Comparison of the compressible class of models and non-local models with the discrete element method for steady fully developed flow of cohesionless granular materials through a vertical channel

Abstract

The flow of granular materials through a vertical channel is examined using the discrete element method (DEM) and the recent continuum models of Henann & Kamrin (Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), Barker et al. (Proc. R. Soc. Lond. A, vol. 473, 2017, p. 20160846), Schaeffer et al. (J. Fluid Mech., vol. 874, 2019, pp. 926–951) and Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, p. R3). The channel is bounded by walls separated by a distance $2 \, W$ in the $x$ -direction. For the DEM, periodic boundary conditions are used in the $z$ - and $y$ - (vertical) directions with no exit at the bottom of the channel. The governing equations reduce to ordinary differential equations in the $x$ -direction. There is a plug layer near the centre and a shear layer near the wall, as observed in experiments. There is a decrease in the solids fraction $\phi$ in the shear layer, except for the models of Barker et al. and Henann & Kamrin. A modification of the latter gives more realistic $\phi$ profiles. The thickness of the shear layer depends on $2\,W$ and the bulk solids fraction $\bar {\phi }$ . For all the models, solutions could not be obtained for some parameter values. An example is the negative fluidity in the model of Henann & Kamrin. The model of Dsouza & Nott predicts much higher normal stresses, possibly because of large contributions from the non-local terms. None of the models specify a complete set of boundary conditions (b.c.). The DEM results suggest that the slip velocity and the wall friction b.c. lead to a slip length and an angle of wall friction that are independent of $2\,W$ . The models are based on extensions of the equations for slow, rate-independent flow. A model that includes collisional effects, such as kinetic theory, should be combined with the present models. A preliminary analysis of the kinetic theory model of Berzi et al. (J. Fluid Mech., vol. 885, 2020, p. A27), shows that it may have undesirable feature.

Different shear regimes in the dense granular flow in a vertical channel

“…Hence the fits of the functional forms for the volume fraction and velocity profiles are not shown here. In Debnath et al (2022), the nonlinear expression (B3) for was used to solve the models of Barker et al (2017) and Schaeffer et al (2019) numerically. However, this does not result in better agreement between the DEM results and the model predictions for and .…”

“…Another issue related to the conduction flux in their model is the following. In the channel flow, if the conduction flux at the interface where φ ad = 0.587 has to be matched, then the slope of the φ profile appears to be discontinuous, which does not agree with the trend predicted by the DEM results (Debnath et al 2022). More work is required to resolve these issues.…”

“…The flow is expected to be unidirectional and fully developed, and the velocity in the vertical direction is a function of only one cross-stream direction, if we assume that the system is infinite in the second cross-stream direction. This flow has been studied in experiments (Savage 1979; Nedderman & Laohakul 1980; Natarajan, Hunt & Taylor 1995; Pouliquen & Gutfraind 1996; Ananda, Moka & Nott 2008) and using theoretical models (Goodman & Cowin 1971; Savage 1979; Gutfraind & Pouliquen 1996; Mohan, Nott & Rao 1997, 1999; Barker, Zhu & Sun 2022; Debnath, Kumaran & Rao 2022). There is consensus that the flow consists of a central plug zone, where the material flows without shearing, and relatively thin layers close to the wall where there is shearing.…”

“…Recently, Debnath et al (2022) compared the predictions of some of these models with simulation results. None of the models predict all the profiles well, and some of the models could not be solved for some values of the parameters used.…”

Extended kinetic theory for granular flow in a vertical chute

“…Despite these and other studies of granular flow through vertical channels and hoppers (Gudhe, Yalamanchili & Massoudi 1994;Natarajan, Hunt & Taylor 1995;Wang, Jackson & Sundaresan 1997) and recent discrete element method (DEM) studies (Zhao et al 2018;González-Montellano, Ayuga & Ooi 2011;Barker, Zhu & Sun 2022;Debnath, Kumaran & Rao 2022a), the flow through a vertical channel still lacks a clear understanding, especially in the rapid flow regime where grains interact through collisions and experience free flight between consecutive encounters. The extended kinetic theory (EKT) permits an understanding of steady, inhomogeneous, collisional, granular shearing flows that range from dilute to dense concentrations, and allows the predictions of profiles of solid volume fraction and particle mean and fluctuation velocity in terms of a small number of measurable parameters.…”