Time-accurate calculation of two-phase granular flows exhibiting compaction, dilatancy and nonlinear rheology

In this paper we propose a time-accurate algorithm for the calculation of two-phase granular flows via two-velocity, two-pressure Eulerian models. Such flows exhibit a number of important characteristics that make their computation particularly challenging. For example, even at constant-density flows the suspended granular phase can compact or expand so that the velocity fields of the two phases are not solenoidal. Also, the granular material typically exhibits a complex and highly nonlinear rheological behaviour. Moreover, the governing equation for the granular volume fraction (often referred to in the literature as “compaction equation” or “particle migration model”) involves stresses as source terms and, therefore it is strongly coupled to the momentum balance laws. Herein we show that this coupling can be a strong destabilising factor in the computations if it is not treated properly. Our algorithm is based on a predictor-corrector time-integration scheme. Further, it employs a generalized double projection method for non-solenoidal velocity fields that results in second-order elliptic equations for the phasial pressures. The efficiency of the proposed algorithm is illustrated via a series of simulations of transient flows with strong concentration gradients, namely, sedimentation of a granular suspension, compaction of a dense sediment bed, and collapse of a submerged granular column. Numerical grid-convergence studies further show that the algorithm exhibits satisfactory convergence rates. Overall, these numerical results indicate that the proposed algorithm is well suited for the simulation of flows of dense suspensions, flows with high volume-fraction variations or material interfaces, as well as for unsteady fluid-solid interaction problems.