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Stability lecture 19

# Stability lecture 19 - Hydrodynamic Stability Lecture 19 ^...

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Hydrodynamic Stability Lecture 19 The Orr Sommerfeld equation replacing ˆ with w φ ( ) ( ) 2 4 2 2 Re IV i U C U φ α φ α φ α φ α φ φ ′′ ′′ ′′ + = For no slip wall B.C.’s are ( ) 0 and 0 at 0, . z z L z φ φ = = = For free slip wall the B.C.’s are ( ) 2 2 0 and 0 at 0, z z L z φ φ = = = . This follows from definition of free-slip (and vorticity). u z free slip w x 0 all along wall 0 at wall w = = Stream function defined by 2 2 , so 0 w u x z u z z ψ ψ ψ = − = = = Approach: Rewrite Orr Sommerfeld in discretized form to solve numerically for ( ) C α , remember C is complex and the imaginary part gives the growth rate. Use 4 2 2 2 2 4 2 2 D D z z z = = Multiply both sides of equation by Re i

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Stability lecture 19 - Hydrodynamic Stability Lecture 19 ^...

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