Description
Many applications involve a disperse population of particles (droplets, bubbles, soots, ...), which need to be accurately simulated. In particular, their velocity dispersion may need to be considered for inertial particles, till the possibility of particle trajectory crossing. Moreover, most of the time, the size polydispersion is important to be taken into account.
These populations can usually be described by a kinetic type equation. Compared to other methods like Monte-Carlo, the moment methods are very attractive for solving this equation, considering the possibility of computational cost reduction for a fixed accuracy. In particular, the quadrature method of moments (QMOM) is an efficient moment method, widely used for the description of the size polydispersion. It uses a Gauss quadrature to compute the unclosed terms. However, it is rarely used for the velocity moments due to the weak hyperbolicity of the underlying system of equations and can also encounter some accuracy limitations in some cases, for size moments.
Here, we will focus on two recently developed quadrature based moment methods: the hyperbolic quadrature method of moments (HyQMOM) and the generalized quadrature method of moments (GQMOM). HyQMOM, first developed in 1D, is a globally hyperbolic velocity moment method, able to describe the velocity dispersion and the particle trajectory crossing. It also has a good behavior when the moment vector tends to the boundary of the moment space, which can append in applications, due to source terms. GQMOM is a generalization of QMOM, allowing the use of a larger number of quadrature points, without increasing the number of moments, thus providing a more accurate moment closure than QMOM at nearly the same computational cost. Both moment methods are based on the properties of the monic orthogonal polynomials Q_n that are uniquely defined by the moments up to order 2n-1.
