Statistical theory of turbulence

In his treatise The local structure of turbulence in an incompressible viscous liquid at very high Reynolds numbers , Kolmogorov [289] considered the elements of free turbulence as random variables, which are in general terms accessible to probability theory. This assumes local isotropic turbulence. Thus the probability distribution law is independent of time, since a temporally steady-state condition is present. For these conditions Kolmogorov postulated two similarity hypotheses  

The laws of statistical distribution for locally-isotropic turbulence are clearly determined by the kinematic viscosity v and the power per unit mass e = P/p -Dimensional analysis gives the following relationship for the linear dimension 2 of a turbulence element  

Energy transfer from larger to smaller turbulence elements is independent of viscosity for all turbulence elements in between with dimensions z. 

On the other hand, the so-called macro-scale of turbulence A is given by the size of the primary eddies and is of the order of magnitude of the stirrer diameter. Thus the precondition for the existence of locally isotropic turbulence is sufficient difference between both scales and in a high Re number [364]. 

The division of the kinetic energy into the individual eddy regions takes the form of an energy spectrum E[k). Only those parts of the spectrum in the region of small eddies are of interest, in which locally-isotropic turbulence is expected. Two regions can be clearly distinguished, for which different relationships apply (see Section 1.4.2.2). 

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Kolmogoroff A (1958) Collected works on the statistical theory of turbulence. Akademic Verlag, Berlin... 

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

The definition of X incorporates the fact that the Lagrangian integral time scale is of the order of Zilw,. The statistical theory of turbulent diffusion outlined in the beginning of Section VIII,B can be used to estimate the functional dependence of g as... 

The statistical theory of turbulent diffusion (Section VIII,B) predicts that the mean square displacement of a fluid particle in, say, the y direction manifests the following behavior ... 

Reynolds (Rl, R2) was one of the earlier investigators to appreciate the random nature of turbulence. The dimensionless parameter bearing his name is widely used as a measure of the physical characteristics of steady, uniform flow. Such a measure is essentially macroscopic and does not describe the local or transient behavior at a point in the stream. In recent years much effort has been devoted to understanding the basic mechanism of momentum transport by turbulence. The early work of Prandtl (P6), Taylor (Tl), Karmdn (Kl), and Howarth (K4) laid a basis for the statistical theory of turbulence which is apparently in reasonable agreement with experiment. More recently Onsager (03), Corrsin (C6), and Kolmogoroff (K10) extended the statistical theory of turbulence to describe the available experimental data in terms of kinetic-energy... 

But in the statistical theory of turbulence the situation is more complicated. 

In this approach, the unsteady processes occurring in turbulent flows are visualized as a combination of some mean process and small-scale fluctuations around it. The typical time variation of fluid velocity at a point in a turbulent flow is shown in Fig. 3.1. In the statistical approach, an instantaneous velocity, U, is visualized as a mean velocity, U (shown by a horizontal line in Fig. 3.1) and fluctuations around it, u. Based on such an approach, the statistical theory of turbulence flows has been developed (see Hinze, 1975 and references cited therein). It has been the basis for most of the engineering modeling of turbulent flow processes. Some of the key concepts of the statistical approach are discussed below. 

The three dimensional flow field in a tank characterized by secondary flow patterns was long inaccessible to theoretical treatment. It is therefore not surprising that it was first tackled by the statistical theory of turbulence [20, 57, 209, 289]. 

In this connection it is important to be able to ascertain the smallest material ball attainable and to estimate the homogenization time, which is thereby required. Mixing or stirring power has to be expended to decrease the diffusion length or decrease the size of the segregated liquid balls. According to the statistical theory of turbulence due to Kolmogorov [143, 289], see Section 1.4.2, the size of the liquid balls can be estimated ... 

A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was introduced by Taylor [159, 160, 161] and [162, 163], and further developed by von Karman [178, 179]. Most of the fundamental ideas and concepts of the statistical turbulence theory were presented in the series of papers published by Taylor in 1935. The two-point correlation function is a central mathematical tool in this theory. Considering the statistics of continuous random functions the complexity of the probability density functions needed in a generalized flow situations was found not tractable in practice. An idealized flow based on the assumption of... 

Dryden HL (1943) A review of the statistical theory of turbulence. Quart Appl... 

Taylor GI (1935) Statistical Theory of Turbulence, I-III. Proc Roy Soc London A151(874) 421-464... 

Taylor GI (1935) Distribution of velocity and temperature between concentric rotating cylindres. Proc Roy Soc London A151(874) 494-512 Taylor GI (1936) Statistical Theory of Turbulence. V. Effect of turbulence on boundary layer. Proc Roy Soc London A156(888) 307-317 Taylor GI (1937) The Statistical Theory of Isotropic Turbulence. Journal of the Aeronautical Sciences 4(8) 311-315... 

This result is identical to (18.25) if b = TL 1, which provides a nice connection between the statistical theory of turbulent diffusion and the basic example considered earlier. 

Many of these theories are quite complex mathematically. The more involved mathematics led one fluid mechanics expert to comment that these-theories confirm one s suspicion that the aim of the statistical theory of turbulence is full employment for mathematicians [12]. 

H. L. Dryden, A review of the statistical theory of turbulence, Q. Appl Math. 1 7-42 (1943). This paper and most of G. I, Taylor s basic papers are contained in S. K. Friedlander and L. Topper, eds.. Turbulence—Classical Papers on Statistical Theory, Interscience, New York, 1961. 

Lin, C.C. Statistical Theories of Turbulence. Princeton University Press 1961. 

Hill, J. C. and C. A. Petty A Statistical Theory of Turbulent Mass Transfer Induced By a Mean Concentration Gradient. PACHEC III, Seoul, Korea, May 8-11, 1983. 

Shinnar R, Church JM. (1960). Statistical theories of turbulence in predicting particle size in agitated dispersions. Ind. Eng. Chem., 52(3) 253-256. 

Taylor GA. (1935) Statistical theory of turbulence. Proc. R. Soc. Lond. A, 151 421 444. Whitaker S. (1972) Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres and for flow in packed beds and mbe bundles. AICHEJ, 18 361-370. 

Mesoscopic statistical theories of turbulence laminar and turbulent transport thermal conductivity, diffusivity effective transport coefficients... 

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