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Let $\mathbb{I\!H}\subset\mathbb{R}^{3}$, with ${Vol}(\mathbb{I\!H})\simL^{3}$, contain a fluid of viscosity $\nu$ and velocity $\mathrm{U}_{i}(x,t)$with $(x,t)\in\mathbb{I\!H}\times[0,\infty)$, satisfying the Navier-Stokesequations with some boundary conditions on $\partial\mathbb{I\!H}$ and evolvingfrom initial Cauchy data. Now let $\mathscr{B}(x)$ be a Gaussian random fielddefined for all $x\in\mathbb{I\!H}$ with expectation$\mathsf{E}\langle\mathscr{B}(x)\rangle=0$, and a Bargmann-Fock binarycorrelation $\mathsf{E}\big\langle\mathscr{B}(x)\otimes\mathscr{B}({y})\big\rangle=\mathsf{C}\exp(-\|{x}-{y}\|^{2}\lambda^{-2})$ with$\lambda\le {L}$. Define a volume-averaged Reynolds number$\mathbf{Re}(\mathbb{I\!H},t)=(|Vol(\mathbb{I\!H})|^{-1}\int_{\mathbb{I\!H}}\|\mathrm{U}_{i}(x,t)\|d\mu({x}){L}/\nu$.The critical Reynolds number is $\mathbf{Re}_{c}(\mathbb{I\!H})$ so thatturbulence evolves within $\mathbb{I\!H}$ for $t$ such that$\mathbf{Re}(\mathbb{I\!H},t)>\mathbf{Re}_{c}(\mathbb{I\!H})$. Let$\psi(|\mathbf{Re}(\mathbb{I\!H},t)-\mathbf{Re}_{c}(\mathbb{I\!H})|)$ be anarbitrary monotone-increasing functional. The turbulent flow evolving within$\mathbb{I\!H}$ is described by the random field $\mathscr{U}_{i}(x,t)$ via a'mixing' ansatz$\mathscr{U}_{i}(x,t)=\mathrm{U}_{i}(x,t)+\beta\mathrm{U}_{i}(x,t)\big\lbrace\psi(|\mathbf{Re}(\mathbb{I\!H},t)-\mathbf{Re}_{c}(\mathbb{I\!H})|)\big\rbrace\mathbb{S}_{\mathfrak{C}}[\mathbf{Re}(\mathbb{I\!H},t)\big]\mathscr{B}(x)$where ${\beta}\ge 1$ is a constant and$\mathbb{S}_{\mathfrak{C}}[\mathbf{Re}(\mathbb{I\!H},t)]$ an indicatorfunction. The flow grows increasingly random if $\mathbf{Re}(\mathbb{I\!H},t)$increases with $t$ 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 Flows with  Bargmann-Fock Random Fields: Stochastically Averaged Navier-Stokes Equations  and Velocity Correlations