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Stability lecture 16 - Hydrodynamic Stability Lecture 16 HW...

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Hydrodynamic Stability Lecture 16 HW Problem: For the Benard problem, prove that for neutral conditions 3 0 fluctuating density and horizontal velocity average of fluctuating density. V u dV k z ρ ρ ∝ − ±²³ ±²³ That is, that the heat flux at the bottom is proportional to the buoyancy flux over the whole domain. Where: 1. The volume integral is over ( ) 0, z h and either. a. Wave period in x and y . b. A finite domain with kinematic B.C. in x, y . (no penetration) 0 u n = i ´ ´ 2. ( ) denotes average over x-y plane at a given z. Hints: steady state, no growth or decay. Same assumptions about density, linear, work with x-y average equation for ρ . Shear term The effect of shear can be either stabilizing or destabilizing – depends on sign of ( which depends on the streamlines. ) 0 1 3 u u ρ Example Pure Convection 1
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Assuming symmetry the Reynolds stress would integrate to zero. Convection with Shear Asymmetric 1 3 1 1 3 3 constant in this case 0 V u x u u u dV x >
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