1.1 Recirculating particle motion

The first real insights into the structure of the recirculation zone were provided by Pouliquen et al. (Reference Pouliquen, Delour and Savage1997) and Pouliquen & Vallance (Reference Pouliquen and Vallance1999), who used a moving camera to approximately measure the lateral recirculating motion of a line of large black crushed fruit stones placed on the surface of a flow of translucent glass beads. Their observations, however, lacked spatial resolution, and further direct experimental observation of the recirculation has been challenging due to its complex time dependence. The recirculation zone propagates quickly downstream at speed $u_{wave}$ as the front advances forward at speed $u_{front}$ , meaning that there is the difficulty of capturing the motion using a camera moving with the recirculation zone. Long chutes are also required before a steady recirculation regime emerges.

An alternative approach is to use the moving-bed flume set-up shown in figure 3, that is similar to that used by Davies (Reference Davies1990). The flume is 104 cm in length with a rough 10 cm wide upward moving conveyor belt positioned between the four stationary vertical walls. The inclination of the channel was set at 19.8° to establish a uniform flow height along the channel. Higher or lower angles were found to cause an accumulation towards the front or rear of the channel, respectively. The belt moves upstream at a velocity $u_{belt}=72~\text{mm}~\text{s}^{-1}$ . This generates the experimental configuration shown schematically in figure 2(b), where the lower layers of the flow are forced upstream by the belt, while the upper layers move downstream under gravity. While this flow is not itself inversely graded, it is mathematically equivalent to the section of an inversely graded avalanche shown in figure 2(a), provided that the segregation and diffusion rates are constant (Thornton & Gray Reference Thornton and Gray2008). The absence of the layer of large particles also allows a steady state to develop within the experimental configuration. Both the experimental configuration and the full problem are assumed to be two-dimensional, meaning that there are no side-wall effects. Just as in the full problem, the large grains in the experimental configuration (figure 2 b) initially segregate upwards and are sheared towards the downstream end of the flume, as shown in the normal exposure photograph in figure 4(a). However, the motion of the belt forces some large grains to be carried upstream, where they subsequently lie below small grains. The large grains resegregate upwards, and once they reach the surface, they are carried back towards the downstream end of the flume. The oblique view in figure 5 looking upstream from the end of the flume clearly shows the accumulated large particles, and resembles the bouldery front shown in figure 1. This moving-bed flume allows the structure of the steady recirculation regime to be examined in greater detail. For example, the long time exposure photograph in 4(b), taken with an exposure time of 133 s, illustrates the time-averaged concentration field of the recirculation zone.

Figure 5. An oblique upstream view from the surface of steady-state coarse-rich front established in the moving-bed flume of figure 3. The large blue and white marbles congregate towards the front of the picture, with the smaller clear glass beads towards the rear.

The individual motion of the particles on the centre line was revealed using refractive index matched scanning (‘RIMS’: Wiederseiner et al. Reference Wiederseiner, Andreini, Epely-Chauvin and Ancey2011a ; Dijksman et al. Reference Dijksman, Rietz, Lorincz, van Hecke and Losert2012; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015). Spherical borosilicate glass beads of density $2230~\text{kg}~\text{m}^{-3}$ and diameters 14 and 5 mm were used, with the volume ratio of large particles to small particles being 2 : 5. As shown in figure 3, the entire flume set-up was submerged in a tank containing a mixture of benzyl-alcohol and ethanol, which acted as the index matched interstitial fluid of viscosity ${\it\mu}=3~\text{mPa}~\text{s}$ , with a fluid density of $995~\text{kg}~\text{m}^{-3}$ . The motor unit for the belt was positioned outside of the tank and drove the belt through a system of chains. A fluorescent dye (rhodamine) was added to the liquid, which was excited by a laser sheet of wavelength 532 nm in a thin plane parallel to the flow direction. As the particles contain no dye, they appear as dark circles on a bright background. The result is a cross-sectional image of the interior of the flow, which is captured through the glass side wall using a high-speed camera. The laser and camera were positioned to capture the section of the flow containing the recirculation zone. The dark circles are tracked over time, with the minimum and maximum diameters used to determine whether that circle corresponds to a small or large particle. The large size ratio between the grains minimised identification errors, although there was a small possibility that a large particle may be mistaken for a small particle. This, however, would only happen if the particle was sliced close to its edge and never moved closer to the plane of the laser. A typical snapshot of the particle motion is shown in figure 6(a), where it can be seen that there are a few large particles in regions of many small particles at the upstream (left) end of the flow. These large particles are seen to move very slowly, compared with the majority of the large particles which recirculate very quickly towards the front. Figure 6(c) shows a time-averaged concentration plot, which was averaged over a 40 min period, with 1 image taken every 2 s. The slow movement of the large particles through the upstream region of small particles lowers the concentration there, and causes the ‘white’ ‘tail’-like region.

