# Introduction to turbulence/Statistical analysis

(Difference between revisions)
 Revision as of 12:07, 21 June 2007 (view source)Jola (Talk | contribs)← Older edit Revision as of 12:15, 21 June 2007 (view source)Jola (Talk | contribs) Newer edit → Line 1: Line 1: - {{Turbulence}} + {{Introduction to turbulence menu}} Much of the study of turbulence requires statistics and stochastic processes, simply because the instanteous motions are too complicated to understand. This should not be taken to mean that the govering equations (usually the Navier-Stokes equations) are stochastic. Even simple non-linear equations can have deterministic solutions that look random. In other words, even though the solutions for a given set of initial and boundary conditions can be perfectly repeatable and predictable at a given time and point in space, it may be impossible to guess from the information at one point or time how it will behave at another (at least without solving the equations). Moreover, a slight change in the intial or boundary conditions may cause large changes in the solution at a given time and location; in particular, changes that we could not have anticipated. Much of the study of turbulence requires statistics and stochastic processes, simply because the instanteous motions are too complicated to understand. This should not be taken to mean that the govering equations (usually the Navier-Stokes equations) are stochastic. Even simple non-linear equations can have deterministic solutions that look random. In other words, even though the solutions for a given set of initial and boundary conditions can be perfectly repeatable and predictable at a given time and point in space, it may be impossible to guess from the information at one point or time how it will behave at another (at least without solving the equations). Moreover, a slight change in the intial or boundary conditions may cause large changes in the solution at a given time and location; in particular, changes that we could not have anticipated. Line 6: Line 6: Further details about statistical analysis in turbulence can be found in: Further details about statistical analysis in turbulence can be found in: - *[[Ensemble average in turbulence|The ensemble and ensemble averages]] + *[[Introduction to turbulence/Statistical analysis/Ensemble average|Ensemble average]] - ** [[Ensemble average in turbulence #The mean or ensemble average| The mean or ensemble average]] + ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Mean or ensemble average|Mean or ensemble average]] - ** [[Ensemble average in turbulence #Fluctuations about the mean | Fluctuations about the mean]] + ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Fluctuations about the mean|Fluctuations about the mean]] - ** [[Ensemble average in turbulence #Higher moments| Higher moments]] + ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Higher moments|Higher moments]] - *[[Probability in turbulence|Probability]] + *[[Introduction to turbulence/Probability|Probability]] - ** [[Probability in turbulence#The histogram and probability density function| The histogram and probability density function]] + ** [[Introduction to turbulence/Statistical analysis/Probability#Histogram and probability density function|Histogram and probability density function]] - ** [[Probability in turbulence#The probability distribution| The probability distribution]] + ** [[Introduction to turbulence/Statistical analysis/Probability#Probability distribution|Probability distribution]] - ** [[Probability in turbulence#Gaussian (or normal) distributions| Gaussian (or normal) distributions]] + ** [[Introduction to turbulence/Statistical analysis/Probability#Gaussian (or normal) distributions|Gaussian (or normal) distributions]] - ** [[Probability in turbulence#Skewness and kurtosis| Skewness and kurtosis]] + ** [[Introduction to turbulence/Statistical analysis/Probability#Skewness and kurtosis|Skewness and kurtosis]] - *  [[Multivariate random vaiables#Multivariate random vaiables| Multivariate random vaiables]] + *  [[Introduction to turbulence/Statistical analysis/Multivariate random vaiables#Multivariate random vaiables|Multivariate random vaiables]] - ** [[Multivariate random vaiables#Joint pdfs and joint moments| Joint pdfs and joint moments]] + ** [[Introduction to turbulence/Statistical analysis/Multivariate random vaiables#Joint pdfs and joint moments|Joint pdfs and joint moments]] - ** [[Multivariate random vaiables#The bi-variate normal (or Gaussian) distribution| The bi-variate normal (or Gaussian) distribution]] + ** [[Introduction to turbulence/Statistical analysis/Multivariate random vaiables#The bi-variate normal (or Gaussian) distribution|The bi-variate normal (or Gaussian) distribution]] - ** [[Multivariate random vaiables#Statistical independence and lack of correlation| Statistical independence and lack of correlation]] + ** [[Introduction to turbulence/Statistical analysis/Multivariate random vaiables#Statistical independence and lack of correlation| Statistical independence and lack of correlation]] - *  [[Estimation from a finite number of realizations]] + *  [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations]] - ** [[Estimation from a finite number of realizations#Estimators for averaged quantities| Estimators for averaged quantities]] + ** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Estimators for averaged quantities|Estimators for averaged quantities]] - ** [[Estimation from a finite number of realizations#Bias and convergence of estimators| Bias and convergence of estimators]] + ** [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations#Bias and convergence of estimators| Bias and convergence of estimators]] - *  [[Generalization to the estimator of any quantity]] + *  [[Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity]] - {| class="toccolours" style="margin: 2em auto; clear: both; text-align:center;" + {{Chapter navigation|Nature of turbulence|Statistical analysis/Ensemble average}} - |- + - | [[Turbulence|Up to turbulence]] | [[Nature of turbulence|Back to nature of turbulence]] + - |} + {{Turbulence credit wkgeorge}} {{Turbulence credit wkgeorge}} - - [[Category: Turbulence]]

## Revision as of 12:15, 21 June 2007

 Nature of turbulence Statistical analysis Reynolds averaged equation Turbulence kinetic energy Stationarity and homogeneity Homogeneous turbulence Free turbulent shear flows Wall bounded turbulent flows Study questions ... template not finished yet!

Much of the study of turbulence requires statistics and stochastic processes, simply because the instanteous motions are too complicated to understand. This should not be taken to mean that the govering equations (usually the Navier-Stokes equations) are stochastic. Even simple non-linear equations can have deterministic solutions that look random. In other words, even though the solutions for a given set of initial and boundary conditions can be perfectly repeatable and predictable at a given time and point in space, it may be impossible to guess from the information at one point or time how it will behave at another (at least without solving the equations). Moreover, a slight change in the intial or boundary conditions may cause large changes in the solution at a given time and location; in particular, changes that we could not have anticipated.

Most of the statistical analyses of turbulent flows are based on the idea of an ensemble average in one form or another. In some ways this is rather inconvenient, since it will be obvious from the definitions that it is impossible to ever really measure such a quantity. Therefore we will spendlast part of this chapter talking about how the kind of averages we can compute from data correspond to the hypotetical ensemble average we wish we could have measured. In later chapters we shall introduce more statistical concepts as we require them. But the concepts of this chapter will be all we need to begin a discussion of the averaged equations of motion in page ???

Further details about statistical analysis in turbulence can be found in:

## Credits

This text was based on "Lectures in Turbulence for the 21st Century" by Professor William K. George, Professor of Turbulence, Chalmers University of Technology, Gothenburg, Sweden.