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Turbulence lecture 8

# Turbulence lecture 8 - Turbulence Lecture 8 If the...

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Turbulence Lecture 8 If the dependence on x is suppressed, and let 0 u = , assume stationarity. Then ( ) ( ) ( ) ( ) ( ) 2 u t u t u t τ ρ τ ρ τ + = = . This is the auto correlation coefficient (normalized auto-covariance). ( ) ( ) ρ τ ρ τ = only depends on τ . From above, Cauchy-Schwartz inequality ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 1 2 1 2 2 1, 0 1 u t u t u t u t u t u t u t ρ τ ρ τ ρ τ ∴≤ = + = Non-dimensional We can now define a power spectral density. ( ) ( ) ( ) ( ) 1 2 i i S e S e d ωτ τω d ω ρ τ τ π ρ τ ω ω −∞ −∞ = = A Fourier Transform pair Now simpler, since ( ) ρ τ is real and symmetric. Complex Conjugate ( ) ( ) ( ) * 1 2 i S e d ωτ S ω ρ τ τ ω π −∞ = = Since ( ) ρ τ is an even function 1

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( ) ( ) ( ) ( ) ( ) * 1 1 2 2 i i S e d e d S ωτ ωτ S ω ρ τ τ ρ τ τ ω ω π π −∞ −∞ = = = = ( ) ( ) * , is real , is symmetric S S S S S S ω ω = = Commonly define Real Transform Pair Auto Correlation ( ) ( ) 0 2 cos S d ρ τ ω ωτ = ω Spectral Density ( ) ( ) 0 1 cos S d ω ρ τ ωτ τ π =
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Turbulence lecture 8 - Turbulence Lecture 8 If the...

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