Adopted simple shear production

Assume the Axiom of Countable Choice for the normalized periodic-torus convention. Let $I\subset \mathbb{R}$ be an open time interval measured in seconds, $L>0$ m, $0<\mathrm{\ell }<L\mathrm{/}2$, and ${\mathbb{T}}_L^3=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$. All spacetime fields below use the ordered domain ${\mathbb{T}}_L^3\times I$ and evaluation $(x,t)$; for a time-indexed spatial field $q$, write $q(x,t):=(q(t))(x)$. Let $u\in C^1(I;C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^3))$ be a periodic divergence-free velocity field measured in m s${}^{-1}$. Let $K:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ be smooth, dimensionless, supported in a compact subset of $B_1(0)$, and satisfy ${\int }_{{\mathbb{R}}^3}K(s)ds=1$. Define

$G_{\mathrm{\ell }}(r)={\sum }_{n\in {\mathbb{Z}}^3}{\mathrm{\ell }}^{-3}K\text{\hspace{0.17em}⁣}\left(\frac{r+Ln}{\mathrm{\ell }}\right),{\bar{u}}_{\mathrm{\ell }}(x,t)={\int }_{{\mathbb{T}}_L^3}{G}_{\mathrm{\ell }}(r)u(x-r,t)dr,$

where $dx$ is volume measure on a period cell. Thus ${\bar{u}}_{\mathrm{\ell }}:{\mathbb{T}}_L^3\times I\to {\mathbb{R}}^3$ is the resolved velocity field, with position in metres and velocity in m s${}^{-1}$. The periodic filter commutes with spatial derivatives, so ${\bar{u}}_{\mathrm{\ell }}$ is divergence-free. Let $k,{\nu }_t\in C^1({\mathbb{T}}_L^3\times I;[0,\mathrm{\infty }))$ be prescribed specific unresolved kinetic-energy and kinematic eddy-viscosity fields, measured in m${}^2$ s${}^{-2}$ and m${}^2$ s${}^{-1}$, respectively. Use $(\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}{)}_{ij}={\mathrm{\partial }}_j{\bar{u}}_{\mathrm{\ell },i}$, let $I_3$ be the $3\times 3$ identity matrix, and define $A:B={\sum }_{i,j=1}^3{A}_{ij}{B}_{ij}$ and $\mathrm{\mid }A{\mathrm{\mid }}_F^2=A:A$ for real $3\times 3$ matrices. Define

$D_{\mathrm{\ell }}=\frac{1}{2}(\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}+(\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}{)}^T)\mathrm{.}$

For the adopted closure, define the modeled specific stress and transfer by ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=2k{I}_3\mathrm{/}3-2{\nu }_t{D}_{\mathrm{\ell }}$ and ${\mathrm{\Pi }}_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=-{\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}$. At a point $(x_0,t_0)\in {\mathbb{T}}_L^3\times I$, suppose the velocity gradient is

$\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}(x_0,t_0)=\left(\begin{array}{ccc}0& S& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$

where $S\in \mathbb{R}$ is the local shear rate in s${}^{-1}$. Put $k_0=k(x_0,t_0)$ and ${\nu }_0={\nu }_t(x_0,t_0)$. Then the modeled transfer at $(x_0,t_0)$ is ${\nu }_0{S}^2$ m${}^2$ s${}^{-3}$. This is a pointwise calculation in the adopted closure model; it asserts no measured value, finite-time evolution, or sign for exact filtered transfer.

Axiom of Countable Choice

positive periodic convolution filter and subfilter stress

periodic filter normalization and derivative commutation

adopted isotropic eddy-viscosity closure