Stability lecture 26

Stability lecture 26 - Hydrodynamic Stability Lecture 26...

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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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Stability lecture 26 - Hydrodynamic Stability Lecture 26...

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