Turbulent flow, statistical description

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. 

Much of the theoretical work in turbulent flows has been concentrated on the description of statistically homogeneous turbulence. In a statistically homogeneous turbulent flow, measurable statistical quantities such as the mean velocity2 or the turbulent kinetic energy are the same at every point in the flow. Among other things, this implies that the turbulence... 

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.,... 

Owing to the complexity of multi-point descriptions, almost all CFD models for complex turbulent flows are based on one-point turbulence statistics. As shown in Section 2.1, one-point turbulence statistics are found by integrating over the velocity sample space, e.g.,... 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

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. 

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). 

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