Geoscience Reference
In-Depth Information
14.3.2 Two-point tensors
Robertson's technique shows that the following two-point tensors have the isotropic
forms (
Batchelor
,
1960
)
a(
x
)b(
x
+
r
)
=
F(r), r
=|
r
|;
c(
x
)u
i
(
x
+
r
)
=
G(r)r
i
;
(14.9)
u
i
(
x
)u
j
(
x
+
r
)
=
H(r)r
i
r
j
+
I(r)δ
ij
;
φ
ij
(
κ
)
=
J(κ)κ
i
κ
j
+
K(κ)δ
ij
,κ
=|
κ
|
.
Here the unknowns
F,G,H,I,J,K
are functions of the magnitude of the vector
on which the tensor depends.
We'll show next that we can determine some of the unknowns in the isotropic ten-
sor forms by using physical constraints such as incompressibility and homogeneity
and geometric constraints such as the symmetries of the tensor.
14.4 Implications of isotropy
14.4.1 Rate of shear production of TKE
Equation (14.5)
for an isotropic second-order tensor indicates that the kinematic
Reynolds stress tensor in isotropic turbulence is
u
j
u
j
3
u
i
u
k
=
δ
ik
,
(14.10)
so the off-diagonal Reynolds stresses vanish. Thus in isotropic, incompressible
turbulence the rate of shear production of TKE,
Eq. (5.45)
, vanishes:
−
u
i
u
k
∂U
i
u
j
u
j
3
δ
ik
∂U
i
u
j
u
j
3
∂U
i
∂x
k
=−
∂x
k
=−
∂x
i
=
0
.
(14.11)
Under isotropy the rate of buoyant production of turbulence also vanishes
(
Problem
14.20)
. Thus, unless it is driven in some other way, such as by stochastic, isotropic
forcing in numerical simulation experiments, constant-density isotropic turbulence
decays because its rate of TKE production vanishes.
14.4.2 Rates of molecular destruction of covariances
The rates of molecular destruction of
u
i
u
k
and TKE, respectively, are
u
i
u
k
:2
ν
∂u
i
∂x
j
∂u
k
∂x
j
=
ν
∂u
i
∂x
j
∂u
i
∂x
j
=
2
νM
ij kj
,
=
νM
ij ij
,
(14.12)
