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 .
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