Day05-slides - 1 Atms 502 CS 505 CSE 566 Numerical Fluid...

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Unformatted text preview: 1 Atms 502 CS 505 CSE 566 Numerical Fluid Dynamics Thu., 7 September 2006 9/29/06 Atms 502 - Fall 2006 2 2 ∆ x grows fastest.. Amplification factor: Stability analysis: Say σ = 1. Then: Plug in: " = 1 # 4 $ sin 2 k % x 2 & ’ ( ) * + " # " 1 2 " = 1 # 4sin 2 $ 2 % & ’ ( ) * L = " # $ = 0 # % = 1 & 4(0) = 1 L = 4 ’ x # $ = ( 2 # % = 1 & 4(0.5) = & 1 L = 2 ’ x # $ = ( # % = 1 & 4( 1 ) = & 3 stable stable unstable 9/29/06 Atms 502 - Fall 2006 3 Operator deFnitions Derivatives: Averaging: • Example: advection w/staggered grids: udT/dx " x f ( x ) = f i + 1/2 # f i # 1/2 $ x ; " 2 x f ( x ) = f i + 1 # f i # 1 2 $ x u x = u i + 1/2 + u i " 1/2 2 ; u 2 x = u i + 1 + u i " 1 2 u u T T T T T 9/29/06 Atms 502 - Fall 2006 4 wave forms You may have seen expressions such as these, describing wave motions: • Units of ω ? • What about x-ct? We’ll be considering the possibility of a complex frequency: So what? u = A cos( kx ) u = A cos[ k ( x " ct )] u = Ae ik ( x " ct ) u = Ae i ( kx " # t ) " = " r + i " i u = Ae i ( kx " # t ) $ u = Ae i kx " # r + i # i ( ) t [ ] $ u = Ae # i t ( ) e i kx " # r t ( ) Amplitude: Exponential growth? 2 9/29/06 Atms 502 - Fall 2006 5 Now look just at time dependence: This is of the form: The above is the oscillation equation Oscillation equation Start with the linear...
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Day05-slides - 1 Atms 502 CS 505 CSE 566 Numerical Fluid...

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