Statistical Turbulence Theory

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. 

In the remainder of this chapter, an overview of the CRE and FM approaches to turbulent reacting flows is provided. Because the description of turbulent flows and turbulent mixing makes liberal use of ideas from probability and statistical theory, the reader may wish to review the appropriate appendices in Pope (2000) before starting on Chapter 2. Further guidance on how to navigate the material in Chapters 2-7 is provided in Section 1.5. 

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. 

A. Yoshizawa. Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Phys. Fluids, 29(7) 2152-2164, 1986. 

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. 

Y. L. Klimontovich, Turbulent Motion and the Structure of Chaos A New Approach to the Statistical Theory of Open Systems, Kluwer, Dordrecht, NL, 1992. 

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... 

Friedlander SK, Topper L (1961) Turbulence Classic papers on statistical theory. Friedlander SK, Topper L (eds). Interscience Publishers, Inc., New York... 

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... 

Friedlander, T., Turbulence Classical Papers on Statistical Theory, London, 1961. 

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. 

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