Turbulent averaging and open closure questions

Let $I\subseteq \mathbb{R}$ be an open time interval, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be an open spatial region, and let $N\ge 1$. For $j=1,\dots ,N$, let $u_j:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ be $C^1$ in $(t,x)$ with continuous second spatial derivatives, let $p_j:I\times \mathrm{\Omega }\to \mathbb{R}$ be $C^1$, and let $f_j:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ be continuous. Suppose $\mathrm{div}{u}_j=0$ and

${\mathrm{\partial }}_t{u}_j+\mathrm{div}(u_j\otimes u_j)+\mathrm{\nabla }{p}_j=\nu \mathrm{\Delta }{u}_j+f_j$

for one common constant $\nu \ge 0$. In the formal SI fluid interpretation, the equation is per unit mass: $u_j$ is velocity (m s${}^{-1}$), $p_j$ is pressure per unit mass (m${}^2$ s${}^{-2}$), $f_j$ is body force per unit mass, or acceleration (m s${}^{-2}$), and $\nu$ is kinematic viscosity (m${}^2$ s${}^{-1}$). These units specify an interpretation of the fields and do not assert physical applicability. Fix dimensionless weights $a_j\ge 0$ with ${\sum }_{j=1}^N{a}_j=1$, define $\bar{u}={\sum }_j{a}_j{u}_j$, $\bar{p}={\sum }_j{a}_j{p}_j$, $\bar{f}={\sum }_j{a}_j{f}_j$, $v_j=u_j-\bar{u}$, and $R={\sum }_j{a}_j{v}_j\otimes v_j$, and use $(u\otimes v{)}_{rs}=u_r{v}_s$ with rowwise tensor divergence. Thus $R$ has units m${}^2$ s${}^{-2}$ under this interpretation. The exact finite weighted identity is

${\mathrm{\partial }}_t\bar{u}+\mathrm{div}(\bar{u}\otimes \bar{u})+\mathrm{\nabla }\bar{p}=\nu \mathrm{\Delta }\bar{u}+\bar{f}-\mathrm{div}R\mathrm{.}$

This is an identity for the specified smooth fields and fixed finite weights. For comparison, choose $t_0\in I$ and a finite $T>0$ such that $[t_0,t_0+T]\subset I$. For each fixed $x\in \mathrm{\Omega }$, the temporal average of the first supplied field over this window is $T^{-1}{\int }_{t_0}^{t_0+T}{u}_1(s,x)ds$; its componentwise ordinary Riemann integral exists because $u_1(\cdot ,x)$ is continuous on the compact window. No equality of this temporal average, the finite weighted mean $\bar{u}$, or a probability-space expectation is assumed or deduced. No long-time limit as $T\to \mathrm{\infty }$ is asserted. The tensor $R$ is the covariance term left by the nonlinear product; a closure prescription is an additional model relation that specifies this unresolved term using selected mean fields or additional variables. The identity itself supplies no such prescription. This remark asserts no measured turbulent flow, statistical averaging theorem, closure law, scaling law, universality result, or existence theorem for turbulent solutions.

finite weighted Reynolds identity and its hypotheses

Flierl and Ferrari, MIT OpenCourseWare 12.820, Chapter 3, §§3.3 and 3.6–3.8, complete 11-page lecture notes

Tennekes and Lumley, A First Course in Turbulence (1972), Preface and Chapter 2, §2.1, full-text digital copy