Exact Reynolds and Mach nondimensionalization

Let $I\subseteq \mathbb{R}$ and $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open, let ${\rho }_0,\mu ,U,L,c>0$, and set $\nu =\mu \mathrm{/}{\rho }_0$, $\mathrm{Re}=UL\mathrm{/}\nu$, and $\mathrm{Ma}=U\mathrm{/}c$. Define $T=Ut\mathrm{/}L$, $X=x\mathrm{/}L$, $I^{\ast }=\{Ut\mathrm{/}L:t\in I\}$, and ${\mathrm{\Omega }}^{\ast }=\{x\mathrm{/}L:x\in \mathrm{\Omega }\}$. For the incompressible branch, let $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ have continuous first time and first and second spatial partial derivatives, and let $p:I\times \mathrm{\Omega }\to \mathbb{R}$ have continuous first spatial partial derivatives. Write $u^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to {\mathbb{R}}^3$ and $p^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to \mathbb{R}$ for the corresponding fields with the same stated regularity. Set $u(t,x)=U{u}^{\ast }(T,X)$ and $p(t,x)=p_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\rho }_0{U}^2{p}^{\ast }(T,X)$, where $p_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is constant. Then the original system is equivalent under this invertible change of variables to

${\mathrm{\partial }}_T{u}^{\ast }+(u^{\ast }\cdot {\mathrm{\nabla }}_X)u^{\ast }=-{\mathrm{\nabla }}_X{p}^{\ast }+{\mathrm{Re}}^{-1}{\mathrm{\Delta }}_X{u}^{\ast },{\mathrm{div}}_X{u}^{\ast }=0.$

For the barotropic branch, let $P:(0,\mathrm{\infty })\to \mathbb{R}$ have two continuous derivatives with $P^{\mathrm{\prime }}({\rho }_0)=c^2>0$. Let $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ have continuous first time and first and second spatial partial derivatives, let $\rho :I\times \mathrm{\Omega }\to (0,\mathrm{\infty })$ have continuous first time and spatial partial derivatives, and let any potential $V$ have continuous spatial partial derivatives through order two. Write $u^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to {\mathbb{R}}^3$, ${\rho }^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to (0,\mathrm{\infty })$, and $p^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to \mathbb{R}$ for the rescaled fields with the corresponding stated regularity. Set $\rho ={\rho }_0{\rho }^{\ast }$ and $P(\rho )=P({\rho }_0)+{\rho }_0{c}^2{p}^{\ast }$; if a potential is present, set $V=U^2{V}^{\ast }$ with $V^{\ast }:{\mathrm{\Omega }}^{\ast }\to \mathbb{R}$. Then the exact transformed equations are

${\mathrm{\partial }}_T{\rho }^{\ast }+{\mathrm{div}}_X({\rho }^{\ast }{u}^{\ast })=0,{\mathrm{\partial }}_T{u}^{\ast }+(u^{\ast }\cdot {\mathrm{\nabla }}_X)u^{\ast }=-{\mathrm{Ma}}^{-2}({\rho }^{\ast }{)}^{-1}{\mathrm{\nabla }}_X{p}^{\ast }-{\mathrm{\nabla }}_X{V}^{\ast }\mathrm{.}$

These are coefficient identities under positive changes of scale. They do not assert a limit as either dimensionless ratio tends to zero or infinity, an error estimate, uniqueness, or identical solutions when geometry or rescaled data differ.

In the optional SI interpretation, $[{\rho }_0]=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $[\mu ]=\mathrm{P}\mathrm{a}\mathrm{s}$, $[U]=[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[L]=\mathrm{m}$, and $[\nu ]={\mathrm{m}}^2{\mathrm{s}}^{-1}$; $\mathrm{Re}$ and $\mathrm{Ma}$ are dimensionless.

The characteristic Reynolds ratio and similarity discussion: Stewart, MIT 8.09, Chapter 6, §6.7, printed pp. 133–134.

Independent Reynolds and Mach parameter treatment: MIT 16.01 Unified Engineering, F5 lecture notes, pp. 1–4.

The formal systems and derivative conventions: Formal parameterized fluid systems and observables.