Fluid equation scalings and dimensionless numbers

Assume Countable Choice. Let $d\in \{1,2,3\}$, $\mathrm{\ell }>0$, and $\nu \ge 0$, and write $Q_{\mathrm{\ell }}={\mathbb{R}}^d\mathrm{/}(\mathrm{\ell }\mathbb{Z}{)}^d$. For a period cell, define $\mathrm{\parallel }f{\mathrm{\parallel }}_{L_{\mathrm{v}\mathrm{o}\mathrm{l}}^2(Q_{\mathrm{\ell }})}=({\int }_{[0,\mathrm{\ell }{]}^d}\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx{)}^{1\mathrm{/}2}$.

First, for $T>0$ and either $\mathrm{\Omega }={\mathbb{R}}^d$ or $\mathrm{\Omega }=Q_{\mathrm{\ell }}$, let $u\in C^{\mathrm{\infty }}([0,T]\times \mathrm{\Omega };{\mathbb{R}}^d)$ and $\pi \in C^{\mathrm{\infty }}([0,T]\times \mathrm{\Omega };\mathbb{R})$ be a velocity-pressure pair solving the unforced incompressible Navier–Stokes system $u_t+(u\cdot \mathrm{\nabla })u=-\mathrm{\nabla }\pi +\nu \mathrm{\Delta }u,\mathrm{div}u=0,$ then for any $\lambda >0$, $u_{\lambda }(t,x)=\lambda u({\lambda }^{2}t,\lambda x),{\pi }_{\lambda }(t,x)={\lambda }^2\pi ({\lambda }^{2}t,\lambda x)$ solves the same system with the same viscosity on ${\mathbb{R}}^d$ or on the torus of period $\mathrm{\ell }\mathrm{/}\lambda$, respectively, for $0\le t\le T\mathrm{/}{\lambda }^2$. On the torus, use the Fourier Leray multiplier $P_0=I$, $P_{\xi }=I-\xi {\xi }^{\mathsf{T}}\mathrm{/}\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for $\xi \mathrm{\ne }0$; the projected equation is preserved as well because $P_{\lambda \xi }=P_{\xi }$. For a cell-volume $L^2$ norm, $\mathrm{\parallel }{u}_{\lambda }(0){\mathrm{\parallel }}_{L_{\mathrm{v}\mathrm{o}\mathrm{l}}^2(Q_{\mathrm{\ell }\mathrm{/}\lambda })}={\lambda }^{1-d\mathrm{/}2}\mathrm{\parallel }u(0){\mathrm{\parallel }}_{L_{\mathrm{v}\mathrm{o}\mathrm{l}}^2(Q_{\mathrm{\ell }})}$.

Second, for any $\alpha ,\beta >0$, if $T>0$, $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ is open, $u\in C^{\mathrm{\infty }}([0,T]\times \mathrm{\Omega };{\mathbb{R}}^d)$, and $\pi \in C^{\mathrm{\infty }}([0,T]\times \mathrm{\Omega };\mathbb{R})$ solve the unforced incompressible Euler system $u_t+(u\cdot \mathrm{\nabla })u=-\mathrm{\nabla }\pi$, $\mathrm{div}u=0$, then on the scaled domain ${\beta }^{-1}\mathrm{\Omega }$, $u_{\alpha ,\beta }(t,x)=\alpha u(\alpha \beta t,\beta x),{\pi }_{\alpha ,\beta }(t,x)={\alpha }^2\pi (\alpha \beta t,\beta x)$ solves the same equation on the correspondingly scaled domain and time interval $0\le t\le T\mathrm{/}(\alpha \beta )$. These transformations are equation symmetries; on a torus they transform its period.

