A Quadrature-based Moment Closure for the Williams Spray Equation

Sprays and other dispersed-phase systems can be described by a kinetic equation containing terms for spatial transport, acceleration, and particle processes (such as evaporation or collisions). In principle, the kinetic description is valid from the dilute (non-collisional) to the dense limit. However, its numerical solution in multi-dimensional systems is intractable due to the large number of independent variables. As an alternative, Lagrangian methods discretize the density function into parcels that are simulated using Monte-Carlo methods. While quite accurate, as in any statistical approach, Lagrangian methods require a relatively large number of parcels to control statistical noise, and thus are computationally expensive. A less costly alternative is to solve Eulerian transport equations for selected moments of the kinetic equation. However, it is well known that in the dilute limit, Eulerian methods have great difficulty describing correctly the moments as predicted by a Lagrangian method. A two-point quadrature-based Eulerian moment closure is developed and tested here for the Williams spray equation. It is shown that the method can successfully handle highly non-equilibrium flows (e.g., impinging particle jets, jet crossing, and particle rebound off walls) that heretofore could not be treated with the Eulerian approach