Stability lecture 23

Stability lecture 23 - Hydrodynamic Stability Lecture 23...

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Hydrodynamic Stability Lecture 23 Stratified shear flows Continued. We derived () () 2 2 22 2 2 2 2 2 Laplacian Plug in normal mode expansion. Note: 0 - Instability 0 - Implies stability ˆˆ ˆ 0 with homo i i ww Uw U U N tx x x C C UC D KwDUUCwNw ∂∂  +∇ ∇+ =   >  −− + =  ±²³ ²´ geneous boundary conditions 0 or 0 as wz z =→ This is the Taylor – Goldstein equation Define vertical displacement η as. () ,, Assume 0 d xzt dz d dz ρ ρη =− Have monotonically decreasing density field. Define Wz and do some convenient variable changes from to vertical displacements: Uz C ˆ w Define () ( ) R e ik x Ct F ze = Note that and w are related by the linearized incompressibility relationship 1
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() 0 with = , , Uw txz xzt z ρ ρρ ρη ′′′ ∂∂∂ ++= This becomes Then in wave number space. ˆ ˆ zt zx z tx ik C F UikF w wik U C F ηρη ηη ∂∂ −− = += −+ =
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Stability lecture 23 - Hydrodynamic Stability Lecture 23...

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