ENERGY ('ASCADE IN LARGE-EDDY SIML'LATIONS
243
where
(4.10)
For a spherically symmetric filter, (4.9) reduces to
t:RS = -47$
1 G(r)J(r)r2 dr.
1
(4.1 1 )
'0
To proceed further we must obtain at least an approximate form for &(r).
This function is a scalar triple correlation having the basic definition
(4.12)
k(r) = (i;(x)i41(x + t1r)>h3.
(il: &1/2x) = f(?U:/(-'.UI>
= 0,
From symmetry rcquirements and the fact that
one can show that k has the small r expansion (Hinze, 1959),
(4.13)
pr3
I;(V$.S
3!
5 !
+ ....
0
0
k(r) =
+
1
this yields forf(r) the expansion
(4.14)
J ( r ) = f$P:r2 + QirEg'r" + *...
Assuming that the U, fluctuations include a portion of the inertial
subrange and the smallest scale in the subrange is 8 A, then k(r) is linear in
that subrange with a known coefficient (Kolmogorov, 1941).
(4.15)
k(r) = -2cr/15ij3
(A 6 r 4 Lo)
whcrc Lo is tlic scale of the energy-containing eddies. The resultant behavior
of,/'(r.) is
(4.16)
.f(r) = -c/ij3
(A 4 r 6 Lo).
Note that using the inertial subrange form off(r) in (4.9) yields cRS = E . This
is no accident. In fact 2Tj!f(r) represents the advective term in the KarmanHowarth equation (von Karmiin and Howarth, 1938), a dynamical equation
for the correlation Qi,(r) = (i7,(x)Ui(s + i, r ) ) , and in the inertial subrange
of the filtered flow this term is solely responsible for the energy loss of Q,,(r).
Thus 2$(r) = -2: for A 4 r 6 L o . Equation (4.15) then follows by using
(4.10).
The corrections to (4.16) for small r however will give cRS < c. This is
shown qualitatively in Fig. 3. The behavior indicated by curve A assumes
that the quadratic term
dominates the small r behavior up to the