Stability lecture 18

Stability lecture 18 - Hydrodynamic Stability Lecture 18...

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Hydrodynamic Stability Lecture 18 New Topic: Chp.’s 2-3 of Drazin Determining the growth rates RR w σ = Example, Kelvin Helmholtz Instability for viscous fluid leading to numerical solution of the Orr-Sommerfeld Equation: ( ) z U 1. Start with Navier-Stokes equations for viscous, incompressible, constant density flow. 2. Assume a base state for a steady mean shear flow. () i UU z = only 3. Linearize 4. Subtract mean flow equation from total flow. 5. Nondimensionalize, drop ( ) ' 's and 's ± . Note that the steady mean flow equation reduces to: 2 2 0 PU v xz ∂∂ −+ = Linearized system for the perturbation is: 222 2 2 1 1. Re 1 2. Re 1 3. Re 4. 0 uU up uuu wU tzxx x y z wwp Uw txz vvp Uv txy uvw xyz  ∂∂ ∂∂ ∂∂∂ ++= + + +  ∂∂∂∂  += + + ++ = 1
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Assume solutions of the form () ± () ( ) ) ) ,,, note phase speed in . ˆ , ˆ , ˆ , ix y t uxyzt uze w Cx vxt vze wxt wze pxt pze αβ ω α +− = = = = = ² ² ² Apply to linearized equations Continuity 22 ˆˆ 1. 0
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Stability lecture 18 - Hydrodynamic Stability Lecture 18...

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