Moment-transport equations

The results shown in Figs. 6 and 7 can be combined to compute the mean mixture fraction ( ) and its variance (c2) from Eqs. (34) and (35), respectively. Example plots are shown in Liu and Fox (2006) and, as expected, they agree with the solution found by solving the moment transport equations directly (Eqs. 28 and 29). 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

The quadrature method of moments (QMOM) is a presumed PDF approach that determines the unknown parameters by forcing the lower-order moments of the presumed PDF to agree with the moment transport equations (McGraw 1997 Barrett and Webb 1998 Marchisio et al. 2003a Marchisio et al. 2003b). As with the multi-environment presumed PDF method discussed in Section 5.10, the form of the presumed PDF is... 

For this case, the mixed-moment transport equations found from (B.5) will be closed. 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Moment methods come in many different variations, but the general idea is to increase the number of transported moments (beyond the hydrodynamic variables) in order to improve the description of non-equilibrium behavior. As noted earlier, the moment-transport equations are usually not closed in terms of any finite set of moments. Thus, the first step in any moment method is to apply a closure procedure to the truncated set of moment equations. Broadly speaking, this can be done in one of two ways. 

Work directly with the unclosed terms in the moment-transport equations to find a functional form to close them (Struchtrup, 2005). For example, a spatial flux involving moment might be closed using a gradient-diffusion model involving... 

Use the transported moments to reconstruct the unknown density function and then compute the unclosed terms in the moment transport equations. For example, in Grad s 13-moment method (Grad, 1949b) the reconstructed density function uses moments up to third order, and the fourth-order moments are found by integration. 

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... 

Many disperse-phase systems involve collisions between particles, and the archetypical example is hard-sphere collisions. Thus, Chapter 6 is devoted to the topic of hard-sphere collision models in the context of QBMM. In particular, because the moment-transport equations for a GBPE with hard-sphere collisions contain a source term for the rate of change of the NDF during a collision, it is necessary to derive analytical expressions for these source terms (Fox Vedula, 2010). In Chapter 6, the exact source terms are derived... 

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. 

The process of finding moment-transport equations starting from the PBE can be continued to arbitrary order. We should note that in most applications the resulting moment-transport equations will not be closed. In other words, the moment-transport equation of order k will involve moments of order higher than k. We will discuss moment-closure methods in Chapters 7 and 8 in the context of quadrature-based moment methods. 

In order to reduce the number of independent variables, we use the definition of the moments in Eq. (2.2) to find the moment-transport equation corresponding to Eq. (2.50) ... 

In general, simulations carried out starting from realizable moment sets should result in realizable moment sets. Nevertheless, the moment-transport equations are integrated numerically with some finite discretization errors. As Wright (2007) clearly reports, most of the problems are caused by the approximation of the convective term, in particular with higher-order discretization schemes, which can turn a realizable set of moments into... 

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