12 - Quadrature-based moment methods for particle-laden flows

Abstract

The method of moments for the solution of a kinetic equation is introduced in this chapter. Generalizations of the kinetic equation for problems involving the evolution of the particle size and strongly coupled heat transfer phenomena are introduced. Linearized forms of the collision operator such as the Bhatnagar–Gross–Krook (BGK) and ellipsoidal statistical BGK models are discussed. The dimensionless form of the Boltzmann kinetic equation is considered to illustrate the role of the Knudsen number on the choice of a computational model for particle-laden flows. Specific forms of the particle velocity distribution are discussed in relation to the Stokes number of the disperse phase. The concept of moment of a multivariate number density function (NDF) is introduced and the physical interpretation of some of the moments of the NDF is discussed. Particular attention is paid to defining the concept of realizability of a moment vector, presenting criteria to determine if a moment vector is realizable on the support where it is defined. Causes of unrealizable moment vectors are discussed and realizable numerical methods to integrate the conservation equations for a moment vector are discussed. The limitations of low-order moment closures are then illustrated, and the need for hyperbolic high-order closures to predict the correct physical phenomena observed in particle-laden flows with inertial particles is explained. A generic formulation of the conditional quadrature method of moments for joint size–velocity distributions is then shown as example of multivariate quadrature-based moment closure. Finally, the second-order anisotropic Gaussian closure is presented.