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Turbulence
Lecture 11
The Reynolds Stresses
There are three stresses in the Reynolds averaged mean momentum equation:
 Viscous

Isotropic

Apparent
j
i
ij
i
U
U
uP
xxx
ρ
∂
∂
∂
+−
−
∂∂∂
u
u
The apparent stress arises from momentum transferred by fluctuating the velocity field.
Consider a Control Volume (
V
)
With C.S.,
A
, and outward normal
.
ˆ
n
Rate of gain of momentum in
V
is due to flow through of
A
.
N
()
ˆ
A
M
UU
n
d
A
t
∂
=−
∂
∫∫
±±
i
±
²³´³µ
momentum
flow rate
per unit
per unit
volume.
area through
A
The j
th
component of the mean
j
M
is
ji
j
i
j
A
i
j
i
V
i
M
uu ndA
t
uu dV
x
∂
+
∂
∂
+
∂
∫∫∫
by the divergence theorem.
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View Full Document So the Reynolds stress
ij
i
uu
x
ρ
∂
−
∂
comes from the mean momentum flux due to the fluctuating
velocity on the boundary
A
()
.
iji
uu n
−
Tensor Properties
Reynolds stress is a 2
nd
order symmetry tensor.
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This note was uploaded on 06/07/2011 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.
 Spring '09
 MEI
 Stress

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