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 p g z ρ = − A Poisson equation for p in terms of ρ . Take of z- component of momentum equation. 2 2 2 2 2 2 2 1 0 0 0 0 0 1 1 g g v w p g t z z z ρ g ρ ρ ρ ρ ρ ρ ρ ρ − ∇ = − = − = − Horizontal Laplacian operator. 2 2 2 1 2 2 2 2 2 2 2 1 0 Take of previous equation. x y K t g K v w K t t t 2 ρ ρ ∇ = +  − ∇ = −   R.H.S is identical to LHS of energy equation. ( ) 2 1 0 0 2 2 2 1 So one equation in one unknown: g w K v w g t t ρ αβ ρ αβ = −  − ∇ =   2 w 1
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