Stability lecture 24

Stability lecture 24 - Hydrodynamic Stability Lecture 24 Go...

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Hydrodynamic Stability Lecture 24 Go back to () ( ) 22 2 2 1. 0 WF N KW F +− = Now multiply by [ ] 12 and integrate over , Fz z ) N 2 1 2 1 2 2 2 2. 0 First term: by parts z z z u z dv WF F N KW Fd z WF Fd z FWF = ′′ = ±²³²´ 2 0 2 1 1 2 2 z z z z WFd z {} 11 2 2 2 2 2 3. 0 Define , which is positive definite and note that Re Re Real part of 3 is 0 zz ri r z N KW Fd z Qz F K F WU C i C U C U C C Qdz N F dz = =+ =− −=  −− =  ∫∫ i C z Imaginary part is: 2 1 20 Take min max If 0 z ir z i r CUCQ d z aU z bU C aC b = = ≤≤ 1
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Then the phase speed of the disturbance has to be between the minimum and maximum of the current speed. This is necessary so that U can change signs on r C [ ] 12 , zz This implies that () ( ) somewhere on , r C = Uz The level at which the current speed equals the phase speed is called the Critical level. 1. At critical level: Wave speed = current speed 2. Taylor – Goldstein eqn. is potentially singular ( 2 UC ) in front of highest derivatives may be zero, and can be singular at critical level.
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This note was uploaded on 06/07/2011 for the course EOC 6934 taught by Professor Staff during the Fall '08 term at University of Florida.

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Stability lecture 24 - Hydrodynamic Stability Lecture 24 Go...

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