Figure 6. (a) An experimental snapshot of the recirculation zone, captured using the moving-bed flume of figure 3 with refractive index matched scanning. The white label indicates the length scale of 14 mm. (b) Structure of the recirculation zone found using DPM simulations. The fixed base particles are shown in grey. Both the experimental and simulation results show several large particles positioned towards the rear, where they are surrounded by many small particles. These large particles are seen to move very slowly, and take a long time to recirculate. (c) Shows the experimental time-averaged concentration field, which was produced by averaging the individual particle positions over a 40 min period, with 1 image every 2 s. The time-averaged concentration field for the simulations was produced by coarse graining all of the particle positions from 749 subsequent time frames, and is shown in (d). Both of the time-averaged concentration plots indicate a ‘tail’ upstream, where the concentration is lower due to the slow motion of a few large grains. This is similar to asymmetric behaviour observed within a linear shear cell (van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015), and motivates a continuum breaking wave structure with an asymmetric flux function, shown in (e) for a cubic flux. The solid lines mark the boundaries of the recirculation zone, with two distinct ‘lens’ and ‘tail’ regions (see § 2). The downstream ‘lens’ region with a strong green hue is where most of the large particles recirculate, whilst the red hue of the upstream ‘tail’ region shows how only a few large particles recirculate through that area. The theory does not account for spatial velocity variations, diffusive remixing or differential particle friction, and finite-size effects are also significant. These may all contribute to the difference in the ‘tail’ structure between the theory and the experiments and simulations. Without calibrating the segregation flux for this particular flow regime, it is remarkable that the asymmetric flux produces a ‘tail’ region, and it is of interest to further understand the asymmetric breaking-wave structure and particle recirculation within it. In all of the above plots, the lower belt moves from right to left, with gravity acting to cause particles to flow downstream towards the right.

It is worthwhile considering what influence the interstitial fluid has on the particle behaviour. The presence of a fluid (rather than air) not only modifies the interstitial pore pressures, but also couples the stress carried by the particles to that carried by the fluid flowing through gaps in the grain matrix (Iverson & LaHusen Reference Iverson and LaHusen1989; Iverson Reference Iverson1997, Reference Iverson2005). This coupling is particularly significant in unsteady flows, since local changes in the particle volume fraction allow large excess pore pressures to develop, which in turn feedback on the granular motion (du Pont et al. Reference du Pont, Gondret, Perrin and Rabaud2003; Muite, Hunt & Joseph Reference Muite, Hunt and Joseph2004; Pailha, Nicolas & Pouliquen Reference Pailha, Nicolas and Pouliquen2008; Pailha & Pouliquen Reference Pailha and Pouliquen2009). However, for steady, dense granular flows such as those sketched in figure 2, the large number of particle–particle contacts mean that frictional interactions are still dominant in determining the rheological behaviour (Ancey, Coussot & Evesque Reference Ancey, Coussot and Evesque1999) even when an interstitial fluid is present. Cassar, Nicolas & Pouliquen (Reference Cassar, Nicolas and Pouliquen2005) showed that, in steady flows submerged in water, at least 75 % of the overburden pressure is borne by the contact network. They also showed that the same rheology used to describe dense steady aerial flows (GDR Midi Reference GDR Midi2004) also applies to immersed flows, with the interstitial fluid changing the time scale of the particle rearrangements. This is consistent with the experimental results of Vallance & Savage (Reference Vallance, Savage, Rosato and Blackmore2000) and the theory of Thornton, Gray & Hogg (Reference Thornton, Gray and Hogg2006) who both showed that the role of the interstitial fluid in flows containing different sized constituents is to modify the segregation time scales. These results would suggest that the physical phenomena observed in the experiments above, with a few large particles recirculating very slowly in regions of small particles, are indicative an underlying asymmetry in the particle motion that occurs whether the flow is dry or submerged. Further experimental work, using techniques such as X-ray tomography (e.g. McDonald, Harris & Withers Reference McDonald, Harris and Withers2012), is needed to compare the particle scale dynamics in dry flows with those containing an interstitial fluid.

