Environmental Engineering Reference
In-Depth Information
Fig. 7.3
Unstable stratification: Bin-average values of
z
0
u
/
z
0
versus empirical stratification
18
−
32
)
h
0
/
U
32
, for the city of Basel with the typical height of build-
parameter Ri
=
(
g
/
32
)(
ings
h
0
∼
14.6m and the neutral-stability roughness length
z
0
≈
1.2
±
0.4 . The bars are standard
1.23 Ri
3/14
, which corresponds to the theoretical 1/3 power-law
errors. The curve is
z
0
u
/
z
0
=
1
+
L
)
1
/
3
dependence:
z
0
u
/
z
0
=
1
+
1.24(
−
h
0
/
decreases; but in the “surface layer” (5
h
0
<z< 10
−
1
h
)it
can be taken height-constant:
mass,
τ
. As z increases,
τ
u
2
∗
τ
≈
τ
|
z
=
5
h
0
≡
, whereas
u
serves as a turbulent
∗
velocity scale:
u
T
∼
.
Given
l
T
and
u
T
, the eddy viscosity,
K
M
∼
u
∗
u
T
l
T
and the velocity gradient,
∂
U
/∂
z
,
become
K
M
=
ku
∗
z
,
(7.1a)
∂
U
/∂
z
=
τ/
K
M
=
u
∗
/
kz
,
(7.1b)
where
k
0.4 is the von Karman constant. Then integrating (7.1b) involves an
integration constant:
U
≈
k
−
1
u
=
ln
z
+
constant, or equivalently
∗
u
z
z
0
u
k
U
=
ln
,
(7.2)
where
z
0
u
(redefined constant of integration) is just the “roughness length”.
1
The
above analysis is not justified in the “roughness layer” (0 <
z
<5
h
0
) and not consis-
tent with the non-slip boundary condition:
U
=
0at
z
=
0. However, if
z
0
u
is known,
1
To achieve better accuracy, (7.2) is often modified by displacing the vertical axis:
U
=
k
−
1
u
∗
ln
(
z
z
u
0
,where
d
0
u
is a fitting parameter called “displacement height”. Our
analyses did not disclose pronounced effect of stratification on
d
0
u
and basically confirmed the
traditional estimate
d
0
u
≈
−
d
u
0
)
/
h
0
.
Search WWH ::
Custom Search