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I know that the ratio of an object's speed to the local sound speed is called the Mach number: $$M = \frac{v}{c_s}$$ where $$c_s = \sqrt{\frac{\gamma kT}{\mu m_p}}$$

I have always thought of the Mach number in a macroscopic sense (e.g., aircraft in the atmosphere) where the ambient sound speed is used (rather than using the object's own average temperature to compute some "internal" sound speed). However, suppose we have a Lagrangian fluid simulation with unresolved macroscopic gas particles as in the case of astrophysical simulations (e.g., each gas particle has a mass of 1000 solar masses).

Can we define the Mach number of an unresolved macroscopic gas particle using its own temperature to compute the sound speed? What would it mean physically to compare the overall bulk velocity of the unresolved particle to its own internal sound speed (the velocity at which sound waves propagate "within" it)?

How would we interpret this "internal" Mach number? Perhaps that if the unresolved particle's velocity is much larger than its own sound speed (given its temperature), we may expect to see complicated substructure (shocks, etc.) if we could increase the resolution of our simulation?

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    $\begingroup$ A Mach number is just a velocity divided by a mechanical wave speed. You can define a particular Mach number more or less however you want. The real question is, what are you trying to accomplish with this Mach number? What phenomenon are you trying to describe by the particle velocity divided by an intrinsic wave speed, rather than the wave speed of the continuum the particle is a part of? $\endgroup$
    – Michael M
    Jan 19 '21 at 13:35
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I have always thought of the Mach number in a macroscopic sense (e.g., aircraft in the atmosphere) where the ambient sound speed is used (rather than using the object's own average temperature to compute some "internal" sound speed).

The speed of sound in a gas is defined as: $$ C_{s}^{2} = \frac{ \partial P }{ \partial \rho } \tag{0} $$ where $P$ is the thermal pressure and $\rho$ is the mass density. In an ideal gas, one can assume the pressure is adiabatic or isothermal or any number of things depending on the system. If we assume the gas is adiabatic, then $P \propto \rho^{\gamma}$ where $\gamma$ is the polytropic index. Then Equation 0 goes to: $$ C_{s}^{2} = \frac{ \gamma P }{ \rho } \tag{1} $$

Can we define the Mach number of an unresolved macroscopic gas particle using its own temperature to compute the sound speed?

In general, most approximations for the thermal pressure explicitly depend upon the temperature. So yes, we often do use the particle temperature to define the speed of sound of the gas.

What would it mean physically to compare the overall bulk velocity of the unresolved particle to its own internal sound speed (the velocity at which sound waves propagate "within" it)?

Objects moving in the fluid faster than the local speed of sound can generate a shock wave. A single particle moving faster than $C_{s}$ will not generate a shock wave, as a shock wave requires the bulk response of the system.

How would we interpret this "internal" Mach number?

The sonic Mach number in a collisionally mediated fluid like Earth's atmosphere is the relevant Mach number as it describes the speed of communication of the system. I am not sure what is meant by "internal" here.

Perhaps that if the unresolved particle's velocity is much larger than its own sound speed (given its temperature), we may expect to see complicated substructure (shocks, etc.) if we could increase the resolution of our simulation?

All fluids are made up of individual particles/molecules, which exhibit velocity distribution functions (VDFs). The VDFs are modeled as being continuous functions like the Maxwellian. In all of these cases, there will be particles with peculiar speeds that exceed the local speed of sound by construction/definition, i.e., if you introduce an artificial cutoff in the VDF at the sound speed, you will have to change/update the sound speed (which will be smaller) and again you will have individual particles moving faster than the sound speed.

However, suppose we have a Lagrangian fluid simulation with unresolved macroscopic gas particles as in the case of astrophysical simulations (e.g., each gas particle has a mass of 1000 solar masses).

You can get shock waves just fine in fluid simulations so long as you properly handle the nonlinear steepening terms (i.e., $\mathbf{V} \cdot \nabla \mathbf{V}$) and so called stiff solutions and you have the proper scale lengths resolved (e.g., mean free path in a collisionally mediated fluid). Technically, I think fluid simulations treat individual particles as massless though, do they not? That is, they only deal with bulk velocity moments not individual particle motions. This is why I say they should be able to handle supersonic motions.

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