Stability lecture 27

Stability lecture 27 - Hydrodynamic Stability Lecture 27 We...

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Hydrodynamic Stability Lecture 27 We derived the Orr-Sommerfeld equation previously. () 22 1 2 Re IV UC U i 4 φ αφ α ′′  −−− = −+  It is a 4 th order ODE, so we need 4 B.C.’s Rigid Walls (top and bottom). 0 at boundaries or Infinite domain ,0 a s Again an eigenvalue Problem for - eigen vector - eigenvalue z C φφ αφφ == →→ Dispersion relations enter in, seek dispersion relation of the form ,,R e 0 FC = Some general results for Inviscid flows. Usually restricted to high Re. We expect the instabilities to occur for Re large. Take 1 0 Re in Orr-Sommerfeld eqn. 1
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Useful for free-shear flows but not for boundary layers. (need full O-S eqn. Near boundaries). Gives: () 2 0 UC U φαφ φ ′′ −−− = Now only a 2 nd order ODE only need 2 B.C.’s B.C.’s 0 αφ = at horizontal boundaries or 0 as z →→ ± Same as the Taylor-Goldstein equation with 2 0 N = . It is called the Rayleigh equation.
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Stability lecture 27 - Hydrodynamic Stability Lecture 27 We...

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