A quadrature-based third-order moment method for dilute gas-particle flows

Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag, and particle–particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows away from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at any Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the Boltzmann equation, the mass, momentum and energy are conserved for arbitrary Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to any application that can be modeled by a kinetic equation (e.g., thermal and non-isothermal flows currently treated using lattice Boltzmann methods), and examples are given from the literature.