Resolved dissipation of adopted eddy viscosity

Assume the Axiom of Countable Choice for the normalized periodic torus convention. Let $I\subseteq \mathbb{R}$ be an open time interval, $L>0$ m, $0<\mathrm{\ell }<L\mathrm{/}2$, and ${\mathbb{T}}_L^3=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$. Spatial coordinates are in metres and time is in seconds. 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 $K:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ be smooth, dimensionless, supported in a compact subset of $B_1(0)$, and satisfy $\int K(s)ds=1$. Define $G_{\mathrm{\ell }}(r)={\sum }_{n\in {\mathbb{Z}}^3}{\mathrm{\ell }}^{-3}K((r+Ln)\mathrm{/}\mathrm{\ell })$ and ${\bar{u}}_{\mathrm{\ell }}(x,t)={\int }_{{\mathbb{T}}_L^3}{G}_{\mathrm{\ell }}(r)u(x-r,t)dr$, using volume measure on a period cell. Let $u\in C^1(I;C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^3))$ be periodic and divergence-free, with constant density ${\rho }_0>0$ kg m${}^{-3}$ and constant dynamic viscosity $\mu \ge 0$ Pa s. Put $\nu =\mu \mathrm{/}{\rho }_0$ m${}^2$ s${}^{-1}$ and let $\pi =p_{\mathrm{d}\mathrm{y}\mathrm{n}}\mathrm{/}{\rho }_0$ be pressure per unit mass in m${}^2$ s${}^{-2}$. Assume the unforced incompressible Newtonian equation

${\mathrm{\partial }}_{t}u+\mathrm{\nabla }\cdot (u\otimes u)+\mathrm{\nabla }\pi =\nu \mathrm{\Delta }u,\mathrm{\nabla }\cdot u=0.$

For this filter define the exact covariance stress and transfer by ${\tau }_{\mathrm{\ell }}={\overline{u\otimes u}}_{\mathrm{\ell }}-{\bar{u}}_{\mathrm{\ell }}\otimes {\bar{u}}_{\mathrm{\ell }}$ and ${\mathrm{\Pi }}_{\mathrm{\ell }}=-{\tau }_{\mathrm{\ell }}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}$. For real $3\times 3$ matrices set $A:B={\sum }_{i,j=1}^3{A}_{ij}{B}_{ij}$ and $\mathrm{\mid }A{\mathrm{\mid }}_F^2={\sum }_{i,j=1}^3\mathrm{\mid }{A}_{ij}{\mathrm{\mid }}^2$.

Use the positive smooth unit-mass periodic convolution filter of width $\mathrm{\ell }$ to define ${\bar{u}}_{\mathrm{\ell }}$. Define the resolved rate-of-strain matrix by

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

The matrix $D_{\mathrm{\ell }}$ has units s${}^{-1}$; let $I_3$ be the $3\times 3$ identity matrix.

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, with units m${}^2$ s${}^{-2}$ and m${}^2$ s${}^{-1}$, respectively. Adopt the modeled specific stress ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=2k{I}_3\mathrm{/}3-2{\nu }_t{D}_{\mathrm{\ell }}$ as the replacement for the exact filtered covariance in the resolved model equation, and define ${\mathrm{\Pi }}_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=-{\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}$. Then, at every $(x,t)\in {\mathbb{T}}_L^3\times I$,

${\mathrm{\Pi }}_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(x,t)=2{\nu }_t(x,t)\mathrm{\mid }{D}_{\mathrm{\ell }}(x,t){\mathrm{\mid }}_F^2\ge 0,{\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}=-2{\nu }_t\mathrm{\mid }{D}_{\mathrm{\ell }}{\mathrm{\mid }}_F^2\le 0.$

If additionally $\nu >0$ and the specific pressure $\pi$ is smooth, the exact filtered-covariance energy theorem applies to this smooth solution. Its resolved-energy balance contains the exact stress contraction $+{\tau }_{\mathrm{\ell }}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}$ on the right-hand side. Replacing that stress slot by ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ in the adopted closure model therefore contributes the displayed nonpositive term to the modeled balance. This is a conditional model deduction; it does not identify the modeled stress with the exact covariance or assert a sign for exact filtered transfer.

Axiom of Countable Choice

constant-density incompressible Newtonian model

positive periodic convolution filter and exact covariance stress

filter derivative commutation and periodic normalization

adopted LES stress-form closure

exact filtered covariance and resolved-energy balance

University of Washington, ME 544, Large-Eddy Simulation lecture notes, §3, printed pp. 10–12, Eqs. (16)–(24), and §4, printed p. 14, Eq. (30)

Jeremy A. Gibbs, LES of Turbulent Flows: Lecture 12, printed pp. 7–8 and 17–24

Glenn Flierl and Raffaele Ferrari, MIT OpenCourseWare 12.820, Turbulence in the Ocean and Atmosphere, Chapter 3, §§3.6–3.8, printed pp. 9–11