Exact Reynolds and Mach nondimensionalization — conditional physical prediction

Use Newtonian spacetime $I\times E$, where $I\subseteq \mathbb{R}$ is a nonempty open interval of absolute time in seconds and $E$ is an oriented three-dimensional Euclidean affine space measured in metres. A Newtonian Cartesian inertial chart is the event map $(t,X)\mapsto (t,x_t(X))$, where $b:E\to {\mathbb{R}}^3$ is an orientation-preserving affine isometry, $w\in {\mathbb{R}}^3$ is constant and $x_t(X)=b(X)+tw$ for every $t\in I$. Write $x$ for its spatial event coordinate; no assumption $0\in I$ is needed for this chart formula. Write $I_3$ for the $3\times 3$ identity matrix and set $\mathrm{SO}(3):=\{Q\in {\mathbb{R}}^{3\times 3}:Q^{\mathsf{T}}Q=I_3,\det Q=1\}$. Inertial-chart changes have the form $t^{\mathrm{\prime }}=t+t_0$, $x^{\mathrm{\prime }}=Qx+vt+a$, where $t_0$ is a constant time translation in seconds, $a\in {\mathbb{R}}^3$ is a constant spatial translation in metres, $v\in {\mathbb{R}}^3$ is a constant relative velocity in metres per second, and $Q\in \mathrm{SO}(3)$ is constant. A frame in this statement means such an adapted chart, not an observer congruence or a basis field. Fix one such chart and a nonempty open spatial coordinate region $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$. All derivatives below are Cartesian derivatives in this fixed chart; the positive coordinate rescaling defined below is not a change of observer.

For a scalar field $f$, ${\mathrm{\nabla }}_{x}f=({\mathrm{\partial }}_{x_1}f,{\mathrm{\partial }}_{x_2}f,{\mathrm{\partial }}_{x_3}f)$.

For a vector field $u=(u_i)$, define $D_{x}u,D:I\times \mathrm{\Omega }\to {\mathbb{R}}^{3\times 3}$, ${\mathrm{div}}_{x}u:I\times \mathrm{\Omega }\to \mathbb{R}$, and ${\mathrm{\Delta }}_{x}u,D_{t}u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ by $(D_{x}u{)}_{ij}={\mathrm{\partial }}_{x_j}{u}_i$, $D=(D_{x}u+(D_{x}u{)}^{\mathsf{T}})\mathrm{/}2$, ${\mathrm{div}}_{x}u={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{u}_j$, $({\mathrm{\Delta }}_{x}u{)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j{x}_j}{u}_i$, and $D_{t}u={\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_x)u$. The fields $D$ and ${\mathrm{div}}_{x}u$ have units ${\mathrm{s}}^{-1}$. Fix constants ${\rho }_0>0$, $\mu >0$, $U>0$, $L>0$, and $c>0$, and define $\nu =\mu \mathrm{/}{\rho }_0$, $\mathrm{Re}=UL\mathrm{/}\nu$, and $\mathrm{Ma}=U\mathrm{/}c$. In SI units, $[u]=[U]=[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[x]=[L]=\mathrm{m}$, $[\rho ]=[{\rho }_0]=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $[p]=[P(\rho )]=[p_{\mathrm{r}\mathrm{e}\mathrm{f}}]=[\sigma ]=\mathrm{P}\mathrm{a}$, $[\mu ]=\mathrm{P}\mathrm{a}\mathrm{s}$, $[\nu ]={\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $[\mathrm{\Theta }]=[X]=[\mathrm{Re}]=[\mathrm{Ma}]=1$; thus the two ratios are dimensionless.

Define the dimensionless coordinates and their domains by $\mathrm{\Theta }=\frac{Ut}{L},X=\frac{x}{L},I^{\ast }=\{\frac{Ut}{L}:t\in I\},{\mathrm{\Omega }}^{\ast }=\{\frac{x}{L}:x\in \mathrm{\Omega }\}\mathrm{.}$ The map $\mathrm{\Phi }:I\times \mathrm{\Omega }\to I^{\ast }\times {\mathrm{\Omega }}^{\ast }$, $\mathrm{\Phi }(t,x)=(Ut\mathrm{/}L,x\mathrm{/}L)$, is bijective with inverse ${\mathrm{\Phi }}^{-1}(\mathrm{\Theta },X)=(L\mathrm{\Theta }\mathrm{/}U,LX)$. The corresponding $X$-derivatives are defined by the same component formulas.

