Stability lecture 11

# Stability lecture 11 - Hydrodynamic Stability Lecture 11...

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Hydrodynamic Stability Lecture 11 Rayleigh-Benard solution continued. Take (momentum equation) i 0 u ∇= i ± Divergence of gradient gives Laplacian. 2 pg z ρ A Poisson equation for p in terms of . Take of z- component of momentum equation. 2 22 2 2 2 2 1 00 0 0 0 11 gg vw p g tz z z g ρρ ∂∂  −∇ ∇ =− ∇ −   Horizontal Laplacian operator. 2 1 2 2 2 1 0 Take of previous equation. xy K t g Kv w K tt t 2 + −∇ −∇∇ = − ∇ R.H.S is identical to LHS of energy equation. () 2 10 0 2 1 So one equation in one unknown: g w w g ραβ αβ 2 w 1

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Non-dimensionalize Length scale: Lh = Time scale: 2 h K = T For diffusion of heat (we’ve dropped temperature). Velocity scale: K h = V ˆ ˆ ; ˆ tx Th u u V == = ± ± ± ± Plug into the equation. () 223 ˆ , etc. multiply result by and drop hats again k uu h hhh kvk = ± ² So equation is for non-dimensional quantities.
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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 11 - Hydrodynamic Stability Lecture 11...

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