General average balance equations and buoyancy closure for a mixture of a continuous fluid and rigid particles

Abstract: The equations of motion of a continuous fluid and rigid, non-Brownian particles that interact through contacts are averaged to derive average balance equations. The averaging procedure is generic, requiring only linearity and commutation with the space and time derivatives. No assumptions about the fluid's rheology and the particles' shapes and compositions are made. We obtain mathematically exact micromechanical expressions for the fluid phase, solid phase, and mixture fields of interest. We demonstrate for a specific example, numerical simulations of sediment transport, based on Direct Numerical Simulations and the Discrete Element Method, that they can be conveniently extracted from such particle-resolved simulations without the need of approximations. In contrast, fields extracted using previous formalisms based on first-order Taylor expansions in the ratio between the particle size and the macroscopic flow scale are in clear disagreement with the exact ones due to a lack of scale separation. This study therefore lays the groundwork for numerically studying the rheology of particulate two-phase flows in systems without scale separation. We also use the simulations to test existing buoyancy closures, that is, decompositions of the fluid-solid interaction force density into a generalized-drag force density and a generalized-buoyancy force density. We find that only one existing closure is consistent with the simulation data and existing experiments. However, it leads to a contradictory fluid momentum balance for certain stationary fluid-particle systems. We resolve this contradiction through a physically-motivated improvement. The improved closure essentially low-pass-filters buoyancy effects of the original closure.