Stability lecture 28

Stability lecture 28 - Hydrodynamic Stability Lecture 28...

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Hydrodynamic Stability Lecture 28 Rayleigh’s Inflection Point Theorem – Continued. Physical Mechanism: Suppose Uz 1, 2 0 on ′′   z Say () 0 background vorticity Vorticity increasing in dd UU dz dz z ζ == < 1
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Rayleigh’s criterion says that U must have extremum on [ ] 12 , zz , but doesn’t say it is a max or min. Fjortoft & Hoilands Refinement – Consider the real part of the depth integrated Rayleigh equation: () {} 22 11 2 2 2 0 r UCU dz dz UC φφ α φ ′′ =− + < ∫∫ From previous theorem we can replace by any value in the range of UU r C min max. to Assume an inflection point exists () () 2 1 2 2 where 0 0 cc c z c z z Uz UUU dz == < A necessary condition for instability is that ( ) [ ] 0 somewhere on , c U −< 0 c on whole interval jets, wakes, mixing layers. The flow is possibly unstable. 2
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For a different velocity profile. Possibly unstable from Rayleigh’s criteria but doesn’t meet second necessary criterion, so inviscidly stable.
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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 28 - Hydrodynamic Stability Lecture 28...

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