Equation (3.55) can now be used for calculating variance for the flow components. Multiplying the terms of this equation with 2u i ' gives
2u
0
i
@ u
0
i
@t þ 2u
0
i u j
@ u
0
i
@x j
þ 2u
0
i u
0
j
@u i
@x j
þ 2u
0
i u
0
j
@ u
0
i
@x j
¼ 2u
0
i d i3
h
0
v
h v
g þ
þ 2f c e ij3 u
0
i u
0
j À 2
u
0
i
q
@p
0
@x i
þ 2u
0
i v
@
2 u
0
i
@x 2
j
þ 2u
0
i
@ u 0
i u 0
j
ð Þ
@x j
ð3:61Þ
Now using the rule for the derivative product to convert 2u i ' ∂u i '/∂t into ∂(u i ')
2 /
∂t, gives
@ u
0
i
ð Þ
2
@t
þ u j
@ u
0
i
ð Þ
2
@x j
þ 2u
0
i u
0
j
@u i
@x j
þ u
0
j
@ u
0
i
ð Þ
2
@x j
¼ 2u
0
i d i3
h
0
v
h v
g þ
þ 2f c e ij3 u
0
i u
0
j À 2
u
0
i
q
@p
0
@x i
þ 2u
0
i v
@
2 u
0
i
@x 2
j
þ 2u
0
i
@ u
0
i u
0
j
À Á
@x 2
j
ð3:62Þ
Applying averages for the terms in Eq. (3.62) we have
@ u
0
i
ð Þ
2
@t þ u j
@ u
0
i
ð Þ
2
@x j
þ 2u
0
i u
0
j
@u i
@x j
þ u
0
j
@ u
0
i
ð Þ
2
@x j
¼ 2u
0
i d i3
h
0
v
h v
g þ
þ 2f c e ij3 u
0
i u
0
j À 2
u
0
i
q
@p
0
@x i
þ 2u
0
i v
@ 2 u
0
i
@x 2
j
þ 2u
0
i
@ u
0
i u
0
j
À Á
@x 2
j
ð3:63Þ
The last term of Eq. (3.63) is zero because the mean for u
0
i is zero. Adding the
term u 02
i @u 0
j =@x j = 0 to the left side of the equation, the last term of this side can be
written as @ u 0
j u 02
i
=@x j
@ u 0
i
ð Þ
2
@t þ u j
@ u 0
i
ð Þ
2
@x j
þ 2u 0
i u 0
j
@u i
@x j
þ
@ u 0
j u 02
i
ð
Þ
@x j
¼
¼ 2d i3 u 0
i
h
0
v
h v
g þ 2f c e ij3 u 0
i u 0
j À 2 0
u i
q
@p0
@x i
þ 2v u 0
i
@ 2 u 0
i
@x 2
j
ð3:64Þ
Equation (3.64) is the general equation for the variance of the wind speed
components, u 0
i
ð Þ
2 . For an expedite characterization of turbulent flow in the
boundary layer this equation may be simplified in several ways. For the last term on
the right-hand side of Eq. (3.64), the following equality can be obtained (Stull
1994):
2u 0
i v
@ 2 u 0
i
@x 2
j
¼ v
@ 2 u 0
i
ð Þ 2
@x 2
j
À 2v
@ u 0
i
@x j
2
ð3:65Þ
From the development of the equation:
3.5 Introduction to Turbulent Motion Equations
47