First-order moment closures

The limiting case where the chemical time scales are all large compared with the mixing time scale r, i.e., the slow-chemistry limit, can be treated by a simple first-order moment closure. In this limit, micromixing is fast enough that the composition variables can be approximated by their mean values (i.e., the first-order moments (0)). We can then write, for example,... 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

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 rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Or, to put it another way, the simplest first-order moment closure is to assume that all scalar covariances are zero ... 

Equation [8] is then substituted into [7] and [6]. Only first-order terms are kept in the wavefunction and in the derived matrix elements, and an average energy approximation is invoked to permit closure of the suras over vibrational states. For the electronic parts of the dipole transition moments, one obtains, for the /th normal mode... 

Knowing the molecular permanent multipole moments and transition moments (or closure moments derived from sum rules, such as (36)), the computation of the first and second order interaction energies in the multipole expansion becomes very easy. One just substitutes all these multipole properties into the expressions (16), (20), (21) and (22), together with the algebraic coefficients (24) (tabulated up to R terms inclusive in ref. , in a somewhat different form ), and one calculates the angular functions (lb) for given orientations of the molecules. 

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. 

In order to solve the moment-transport equation, a closure must be provided for Using the QMOM, the first IN moments can be used to find N weights and N abscissas a- The flux function can then be closed using... 

The first derivative term vanishes due to the identification Qo = (Q)- Terms above a certain order (Q") are approximated by products of the lower order moments using the closure procedure described in detail in Refs. [56,58]. 

The key to the MOM is that the lower-order moments can be tracked directly without requiring further information of the number density distribution. The standard MOM accomplishes this achievement by utilizing equations formulated for the evolution of the moments in closed form. For this reason the closure relations employed can only be functions of the moments themselves. It is thus noted that it is necessary to limit the dependency of the growth velocity to a linear function of the size property as well as the expansion of the source terms to the -th order moment k < j) in order to close the set of the -th moment [90, 187]. It is assumed that the moment integration of the source term functions can be expanded in terms of the first k moments of the distribution. This inherent closure requirement is a severe restriction of the method. 

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