In the first, force-free branch, let $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ have continuous first time partial derivatives and continuous first and second spatial partial derivatives, and let the pressure field $p:I\times \mathrm{\Omega }\to \mathbb{R}$ be continuous with continuous first spatial partial derivatives. Let the density field $\rho :I\times \mathrm{\Omega }\to (0,\mathrm{\infty })$ be the constant ${\rho }_0$. Assume ${\mathrm{div}}_{x}u=0$ and the selected incompressible Newtonian equations $\sigma =-p{I}_3+2\mu D,{\rho }_0{D}_{t}u={\mathrm{div}}_x\sigma ,({\mathrm{div}}_x\sigma {)}_i={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{\sigma }_{ij},$ where $\sigma :I\times \mathrm{\Omega }\to {\mathbb{R}}^{3\times 3}$ is stress, $p$ is the constraint multiplier, not an equation-of-state pressure, and the tension-positive stress convention identifies $\sigma n$ as traction on an outward unit normal $n$. Define $u^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to {\mathbb{R}}^3$ and $p_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to \mathbb{R}$ by $u^{\ast }(\mathrm{\Theta },X)=\frac{1}{U}u({\mathrm{\Phi }}^{-1}(\mathrm{\Theta },X)),p_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }(\mathrm{\Theta },X)=\frac{p({\mathrm{\Phi }}^{-1}(\mathrm{\Theta },X))-p_{\mathrm{r}\mathrm{e}\mathrm{f}}}{{\rho }_0{U}^2},$ for a constant reference pressure $p_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathbb{R}$. The scaled fields are dimensionless, $[u^{\ast }]=[p_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }]=1$, and have respectively the same stated velocity and spatial-pressure regularity. Equivalently, $u(t,x)=U{u}^{\ast }(\mathrm{\Theta },X)$ and $p(t,x)=p_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\rho }_0{U}^2{p}_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }(\mathrm{\Theta },X)$. The governing equations become exactly ${\mathrm{\partial }}_{\mathrm{\Theta }}{u}^{\ast }+(u^{\ast }\cdot {\mathrm{\nabla }}_X)u^{\ast }=-{\mathrm{\nabla }}_X{p}_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }+{\mathrm{Re}}^{-1}{\mathrm{\Delta }}_X{u}^{\ast },{\mathrm{div}}_X{u}^{\ast }=0.$

In the separate, unforced barotropic Euler branch, let $\rho :I\times \mathrm{\Omega }\to (0,\mathrm{\infty })$ and $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ have continuous partial derivatives through total order two, and let $P:(0,\mathrm{\infty })\to \mathbb{R}$ have continuous derivatives through order two, with pressure measured in pascals. Set the potential $V:\mathrm{\Omega }\to \mathbb{R}$ to the zero function. Assume ${\mathrm{\partial }}_t\rho +{\mathrm{div}}_x(\rho u)=0,\rho D_{t}u+{\mathrm{\nabla }}_{x}P(\rho )=0,$ and assume at the chosen reference density that $P^{\mathrm{\prime }}({\rho }_0)=c^2>0$, where $[P^{\mathrm{\prime }}({\rho }_0)]={\mathrm{m}}^2{\mathrm{s}}^{-2}$. Define ${\rho }^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to (0,\mathrm{\infty })$, $u^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to {\mathbb{R}}^3$, and $p_{\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to \mathbb{R}$ by ${\rho }^{\ast }(\mathrm{\Theta },X)=\frac{\rho ({\mathrm{\Phi }}^{-1}(\mathrm{\Theta },X))}{{\rho }_0},u^{\ast }(\mathrm{\Theta },X)=\frac{u({\mathrm{\Phi }}^{-1}(\mathrm{\Theta },X))}{U},p_{\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }(\mathrm{\Theta },X)=\frac{P({\rho }_0{\rho }^{\ast }(\mathrm{\Theta },X))-P({\rho }_0)}{{\rho }_0{c}^2}\mathrm{.}$ These scaled fields are dimensionless, $[{\rho }^{\ast }]=[u^{\ast }]=[p_{\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }]=1$, and have continuous partial derivatives through order two. The exact transformed mass and momentum equations are ${\mathrm{\partial }}_{\mathrm{\Theta }}{\rho }^{\ast }+{\mathrm{div}}_X({\rho }^{\ast }{u}^{\ast })=0,{\mathrm{\partial }}_{\mathrm{\Theta }}{u}^{\ast }+(u^{\ast }\cdot {\mathrm{\nabla }}_X)u^{\ast }=-{\mathrm{Ma}}^{-2}({\rho }^{\ast }{)}^{-1}{\mathrm{\nabla }}_X{p}_{\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }\mathrm{.}$

