Coarse-graining particulate two-phase flow

To acquire the ability to numerically study the rheology of particulate two-phase flows that lack scale separation, we present a general method to average or coarse-grain the equations of motion of a mixture of a continuous fluid of arbitrary rheology and non-Brownian particles, interacting via contacts, of arbitrary shapes and compositions. It universally covers ensemble and typical spatio-temporal averaging procedures and overcomes two shortcomings of existing methods. First, the derived micromechanical expressions for the coarse-grained fields are mathematically exact and formulated in a manner that allows a computationally cheap extraction from Direct Numerical Simulation-Discrete Element Method (DNS-DEM) simulations, avoiding the unlimited-order derivatives appearing in previous exact formulations. Second, the microscopic volume fraction of each particle is its corresponding indicator function, rather than the traditional volume-weighted delta distribution at its center of mass, to ensure that the resulting macroscopic fluid and solid volume fractions add precisely to unity. This leads to an additional contact stress contribution not seen in standard coarse-grained expressions for granular matter, and, for non-spherical particles, to particle-rotational contributions to translational solid phase balance equations. Many implementations of DNS-DEM simulations are based on the Immersed Boundary Method (IBM), for which modifications of the coarse-graining method are necessary due to certain peculiarities of the IBM, such as the replacement of the particles' interiors by pseudo-fluid. We therefore derive a mathematically exact adaptation of the method that is applicable to the IBM, implement it to obtain coarse-grained fields from IBM-based sediment transport simulations, and validate this adaptation and its implementation.