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Let's start with euler equations: https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)#Euler_equations

They do describe the system that I want, but they produce wave-like processes that have finite propagation speed and hence severely limit time stepping during numerical simulation. However, in my case I don't really need the wave part, I simulate relatively slow flows and I'd like to assume that sound speed in my flow is infinite. This gets into the area of Navier-Stokes equation, but I have not found a formulation of compressible navier-stokes with temperature terms.

I'm pretty sure there has to be a standard name for a system of equation that governs a process like this. Can anybody please suggest what to look for?

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  • $\begingroup$ What is the specific system? $\endgroup$ – Chet Miller Dec 27 '20 at 13:36
  • $\begingroup$ Well, if I knew the name I would not have asked for it. Problem is, I don't know what the name of this system is. $\endgroup$ – Suslik Dec 27 '20 at 13:51
  • $\begingroup$ Can you describe the physical system in words? $\endgroup$ – Chet Miller Dec 27 '20 at 14:18
  • $\begingroup$ Isnt that in the OP? Low Mach number time-dependent compressible flow equations with ideal gas thermodynamics. $\endgroup$ – Suslik Dec 27 '20 at 14:39
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Ah, I figured it out. Turns out, it's literally called "low-mach number compressible fluid dynamics". Since sound speed is way higher than advection speed in this case, it's very inefficient to solve these governing equations as hyperbolic ones because time step is limited by sound speed. Instead, it's possible to re-formulate the dynamic part as parabolic and solve it in predictor-corrector fashion where dynamic pressure is calculated from Poisson equation, in this case time step is defined by advection speed and not the sound speed. Examples of numerical methods for solving these are Simple and PISO.

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