Turbulent flow transport theory

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

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 recent advances in turbulence theory have been so rapid and the experimental investigations sufficiently extensive that it is beyond the scope of the present discussion to consider these matters in detail. However, before the influence of turbulence upon thermal and material transport is considered, it is worth while to describe briefly some of the characteristics of turbulent flow from an experimental point of view. The description will be limited to the characteristics of steady uniform flow... 

Fallis (FI) considered thermal transport in transitional and turbulent boundary flows and supplied a reasonable analysis of this difficult problem which is in agreement with the work of Eber (El) and the theory of Eckert and Drewitz (E2). Callaghan (Cl) contributed to the analogies between thermal and material transport in turbulent flow with particular emphasis upon the behavior near and in the boundary layer. The effect... 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

Review of the theory of turbulent flow and its relation to sediment-transportation." Trans. Am. Geophys. Union, Hydrology Sec., 14th ann. meeting, pp. 487-491. 

A consistent theory of turbulent flow in horizontally homogeneous canopies has been developed over the last twenty five to thirty years. This theory asserts that the origin of the large energy-containing eddies that dominate canopy TKE and transport is an... 

The concept of the full PDF approaches is to formulate and solve additional transport equations for the PDFs determining the evolution of turbulent flows with chemical reactions. These models thus require modeling and solution of additional balance equations for the one-point joint velocity-composition PDF. The full PDF models are thus much more CPU intensive than the moment closures and hardly tractable for process engineering calculations. These theories are comprehensive and well covered by others (e.g., [8, 2, 26]), thus these closures are not examined further in this book. 

In chap 1 a survey of the elements of transport phenomena for single phase multicomponent mixtures is given. This theory serves as basis for the development of most chemical engineering models as well as the multiphase flow concepts to be presented in the following chapters. The first part of the chapter considers laminar single phase flows for multicomponent mixtures. In the second part of the chapter the governing equations are applied to turbulent flows. 

In 1924 Lewis and Whitman 1 suggested that the film theory model could be applied to both die gas and liquid phases during gas absorption. This two-film theory has hed extensive use in modeling steady-state transport between two phases. Transferor species A occurring between a gas phase and liquid phase, each of which may be in turbulent flow, can be described by the individual rate expressions bstween the bulk of each phase and the interface. 

The transition from a turbulent flow regime with advective and eddy transport to a small scale dominated by viscosity and diffusional transport is apparent when an impermeable solid-water interface such as the sediment surface is approached (Fig. 5.4). According to the classical eddy diffusion theory, the vertical component of the eddy diffusivity, E, decreases as a solid interface is approached according to E = A v where A is... 

In considering the transport of a species from a fluid in turbulent flow toward a solid surface, for example, an electrochemically active species to an electrode, Nemst assumed that the transport was governed by molecular diffusion through a stagnant film of fluid of thickness 5. This model, although having questionable physical relevance, is quite useful for correlating effects such as the influence of chemical reaction on mass transfer. A few simple examples of the use of film theory to describe mass transfer in the presence of chemical reaction are considered here. 

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