Turbulence lecture 25

Turbulence lecture 25 - Turbulence Lecture 25 Spectral...

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Turbulence Lecture 25 Spectral Cascade Low wave number High wave number () () () () ( ) 2 3 2 * , ˆ 1 ˆ ˆ ik x ii k L ik x L ix r ik x ij i j k ux uke uk uxe d x L uxu xr uku e e + = = += i ±± i Ai i ± A ± ± ± A ± ± If homogeneous, this has to be independent of ± x . ( ) ( ) r i k x ik x i r ee e e +− Ai A i iA ± ± i ** ˆˆ δ = A A ± ukuk ± k This can be shown for a homogeneous process. Fourier amplitudes are statistically uncorrelated, but they are not statistically independent. 1

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() () () () () () * 2 * 6 2 ˆˆ 1 = = ∫∫ i ± ± ± ±± ± ± ± ± ± ± ²³´³µ ij ik r ij i j k L ik y x ij L Rxy Rr ukuk e uxuy e d x d y L / two triple integrals Change or →=− ± yry xx ± y 6 1 = i ± ± ± ik r ij R r e drdx L Integral over 3 is ± dx L 2 * 3 2 1 = i ± ± L ik r i j L Rre d r L Take limit as , assume →∞ L 2 3 2 3 * 3 ij 1 lim is bounded Define. lim 2 " Energy Spectrum Tensor." 1 2 π −∞  Φ=   i ± ± i ± ± ± L ik r ij L L ij i
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This note was uploaded on 06/07/2011 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.

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Turbulence lecture 25 - Turbulence Lecture 25 Spectral...

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