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Hydrodynamic Stability
Lecture 29
Consider the Reynolds stress of a perturbation….
()
2
1
Re
0 & B.C.'s.
Assume 2D, periodic in .
,
where is averaged over wavelength so
changes as
grows.
uu u
p
u
t
u
x
uxt
u u
u
u
u
u
∂
+∇=
−
∇+ ∇
∂
∇=
′
=+
′
i
±± ±
±
i
±
±±
± ±
±
Analagous to Reynolds avg. in turbulence.
Form an equation for
by averaging eqns. for
±
uu
.
u
 initially the basic flow function of
( )
,
zt
averaged in
x
already.
±
Take the product
2
... gives an energy equation for
.
2
i
u
t
′
∂
′′
+
∂
i
Integrate over one wavelength on
[ ]
12
and
,
x
zz
() ()
2
22
rate of change of KE
exchange of perturbation KE with
this last term is the rate of dissipation
background flow
of perturbat
11
2R
e
dU
w
u
u
w
dxdz
u w
dxdz
dxdz
td
z
∂∂
′
′
+=
−
− −
∫∫
²³³³´ ³ ³ ³µ
²³³³´³³³µ
ion KE 0
<
x
z
∂
∂
²³³³³´³³³³µ
The instability can grow
only if
the first term on RHS is positive.
A lot depends on this term, the exchange of perturbation energy with the mean flow.
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 Fall '08
 Staff

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