Turbulence lecture 7

# Turbulence lecture 7 - Turbulence Lecture 7...

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Turbulence Lecture 7 Terminology (assuming 0 uv == ) standard deviation of . standard deviation of . Correlation: - Correlation coefficient. u v u v uv C σ σσ = = = Cauchy – Schwartz inequality () 1 2 22 This says , 1 , 1 means perfect correlation , means perfect anti-correlation (exactly out of phase) , 0 perfectly uncorrelated uv u v Cxt   = =− = ± ± ± ± Example: Joint Gaussian. PDF , Puv , 2 u 1 21 xt v C πσσ = ± 2 1 exp C uu 2 vv uC u v v σσσ −+  Assuming 0 i.e., uuu 1

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This is the equation of an ellipse. Ellipses – general form. Axes parallel to coordinate axes: 22 0 Ax Cy Dx Ey F ++++ = General Form: axes oblique to coordinate axes: 0 Ax Bxy Cy Dx Ey F +++++ = So the slope of the oblique axis is 2 uv C σσ if you rotate your axis. 2
Then in the ** * * coordinate system 0 uv C σσ == () * * ,0 uvP u v dudv = ∫∫ So rotation can give statistical uncorrelation for Gaussian Joint PDF. Statistical independence implies statistical uncorrelation.

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Turbulence lecture 7 - Turbulence Lecture 7...

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