Introduction to turbulence/Statistical analysis
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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. | ||
+ | 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 <font color="orange">page ???</font> | ||
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- | + | *[[Introduction to turbulence/Statistical analysis/Ensemble average|Ensemble average]] | |
+ | ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Mean or ensemble average|Mean or ensemble average]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Fluctuations about the mean|Fluctuations about the mean]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Ensemble average#Higher moments|Higher moments]] | ||
+ | *[[Introduction to turbulence/Statistical analysis/Probability|Probability]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Probability#Histogram and probability density function|Histogram and probability density function]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Probability#Probability distribution|Probability distribution]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Probability#Gaussian (or normal) distributions|Gaussian (or normal) distributions]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Probability#Skewness and kurtosis|Skewness and kurtosis]] | ||
+ | * [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Multivariate random vaiables|Multivariate random vaiables]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Joint pdfs and joint moments|Joint pdfs and joint moments]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#The bi-variate normal (or Gaussian) distribution|The bi-variate normal (or Gaussian) distribution]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/Multivariate random variables#Statistical independence and lack of correlation| Statistical independence and lack of correlation]] | ||
+ | * [[Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations|Estimation from a finite number of realizations]] | ||
+ | ** [[Introduction to turbulence/Statistical analysis/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#Bias and convergence of estimators| Bias and convergence of estimators]] | ||
+ | * [[Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity|Generalization to the estimator of any quantity]] | ||
- | + | {{Turbulence credit wkgeorge}} | |
- | + | {{Chapter navigation|Nature of turbulence|Reynolds averaged equations}} |
Latest revision as of 18:06, 25 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
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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:
- Ensemble average
- Probability
- Multivariate random vaiables
- Estimation from a finite number of realizations
- Generalization to the estimator of any quantity
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.