Introduction to turbulence

**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!

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Latest revision as of 12:34, 15 March 2012

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

Statistical analysis

Reynolds averaged equations and the turbulence closure problem

Turbulence kinetic energy

Stationarity and homogeneity

Homogeneous turbulence

Free turbulent shear flows

Wall bounded turbulent flows

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.

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-==What is Turbulence?==
+{{Introduction to turbulence menu}}
-Turbulence is that state of fluid motion which is characterized by apparently random and chaotic three-dimensional [[vorticity]]. When turbulence is present, it usually dominates all other flow phenomena and results in increased energy dissipation, mixing, heat transfer, and drag.
+__NOTOC__
-For a long time scientists were not really sure in which sense turbulence is 'random', but they were pretty sure it was. Like any one who is trained in physics, we believe the flows we see around us must be the solution to some set of equations which govern. (This is after all what mechanics is about- writing equations to describe and predict the world around us) But because of the nature of the turbulence, it wasn't clear whether the equations themselves had some hidden randomness, or just the solutions. And if the latter, was it something the equations did to them, or a consequence of the intial conditions
+'''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)'''
-===Why Study Turbulence?===
+== [[Introduction to turbulence/Nature of turbulence|Nature of turbulence]] ==
-There really are the two reasons for studying turbulence- engineering and physics! And they are not necessarily complementary, atleast in the short run.
+* [[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?]]
-Certainly a case can be made that we don't know enough about the turbulence to even start to consider engineering problems. To begin with, we always have fewer equations that unknowns in any attempt to predict anything other than the instantaneous motions. This is the famous [[turbulence closure problem]].
+== [[Introduction to turbulence/Statistical analysis|Statistical analysis]] ==
-Of course, closure is not a problem when performing a so called [[DNS simulation]] (Direct Numerical Simulations) in which we numerically produce the instantaneous motions in a computer using the exact equations governing the fluid. Unfortunately we won't be able to perform such simulations for real engineering problems until atleast a few hundred generations of computers have come and gone. And this wo't really help us too much, since even when we now perform a DNS simulation of a really simple flow, we are already overwhelmed by the amount of data and its apparent random behaviour. This is because without some kind of theory, we have no criteria for selecting from it in a single lifetime what is important.
+*  [[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]]
-The engineer's counter argument to the scientist's lament above is:
+== [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaged equations and the turbulence closure problem]] ==
-* airplanes must fly,
+* [[Introduction to turbulence/Reynolds averaged equations#Equations governing instantaneous fluid motion|Equations governing instantaneous fluid motion]]
-* weather must be forecast,
+* [[Introduction to turbulence/Reynolds averaged equations#Equations for the average velocity|Equations for the average velocity]]
-* sewage and water management systems must be built
+* [[Introduction to turbulence/Reynolds averaged equations#The turbulence problem|The turbulence problem]]
-* society needs ever more energy-efficient hardware and gadgets.
+* [[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]]
-Thus the engineer argues, no matter the inadequate state of our knowledge, '''we have the responsibilty as engineers to do the best we can with what we have'''. Who, considering the needs, could seriously argue with this? Almost incredibly - some physicists do!
+== [[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]]
-It seems evindent then that there must be at least two levels of assault on turbulence. At one level, the very nature of turbulence must be explored. At the other level, our current state of knowledge- however inadequate it might be- must be stretched to provide engineering solutions to real problems.
+== [[Introduction to turbulence/Stationarity and homogeneity|Stationarity and homogeneity]] ==
+* [[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]]
-===The cost of our ignorance===
+== [[Introduction to turbulence/Homogeneous turbulence|Homogeneous turbulence]] ==
-It is difficult to place a price tag on the cost of our limited understanding of turbulence, but it requires no imagination at all to realize that it must be enormous. Try to estimate, for example, the aggregate cost to society of our limited turbulence prediction abilities which result in inadequate weather-forecasts alone. Or try to place a value on the increased cost to the consumer need of the designer of virtually every fluid-thermal system-from heat exchangers to hypersonic planes- to depend on empiricism and experimentation, with the resulting need for abundant safety factors and non-optimal performance by all but the crudest measures.Or consider the frustration to engineers and cost to management of  the never-ending need for 'code-validation' experiments every time a new class of flows is encounteredor major design change is contemplated. The whole idea of 'codes' in the first place was to be able to evaluate designs wihtout having to do experiments or build prototypes.
+* [[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 ]]
-===What do we really know for sure?===
+== [[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]]
-Turbulence is a subject on which still studies are going on. We really don't know a whole lot for sure about turbulence. And worse, we even disagree about what we think we know! There are indeed some things some researchers think we understand pretty well - like for example the kolmogorov similarity theory for the dissipative scales and the Law of the Wall for wall-bounded flows. These are based on assumptions and logical constructions about how we believe turbulence behaves in the limit of infinite Reynolds number. But even these ideas have never been tested in controlled laboratory ecperiments in the limits of high Reynolds number, because no one has ever had the large scale facilities required to do so.
+== [[Introduction to turbulence/Wall bounded turbulent flows|Wall bounded turbulent flows]] ==
-It seems to be a characteristic of humans(and contrary to popular beleif, scientists and engineers are indeed human) that we tend to accept ideas which have been around a while as fact, instead of just working hypotheses that are still waiting to be tested. One can reasonably argue that the acceptance of most ideas in turbulence is perhaps more due to the time lapsed since they were proposed and found to be in resonable agreement with limited data base, than that they have been subjected to experimental tests over the range of their assumed validity. Thus it might be wise to view most 'established' laws and theories of turbulence as more like religious creeds than matters of fact.
+* [[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]]
-The whole situation is a bit analogous to the old idea that the sun and stars revolved around the earth - it was a fine idea, and even good today for navigational purposes. The only problem was that one day someone (Copernicus, Brahe and Galileo among them) looked up and realized it wasn't true. So it may be with a lot of what we believe today to be true about turbulence - some day you may be the one to look at evidence in a new way and decide that things we thought to be true are wrong.
-==Credits==
-'''This text was based on "Introduction to Turbulence" by Professor William K.George, Chalmers University of Technology, Sweden.'''
+{{Turbulence credit wkgeorge}}
+[[Category: Turbulence]]