2 Overview

Khan et al. (Reference Khan, Verma, Hankare, Kumar and Kumar2020) study granular shocks experimentally, using a gravity-driven flow of grains through a thin glass channel, 900 mm long by 310 mm wide, 5 mm deep, and inclined at $\unicode[STIX]{x1D719}\approx 33^{\circ }{-}80^{\circ }$ to the horizontal. During an experiment, ∼7 kg of 0.125 mm diameter glass beads are released uniformly across the chute width, forming a thin accelerating sheet of granular flow, about 2.5 mm deep and reaching speeds of 1–3 m s^-1. At the midpoint of the chute, two triangular obstacles alongside one another, with vertices pointing towards the oncoming flow, obstruct and divert the flowing grains, creating shocks in the granular flow (figure 1a).

Figure 1. Adapted from Khan et al. (Reference Khan, Verma, Hankare, Kumar and Kumar2020). (a) A rapid granular flow (direction indicated by arrows) impinging onto two triangular obstacles. Oblique granular shocks separate undisturbed flow from denser and thicker flow (which appears darker). (b) Shocks from blunter obstacles are detached (i.e. they do not contact the obstacle apex) and collide to form a very dense granular streak. (c–f) The changing morphology of granular shocks and dense streak as the chute inclination angle $\unicode[STIX]{x1D719}$ is increased from 33^° to 80^°.

Across these shocks, the flow rapidly becomes both denser and thicker, thus combining aspects of both classical free-surface and compression shocks. Depending on the shape and spacing of the obstacles, and inclination angle of the chute, which determines the speed of the incoming flow, Khan et al. demonstrate that a wide variety of steady and unsteady shock structures can form.

When the obstacles are sufficiently widely spaced, the flow around each can be treated separately. As in supersonic compressible fluid flows, the flow depends qualitatively on the angle at the apex of each obstacle. When this angle is small, oblique shocks form at this apex (figure 1a), whereas for larger angles the shock is detached from the obstacle, forming some distance upstream of the apex (figure 1b).

When the chute inclination angle $\unicode[STIX]{x1D719}$ is small (figure 1c), a region of slowly moving densely packed grains collects around the apex of each obstacle (Amarouchene, Boudet & Kellay Reference Amarouchene, Boudet and Kellay2001). The focus of Khan et al.’s work, however, is the flow in between two relatively closely spaced obstacles, where the shocks and associated thicker stream of grains diverted from each obstacle collide. The collision of these two streams forms a region of very dense, near-incompressible granular flow (with grain volume fraction close to the maximum ≈0.6) that is in frictional contact with both the lower and upper surfaces of the chute. The energy loss associated with the inelastic collision of grains in this region promotes the formation of a persistent dense cluster, within an otherwise energetic granular flow (Kudrolli, Wolpert & Gollub Reference Kudrolli, Wolpert and Gollub1997). Grains at the top of this dense region rest on those by those below, forming a granular pile (which is likely further supported by friction with the chute walls, as in Taberlet et al. (Reference Taberlet, Richard, Valance, Losert, Pasini, Jenkins and Delannay2003)) but grains at the very bottom of the dense region fall under gravity to form a dense granular streak. As the inclination angle of the chute is increased (figure 1c–f), this shock interaction region becomes the densest and slowest-moving part of the flow. The presence of such a dense, liquid-like region, suspended away from the obstacle by dilute gas-like flow, is reminiscent of the so-called granular Leidenfrost effect (Eshuis et al. Reference Eshuis, van der Weele, van der Meer and Lohse2005), in which a dense granular flow is suspended above a vibrating boundary by energetic grains in a dilute gas-like state.

As the inclination angle $\unicode[STIX]{x1D719}$ increases, the upstream edge of this dense region of grains changes in shape from pointed (figure 1c,d) to a blunt, flat-topped shape (figure 1e,f). For parameters close to the transition between these two regimes (figure 1e), Khan et al. (Reference Khan, Verma, Hankare, Kumar and Kumar2020) observe that the dense collision region oscillates from side to side, bouncing between the two obstacles with a frequency of approximately 3 Hz. This instability leaves oscillations in the dense streak of grains downstream.

3 Future

The granular flows observed by Khan et al. (Reference Khan, Verma, Hankare, Kumar and Kumar2020) present a significant challenge for theoretical continuum models for granular flow, exhibiting both dilute and dense regions of flow, a strong influence of boundaries, and flow features potentially comparable in size to the mean free path of grains. Some insight into the role of granular rheology in these flows can be gained by comparing the experimental results with solutions of possibly the simplest analogous shock-forming system, the non-dimensional shallow-water equations (Tan Reference Tan1992). These hyperbolic equations express the conservation of mass, $\unicode[STIX]{x2202}_{t}h+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h\bar{\boldsymbol{u}})=0$, and the conservation of momentum, $\unicode[STIX]{x2202}_{t}(h\bar{\boldsymbol{u}})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(h\bar{\boldsymbol{u}}\bar{\boldsymbol{u}}+\unicode[STIX]{x1D644}h^{2}/2)=0$, of a thin layer of inviscid, incompressible fluid, of thickness $h$ and depth-averaged (two-dimensional) velocity $\bar{\boldsymbol{u}}$. Granular physics is entirely neglected in these equations, yet numerical solutions of flows past triangular obstacles (figure 2) show some qualitatively similar shock interactions to those observed by Khan et al., illustrating the fundamental importance of mass and momentum conservation in determining these structures. But the simple shallow-water model fails, of course, to capture much of the complexity of Khan et al.’s flows, including large density variations, oscillatory shocks and the ‘indent’ in the detached shock near the apex of each obstacle at high inclinations (figure 1f), which must be attributed to specifics of granular compression and shear rheology. A unified theoretical description of these observations, and more generally of granular flows with both gas-like and liquid-like regions, remains an important challenge for the granular modelling community.

Figure 2. Steady numerical solutions of the shallow-water equations for uniform flow past triangular obstacles, showing flow thickness (shading) and streamlines. Discontinuities in thickness indicate the formation of shocks. The shock structures depend strongly on the inflow Froude number $Fr=|\bar{\boldsymbol{u}}|/\sqrt{h}$, and are reminiscent of several regimes observed by Khan et al. (Reference Khan, Verma, Hankare, Kumar and Kumar2020).