3.5.2.2 Equation for Linear Momentum Conservation
Equation (3.53) results from the generalization of Eq. (3.48) for linear momentum
conservation for turbulent flow. It considers the decay of velocity components, the
average values, and fluctuations to quantify the increase in motion quantities
formed in the control volume because of mechanical or thermal turbulence.
Then, expanding the velocity and pressure variables into the mean and turbulent
components gives:
@ u i þ u
0
i
ð
Þ
@t
þ u j þ u
0
j
@ u i þ u
0
i
ð
Þ
dx j
¼ Àd i3 g À
h
0
v
h v
þ f c e ij3 u j þ u
0
j
À
À
1
q
@ p þ p
0
ð
Þ
@x i
þ v
@
2 u i þ u
0
i
ð
Þ
@x 2
j
ð3:54Þ
or
@u i
@t þ
@u
0
i
@t þ u j
@u i
@x j
þ u j
@u
0
i
@x j
þ u
0
j
@u i
@x j
þ u
0
j
@u
0
i
@x j
¼ Àd i3 g þ d i3 g
h v 0
h v
þ
þ f c e ij3 u j þ f c e ij3 u
0
j À
1
q
@p
@x i
À
1
q
@p
0
@x i
þ v
@
2 u i
@x 2
j
þ v
@
2 u
0
i
@x 2
j
ð3:55Þ
Averaging Eq. (3.55) gives u i
@u i
@t þ
@u 0
i
@t þ u j
@u i
@x j
þ u j
@u
0
i
@x j
þ u
0
j
@u i
@x j
þ u
0
j
@u
0
j
@x j
¼ Àd i3 g þ d i3 g
h
0
v
h v
þ
þ f c e ij3u j þ f c e ij3 u 0
j À
1
q
@p
@x i
þ
1
q
@p 0
@x i
þ v
@ 2 u j
@x 2
j
þ v
@ 2 u
0
j
@x 2
j
ð3:56Þ
The means for the terms with fluctuations are zero, so that Eq. (3.55) can be
written as
@ u i
@t
þ u j
@ u i
@x j
þ u 0
J
@u 0
l
@x j
¼ À@ i3 g þ f c e l3 u j À
1
q
@ p
@x i
þ v
@
2
u i
@x j
ð3:57Þ
Then, by multiplying the continuity equation with the turbulent motions
Eq. (3.53) for u i ', and using the means gives
u
0
j
@u
0
j
@x j
¼ 0
ð3:58Þ
and by adding Eq. (3.58) to the third term on the left of Eq. (3.57) gives (Stull
1994):
@u i
@t
þ u j
@u i
@x j
þ
@ u 0
i u 0
j
@x j
¼ d i3 g þ f c e ij3 u j À
1
q
@p
@x i
þ v
@
2 u i
@x 2
j
ð3:59Þ
3.5 Introduction to Turbulent Motion Equations
45