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Stability lecture 8

Stability lecture 8 - Lecture 8 Instabilty Kelvin-Helmholtz...

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Unformatted text preview: Lecture 8 Instabilty: Kelvin-Helmholtz Continued Plug these into 2 Kinematic and 1 dynamic B.C. ( ) ( ) 1. i x my t e e σ + A ( ) ( ) 1 i x my t kB A e e σ σ + − = A ( ) ( ) 1 2 i x my t Ui A e e σ + − A A 2 2 1 1 1 2 2 2 1 2. + 2 1 1 3. 2 2 kB A Ui A B gA Ui B B gA Ui B K TA σ ρ σ ρ σ = + − − + − − − = A A A Three homogeneous, linear equations for 1 2 , , A B B MA = ¡ ¡ ¡ ¡ These have a non-trivial solution iff the determinant of the matrix is zero. i.e. det . M = ¡ ¡ ( ) 1 2 2 1 1 2 2 1 2 1 2 1 1 2 2 A Ui k MA Ui k B k T g Ui Ui B σ σ ρ ρ ρ σ ρ σ − = + − + − − − + + A A ¡ ¡ A A = ( ) 2 2 2 2 1 2 2 1 1 1 det 2 2 M Ui k Ui k k g k T ρ σ ρ σ ρ ρ − = − + + + − +...
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Stability lecture 8 - Lecture 8 Instabilty Kelvin-Helmholtz...

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