Initial values are data on a slice $t=t_0\in I$; boundary values are data on a chosen subset $\mathrm{\Gamma }\subseteq I\times \mathrm{\partial }\mathrm{\Omega }$ wherever the relevant traces exist. Extend $\mathrm{\Phi }$ to $I\times {\mathbb{R}}^3$ by the same coordinate formula and put $S^{\ast }=\mathrm{\Phi }(S)$ for any such initial or boundary data set $S$. If $g_u:S\to {\mathbb{R}}^3$, $g_p:S\to \mathbb{R}$, $g_{\rho }:S\to (0,\mathrm{\infty })$, and $g_P:S\to \mathbb{R}$ are prescribed velocity, incompressible pressure-multiplier, density, and barotropic-pressure data, respectively, define $g_u^{\ast }:S^{\ast }\to {\mathbb{R}}^3$, $g_{p,\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }:S^{\ast }\to \mathbb{R}$, $g_{\rho }^{\ast }:S^{\ast }\to (0,\mathrm{\infty })$, and $g_{P,\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }:S^{\ast }\to \mathbb{R}$ by $g_u^{\ast }(\mathrm{\Phi }(t,x))=\frac{g_u(t,x)}{U},g_{p,\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }(\mathrm{\Phi }(t,x))=\frac{g_p(t,x)-p_{\mathrm{r}\mathrm{e}\mathrm{f}}}{{\rho }_0{U}^2},g_{\rho }^{\ast }(\mathrm{\Phi }(t,x))=\frac{g_{\rho }(t,x)}{{\rho }_0},g_{P,\mathrm{b}\mathrm{a}\mathrm{r}}^{\ast }(\mathrm{\Phi }(t,x))=\frac{g_P(t,x)-P({\rho }_0)}{{\rho }_0{c}^2}\mathrm{.}$ Here $g_p$ and $g_{p,\mathrm{i}\mathrm{n}\mathrm{c}}^{\ast }$ apply to the incompressible branch, while $g_{\rho }$ and $g_P$ apply to the barotropic branch; a prescribed wall-velocity field uses the same velocity map. The spatial dilation takes each boundary trace portion to its image and preserves its outward Euclidean unit normal wherever that normal exists. For momentum density $m:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$, with units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, define $m^{\ast }:I^{\ast }\times {\mathrm{\Omega }}^{\ast }\to {\mathbb{R}}^3$ by $m^{\ast }(\mathrm{\Phi }(t,x))=\frac{m(t,x)}{{\rho }_{0}U}={\rho }^{\ast }(\mathrm{\Phi }(t,x))u^{\ast }(\mathrm{\Phi }(t,x)),$ where ${\rho }^{\ast }(\mathrm{\Phi }(t,x))=1$ in the constant-density branch. Thus $m^{\ast }$ is dimensionless. Whenever the integrals exist, total momentum on corresponding spatial regions $A\subseteq \mathrm{\Omega }$ and $A^{\ast }=A\mathrm{/}L$ satisfies ${\int }_A\rho udx={\rho }_{0}U{L}^3{\int }_{A^{\ast }}{\rho }^{\ast }{u}^{\ast }dX\mathrm{.}$ The total momentum has units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, as does its scale factor ${\rho }_{0}U{L}^3$. Because $\mathrm{\Phi }$ and each field-value scaling are invertible, these formulas carry any already-compatible initial or boundary data to the corresponding scaled data; they assert no compatibility or solvability beyond the selected model and its separate data assumptions. Thermal evolution and heat-flux scaling are not part of this calculation. The coefficient identities do not assert an incompressible or inviscid limit, an error bound, uniqueness, identical solutions for different geometries or rescaled data, or a turbulence threshold. They are conditional predictions of the adopted model formulations and are not experimental observations or proof that either model describes a material.

Iain Stewart, MIT 8.09, Chapter 6, §6.7, Reynolds number and similarity, printed pp. 133–134

David Tong, Fluid Mechanics, Part II, §§3.3.1–3.3.2, Reynolds number and scaling, printed pp. 72–76

David Tong, Fluid Mechanics, Part III, §4.5.3, Mach number definition, printed p. 166

MIT 16.01 Unified Engineering, F5 lecture notes, Effects of Reynolds Number and Mach Number, pp. 1–4