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

TL;DR

The work develops a stochastic closure for turbulence by mixing a smooth Navier–Stokes flow $U_a(x,t)$ with a Bargmann–Fock Gaussian field ${\mathscr{B}}(x)$ to form a randomized turbulent velocity ${\mathscr{U}}_a(x,t)$. It builds a rigorous probabilistic framework based on stochastic averaging, deriving stochastically averaged Navier–Stokes equations and identifying transformed or 'boosted' NS forms that preserve the mean flow while amplifying turbulence via a Reynolds-number-controlled mixing term. The paper then develops a hierarchy of velocity correlations, from binary to triple and higher-order moments, and derives PDEs for the Reynolds stresses ${\bm{\mathsf{T}}}_{ab}$, as well as energy integrals and vorticity statistics, including stochastic vorticity and vortex tangles. A pressure-gradient extension is provided, yielding SA-NS equations with turbulent pressure contributions, and the framework is extended to homogeneous/isotropic settings and Hopf-like functional constructs. Although exploratory and heuristic, the approach offers a mathematically tractable stochastic closure that links turbulent statistics to the Reynolds control parameter and provides a path toward rigorous, computable descriptions of turbulence via regulated GRFs.

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 $ν$ 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 $\mathscr{B}(x)$ on $\mathfrak{D}$ with Bargmann--Fock correlation $\mathbb{E}\langle\mathscr{B}(x)\otimes\mathscr{B}(y)\rangle=\mathsf{C}\exp(-|x-y|^{2}λ^{-2})$, where $λ\le \mathrm{L}$. For the volume-averaged Reynolds number $\mathbf{Re}(\mathfrak{D},t)=(|\mathrm{Vol}(\mathfrak{D})|^{-1}\int_{\mathfrak{D}}|U_{a}(x,t)|dμ(x))\mathrm{L}/ν$, 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 $α\ge 1$, $ψ$ monotone increasing, and $\mathbb{I}_{\mathcal{S}}$ active only for $\mathbf{Re}>\mathbf{Re}_{c}$. The construction preserves the mean flow, $\mathbb{E}\langle\mathscr{U}_{a}(x,t)\rangle=U_{a}(x,t)$, 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 $\mathsf{T}_{ab}(x,y;t)=\mathbb{E}\langle\mathscr{U}_{a}(x,t)\otimes\mathscr{U}_{b}(y,t)\rangle$ 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.

Figures (3)

• Figure 1: Volume-averaged Reynolds number within domain $\bm{\mathfrak{D}}$, with smooth fluid flow $U_{a}(x,t)$ and viscosity $\nu$
• Figure 4: Transition of a smooth flow $U_{a}(x,t)$ to a turbulent flow ${\mathscr{U}}_{a}(x,t)$ as $\bm{\mathrm{R}e}(\bm{\mathfrak{D}},t)$ exceeds $\bm{\mathrm{R}e}_{c}(\bm{\mathfrak{D}})$. The turbulent velocity fluctuates randomly about the mean.
• Figure 5: Evolution of vortex tangles or correlations within $\bm{\mathfrak{D}}$ with increasing $\bm{\mathrm{Re}}(\bm{\mathfrak{D}},t)$.

Theorems & Definitions (99)

• Proposition 1.3
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9