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PhD course require own
reading, view lectures as overview to get main points,
read details in book
\partial/\partial x
-> i\alpha, \partial /\partial y -> D, \partial/\partial z -> i\beta
Continuity equation and
equation for normal vorticity
Re_L corresponds to a
3D wave at a larger Reynolds number
Instability of normal
vorticity, v: given function
Frobenious series is a
form of a generalized series expansion
Generalized Airy
equation
Approximation of
continuous spectrum: eigenmodes which oscillates as y->\infty
Look for oscillating
solutions in free-stream
\beta = 0, due to
Squire
Leave time derivative
in
Add enough of the
homogeneous solution to satisfy BC
Top result shown by
similar method used in Rayleigh derivation
Note that driving term
is on adj. OS, implying \zeta=0
We can just as well
work with expansion coefficients
\sigma_1: largest
singular value, v_1: optimal disturbance
Max transient growth
for low \alpha
integral: Laplace
tranform
Large resolvent \sim
sensitive eigenvalues
Bottom right: large
response to forcing with frequency
\omega=0
Natural time scale from
Taylor expansion (similar to BL scaling), 3D problem
have two parameters: \alpha R and k
Normal (commutes with
its adjoint) ó orthogonal eigenvaules
Scaling valid also for
small \alpha R