Geoscience Reference
In-Depth Information
6.3.2 TKE budgets for the resolvable and subfilter-scale motions
6.3.2.1 The decomposition of TKE
The decomposition
u
i
=
u
i
+
u
i
implies that TKE has three components,
(u
i
+
u
i
)(u
i
+
u
i
)
u
i
u
i
2
u
i
u
i
2
=
=
(
TKE of resolvable-scale motion
)
2
u
i
u
i
+
(
joint TKE
)
u
i
u
i
2
+
(
TKE of subfilter-scale motion
).
(6.28)
The cross term
u
i
u
i
vanishes with a wave-cutoff filter because the resolvable
and subfilter-scale fields have no Fourier modes in common and, hence, are
uncorrelated. This is a mathematical property of the Fourier representation of
homogeneous fields that is directly related to the property (discussed in
Chapter 2
,
Section 2.5.1
) that Fourier coefficients of different wavenumbers are uncorrelated.
We make th
is cu
rre
nt pr
op
erty
plausible in
Appendix 6.1
.
Thus, the wave-cutoff
filter yields
u
i
u
i
u
i
u
i
,
neatly decomposing the TKE into what we shall
call the
resolvable-scale
TKE and the
subfilter-scale
TKE.
u
i
u
i
+
=
6.3.2.2 The resolvable-scale TKE budget
Multiplying the filtered Navier-Stokes
equation (6.23)
by
u
i
and ensemble
averaging yields
1
1
2
(u
i
u
i
)
,t
=−
ρ
p
,i
u
i
+
β
i
u
i
−
u
i
(u
i
u
j
)
,j
.
(6.29)
Using the property
u
i,i
=
0 the pressure covariance here can be rewritten as
(p
r
u
i
)
,i
,
which vanishes by homogeneity. Thus, the resolvable-scale TKE budget of our
equilibrium, homogeneous turbulence is simply
1
2
(u
i
u
i
)
,t
=
β
i
u
i
−
u
i
(u
i
u
j
)
,j
=
0
.
(6.30)
Comparing
Eq. (6.30)
with the TKE
equation (6.17)
for the full motion shows
that the term
u
i
(u
i
u
j
)
,j
plays the same role for resolvable-scale TKE that viscous
dissipation plays for TKE.
As the filter scale
→
0, the resolvable-scale velocity approaches its unfil-
tered counterpart (
u
i
→
u
i
) and the rate-of-loss term in
Eq. (6.30)
vanishes by
homogeneity:
=
−
(u
i
u
i
u
j
)
,j
2
u
i
(u
i
u
j
)
,j
−
→−
u
i
(u
i
u
j
)
,j
=
0
.
(6.31)
