Statistical description of turbulent flow

Given their complexity and practical importance, it should be no surprise that different approaches for dealing with turbulent reacting flows have developed over the last 50 years. On the one hand, the chemical-reaction-engineering (CRE) approach came from the application of chemical kinetics to the study of chemical reactor design. In this approach, the details of the fluid flow are of interest only in as much as they affect the product yield and selectivity of the reactor. In many cases, this effect is of secondary importance, and thus in the CRE approach greater attention has been paid to other factors that directly affect the chemistry. On the other hand, the fluid-mechanical (FM) approach developed as a natural extension of the statistical description of turbulent flows. In this approach, the emphasis has been primarily on how the fluid flow affects the rate of chemical reactions. In particular, this approach has been widely employed in the study of combustion (Rosner 1986 Peters 2000 Poinsot and Veynante 2001 Veynante and Vervisch 2002). 

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. 

Despite its widespread use in the statistical description of turbulent reacting flows, the one-point joint velocity PDF does not describe the random velocity field in sufficient detail to understand the physics completely. For example, the one-point description tells us nothing about the statistics of velocity gradients, e.g.,... 

The ultimate descriptio we would like to have of turbulent flow would be an explicit expression for V, 1/, and Vy, and as functions of time and position. Then we could predict the average and the fluctuating velocities at any point and any time. Currently it seems impossible to make such a description the problem Is much too complex. The next best thing is a statistical description of the flow, i.e., what fraction of the time V, v, Vy, etc., have certain values. So far most of the experimental and theoretical work done on turbulence has been directed at these statistical properties of the,flow. Below we give a set of definitions which are widely used in the turbulence literature to describe such statistical properties of the flow and some experimental values of the quantities so defined. 

Early emphasis in turbulence research was on the statistical analysis and description of turbulent flow fields. Theoretical advances in the field have been complemented by experiments that included pressure measurements and by the point measurement technique of hot wire anemometry (HWA). The intrusive nature of this latter technique has precluded its use in some experiments, whereas in others, corrections have been introduced to the measurement results. 

The idea that the correlation functions and other statistical moments of the fluid mechanical fields must be recognized as the fundamental characteristics of turbulence was first stated by Keller and Friedmann ([81] see also [8], Sect. 2.2 [112], Chap. 2), who proposed a general method of obtaining the differential equations for the moments of arbitrary order for the description of turbulent flow. The determination of... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

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