Introduction to turbulence

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-	# [[What is Turbulence?]]	+	{{Introduction to turbulence menu}}
-	# [[The elements of statistical analysis]]	+
-	# [[Reynoldas averaged equations]]	+
		+	__NOTOC__
-	==2.4 The Origins of Turbulence==	+	'''This section is currently undergoing heavy reorganization and editing. Please excuse any errors or unfinished parts. --[[User:Jola|Jola]] 07:01, 21 June 2007 (MDT)'''
-	Turbulent flows can often be observed to arise from laminar flows as the Reynolds number, (or someother relevant parameter) is increased. This happens because small disturbances to the flow are no longer damped by the flow, but begin to grow by taking energy from the original laminar flow. This natural process is easily visualized by watching the simple stream of water from a faucet (or even a pitcher). Turn the flow on very slow (or pour) so the stream is very smooth initially, at least near the outlet. Now slowly open the faucet (or pour faster) abd observe what happens, first far away, then closer to the spout. The surface begins to exhibit waves or ripples which appear to grow downstream . In fact, they are growing by extracting energy from the primary flow. Eventually they grow enough that the flow breaks into drops. These are capillary instabilities arisiing from surface tension, but regardless of the type of instability, the idea is the same -small (or infinitesimal ) disturbances have grown to disrupt the serenity (and simplicity) of laminar flow.	+	== [[Introduction to turbulence/Nature of turbulence|Nature of turbulence]] ==
-	The manner in which the instabilities grow naturally in a flow can be examined using the equations we have already developed above. We derived them by decomposing the motion into a mean and fluctuating part. But suppose instead we had decomposed the motion into a base flow part (the initial laminar part) and into a disturbance which represents a fluctuating part superimposed on the base flow. The result of substituting such a decomposition into the full Navier-Stokes equations and averaging is precisely that given by equations 2.13 and 2.15. But the very important difference is the additional restriction that what was previously identified as the mean (or averaged ) motion is now also the base or laminar flow.	+	* [[Introduction to turbulence/Nature of turbulence#The turbulent world around us|The turbulent world around us]]
		+	* [[Introduction to turbulence/Nature of turbulence#What is turbulence?|What is turbulence?]]
		+	* [[Introduction to turbulence/Nature of turbulence#Why study turbulence?|Why study turbulence?]]
		+	* [[Introduction to turbulence/Nature of turbulence#The cost of our ignorance|The cost of our ignorance]]
		+	* [[Introduction to turbulence/Nature of turbulence#What do we really know for sure?|What do we really know for sure?]]
-	Now if the base flow is really laminar flow (which it must be by our original hypothesis), then our averaged equations governing the base flow must yield the same mean flow as the original laminar flow on which the disturbances was superimposed. But this can happen only if these new averaged equations reduce to '''exactly''' the same lamiane flow equations without any evidence of a disturbance. Clearly from equations 2.13 and 2.15, this can happen ''only if all the Reynolds stress terms vanish identically!'' Obviously this requires that the disturbances be infintesimal so the extra terms can be neglected - hence our interest in infinitesimal disturbances.	+	== [[Introduction to turbulence/Statistical analysis|Statistical analysis]] ==
-	So we hypothesized a base flow which was laminar and showed that it is unchanged even with the imposition of infintesimal disturbances on it - ''but only as long as the disturbances'' '''remain''' ''infinitesimal!'' What happens if the disturbance starts to grow? Obviously before we conclude that all laminar flows are laminar forever we better investigate whether or not these infinitesimal disturbances can grow to ''finite'' size. To do this we need an equation for the fluctuation itself.	+	*  [[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 variables]]
		+	** [[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]]
-	more to come soon.............	+	== [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaged equations and the turbulence closure problem]] ==
-	==Credits==	+	* [[Introduction to turbulence/Reynolds averaged equations#Equations governing instantaneous fluid motion|Equations governing instantaneous fluid motion]]
-	'''This text was based on "Introduction to Turbulence" by Professor William K.George, Chalmers University of Technology, Sweden.'''	+	* [[Introduction to turbulence/Reynolds averaged equations#Equations for the average velocity|Equations for the average velocity]]
		+	* [[Introduction to turbulence/Reynolds averaged equations#The turbulence problem|The turbulence problem]]
		+	* [[Introduction to turbulence/Reynolds averaged equations#Origins of turbulence|Origins of turbulence]]
		+	* [[Introduction to turbulence/Reynolds averaged equations#Importance of non-linearity|Importance of non-linearity]]
		+	* [[Introduction to turbulence/Reynolds averaged equations#Turbulence closure problem and eddy viscosity|Turbulence closure problem and eddy viscosity]]
		+	* [[Introduction to turbulence/Reynolds averaged equations#Reynolds stress equations|Reynolds stress equations]]
		+	== [[Introduction to turbulence/Turbulence kinetic energy|Turbulence kinetic energy]] ==
		+	* [[Introduction to turbulence/Turbulence kinetic energy#Fluctuating kinetic energy|Fluctuating kinetic energy]]
		+	* [[Introduction to turbulence/Turbulence kinetic energy#Rate of dissipation of the turbulence kinetic energy|Rate of dissipation of the turbulence kinetic energy]]
		+	* [[Introduction to turbulence/Turbulence kinetic energy#Kinetic energy of the mean motion and production of turbulence|Kinetic energy of the mean motion and production of turbulence]]
		+	* [[Introduction to turbulence/Turbulence kinetic energy#Transport or divergence terms|Transport or divergence terms]]
		+	* [[Introduction to turbulence/Turbulence kinetic energy#Intercomponent transfer of energy|Intercomponent transfer of energy]]
-	----	+	== [[Introduction to turbulence/Stationarity and homogeneity|Stationarity and homogeneity]] ==
-	<i> Return to [[Turbulence modeling]] </i>	+	* [[Introduction to turbulence/Stationarity and homogeneity#Processes statistically stationary in time|Processes statistically stationary in time]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Autocorrelation|Autocorrelation]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Autocorrelation coefficient|Autocorrelation coefficient]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Integral scale|Integral scale]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Temporal Taylor microscale|Temporal Taylor microscale]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Time averages of stationary processes|Time averages of stationary processes]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Bias and variability of time estimators|Bias and variability of time estimators]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Random fields of space and time|Random fields of space and time]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Multi-point statistics in homogeneous field|Multi-point statistics in homogeneous field]]
		+	* [[Introduction to turbulence/Stationarity and homogeneity#Spatial integral and Taylor microscales|Spatial integral and Taylor microscales]]
		+
		+	== [[Introduction to turbulence/Homogeneous turbulence|Homogeneous turbulence]] ==
		+
		+	* [[Introduction to turbulence/Homogeneous turbulence#A first look at decaying turbulence |A first look at decaying turbulence ]]
		+	* [[Introduction to turbulence/Homogeneous turbulence#The dissipation equation and turbulence modelling |The dissipation equation and turbulence modelling ]]
		+	* [[Introduction to turbulence/Homogeneous turbulence#A second look at simple shear flow turbulence |A second look at simple shear flow turbulence ]]
		+
		+	== [[Introduction to turbulence/Free turbulent shear flows|Free turbulent shear flows]] ==
		+	* [[Introduction to turbulence/Free turbulent shear flows#Introduction|Introduction]]
		+	* [[Introduction to turbulence/Free turbulent shear flows#The averaged equations|The averaged equations]]
		+	* [[Introduction to turbulence/Free turbulent shear flows#Two-dimensional Turbulent Jets|Two-dimensional Turbulent Jets]]
		+	* [[Introduction to turbulence/Free turbulent shear flows#Other free shear flows|Other free shear flows]]
		+
		+	== [[Introduction to turbulence/Wall bounded turbulent flows|Wall bounded turbulent flows]] ==
		+
		+	* [[Introduction to turbulence/Wall bounded turbulent flows#Introduction|Introduction]]
		+	* [[Introduction to turbulence/Wall bounded turbulent flows#Review of laminar boundary layers|Review of laminar boundary layers]]
		+	* [[Introduction to turbulence/Wall bounded turbulent flows#The "outer" turbulent boundary layer|The "outer" turbulent boundary layer]]
		+	* [[Introduction to turbulence/Wall bounded turbulent flows#The “inner” turbulent boundary layer|The “inner” turbulent boundary layer]]
		+	* [[Introduction to turbulence/Wall bounded turbulent flows#The viscous sublayer|The viscous sublayer]]
		+
		+
		+
		+	{{Turbulence credit wkgeorge}}
		+
		+	[[Category: Turbulence]]

