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Stability lecture 31

# Stability lecture 31 - Hydrodynamic Stability Sturm...

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Hydrodynamic Stability Sturm Liouville Eigenvalue problem. Consider an equation ( ) ( ) ( ) ( ) ( ) 1 2 0 with 0 d d p x y q x s x y dx dx y x y x λ + = = = where y is the eigenfunction. If everywhere in [ ] 1 2 , x x we have ( ) ( ) 0 and 0 p x s x > , then there exists eigenvalues λ and all the eigenvalues are real and furthermore if ( ) q x has one sign on the interval [ ] 1 2 , x x then all the λ have that sign, but if changes sign, then there exists eigenvalues both ( q x ) 0 and 0 λ λ > < . 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. ( ) ( ) 2 2 1 2 Re background flow complex phase speed IV U C U i U C 4 φ α φ φ φ α φ α φ α ′′ ′′ ′′ = + B.C.’s – No slip, rigid walls 0 Free stream , 0 as z αφ φ αφ φ = = ′ → → ∞

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