Stability lecture 29

Stability lecture 29 - Hydrodynamic Stability Lecture 29...

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Hydrodynamic Stability Lecture 29 Consider the Reynolds stress of a perturbation…. () 2 1 Re 0 & B.C.'s. Assume 2-D, periodic in . , where is averaged over wavelength so changes as grows. uu u p u t u x uxt u u u u u u +∇= ∇+ ∇ ∇= =+ i ±± ± ± i ± ±± ± ± ± Analagous to Reynolds avg. in turbulence. Form an equation for by averaging eqns. for ± uu . u - initially the basic flow function of ( ) , zt averaged in x already. ± Take the product 2 ... gives an energy equation for . 2 i u t  ′′ +   i Integrate over one wavelength on [ ] 12 and , x zz () () 2 22 rate of change of KE exchange of perturbation KE with this last term is the rate of dissipation background flow of perturbat 11 2R e dU w u u w dxdz u w dxdz dxdz td z ∂∂  += − −   ∫∫ ²³³³´ ³ ³ ³µ ²³³³´³³³µ ion KE 0 < x z ²³³³³´³³³³µ The instability can grow only if the first term on RHS is positive. A lot depends on this term, the exchange of perturbation energy with the mean flow.
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Stability lecture 29 - Hydrodynamic Stability Lecture 29...

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