Stability lecture 4

Stability lecture 4 - Lecture 4 Governing equation 1 u + u...

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Lecture 4 Governing equation () 1 or 0 uu u pg t u α ρ +∇= −∇− ∇= i ±± ± ± ± i ± Note that in this inviscid formulation, a shear along the interface is allowable. 1. Assume flow is irrotational except at interface. Therefore ui , 1,2 i = ± i u i φ =∇ a velocity potential ± so 2 0 ii u i ± Mathematically Assume irrotational – momentum equation can be integrated to give Bernoulli equation. This gives p in terms of u i ² . Makes sure continuity equation is satisfied. 1. and B. C.’s 2 0 i 2. Assume small amplitudes, then linearize, say 1 for a x η ³ . Gives Laplace’s equation for with linear B.C.’s. Linearizing: Assume small amplitude 1 ³ compared to a wavelength. 1 i u ³ compared to other velocities that would arise in the problem at finite time. ± 1
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B.C. 1 11 wu zt η φη ∂∂ == +∇ i ± linearize 2 ε ² neglect products of perturbations Use a Taylor Series 2 1 2 0 0 w zz φφ =+ linearize .... + 2 Boundary Conditions 2. 12 00 z φ ηφ For inviscid, irrotational flow the momentum equation can be integrated to give
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Stability lecture 4 - Lecture 4 Governing equation 1 u + u...

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