---

Latest revision as of 12:34, 15 March 2012

Introduction to turbulence

Nature of turbulence

Statistical analysis

Ensemble average Probability Multivariate random var ... Estimation from a finite ... Generalization to the esti ...

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!

This section is currently undergoing heavy reorganization and editing. Please excuse any errors or unfinished parts. --Jola 07:01, 21 June 2007 (MDT)

Nature of turbulence

• The turbulent world around us
• What is turbulence?
• Why study turbulence?
• The cost of our ignorance
• What do we really know for sure?

Statistical analysis

• Ensemble average
  • Mean or ensemble average
  • Fluctuations about the mean
  • Higher moments
• Probability
  • Histogram and probability density function
  • Probability distribution
  • Gaussian (or normal) distributions
  • Skewness and kurtosis
• Multivariate random variables
  • Joint pdfs and joint moments
  • The bi-variate normal (or Gaussian) distribution
  • Statistical independence and lack of correlation
• Estimation from a finite number of realizations
  • Estimators for averaged quantities
  • Bias and convergence of estimators
• Generalization to the estimator of any quantity

Reynolds averaged equations and the turbulence closure problem

• Equations governing instantaneous fluid motion
• Equations for the average velocity
• The turbulence problem
• Origins of turbulence
• Importance of non-linearity
• Turbulence closure problem and eddy viscosity
• Reynolds stress equations

Turbulence kinetic energy

• Fluctuating kinetic energy
• Rate of dissipation of the turbulence kinetic energy
• Kinetic energy of the mean motion and production of turbulence
• Transport or divergence terms
• Intercomponent transfer of energy

Stationarity and homogeneity

• Processes statistically stationary in time
• Autocorrelation
• Autocorrelation coefficient
• Integral scale
• Temporal Taylor microscale
• Time averages of stationary processes
• Bias and variability of time estimators
• Random fields of space and time
• Multi-point statistics in homogeneous field
• Spatial integral and Taylor microscales

Homogeneous turbulence

• A first look at decaying turbulence
• The dissipation equation and turbulence modelling
• A second look at simple shear flow turbulence

Free turbulent shear flows

• Introduction
• The averaged equations
• Two-dimensional Turbulent Jets
• Other free shear flows

Wall bounded turbulent flows

• Introduction
• Review of laminar boundary layers
• The "outer" turbulent boundary layer
• The “inner” turbulent boundary layer
• The viscous sublayer

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.