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

TL;DR

This paper introduces a stochastic model for turbulent fluid flows by mixing smooth velocity fields with Gaussian random fields weighted by Reynolds number, providing a new closure approach for Navier-Stokes turbulence analysis.

Contribution

It proposes a novel Reynolds-weighted mixing ansatz that preserves mean flow while modeling turbulence growth and offers a stochastic closure for Navier-Stokes equations.

Findings

Provides a stochastic turbulence model with Reynolds-dependent growth

Defines a Hopf-like functional for circulation statistics

Enables analysis of higher moments and correlations in turbulence

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 $\mathscr{B}(x)$ on $\mathfrak{D}$ with Bargmann--Fock correlation $\mathbb{E}\langle\mathscr{B}(x)\otimes\mathscr{B}(y)\rangle=\mathsf{C}\exp(-|x-y|^{2}\lambda^{-2})$, 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 \[\mathscr{U}_{a}(x,t)=U_{a}(x,t)+\alpha U_{a}(x,t)\psi(|\mathbf{Re}(\mathfrak{D},t)-\mathbf{Re}_{c}(\mathfrak{D})|)\mathbb{I}_{\mathcal{S}}[\mathbf{Re}(\mathfrak{D},t)]\mathscr{B}(x)\] 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, $\mathbb{E}\langle\mathscr{U}_{a}(x,t)\rangle=U_{a}(x,t)$, 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.