Geoscience Reference
In-Depth Information
Figure 15.3 A schematic of the longitudinal and transverse correlation functions.
From
Batchelor
,
1960
.
15.4.2.2 The correlation tensor
Batchelor
(
1960
) was evidently the first to apply this analysis to the correlation
tensor
R
ij
in isotropic turbulence. As a second-order tensor function of a vector, it
has the isotropic form
R
ij
(
r
)
=
u
i
(
x
)u
j
(
x
+
r
)
=
α(r)r
i
r
j
+
β(r)δ
ij
,
(15.58)
with
α(r)
and
β(r)
to be determined. He expressed them in terms of the longitudinal
and transverse correlations
f
and
g
, respectively, sketched in
Figure 15.3
:
u
2
f(r)
α(r)r
2
u
2
g(r)
=
R
11
(r,
0
,
0
)
=
+
β(r)
;
=
R
11
(
0
,r,
0
)
=
β(r),
(15.59)
1
u
2
. Solving
(15.59)
for
α
and
β
yields
where by isotropy
u
p
=
u
n
=
3
u
i
u
i
=
u
2
f
,
−
g
u
2
g.
α
=
β
=
(15.60)
r
2
Thus, in terms of
f
and
g
the isotropic form of
R
ij
is
u
2
f(r)
g(r)δ
ij
.
−
g(r)
R
ij
(
r
)
=
r
i
r
j
+
(15.61)
r
2
The two unknown functions
f
and
g
are related through incompressibility. We
have
∂u
i
∂x
i
=
∂R
ij
∂r
j
0
=
,
(15.62)
and since
∂
∂r
i
=
∂r
∂r
i
∂
∂r
=
r
i
r
∂
∂r
,
r
2
=
r
i
r
i
,
(15.63)
this yields the constraint
ξ
∂α
1
r
∂β
∂r
=
∂r
+
4
α
+
0
.
(15.64)
