Collision source term

In this section we proceed deriving an explicit expression for the collision source term ( )coiiision In (2.78). 

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. 

Keeping only terms up to first-order spatial derivatives in Eq. (6.19) (i.e. up to order dp overall), the collision source term and collisional-fiux term are... 

Using the analytical expressions for the integrals over the collision angles, we can now rewrite the collision source term in Eq. (6.21) as the sum of two contributions. 

At this point, the remaining integrals in the collision source terms are with respect to the velocity density functions. We will now consider the special case in which ij/ represents the integer moments with respect to the velocity components. 

These coefficients, and the summations in Eq. (6.60), are most easily computed using a symbolic math program. Some examples of co, v, g) for selected moments up to fifth order are given in Tables 6.1-6.9. For clarity, in these tables we have denoted the velocity difference vector by g = gi,g2,g3) and g =g + gl+gy The final expressions for the collision source terms for integer moments of order j = h+h + h can now be written in the form of Eq. (6.54) as... 

We can now express the collision source term and collisional-flux term for the integer moments of order y as... 

The utility of the kinetic model is most evident when evaluating the collision source terms for the moments. As noted earlier, Eq. (6.109) is closed but results in complicated polynomial expressions for higher-order moments. In contrast, the kinetic model... 

On the left-hand side of this equation, the collisional flux, deflned by Eq. (6.70), appears. On the right-hand side, the collision source terms, deflned by Eqs. (6.68) and (6.69), appear. The kinetic fluxes (or free transport) correspond to the moments. ... 

Summing together the three expressions in Eq. (6.143) and dividing by three gives the collision source term for the granular temperature ... 

These equations have the same form as in the monodisperse case, expect that they contain additional terms due to collisions between unlike particles. Note that such collisions affect both the spatial flux (e.g. 0/3/3/3,12) and the collision source terms (e.g. C/3/3/3,12). Because Eq. (6.148) can be found from Eq. (6.147) by a permutation of the indices, hereafter we will consider only Eq. (6.147). 

As with the first-order moments, this expression has contributions due to the kinetic and collisional fluxes on the left-hand side, and due to the collision source terms on the right-hand side. The contributions due to like-particle collisions (Oi,200,11 and C2oo,ii) have the same forms as in the case of monodisperse particles described above. We will thus look briefly at the terms due to unlike-particle collisions. 

The term in Eq. (6.167) can be rewritten in closed form using integer moments of up to third order. On the other hand, the right-hand side of Eq. (6.168) cannot be written in terms of any finite set of integer moments and must therefore be closed (e.g. using quadrature). The collision source term C2oo,i2 has four contributions ... 

The function ffjl is derived analytically from the hard-sphere-collision integral, and readers interested in the exact forms are referred to Tables 6.1-6.3 of Chapter 6. One crucial issue is the description of the equilibrium distribution with QBMM. In fact, since the nonlinear collision source terms that drive the NDF and its moments to the Maxwellian equilibrium are approximated, the equilibrium is generally not perfectly described. The error involved is generally very small, and is reduced when the number of nodes is increased, but can be easily overcome by using some simple corrections. Details on these corrections for the isotropic Boltzmann equation test case are reported in Icardi et al. (2012). 

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