euler equations

I'd like to simulate strong shock (i.e., Rankine Hugoniot conditions) on inviscid condition, using the non-conservative Euler equations and I don't know if I should use the relation, $$c^2=\left(\frac{\partial p}{\partial\rho}\right)_s$$ for the speed of sound. Are we able to use this classical sound speed formula on Navier-Stokes (compressible viscious flow) equations?

• It's a relation based on the equation of state, can you explain why you think it wouldn't be valid for both? – Kyle Kanos Aug 6 '17 at 16:15
• @KyleKanos thank for your answer. Well, i'd like to simulate strong shock (Rankine Hugoniot) and i don't know if i should use or not this formula which includes the isentropic derivative on terms of Pressure and Density? Between, i'd like to use the non-conservative form of Euler equation. – Abdoulaye ndiongue Aug 6 '17 at 18:02
• If I'm not wrong, the Navier-Stokes equations are actually conservation of Energy, mass and momentum (besides equations of state), so if you add the corresponding terms (like work done in the energy equation, and so on), I don't think it's wrong. – FGSUZ Aug 6 '17 at 21:24

## 1 Answer

The speed of sound can be derived from Navier-Stokes equations (cf. Ron Maimon's answer here) or (equivalently) through Bernoulli's equation (cf. Genick Bar-Meir's Fundamentals of compressible Flow Mechanics online text); I've also seen the ideal gas form derived from thermodynamic principles, but cannot find a source at the moment.

So it appears that the speed of sound is independent of the choice of fluid dynamics pictures considered.

Anecdotally, I went through some of the academic-use hydrodynamics codes I have on hand (most of which are for conservative Eulerian hydrodynamics for astrophysical purposes) and the ones that had functions for the speed of sound (some did it in-place, which makes it harder to find using grep) used the explicit forms (e.g., $c_\text{ideal gas}^2=\gamma p/\rho$). So you should be on safe grounds here.