Stability lecture 11

Stability lecture 11 - Hydrodynamic Stability Lecture 11...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

Stability lecture 11 - Hydrodynamic Stability Lecture 11...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online