Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.14925 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914159166226432 |
|---|---|
| author | Miller, Steven D |
| author_facet | Miller, Steven D |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14925 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Turbulent Fluid Mechanics Via Nonlinear Mixing Of Smooth Velocity Flows With Reynolds-Weighted Random Fields Miller, Steven D Mathematical Physics 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. |
| title | A Turbulent Fluid Mechanics Via Nonlinear Mixing Of Smooth Velocity Flows With Reynolds-Weighted Random Fields |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2211.14925 |