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I can see two ways of non-dimensionalizing Navier Stokes equation. One of them is from Wikipedia where one can see that the non-dimensionalization is performed by taking $p^*=\frac{p}{\rho U^2}$ where $U$ is some characteristic fluid speed. As a consequence, the Reynolds number appears in the non-dimensionalized equation. However, I could also have set (more appropriately) $p^*=\frac{p}{\rho c_s^2}$ where $c_s$ is the speed of sound in the fluid. In this case, both the Mach number and the Reynolds number would appear in the non-dimensionalized equation, with the Mach number appearing as $\frac{\nabla^*p*}{M^2}$. Which is the more appropriate way of doing this?

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In the section you are referring to, the article only considers incompressible Navier-Stokes, i.e. with: $$ \nabla \cdot v $$ You therefore don't have a speed of sound. More precisely, using: $$ c = \sqrt{\frac{K}{\rho}} $$ with $K$ the compressibility, the incompressible limit is when $K\to\infty$, or equivalently $c\to\infty$.

If the fluid were compressible, then yes your objection is legitimate. In general, non-dimensionalizing equations is ambiguous when you have dimensionless numbers since you can change units by multiplying by powers of these dimensionless numbers. For incompressible Navier Stokes, it's only $Re$, but for a compressible one, you'll also have $Re,M$. To solve the ambiguity, you typically have a certain limit in mind.

For example, the choice of $p^*=\frac{p}{\rho U^2}$ can be justified in the turbulent incompressible limit. You want to choose dimensionless variables that are independent of $\nu$ (since $\nu\to0$) and of $K$ (since $K\to\infty$). Another way to view this is that you are sending your dimensionless numbers to $0$ or $\infty$. In this case, this uniquely fixes the choices of dimensionless units. It all depends on the physical phenomena you are interested in.

Hope this helps.

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