Hydrodynamic Stability
Lecture 27
We derived the OrrSommerfeld equation previously.
(
)
2
2
1
2
Re
IV
U
C
U
i
4
φ
α φ
φ
φ
α φ
α φ
α
′′
′′
′′
−
−
−
=
−
+
It is a 4
th
order ODE, so we need 4 B.C.’s Rigid Walls (top and bottom).
0 at boundaries
or Infinite domain
,
0 as
Again an eigenvalue Problem for
 eigen vector
 eigenvalue
z
C
αφ
φ
αφ φ
φ
′
=
=
′ →
→ ∞
Dispersion relations enter in, seek dispersion relation of the form
(
)
,
,Re
0
F C
α
=
Some general results for Inviscid flows.
Usually restricted to high Re.
We expect the instabilities to occur for Re large.
Take
1
0
Re
→
in OrrSommerfeld eqn.
1
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Useful for freeshear flows but not for boundary layers. (need full OS eqn. Near boundaries).
Gives:
(
)
(
)
2
0
U
C
U
φ
α φ
φ
′′
′′
−
−
−
=
Now only a 2
nd
order ODE only need 2 B.C.’s
B.C.’s
0
αφ
=
at horizontal boundaries or
0 as
z
αφ
→
→ ±∞
Same as the TaylorGoldstein equation with
2
0
N
=
.
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 Fall '08
 Staff
 Fundamental physics concepts, Necessary and sufficient condition, dz, Z1, z2

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