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Hydrodynamic Stability
Lecture 26
Stability of Parallel shear flows (or nearly parallel).
Solve the initial value problem.
Assume linear equations describing a fluid system.
,,,
Lz
tzx
∂∂∂
=
0
u
)
With homogeneous B.C.’s and I.C.’s
()
(
,,0
,
uxz
f xz
=
Assume bounded domain in
z
.
Method of solution
Coefficients don’t depend on
x
or
t
.
Fourier transform in
x.
Laplace transform in
t
.
Results in an inhomogeneous.
Ordinary Differential Equation (ODE) in
( )
ˆ
for
, ,
zuz
α
σ
Solved in principle by Green’s functions.
N
Green's fxn
I.C.
ˆ
ˆ ,,
,
,
,
uz
G
z
z f z
d
ασ
′′
=−
∫
±²³²´
z
′
Inverting the Laplace transform, may require Bromwich contour to deal with singularities in the
Green’s function.
1
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,,
P
o
l
e
s
Discrete modes and Normal modes
+
Branch points/cuts
unbounded operator
continuous spectrum
do to singularities in the ODE.
uz
t
α
=
∑
∑
±
In general, representation in terms of normal modes (solutions obtained so far) is not complete.
The continuous spectrum leads to algebraic dependence on
t
.
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 Fall '08
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