A Gauss/Anti-Gauss Quadrature Method of Moments Applied to Population Balance Equations with Turbulence-Induced Nonlinear Phase-Space Diffusion

Turbulent dispersed multiphase systems are of paramount importance in numerous physical applications. In terms of the numerical modeling of such systems, crucial aspects are, among others, the mathematical representation of dispersed particles and the effect of fluid turbulence on those particles. The former can be classified in terms of the level of detail as microscale, mesoscale or macroscale approach. A mesoscale model provides a statistical description of the observed particle population, taking the form of a number density function (NDF) with respect to a vector of variables in phase space representing particle properties such as size and velocity. The evolution of the particle phase is then governed by a transport equation for the NDF, referred to as population balance equation (PBE). Quadrature-based moment methods are, instead, based on the macroscopic description in terms of a set of moments associated with the NDF. The governing moment equations can be derived directly from the underlying PBE and the unclosed integral terms arising due to an unknown NDF are closed using Gaussian quadrature rules. One way to model the effect of a turbulent fluid velocity field on particle velocities in the mesoscopic description is a diffusion term in the PBE as a result of random microscale fluctuations. Considerations on the microscopic behavior concerning the momentum exchange between fluid and dispersed particles suggest that this diffusion term is nonlinear and nonsmooth. This will, combined with the common quadratic drag law, be revealed to entail large approximation errors when using Gaussian quadrature rules for closure. In the present work, we propose a modification of the quadrature method of moments (QMOM), namely the Gauss/anti-Gauss-QMOM (GaG-QMOM), making use of anti-Gaussian quadrature formulae to reduce the large errors due to this particular form of diffusion. We investigate this method in a series of simple univariate test cases with analytical reference solutions. Moreover, we propose a realizability-preserving variation of the strong-stability preserving Runge-Kutta. (RKSSP) schemes that is suited to problems involving phase-space diffusion. Besides the observation that the modified second-order RKSSP scheme is an adequate realizability-preserving alternative to the standard RKSSP scheme, the numerical results reveal that, compared to the standard QMOM, our suggested approach can reduce the large errors by one to two orders of magnitude