Stability lecture 32

Stability lecture 32 - Hydrodynamic Stability Lecture 32...

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Hydrodynamic Stability Lecture 32 Inviscid instability of steady rotating flows Coordinate system – cylindrical () ,, or , , rz u v w vvv θ Incompressible, Inviscid Governing Equations 2 1 1. radial momentum 11 2. Asmuthial momentum 1 3. axial momentum 4. 1 5. 0 Du v p Dt r r Du v p V Dt r r Dw p Dt z Dv uw Dt t r r z uu v w rrr z ρ ρθ −= += =− ∂∂ ∂ ∂ =+ + + ∂∂ ++ + = 1
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Conservation of mass. B.C.’s – assume steady state. () () 0, , uv r w r =  ± 1-5 reduce, for the mean flow to 2 1 vp rr ρ −= . A balance between centripital force and pressure force. Kinematic B.C. is satisfied for concentric cylinder geometry. Rayleigh’s Heuristic Argument: () with 0 rewrite 2 11 or 1 0 for axisymmetric flow. is angular momentum. 0 says Angular momentum/unit mass is conser wr rv rv v v v v uv p ru w r v u r v w tr r z r t r r z Dp rv Dt rv D rv Dt 1 θ θρ ρθ = ∂∂ ∂∂ ∂ ∂  ++ + += + + + =  ∂∂∂ ∂  =− = = ved following the flow.
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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 32 - Hydrodynamic Stability Lecture 32...

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