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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.14925 |
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Table of Contents:
- 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.