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Stability lecture 9 - Lecture 9 Kelvin-Helmholtz Solution...

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Lecture 9 Kelvin-Helmholtz Solution Continued For a given k , Re ( ) R σ σ = is greatest for k = A , most unstable disturbances are 2-D. For a homogeneous fluid. 1 2 0 , 0, 0 T C ρ ρ = = = 1 2 U σ = ± A Stability condition U 2 0 > Any shear is unstable! If has a minimum at ( ) 2 2 1 0 , C k ρ ρ > ( )( ) 2 2 1 1 2 2 2 1 1 2 crit 2 2 1 2 4 n k g T gT U ρ ρ ρ ρ ρ ρ ρ ρ = + = This tells how strong a shear is required to overcome stable stratification Physical Insight Approximate a vortex sheet by a row of point vortices, and assume the interface is displaced. 1
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Point vortices have induced velocity fields 2 r π Γ , look at induced velocities in perturbed state. Leads to steepening A pressure argument: Where the flow accelerates, the pressure drops (think of channel flow over a wavy wall). 2
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This produces a positive feedback, as the disturbance grows, the pressure force grows and feeds more energy into the disturbance. Consider case of two vortex sheets. One vortex sheet Now, consider two sheets.
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