Finally, let $U,L,{\rho }_{\ast },\eta >0$ be physical scales in SI units, where $U$ is in m/s, $L$ in m, ${\rho }_{\ast }$ is a constant positive reference density in kg/m${}^3$, and dynamic viscosity $\eta$ is in Pa$\cdot$s. Set kinematic viscosity $\nu =\eta \mathrm{/}{\rho }_{\ast }$, in m${}^2$/s. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be an open dimensionless spatial region and $I\subseteq \mathbb{R}$ an open time interval; set ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=L\mathrm{\Omega }$ and $I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(L\mathrm{/}U)I$. Take $u,v\in C^{\mathrm{\infty }}(\mathrm{\Omega }\times I;{\mathbb{R}}^d)$ and $\pi ,\delta {\rho }^{\mathrm{\prime }}\in C^{\mathrm{\infty }}(\mathrm{\Omega }\times I;\mathbb{R})$; the scaled fields $u,v,\pi ,\delta {\rho }^{\mathrm{\prime }}$ are dimensionless. Define physical velocity maps $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}},v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};{\mathbb{R}}^d)$, physical-pressure maps $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}},{\pi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};\mathbb{R})$, and density-perturbation map $\delta \rho \in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};\mathbb{R})$ by $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X,t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})=Uu(X\mathrm{/}L,U{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}L),p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X,t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})={\rho }_{\ast }{U}^2\pi (X\mathrm{/}L,U{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}L),$ $v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X,t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})=Uv(X\mathrm{/}L,U{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}L),\delta \rho (X,t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})={\rho }_{\ast }\delta {\rho }^{\mathrm{\prime }}(X\mathrm{/}L,U{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}L)\mathrm{.}$ Adopt the constant-reference-density kinematic-pressure convention ${\pi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\rho }_{\ast }$; thus ${\pi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=U^2\pi$ has units m${}^2$/s${}^2$, while $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ has units Pa. If $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}},{\pi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ satisfy the constant-density incompressible system ${\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+(u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cdot {\mathrm{\nabla }}_X)u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-{\mathrm{\nabla }}_X{\pi }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+\nu {\mathrm{\Delta }}_X{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}},{\mathrm{div}}_X{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=0,$ then the stated substitution gives the dimensionless system ${\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_x)u=-{\mathrm{\nabla }}_x\pi +R{e}^{-1}{\mathrm{\Delta }}_{x}u,{\mathrm{div}}_{x}u=0,$ where $Re=UL\mathrm{/}\nu ={\rho }_{\ast }UL\mathrm{/}\eta$. Let $p_{\mathrm{e}\mathrm{o}\mathrm{s}}:(0,\mathrm{\infty })\to \mathbb{R}$ be a differentiable equation-of-state pressure map, measured in Pa, at ${\rho }_{\ast }$, with $p_{\mathrm{e}\mathrm{o}\mathrm{s}}^{\mathrm{\prime }}({\rho }_{\ast })>0$ in m${}^2$/s${}^2$, and set the sound speed $c_{\ast }=\sqrt{p_{\mathrm{e}\mathrm{o}\mathrm{s}}^{\mathrm{\prime }}({\rho }_{\ast })}$ in m/s. For the linear acoustic tangent system, let the physical fields $\delta \rho :{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to \mathbb{R}$ and $v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathbb{R}}^d$ satisfy ${\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\delta \rho +{\rho }_{\ast }{\mathrm{div}}_X{v}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=0,{\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}{v}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+(c_{\ast }^2\mathrm{/}{\rho }_{\ast }){\mathrm{\nabla }}_X\delta \rho =0.$ Here $\delta \rho$ has units kg/m${}^3$. The scales $X=Lx$, $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(L\mathrm{/}U)t$, $\delta \rho ={\rho }_{\ast }\delta {\rho }^{\mathrm{\prime }}$, and $v_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=Uv$ give the momentum coefficient $M{a}^{-2}$, where $Ma=U\mathrm{/}{c}_{\ast }$. Both $Re$ and $Ma$ are dimensionless. These are chain-rule identities and assert no continuum limit or empirical similarity.

Periodic Leray and pressure maps

Tong, Fluid Mechanics, Chapter 2, §3.3.1

Tong, Fluid Mechanics, Chapter 3, §4.4.4