242
A. LfONhRI>
Assumiiig that the influencc of the liiolecular viscosity term is negligible.
energy loss of the large scales will occur through the action of the nonlinear.
resohable-scale and siibgrid scale terms t 7 ( i i iij )Pxj and ?ri,i/?.vj, respectively. The rate of energy cascade due to the former we denote cW and to thc
latter. r5sos. If I: is the total loss rate. then
(4.1)
I: = I:RS t tk(;s
Multiplying Eq. (3.X) by ui and volume averaging we find that
(4-2 1
ERS = (Ili f7(idiuj)/?xj)
(4.3)
pSas = (idi ?tij/?sj>
wlictc ( :. dcnotes voluine averaging
(4.4)
( h ) = L'.h ( x ) d x
J"
Wc concentrate an the evaluation of cRS. In terms of the triple correlation
tensor, defined by
(4.5)
s i k . j ( s ) = < u i ( x ) ~ k ( x ) i r j ( x + 6)).
(ill1 overbar denotes B quantity corresponding to the filtered flow field 7ii) cRS
can he written as
(4.6)
Rtcausc of the presence of the lilter G. the behavior of Si,,i only in the
donisin IS 1 5 A is important. We assume that the velocity fluctuations in
this range of scales arc homogeneous and isotropic in which case Sl&) can
bc written in terms of a single scalar function Qr) (I' = 15 I ) (Hinze, 1059).
(4.7)
wherc (I' is the filtered turbulence intensity
(4.8)
42 = ( i i l G i ) .
Performing the differentiations and contractions required by (4.6). we find
that