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Hydrodynamic Stability
Lecture 30
Consider the structure of a growing disturbance in a parallel shear flow.
Streamlines are useful for visualization.
Streamline pattern near critical level.
[]
()
,,
uUuw
zx
ψ
ψψ
∂∂
′′
′
′
=+
=
+ −
±
±
Transform to coordinate system moving with speed
C
.
r
{}
() (
)
2
2
1
H.O.T.
2
Re
for a fixed .
Near critical level, equation for streamline is:
1
cos
...
Const.
2
Plot this 
B
B
r
z
rB
z
z
ix
BB
B
UC uw
U
C
dz
u
z
z
Az
e
t
Uzz
A z
x
α
ψφ
φα
γ
′
′
−+
=
+
′
=− =
′ =
′
+
+
=
∫
±
±
i
Cat’s eye pattern, right near critical level
1
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View Full DocumentClasses of solutions to the Rayleigh equation.
1. For neutral modes – solutions using Drazin’s Method
a. Classes of exact solutions:
hyperbolic profiles, etc.
2. Solutions for broken line profiles
3. Asymptotic methods, e.g. small
, large
α
λ
4. Numerical methods (for pretty arbitrary profiles).
Spatial Modes
()
Consider
where
where
is real,
is complex.
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 Fall '08
 Staff

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