Discrete particle method (DPM) simulations of a moving bed-flume set-up were also performed using the MercuryDPM code (MercuryDPM.org; Thornton et al. Reference Thornton, Krijgsman, Fransen, Gonzalez, Tunuguntla, ten Voortwis, Luding, Bokhove and Weinhart2013a ,Reference Thornton, Krijgsman, te Voortwis, Ogarko, Luding, Fransen, Gonzalez, Bokhove, Imole and Weinhart b ). A dry bidisperse mixture of spherical particles was used, with all of the particles of the same (non-dimensional) density ${\it\rho}^{\ast }={\rm\pi}/6$ , but of two different (non-dimensional) diameters, $d^{s}=1$ and $d^{l}=2.4$ , for small and large particles, respectively. All of the simulation parameters were non-dimensionalised so that $g=1$ . A frictional spring-dashpot model (Cundall & Strack Reference Cundall and Strack1979; Weinhart et al. Reference Weinhart, Thornton, Luding and Bokhove2012) with linear elastic and linear dissipative contributions was used for both the normal and tangential forces. The tangential force models the effects of particle surface roughness, and its spring stiffness was taken to be $2/7$ of the spring stiffness for the normal direction. The tangential force also truncates so that it is always less than $1/2$ of the normal force. The particles all had the same coefficient of restitution $r_{c}=0.1538$ , which was chosen to be less than typical known values for glass $({\sim}\!0.9)$ in order to model the dissipative effects of the interstitial fluid removing energy from the system. The contact time for all head on collisions was fixed at 0.0054, with the collision properties chosen to be different for small/small, small/large and large/large collisions so that both the contact time and the coefficient of restitution were the same even in the mixed case. Further details of the precise DPM implementation may be found in Thornton et al. (Reference Thornton, Weinhart, Luding and Bokhove2012b ) and Weinhart et al. (Reference Weinhart, Thornton, Luding and Bokhove2012). The simulations were conducted in a box of length $300d^{s}$ with fixed end walls and width $8.4d^{s}$ . The side walls were periodic in order to bring the simulations closer to the assumptions of the analytic model in figure 2(b), which is two-dimensional and has no side-wall effects. A small inclined wall was placed between the base and the vertical upstream wall in order to prevent small particles being crushed by the wall or shooting away from it. This was seen to only affect the dynamics very close to the wall, and did not affect the recirculation zone. A rough moving base was created in several steps. Firstly, particles of diameter $d^{b}=1.7$ were stuck randomly to a horizontal plate. Particles of diameter $d^{b}$ were slowly dropped onto this plate and allowed to settle. Once a thick layer of height $12d^{b}$ was produced, a slice of particles was taken whose centres lay between $9.3d^{b}$ and $11d^{b}$ . These particles were endowed with infinite mass and inclined at an angle of $23^{\circ }$ to form the base for the moving-bed flume simulations. The layer is thick enough to ensure that no flowing particles can fall through the rough base during the simulations. More details of this base creation process can be found in Weinhart et al. (Reference Weinhart, Thornton, Luding and Bokhove2012) whereas a detailed description of different bed creation methods and their effect on the macroscopic friction experienced by the flow can be found in Thornton et al. (Reference Thornton, Weinhart, Luding. and Bokhove2012a ). Before each time step ${\rm\Delta}t=10^{-4}\sqrt{d^{s}/g}$ , the base was moved upstream by a distance $u_{belt}{\rm\Delta}t=1.5\times 10^{-4}d^{s}$ . The system was allowed to evolve until a steady recirculation zone was formed.

