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I'm reading an article from Shiyi Chen et.al named "On boundary conditions in lattice Boltzmann methods". In the first section they proceed to the well known Chapman-Enskog expansion in order to derive the Navier-Stokes equation from the Lattice Boltzmann Equation (in the D2Q9 model): $\frac{\partial f_i}{\partial t} + e_i.\nabla f_i = \Omega_i$, where $f_i$ is the velocity discretized distribution function, $e_i$ is a unit vector and $\Omega_i$ is the BGK collision operator. After they have proceeded to the Chapman expansion, they state that:
"If, in addition, a low Mach number (Ma) assumption is invoked as the nearly incompressible limit is approached, i.e., $Ma=u/C_s$ (where $C_s$ is the speed of sound) and $u' \sim \rho' \sim p' \sim O(Ma^2)$ [...] Here, the primed quantities denote fluctuations".

I don't really understand how do they come to $u' \sim \rho' \sim p' \sim O(Ma^2)$. Can someone explains me why do they have this relation ?

Have a good day.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented 13 hours ago
  • $\begingroup$ I have just edited it, I hope it is more clear than before. thanks $\endgroup$
    – Tristan Millet
    Commented 13 hours ago

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