In this equation, the terms on the right are negative when there is kinetic energy
loss and positive when there are gains. Term VII is the loss by viscous dissipation
of turbulence whenever the TKE is non-zero. Eq. (3.89) can be used to analyze the
various components terms of the TKE energy balance in the surface boundary layer,
where these terms are of greater magnitude, and therefore easier to measure, as can
be seen in Fig. 3.1. Under conditions of thermal neutrality, without energy production by buoyancy effects and when the pressure redistribution components add
up to zero, turbulent mechanical production in Eq. (3.89) is balanced by dissipation
viscous, e (Kaimal and Finnigan 1994):
e ¼ À À
u
0
1 u
0
3
@
À
u 1
@x 3
ð3:90Þ
Under steady-state conditions @
À
e =@t
À
Á ¼ 0, Eq. (3.89) can be written as the sum
of dimensionless parameters. Multiplying the terms of this equation by
k z À d
ð
Þ=u
3
Ã
À
Á
gives (Kaimal and Finnigan 1994):
0 ¼ À
z À d
L
À
kðz À dÞ
u 3
Ã
d u 0
3 e
À Á
@x 3
À
kðz À dÞ
u 3
Ã
1
q
@ u 0
3 p 0
À
Á
@x 3
þ
kðz À dÞ
u Ã
@u 1
@x 3
À
II
III
IV
V
À
k(z À dÞe
u 3
Ã
VI
ð3:91Þ
0 ¼ À
z À d
L
À / t þ / p þ / M À / e
II
III IV
V VI
ð3:92Þ
wherein L is the Monin–Obukhov length, and the remaining terms are given by
(II)
z À d
L
¼ n ¼ À
g
h
u 0
3 h
0
u 3
à =k z À d
ð
Þ
ð3:93Þ
(III)
/ t ¼
Àn n 0
0
n [ 0
&
ð3:94Þ
3.5 Introduction to Turbulent Motion Equations
57