A Turbulent Fluid Mechanics Via Nonlinear Mixing Of Smooth Velocity Flows With Reynolds-Weighted Random Fields

Abstract

We consider a finite-volume domain $\mathfrak{D}\subset {\mathbb{R}}^3$ of size $\mathrm{Vol}(\mathfrak{D})\sim {\mathrm{L}}^3$ containing a viscous fluid of kinematic viscosity $\nu$ with velocity field $U_a(x,t)$ satisfying the Navier--Stokes equations with prescribed boundary data. We introduce a zero-centred homogeneous-isotropic Gaussian field $\mathcal{B}(x)$ on $\mathfrak{D}$ with Bargmann--Fock correlation $\mathbb{E}⟨\mathcal{B}(x)\otimes \mathcal{B}(y)⟩=\mathsf{C}\exp (-|x-y{|}^2{\lambda }^{-2})$, where $\lambda \le \mathrm{L}$. For the volume-averaged Reynolds number $\mathbf{Re}(\mathfrak{D},t)=(|\mathrm{Vol}(\mathfrak{D}){|}^{-1}{\int }_{\mathfrak{D}}|U_a(x,t)|d\mu (x))\mathrm{L}/\nu$, let ${\mathbf{Re}}_c(\mathfrak{D})$ denote the critical threshold for turbulence. We propose a Reynolds-weighted mixing ansatz for a turbulent velocity field \[\mathscr{U}_{a}(x,t)=U_{a}(x,t)+αU_{a}(x,t)ψ(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]\mathscr{B}(x)\] with $\alpha \ge 1$, $\psi$ monotone increasing, and ${\mathbb{I}}_{\mathcal{S}}$ active only for $\mathbf{Re}>{\mathbf{Re}}_c$. The construction preserves the mean flow, $\mathbb{E}⟨{\mathcal{U}}_a(x,t)⟩=U_a(x,t)$, while allowing turbulence intensity to grow with the control parameter $\mathbf{Re}$. This provides a tentative stochastic closure for Navier--Stokes, enabling Reynolds-type correlations ${\mathsf{T}}_{ab}(x,y;t)=\mathbb{E}⟨{\mathcal{U}}_a(x,t)\otimes {\mathcal{U}}_b(y,t)⟩$ and higher moments. For test functions $f$ and curves $\Im \subset \mathfrak{D}$ we define a Hopf-like functional \[\mathbb{H}[\mathscr{U}_{a},t]=\mathbb{E}\bigg\langle\exp\bigg(i\int_{\Im}f(x,t)\mathscr{U}_{a}(x,t)dx^{a}\bigg)\bigg\rangle\] encoding circulation statistics generated by the mixing ansatz.

57 pages, 5 figures. Notational corrections and improvements