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Second level
Third level
Fourth level
Fifth level
Same dispersion
relation as in spatial case. i.e. \omega=\alpha U – i \alpha^2\epsilon
Expansions assumed
around neutral curve, then small imaginary dw can be related
to small imaginary da
Note that modes with
negative \alpha_i here represent decaying upstream propagating
modes
TS mode is damped in
left figure
Taking into account
non-parallel effects for TS-waves. Assume that U(x,y) and
V(x,y) is the base flow.
Normalization condition
is integral of weighted x-variation
Generalizations of
optimals to spatially developing, non-parallel flow
3D disturbances that
corresponds to streaks and streamwise vorticies
Expand u_0 in
eigenfunctions of A*A, operating repeatedly with A*A will result bring out \lambda_1
Identify the action of
the adjoint in the integrals resulting from the integration by parts