Geoscience Reference
In-Depth Information
τ
ik
s
kl
˜
s
ij
r
u
j
∂x
k
+
∂
˜
u
i
p
r
ρ
˜
2
e
3
˜
∂
˜
1
3
δ
ij
τ
kl
˜
p
ρ
˜
˜
s
ij
−
τ
jk
s
ij
+
∂x
k
−
−
+
δ
ik
(
p
r
.
p
r
δ
jk
(
p
r
−
3
δ
ij
(
1
ρ
∂
∂x
k
2
p)
r
u
j
˜
p)
r
u
i
˜
p)
r
u
k
˜
+
u
j
˜
˜
−˜
+
u
i
˜
˜
−˜
u
k
˜
˜
−˜
(6.71)
The terms on the left side represent local time change and advection by the resolved
velocity. The first term on the right is a divergence, so it integrates to zero over
the flow; this is a transport term. The second and third terms on the right represent
interactions between
τ
ij
and the resolved strain rate. The first of this pair, a
gradient
production term, produces
τ
ij
that is aligned with this strain rate. The second of
this pair is a
tilting
production term that reorients
τ
ij
. The fourth and fifth terms are
pressure destruction, and the sixth term is pressure transport.
6.6.2.2 TKE
Lilly
(
1967
) also derived the equation for the evolution of
e
−
˜
u
i
/
2
,
u
i
)
r
u
i
˜
=
(
u
i
˜
˜
the TKE of the subfilter-scale motion. It reads
(
τ
ij
2
u
i
)
r
u
k
( u
i
u
i
)
r
2
∂e
∂t
+˜
∂e
∂x
k
=
∂
∂x
k
u
k
˜
˜
u
i
˜
u
k
s
ij
−
u
i
(
u
i
)
r
˜
−
−˜
u
k
˜
˜
2
(6.72)
u
k
˜
p
r
ρ
p)
r
ρ
(
u
k
˜
˜
˜
u
i
˜
u
i
˜
u
k
+
+˜
−
−˜
.
On the left are local time change and advection; on the right are shear production,
a pair of transport terms, and viscous dissipation.
6.6.2.3 Scalar flux
The conservation equation for the subfilter-scale scalar flux
f
i
, which appears in
Eq. (6.64)
,is(
Wyngaard
,
2004
;
Hatlee and Wyngaard
,
2007
)
∂f
i
∂t
+˜
∂f
i
∂x
j
u
j
(
u
j
∂
∂x
j
u
j
)
r
u
i
)
r
u
j
−˜
c
r
(
u
j
)
r
u
i
(
u
j
)
r
c
r
u
i
˜
+
c
˜
u
i
˜
˜
−
(
c
˜
˜
˜
u
i
˜
˜
−˜
c
˜
˜
+
2
˜
˜
r
,
∂x
i
(
c
r
=−
u
i
c
r
∂x
j
+
c
r
∂x
i
f
j
∂
˜
1
ρ
∂
R
ij
∂
˜
1
ρ
p
∂
c
∂x
i
˜
p
r
∂
˜
c)
r
p
r
+
p
˜
˜
−˜
˜
∂x
j
−
˜
−˜
u
j
)
r
u
i
˜
u
j
.
R
ij
=
(
u
i
˜
˜
−˜
(6.73)
The interpretation of the terms here is analogous to that for deviatoric stress: on
the left side are local time change, advection by the resolved velocity, turbulent
