A numerical method for two-phase flows of dense granular mixtures

Projection methods for two velocity–two pressure models for flows of heterogeneous mixtures

“…This can be done in a straightforward manner in two steps. First, by combining the product differentiation identity for p f ∇φ s and the saturation relation (7) and second, by dividing by φ f , which by (8) is strictly and uniformly bounded away from zero, and by using (4). Upon doing so, we arrive at,…”

“…This problem can be bypassed by noting that the velocity of the mixture u s φ s + u f φ f is divergence-free; in order to see this, it suffices to combine (4) with (2) and the saturation relation (7). Consequently, one could opt for solving the governing equations of the solid phase, (2) and (3), and those of the mixture.…”

“…This can jeopardize the performance of a projection method because, in general, p f is not bounded in the L ∞ -norm; see, for example, [19] for theoretical and [7] for numerical evidence. Nevertheless, as shown above, a combination of algebraic and differential operations allows for casting (5) in a form where p f appears only as a gradient.…”

“…In recent years there has been an increasing interest in the algorithm development of two-velocity, two-pressure models; see, for example, the recent discussions of [6,7] that focus on the numerical integration of such models for granular flows. However, despite these efforts, relevant literature remains notably restrained and our understanding of the related numerical analysis is rudimentary.…”

“…Based on this analysis, we propose a two-phase generalization of the classical Chorin-Temam method, suitably tailored to the structure of the equations at hand. Following the earlier work of [7], the proposed algorithm is coupled with a regularization method that, besides its ease of implementation, can efficiently handle strong material interfaces with varying topology. In the last part of this study, the performance and efficacy of the proposed algorithm are assessed via the numerical simulation of a Poiseuille and Couette flow of a simple fluid over and through granular beds and of a transient flow of a simple fluid over two-dimensional dunes.…”