Figure 6(b) shows a snapshot from the simulations, which have a very similar structure to the experimental results: most large particles recirculate quickly at the front but a few large particles recirculate slowly at the rear. This behaviour is also evident in the time- and width-averaged concentration plot shown in figure 6(d), which was produced by employing the micro–macro coarse-graining technique (Goldhirsch Reference Goldhirsch2010; Weinhart et al. Reference Weinhart, Hartkamp, Thornton and Luding2013) on the individual particle positions from 749 subsequent time steps. The new extension by Tunuguntla, Thornton & Weinhart (Reference Tunuguntla, Thornton and Weinhart2015), based on a mixture theory formulation (Morland Reference Morland1992), allowed the (partial) densities for the bulk ( ${\it\rho}$ ), small ( ${\it\rho}^{s}$ ) and large particles ( ${\it\rho}^{l}$ ) to be separately extracted, with the small particle concentration defined as ${\it\rho}^{s}/{\it\rho}$ , i.e. the local small particle material density over the local granular material density. The coarse-graining method used two-dimensional Gaussian functions at each of the particle positions and generated the continuum field at every point in space; however, for ease of computing, the data is shown on a $250\times 250$ grid. As was seen in the experimental concentration field in figure 6(c), the slow moving large particles have lowered the upstream concentration and produced a white ‘tail’ protruding backwards from the main region of recirculation. This qualitative similarity between the concentration field of the simulations that were laterally periodic (figure 6 d) and the concentration field of the experiments (figure 6 c) indicates that there are only minimal effects arising from the side walls and justifies the two-dimensional approximation of the analytic solution. Dry simulations, using a much higher restitution coefficient, also gave a similar concentration field, indicating that the behaviour is not an artefact of the presence of the fluid nor the exact particle properties. Despite the fact that no attempt was made to calibrate the simulations and experiments, both show very similar behaviour using different sized particles in different sized flumes. The presence of the ‘tail’, in which large particles recirculate very slowly through regions of many small particles, points towards a fundamental asymmetry in the interactions between the large and small particles. Recently, van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) uncovered a similar asymmetry in a linear shear cell, and showed how the asymmetry could be modelled using a continuum approach.

1.2 Continuum segregation equation for bidisperse mixtures

Non-dimensional continuum models for segregation in bidisperse mixtures (e.g. Bridgwater, Foo & Stephens Reference Bridgwater, Foo and Stephens1985; Savage & Lun Reference Savage and Lun1988; Bridgwater Reference Bridgwater and Mehta1994; Dolgunin & Ukolov Reference Dolgunin and Ukolov1995; Gray & Thornton Reference Gray and Thornton2005; Gray & Chugunov Reference Gray and Chugunov2006; Thornton et al. Reference Thornton, Gray and Hogg2006; May, Shearer & Daniels Reference May, Shearer and Daniels2010) all share a similar advection–diffusion structure

