Isotropic eddy viscosity and unresolved energy model

Assume the Axiom of Countable Choice for the normalized periodic torus convention. Let $I\subseteq \mathbb{R}$ be an open time interval, let $L>0$ m, and let ${\mathbb{T}}_L^3=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$ have volume measure $dx$ on a period cell and normalized measure $d{\mu }_L=L^{-3}dx$. Coordinates $x,r,L,\mathrm{\ell }$ are measured in metres and time $t$ 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 ${\int }_{{\mathbb{R}}^3}K(s)ds=1$. For $0<\mathrm{\ell }<L\mathrm{/}2$, define

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

The kernel $G_{\mathrm{\ell }}$ has units m${}^{-3}$ and filtering preserves the units of $q$.

The physical base model is the smooth, unforced, constant-density incompressible Newtonian branch: ${\rho }_0>0$ kg m${}^{-3}$ and dynamic viscosity $\mu \ge 0$ Pa s are constant, $u\in C^1(I;C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^3))$ is periodic with $\mathrm{\nabla }\cdot u=0$ and velocity units m s${}^{-1}$, and the pressure multiplier is a periodic scalar field $p_{\mathrm{d}\mathrm{y}\mathrm{n}}\in C^1({\mathbb{T}}_L^3\times I;\mathbb{R})$ in Pa. Let $I_3$ be the $3\times 3$ identity matrix and set $D(u)=(\mathrm{\nabla }u+(\mathrm{\nabla }u{)}^T)\mathrm{/}2$; the dynamic Cauchy stress is $\sigma =-p_{\mathrm{d}\mathrm{y}\mathrm{n}}{I}_3+2\mu D(u)$, and $\nu =\mu \mathrm{/}{\rho }_0$ m${}^2$ s${}^{-1}$. The pressure per unit mass in the filtered equation is the filtered scalar field ${\pi }_{\mathrm{\ell }}={\overline{p_{\mathrm{d}\mathrm{y}\mathrm{n}}}}_{\mathrm{\ell }}\mathrm{/}{\rho }_0\in C^1({\mathbb{T}}_L^3\times I;\mathbb{R})$ in m${}^2$ s${}^{-2}$, where ${\overline{p_{\mathrm{d}\mathrm{y}\mathrm{n}}}}_{\mathrm{\ell }}(x,t)={\int }_{{\mathbb{T}}_L^3}{G}_{\mathrm{\ell }}(r)p_{\mathrm{d}\mathrm{y}\mathrm{n}}(x-r,t)dr$, and $\mathrm{\nabla }{\pi }_{\mathrm{\ell }}$ denotes its spatial gradient. The isotropic modeled term $2k{I}_3\mathrm{/}3$ remains in ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ below; it is not absorbed into ${\pi }_{\mathrm{\ell }}$.

For the filtered velocity ${\bar{u}}_{\mathrm{\ell }}$, define its rate-of-strain tensor 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${}^{-1}$. Use the Frobenius contraction $A:B={\sum }_{i,j=1}^3{A}_{ij}{B}_{ij}$ for real $3\times 3$ matrices.

Adopt the following large-eddy simulation (LES) stress-form closure. A closure here means a specified modeled stress substituted for the exact filtered covariance in the resolved momentum equation. Let $k,{\nu }_t\in C^1({\mathbb{T}}_L^3\times I;[0,\mathrm{\infty }))$ be prescribed model fields, where $k$ represents specific unresolved kinetic energy in m${}^2$ s${}^{-2}$ and ${\nu }_t$ is kinematic eddy viscosity in m${}^2$ s${}^{-1}$. The modeled specific subfilter stress and its modeled transfer are

${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=\frac{2k}{3}{I}_3-2{\nu }_t{D}_{\mathrm{\ell }},{\mathrm{\Pi }}_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}=-{\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}:\mathrm{\nabla }{\bar{u}}_{\mathrm{\ell }}\mathrm{.}$

The closed filtered momentum equation uses the sign convention

${\mathrm{\partial }}_t{\bar{u}}_{\mathrm{\ell }}+\mathrm{\nabla }\cdot ({\bar{u}}_{\mathrm{\ell }}\otimes {\bar{u}}_{\mathrm{\ell }})+\mathrm{\nabla }{\pi }_{\mathrm{\ell }}=\nu \mathrm{\Delta }{\bar{u}}_{\mathrm{\ell }}-\mathrm{\nabla }\cdot {\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}},\mathrm{\nabla }\cdot {\bar{u}}_{\mathrm{\ell }}=0.$

Both terms in ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ have units m${}^2$ s${}^{-2}$, and ${\mathrm{\Pi }}_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ has units m${}^2$ s${}^{-3}$. This is an adopted constitutive form, not a derivation of a physical law: $k$ and ${\nu }_t$ must be supplied as model inputs or by separate model equations. The prescription does not assert that ${\tau }_{\mathrm{\ell }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ equals the exact covariance ${\overline{u\otimes u}}_{\mathrm{\ell }}-{\bar{u}}_{\mathrm{\ell }}\otimes {\bar{u}}_{\mathrm{\ell }}$, that the modeled tensor is positive semidefinite, or that the closure is empirically adequate in every flow regime. No absence of backscatter for the exact filtered stress is asserted.

Axiom of Countable Choice

constant-density incompressible Newtonian model

positive periodic convolution filter and exact subfilter stress

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