Geoscience Reference
In-Depth Information
Each of two terms contributing to
I
is a contraction of a kinematic stress tensor
and a strain-rate tensor. This is a familiar form: we saw it in the shear-production
term of the TKE budget,
Eq. (5.42)
, where it represents the mean rate of loss of
kinetic energy of the mean flow by transfer to the turbulence. We saw it also in the
viscous dissipation term of the TKE budget
(Problem 1.7)
, where it represents the
mean rate of loss of TKE through conversion into internal energy. Guided by those
examples, and seeing its role in
Eq. (6.40)
, we interpret
I
as the net mean rate of
loss of resolvable-scale TKE due to two effects: t
he work
ing of the subfilter-scale
stresses against the resolvable-scale strain rate
u
i
u
j
s
ij
, and the working of the
−
resolvable-scale stresses against the subfilter-scale strain rate
u
i
u
j
s
ij
.
6.3.4 The flow of turbulence kinetic energy from larger scales to smaller
By adding themwe could summarize our TKE budgets
(6.40)
and
(6.41)
in a single
equation:
u
i
u
i
u
i
u
i
1
2
1
2
1
2
(u
i
u
i
)
,t
=
β
i
u
i
−
,t
+
,t
=
.
(6.44)
But this does not reveal the subtlety of the full set. The TKE of the large-scale
turbulence field is kept in balance not by viscous dissipation, as the summary
equation (6.44)
might suggest, but by its mean rate of TKE transfer to the subfilter
scales. In steady turbulence these subfilter scales, in turn, dissipate TKE at the same
mean rate per unit mass.
Alarge-
R
t
turbulent velocity field has a wide range of eddies that are small
compared to
but large compared to
η
. These eddies contain negligible TKE,
carry negligible flux, contribute negligibly to TKE production, and do negligible
viscous dissipation. But they are an essential link between the energy-containing and
dissipative eddy ranges. They receive their energy by transfer from the resolvable
eddies and in turn transfer energy to subfilter eddies. This range of intermediate
eddies, called the inertial subrange, has a mean rate of transfer of kinetic energy
per unit mass through it that is independent of scale and numerically equal to
.We
call this
spectral
energy transfer.
6.3.5 The flow of scalar variance from larger scales to smaller
Imagine our forced, equilibrium, homogeneous turbulence advecting a conserved
scalar having a stochastic, homogeneous, stationary source term
S(
x
,t)
chosen such
that the scalar field is homogeneous and steady
(Problem 6.13)
.If
S
is confined
to the larger scales and passes perfectly through the filter, the resolved part of the
scalar follows
c
,t
+
(u
j
c)
,j
=
γc
,jj
+
S.
(6.45)
