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Unformatted text preview: Turbulence Lecture 8 If the dependence on x is suppressed, and let u = , assume stationarity. Then ( ) ( ) ( ) ( ) ( ) 2 u t u t u t + = = . This is the auto correlation coefficient (normalized auto-covariance). ( ) ( ) = only depends on . From above, Cauchy-Schwartz inequality ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 1 2 1 2 2 1, 1 u t u t u t u t u t u t u t = + = Non-dimensional We can now define a power spectral density. ( ) ( ) ( ) ( ) 1 2 i i S e S e d d = = A Fourier Transform pair Now simpler, since ( ) is real and symmetric. Complex Conjugate ( ) ( ) ( ) * 1 2 i S e d S = = Since ( ) is an even function 1 ( ) ( ) ( ) ( ) ( ) * 1 1 2 2 i i S e d e d S S = = = = ( ) ( ) * , is real , is symmetric...
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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.
- Spring '09