Moment-transport equation turbulence

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. In this chapter the many possible number-density functions (NDF), formulated with different choices for the internal coordinates, are presented, followed by an introduction to the PBE in their various forms. The chapter concludes with a short discussion on the differences between the moment-transport equations associated with the PBE, and those arising due to ensemble averaging in turbulence theory. 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

The next category of turbulence closures, i.e., impl3ung to be more accurate than the very simple algebraic models, is a hierarchy of turbulent models based on the transport equation for the fluctuating momentum field. These are the first-order closure models, i.e., those that require parameterizations for the second moments and the second-order closure models, i.e., those that... 

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. 

This implies that higher order moments are introduced, thus the system of PDEs cannot be closed analytically. It is possible to show that similar effects will occur for the other source terms as well. This problem limits the application of the exact method of moments to the particular case where we have constant kernels only. In other cases one has to introduce approximate closures in order to eliminate the higher order moments ensuring that the transport equations for the moments of the particle size distribution can be expressed in terms of the lower order moments only (i.e., a modeling process very similar to turbulence modeling). 

Compared with Equation 4.6, Equation 4.12 contains the term -pUi Uj, the so-called Reynolds stress, which represents the effect of turbulence and must be modeled by the CFD code. Limited computational resources restrict the direct simulation of these fluctuations, at least for the moment. Therefore the transport equations are commonly modified to account for the averaged fluctuating velocity components. Three commonly applied turbulence modeling approaches have been used in the CFD model of spray drying system, i.e., k-Q model (Launder and Spalding 1972, 1974), RNG k-e model (Yakhot and Orszag 1986), and a Reynolds stress model (RSM) (Launder et al. 1975). 

The parameters Cq, C, Ce, and are model constants and need to be specified [30,31]. The same goes for the pressure dilatation term lid [31,32]. The transport equations for all of the SGS moments are readily obtained by integration of this Fokker-Planck equation. This provides a complete statistical description of turbulence. The idea is to find methods that could take advantage of quantum resources in order to speed up these calculations, at least polyno-mially in the number of variables. Because of the size of the problem typically considered, such a speedup could transform the way these problems are treated in engineering providing solutions to problems many orders of magnitude faster than are possible with classical computers. 

In the case of turbulent advection velocity, the transported quantity in the PBE (i.e. the NDF) fluctuates around its mean value. These fluctuations are due to the nonlinear convection term in the momentum equation of the continuous phase. In turbulent flows usually the Reynolds average is introduced (Pope, 2000). It consists of calculating ensemble-averaged quantities of interest (usually lower-order moments). Given a fluctuating property of a turbulent flow f>(t,x), its Reynolds average at a fixed point in time and space can be written as... 

In the last decade, we have found a rapid development of the method known as the lattice Boltzmann equation (LBE) method. Because of its physical soundness and outstanding amenability to parallel processing, the LBE method has been successfully applied for the simulation of a variety of flow- and mass-transport situations, including flows through porous media, turbulence, advective diffusion, multiphase and reactive flows, to name but a few. In particular, an effective numerical approach to simulate advective-diffusion transport (the moment-... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

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