Stability lecture 31

Stability lecture 31 - Hydrodynamic Stability Sturm...

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Hydrodynamic Stability Sturm Liouville Eigenvalue problem. Consider an equation () () () 12 0 with 0 dd px y qx sx y dx dx yx λ  +−   == = where y is the eigenfunction. If everywhere in [ ] , x x we have () ( ) 0 and 0 sx >≥ , then there exists eigenvalues and all the eigenvalues are real and furthermore if ( ) has one sign on the interval [ ] , x x then all the have that sign, but if changes sign, then there exists eigenvalues both ( ) 0 and 0 >< . Moreover the corresponding eigenfunctions are complete over space regardless of the sign. 2 L References: Morse & Feshbach, vol. 1 – pg 719. If all the eigenvalues are real, the flow is stable. Lecture 31 Effects of viscosity The basic equation is the Orr-Sommerfeld eqn. 22 1 2 Re background flow complex phase speed IV UC U i U C 4 φ αφ α ′′ −−− = −+ B.C.’s – No slip, rigid walls 0 Free stream , 0 as z φφ αφφ →→ This is an eigenvalue problem for a given U , Re, with a complex eigenvalue C ω = 4 th order in z , 4 B.C.’s. It is not singular at , 0, ir C = .
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This note was uploaded on 06/07/2011 for the course EOC 6934 taught by Professor Staff during the Fall '08 term at University of Florida.

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Stability lecture 31 - Hydrodynamic Stability Sturm...

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