Geoscience Reference
In-Depth Information
In this limit the viscous term in the
u
i
equation of motion
(6.23)
is no longer
negligible. The term it produces in the resolvable TKE budget
(6.30)
converges to
−
and
Eqs. (6.17)
and
(6.30)
become identical.
6.3.2.3 The subfilter-scale TKE budget
Subtracting
Eq. (6.22)
for
u
i
from
Eq. (6.16)
for
u
i
yields the equation for
u
i
:
1
u
i,t
+
(u
i
u
j
)
,j
ρ
p
,i
+
νu
i,jj
.
=−
(6.32)
Multiplying by
u
i
, ensemble averaging, and rewriting the viscous term yields the
subfilter-scale TKE budget:
1
2
(u
i
u
i
)
,t
=−
ν
2
(u
i
u
i
)
,jj
−
u
i
(u
i
u
j
)
,j
+
νu
i,j
u
i,j
.
(6.33)
By using
u
i
=
u
i
we can write the final term, molecular destruction, as
u
i
−
νu
i,j
u
i,j
2
νu
i,j
u
i,j
+
νu
i,j
u
i,j
.
=
νu
i,j
u
i,j
−
(6.34)
The first term on the right is
. The second and third terms have had the most
dissipative eddies filtered out of one and both of their factors, respectively. As a
result these terms are quite small at large
R
t
, so we neglect them and write
νu
i,j
u
i,j
νu
i,j
u
i,j
=
.
(6.35)
This allows us to neglect the molecular diffusion term in
Eq. (6.33) (Problem 6.15)
.
Equation (6.33)
then becomes
1
2
(u
i
u
i
)
,t
=−
u
i
(u
i
u
j
)
,j
−
.
(6.36)
This is the simplified subfilter-scale TKE budget. The first term on the right side
must be a source term that balances viscous dissipation.
Let's summarize the TKE budgets for our equilibrium, homogeneous, large-
R
t
turbulence:
1
2
(u
i
u
i
)
,t
=
β
i
u
i
−
u
i
(u
i
u
j
)
,j
Resolvable-scale TKE:
=
0
,
(6.37)
1
2
(u
i
u
i
)
,t
=−
u
i
(u
i
u
j
)
,j
−
Subfilter-scale TKE:
=
0
,
(6.38)
1
2
(u
i
u
i
)
,t
=
TKE:
β
i
u
i
−
=
0
.
(6.17)
