Geoscience Reference
In-Depth Information
If we now integrate
Eq. (5.51)
over a volume the divergence terms on the right
become integrals over the bounding surface. In a homogeneous flow the first surface
integral vanishes and we have
∂
∂t
U
i
U
i
2
1
ρ
dV
=−
PU
n
dσ
+
u
i
u
j
S
ij
dV.
(5.52)
volume
surface
volume
Equation (5.52)
says that the time rate of change of volume-integrated MKE is due
to an imbalance between its rate of gain through mean pressure differences and its
rate of loss by working against turbulent stress in producing TKE
(Problem 5.22)
.
The latter is ultimately balanced by the rate of gain of internal energy of the fluid
by viscous dissipation of TKE in the smallest eddies.
5.5.3 Insights into pressure covariances
5.5.3.1 Role in budgets of stress and scalar flux
In a
steady, horizo
nta
lly homogeneous boundary-layer flow the budgets
(5.41)
of
uw
and
(5.40)
of
cw
reduce to balances of mean-gradient production, turbulent
transport, and pressure-gradient interaction:
u
∂p
,
∂uw
2
∂z
∂uw
∂t
w
2
∂U
1
ρ
w
∂p
∂x
=
0
=−
∂z
−
−
∂z
+
(5.53)
c
∂p
∂z
.
∂cw
2
∂z
∂cw
∂t
w
2
∂C
1
ρ
=
0
=−
∂z
−
−
Turbulent transport integrates to zero over the whole flow, so the steady, global
balance is between mean-gradient production and pressure destruction. This was
first confirmed observationally in the 1968 Kansas experiments with temperature
as the scalar; the pressure covariances were inferred from the imbalance of the
measured terms (
Wyngaard
et al
.
,
1971
). It has since been verified by direct mea-
surements of the pressure covariance (
Wilczak and Bedard
,
2004
). Physically, this
pressure covariance can be interpreted as a rate of production of opposite-signed
flux (
Question 5.12
).
5.5.3.2 TKE budget
Equation (5.41)
contains the component TKE equations for a horizontally homo-
geneous, quasi-steady boundary-layer flow. They are particularly revealing if we
rewrite the pressure covariance as
∂p
∂x
α
=
∂
∂x
α
u
α
p
p
∂u
α
u
α
−
∂x
α
,α
=
1
,
2
,
3
,
no sum on
α,
(5.54)
