# Two ways of non-dimensionalizing Navier Stokes equation

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?

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.