Large-eddy dynamics, the energy cascade, and large-eddy simulation - Turbulence in the Atmosphere - page 132

u i u j or ( u i u j ) r is calculated as the solution of the filtered
equations (6.64) or (6.67) proceeds. If the filter scale is much less than the turbu-
lence scale , these can contain virtually all of the turbulent flux. This can give LES
a strong advantage over computational approaches that model the turbulent flux.
The subfilter-scale fluxes in LES are important, even in cases where they aremuch
smaller than the resolved fluxes, because they are intimately involved in interscale
transfer (Problem 6.16) . Thus, before the set (6.64) or (6.67) can be solved its
subfilter-scale fluxes must be represented through a subfilter-scale model .

In LES the quantity

6.6.2 Conservation equations for subfilter-scale quantities
One can derive the conservation equations for the subfilter-scale fluxes from their
definitions in Eq. (6.64) or (6.67) . We'll work with (6.64) and assume the Reynolds
number of the motion at the cutoff scale is large enough that the molecular diffusion
terms are negligible.

6.6.2.1 Stress
It is convenient to work with the modified equation of motion

p r
ρ + 3 e

2

u i,t + u i ˜ u j ) ,j τ ij,j , ˜ ( ˜ =−

,i +

2
3 eδ ij ;

where τ ij u i ˜ u j − u j ) r u i ) r u i ˜ u i . =˜ ( u i ˜ ˜ + 2 e = ( u i ˜ ˜ −˜ (6.69)

We use the superscript d because τ ij is a deviatoric kinematic stress tensor, the
difference between the kinematic subfilter-scale stress tensor u i ˜ u j − u j ) r and its ˜ ( u i ˜ ˜

isotropic form 2 eδ ij / 3; e is the subfilter-scale TKE. This deviatoric representation

facilitates the modeling of subfilter-scale stress.
The basis of the derivation is the property

−

r

(

u j

u j
∂t −˜

∂ ˜

u i ∂

∂t

u i ∂

u j
∂t +˜

˜

u j ∂

u i
∂t

˜ ∂ ˜

u j ) r u i ˜ u i u j u i ˜ ˜ −˜ = ˜ −˜

∂t ,

(6.70)

u i . The result for τ ij ,asdefined
in Eq. (6.69) ,is( Lilly , 1967 ; Hatlee and Wyngaard , 2007 )

in which one then uses the evolution equation for

(

∂τ ij
∂t +˜

∂τ ij
∂x k =

∂
∂x k

u k u k ) r u i ( u k ) r u j ( u k ) r u k ( u j ) r u i ˜ u j ˜ u k

u i ˜ ˜ u j ˜ −˜ u j ˜ ˜ −˜ u i ˜ ˜ −˜ u i ˜ ˜ + 2 ˜

u k

u k r u k u l r

2 ˜ u l 2

δ ij
3

u l ˜ u l ( u k ) r − ˜ − 2 ˜ u l ˜ ˜ −˜ ˜ + ˜