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Let $\mathfrak{D}\subset\mathbb{R}^{3}$, with ${Vol}(\mathfrak{D})\simL^{3}$, contain an incompressible fluid of viscosity $\nu$ and velocity$\mathrm{U}_{i}(x,t)$ with $(x,t)\in\mathfrak{D}\times[0,\infty)$, satisfyingthe Navier-Stokes equations with some boundary conditions on$\partial\mathfrak{D}$ and evolving from initial Cauchy data. Now let$\mathscr{B}(x)$ be a Gaussian random field defined for all $x\in\mathfrak{D}$with expectation $\mathbb{I\!E}\langle\mathscr{B}(x)\rangle=0$, and aBargmann-Fock binary correlation $\mathbb{I\!E}\big\langle\mathscr{B}(x)\otimes\mathscr{B}({y})\big\rangle=\mathsf{C}\exp(-\|{x}-{y}\|^{2}\lambda^{-2})$ with$\lambda\le {L}$. Define a volume-averaged Reynolds number$\mathbf{Re}(\mathfrak{D},t)=(|Vol(\mathfrak{D})|^{-1}\int_{\mathfrak{D}}\|\mathrm{U}_{i}(x,t)\|d\mu({x}){L}/\nu$.The critical Reynolds number is $\mathbf{Re}_{c}(\mathfrak{D})$ so thatturbulence fully evolves within $\mathfrak{D}$ for $t$ such that$\mathbf{Re}(\mathfrak{D},t)>\mathbf{Re}_{c}(\mathfrak{D})$. Let$\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)$ be anarbitrary monotone-increasing weighting functional. The turbulent flow evolvingwithin $\mathfrak{D}$ is described by the random field $\mathscr{U}_{i}(x,t)$via a 'mixing' ansatz$\mathscr{U}_{i}(x,t)=\mathrm{U}_{i}(x,t)+\beta\mathrm{U}_{i}(x,t)\big\lbrace\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\big\rbrace\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)\big]\mathscr{B}(x)$ where${\beta}\ge 1$ is a constant and$\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]$ an indicator function.The flow grows increasingly random if $\mathbf{Re}(\mathfrak{D},t)$ increaseswith $t$ so that this is a 'control parameter'. The turbulent flow$\mathscr{U}_{i}(x,t)$ is a solution of stochastically averaged N-S equations.Reynolds-type velocity correlations are estimated.

A Turbulent Fluid Mechanics via Nonlinear 'Mixing' of Smooth Velocity  Flows With Weighted Random Fields: Stochastically Averaged Navier-Stokes  Equations and Velocity Correlations