Quadrature-based moment methods

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. 

Moment closures for the GPBE Quadrature-based moment methods... 

The reconstruction of the NDF using quadrature-based moment methods (QBMM) is described in detail in Chapter 3. In comparison with the moment closures introduced above, QBMM have the following similarities and advantages. 

For all other cases, it will be necessary to solve the moment-transport equations derived from the GPBE as described in Chapter 4. In Chapter 8 the numerical algorithms used to find approximation solutions to the GPBE using quadrature-based moment methods are presented in detail. 

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. 

For quadrature-based moment methods, the collision terms for integer moments are needed. For this special case, the method developed in Fox Vedula (2010) leads to closed expressions for the integrals in terms of finite sums. For integer moments, we have il/ y) = Vj 2 Vg and the terms in the sums over n in Eqs. (6.55)-(6.57) are zero for n > max(/i, /2. h)- For this particular case, we will define for m = 0 by... 

Einally, we can note that Eq. (6.69) will be closed in terms of the moments of order 7 -I- 1 and their spatial gradients. This result is interesting because closure of the moments of order 7 -1- 1 given the moments of order 7 using quadrature-based moment methods is usually quite accurate. Using a symmetric change of variables, it is straightforward to show... 

The analytical expressions for will be used directly with quadrature-based moment methods to evaluate the collision source and collisional-flux terms for each integer moment. The numerical implementation of these terms in the context of quadrature is discussed in Section 6.5. 

The other two collision source vectors, and can be evaluated using the definitions in Eqs. (6.104) and (6.106). As mentioned earlier, will be closed in terms of the moments of order two and lower, and their gradients. In contrast, C will not be closed in terms of any finite set of moments. Nevertheless, it can be approximated using quadrature-based moment methods as described in Section 6.5. In the fluid-particle limit d d2), neither CI2 i or C will contribute terms involving spatial gradients of the fluid properties (i.e. buoyancy, lift, etc.) to the fluid-phase momentum equation. As mentioned earlier, such terms result from the model for gapi i-n) and would appear, for example, on using the expression in Eq. (6.81). With the latter, Eq. (6.161) becomes... 

The moment-transport equations discussed above become more and more complicated as the order increases. Moreover, these equations are not closed. In quadrature-based moment methods, the velocity-distribution function is reconstructed from a finite set of moments, thereby providing a closure. In this section, we illustrate how the closure hypothesis is applied to solve the moment-transport equations with hard-sphere collisions. For clarity, we will consider the monodisperse case governed by Eq. (6.131). Formally, we can re-express this equation in conservative form ... 

In summary, we have demonstrated in this section how quadrature-based moment methods can be used to evaluate the terms in the moment-transport equations arising from collisions. The principal observation is that it suffices to know the functional forms for the terms which are derived and tabulated in Section 6.1. We also observed that, unlike traditional moment closures, the closures developed in this section are applicable to highly non-equilibrium flows. 

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