Stability lecture 9

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 = is greatest for k = A , most unstable disturbances are 2-D. For a homogeneous fluid. 12 0 , 0, 0 TC ρ === 1 2 U A Stability condition U 2 0 > Any shear is unstable! If has a minimum at 2 21 0 Ck ρρ > 2 1 2 2 2112 crit 22 4 n kg T gT U  =    −+  =   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. This is a 2-D Jet. There are two modes possible.
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Stability lecture 9 - Lecture 9 Kelvin-Helmholtz Solution...

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