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Effect of Turbulent Fluctuations on Mean Flow: ReynoldsAveraging
In a turbulent flow, we can define the mean, steady flow as:
0
1
(, ,) l
im
(, ,,)
T
T
uxyz
uxyztd
t
T
→∞
=
∫
This allows us to split the flow properties into a mean and a fluctuating part:
mean
turbulent
part
fluctuating
part
(,,,)
(,,)
uxyzt
uxyz u xyzt
vxyzt
vxyz v xyzt
wxyzt
wxyz
w xyzt
pxyzt
pxyz
p xyzt
′
=+
′
′
′
±²³ ²
´±
²
³
²
´
Note: the mean of
u
′
is zero:
{}
00
11
lim
lim
TT
uu u
u
u dt
u dt
→∞
→∞
′
−=
′
∫∫
0
1
lim
T
T
u
udt
u
u
T
→∞
′
∫
±²³²´
⇒
N
0
uuu
=
′
⇒
0
u
′
=⇐
mean of fluctuations is zero.
Now, we will develop equations which govern the mean flow and try to develop
some insight into how the fluctuations alter the mean flow equations.
Let’s start with incompressible flow and look at the
−
x
momentum:
−
x
momentum:
222
1
u
p
u
u
u
uvw
txyz
x
x
y
z
ν
ρ
∂
∂
∂
∂
∂
∂∂∂
+++
=
−
+
+
+
∂
∂
∂
∂∂
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 Fall '03
 willcox
 Dynamics, Aerodynamics

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