Turbulence lecture 5

Turbulence lecture 5 - Turbulence Lecture 5 Stochastic...

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Turbulence Lecture 5 Stochastic Tools Consider the wake flow experiment G u Generating a turbulent wake, measure speed at u at time t at location * * x obtain , then repeat the whole process in another experiment, obtain at same ( ** 1 , uxt * ) ) ( , 2 x and t in general u . Could repeat this (in theory) many times. A given experiment is called a “realization”. * 1 u 2 A collection of experiments is called an ensemble. Lets say we have N realizations. We can define an ensemble average. () ( 1 1 , N N i i ux t u x t N = = ±± ) , Now if the ensemble average usually converges to the probability average or mean. N →∞ 1 1 ,l i m , N i N i u xt N →∞ = = This average depends, of course on , x t ± Often we consider time averages, less often, spatial averages. These averages are not useful unless the process is statistically stationary. Ensemble averages are always defined. A useful function is the window funtion 1
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() ** ,; , , 1 0 uu ux t uux t u u otherwise χ  ∆=  ≤≤ + = ± ± , t ±
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Turbulence lecture 5 - Turbulence Lecture 5 Stochastic...

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