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

We consider a finite-volume domain $\mathfrak{D}\subset {\mathbb{R}}^3$ of size $\mathrm{V}\mathrm{o}\mathrm{l}(\mathfrak{D})\sim {\mathrm{L}}^3$ 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 $\mathbb{E}⟨\mathcal{B}(x)\otimes \mathcal{B}(y)⟩=\mathsf{C}\exp (-\mathrm{\mid }x-y{\mathrm{\mid }}^2{\lambda }^{-2})$, where $\lambda \le \mathrm{L}$. For the volume-averaged Reynolds number $\mathbf{R}\mathbf{e}(\mathfrak{D},t)=(\mathrm{\mid }\mathrm{V}\mathrm{o}\mathrm{l}(\mathfrak{D}){\mathrm{\mid }}^{-1}{\int }_{\mathfrak{D}}\mathrm{\mid }{U}_a(x,t)\mathrm{\mid }d\mu (x))\mathrm{L}\mathrm{/}\nu$, let ${\mathbf{R}\mathbf{e}}_c(\mathfrak{D})$ denote the critical threshold for turbulence. We propose a Reynolds-weighted mixing ansatz for a turbulent velocity field $${\mathcal{U}}_a(x,t)=U_a(x,t)+\alpha U_a(x,t)\psi (\mathrm{\mid }\mathbf{R}\mathbf{e}(\mathfrak{D},t)-{\mathbf{R}\mathbf{e}}_c(\mathfrak{D})\mathrm{\mid }){\mathbb{I}}_{\mathcal{S}}[\mathbf{R}\mathbf{e}(\mathfrak{D},t)]\mathcal{B}(x)$$ with $\alpha \ge 1$, $\psi$ monotone increasing, and ${\mathbb{I}}_{\mathcal{S}}$ active only for $\mathbf{R}\mathbf{e}>{\mathbf{R}\mathbf{e}}_c$. 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 $\mathbf{R}\mathbf{e}$. 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 $\mathrm{\Im }\subset \mathfrak{D}$ we define a Hopf-like functional $$\mathbb{H}[{\mathcal{U}}_a,t]=\mathbb{E}⟨\exp (i{\int }_{\mathrm{\Im }}f(x,t){\mathcal{U}}_a(x,t)d{x}^a)⟩$$ encoding circulation statistics generated by the mixing ansatz.