A filtered coarse-grain Euler-Lagrange formulation for simulating fluidized polydisperse particles

Coarse-grain simulations of particle-laden flows to industrially relevant scales are unable to account for every particle in the system. A common approach to reduce the computational cost is to lump particles into parcels. Ad-hoc corrections are often employed to counter the effects of coarse-grain approximations. They typically fail to converge to the underlying deterministic equations in the limit the number of particles within the parcel approaches unity. In this work, a rigorous formulation of the filtered Eulerian-Lagrangian equations are presented. While exact, the equations result in unclosed terms, notably a sub-filtered drag force. Parcel collisions are handled using soft-sphere approach, modifying the coefficient of restitution based on the number of particles per parcel. The unclosed terms are informed by highly resolved (deterministic) Euler-Lagrange simulations of unbounded moderately dense gas-solid flows. Variation in particle size and velocity within each parcel is quantified. The relative contribution of the sub-filtered drag is found to increase with the number of particles per parcel, and depend on the local filtered volume fraction and Reynolds number. Symbolic regression is employed to obtain closed-form algebraic models.