Hydrodynamic Stability
Lecture 19
The Orr Sommerfeld equation replacing
ˆ with
w
φ
(
)
(
)
2
4
2
2
Re
IV
i
U
C
U
φ
α φ
α φ
α
φ
α φ
φ
′′
′′
′′
−
+
=
−
−
−
For no slip wall B.C.’s are
( )
0 and
0 at
0,
.
z
z
L
z
φ
φ
∂
=
=
=
∂
For free slip wall the B.C.’s are
( )
2
2
0 and
0 at
0,
z
z
L
z
φ
φ
∂
=
=
=
∂
.
This follows from definition of freeslip (and vorticity).
u
z
∂
∂
free slip
w
x
∂
−
∂
0 all along wall
0
at wall
w
=
=
Stream function defined by
2
2
,
so
0
w
u
x
z
u
z
z
ψ
ψ
ψ
∂
∂
= −
=
∂
∂
∂
∂
=
=
∂
∂
Approach:
Rewrite Orr Sommerfeld in discretized form to solve numerically for
(
)
C
α
, remember
C
is
complex and the imaginary part gives the growth rate.
Use
4
2
2
2
2
4
2
2
D D
z
z
z
∂
∂
∂
=
=
∂
∂
∂
Multiply both sides of equation by
Re
i
−
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 Fall '08
 Staff
 Fluid Dynamics, Orr Sommerfeld, Orr Sommerfeld equation, free slip, slip wall, slip wall B.C.

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