Stability lecture 30

Stability lecture 30 - Hydrodynamic Stability Lecture 30...

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Hydrodynamic Stability Lecture 30 Consider the structure of a growing disturbance in a parallel shear flow. Streamlines are useful for visualization. Streamline pattern near critical level. [] () ,, uUuw zx ψ ψψ ∂∂  ′′ =+ = + −   ± ± Transform to coordinate system moving with speed C . r {} () ( ) 2 2 1 H.O.T. 2 Re for a fixed . Near critical level, equation for streamline is: 1 cos ... Const. 2 Plot this - B B r z rB z z ix BB B UC uw U C dz u z z Az e t Uzz A z x α ψφ φα γ −+ = + =− = ′ = + + = ± ± i Cat’s eye pattern, right near critical level 1
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Classes of solutions to the Rayleigh equation. 1. For neutral modes – solutions using Drazin’s Method a. Classes of exact solutions: hyperbolic profiles, etc. 2. Solutions for broken line profiles 3. Asymptotic methods, e.g. small , large α λ 4. Numerical methods (for pretty arbitrary profiles). Spatial Modes () Consider where where is real, is complex.
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Stability lecture 30 - Hydrodynamic Stability Lecture 30...

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