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 () 24 2 2R e IV iU C U αφ α  ′′ −+ = −−−  For no slip wall B.C.’s are 0 and 0 at 0, . z zL z == = For free slip wall the B.C.’s are 2 2 0 and 0 at 0, z z = . 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 wu x z u zz ψ ∂∂ =− = Approach: Rewrite Orr Sommerfeld in discretized form to solve numerically for ( ) C , remember C is complex and the imaginary part gives the growth rate. Use 42 2 22 2 DD z  ∂∂∂   Multiply both sides of equation by Re i Gives: 22 22 2 2 2 2 2 Re i U D U D UC D  −− − − −= 
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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 19 - Hydrodynamic Stability Lecture 19 ^...

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