(1.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial {\it\phi}}{\partial t}}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\phi}\boldsymbol{u})-{\displaystyle \frac{\partial }{\partial z}}(S_{r}F({\it\phi}))={\displaystyle \frac{\partial }{\partial z}}\left(D_{r}{\displaystyle \frac{\partial {\it\phi}}{\partial z}}\right), & & \displaystyle\end{eqnarray}$$

where the $z$ coordinate is the upward pointing normal to the flume bed, the $x$ coordinate points down the flume and the $y$ coordinate points horizontally across the flume bed. The bulk velocity field $\boldsymbol{u}=(u,v,w)$ has components in the above directions, the small particle concentration is ${\it\phi}$ , and $S_{r}$ and $D_{r}$ are the non-dimensional segregation and diffusive-remixing coefficients, respectively. As the typical length and height of the avalanche are $L$ and $H$ , and magnitudes of the downstream and segregation velocities are $U$ and $Q$ , the non-dimensional segregation coefficient $S_{r}=QL/(HU)$ represents the ratio of the typical segregation time scale $Q/H$ to the typical downstream transport time scale $U/L$ . Similarly, the non-dimensional diffusion coefficient $D_{r}=DL/(H^{2}U)$ represents the ratio of the typical diffusion time scale $D/H^{2}$ to the typical downstream transport time scale $U/L$ , with $D$ being the diffusivity between the two particle species. The large particle concentration is $1-{\it\phi}$ since the solids volume fraction is assumed to be uniform and constant throughout the flowing layer (Rognon et al. Reference Rognon, Roux, Naaim and Chevoir2007). The first term on the left-hand side in (1.1) describes the temporal evolution, whilst the second term describes the advection with the bulk flow. The segregation is captured by the third term, with $F({\it\phi})$ the segregation flux and the negative sign indicating that there is a net motion of small particles downwards. The segregation flux is often assumed to be the product of the small and large particle concentrations,

(1.2) $$\begin{eqnarray}\displaystyle F({\it\phi})={\it\phi}(1-{\it\phi}), & & \displaystyle\end{eqnarray}$$

and has the property that segregation ceases when the concentration reaches zero (pure large phase) or unity (pure small phase). The right-hand side of equation (1.1) reduces the sharp concentration shocks that develop between the two species, and models the diffusion of one species into the other that results from the random-walk-like behaviour of the grains. In many flows, this is small compared to the segregation (Gray & Hutter Reference Gray and Hutter1997; Dasgupta & Manna Reference Dasgupta and Manna2011; Wiederseiner et al. Reference Wiederseiner, Andreini, Epely-Chauvin, Moser, Monnereau, Gray and Ancey2011b ; Thornton et al. Reference Thornton, Weinhart, Luding and Bokhove2012b ) and so the non-diffuse solution in which $D_{r}=0$ is a useful approximation, with (1.1) reducing to a scalar hyperbolic equation. A full review of the derivation, history and applications of (1.1) can be found in Gray, Gajjar & Kokelaar (Reference Gray, Gajjar and Kokelaar2015).

1.3 Asymmetry between large and small particle motion

Recent experiments by Golick & Daniels (Reference Golick and Daniels2009) and van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) have uncovered an underlying asymmetry in the behaviour of large and small grains during segregation, with a characteristic dependence on the local relative volume fraction of small particles. Within their annular ring shear experiments, Golick & Daniels (Reference Golick and Daniels2009) inferred that large particles were segregating very slowly in regions of many small particles, but were not able to further explain this observation. Using a classical linear shear cell (Bridgwater Reference Bridgwater1976) and the ‘refractive index matched scanning technique’ (Wiederseiner et al. Reference Wiederseiner, Andreini, Epely-Chauvin and Ancey2011a ; Dijksman et al. Reference Dijksman, Rietz, Lorincz, van Hecke and Losert2012), experiments by van der Vaart et al. quantified on both bulk and particle scales how large particles rise slower in regions of many small particles compared to small particles percolating down through a region of many large particles. They also showed that the large particle velocity displayed a peak at approximately ${\it\phi}=0.55$ , proving that the coarse grains rise quickest as a group. Gajjar & Gray (Reference Gajjar and Gray2014) showed that the normal constituent velocities associated with the segregation equation (1.1) are

(1.3a,b ) $$\begin{eqnarray}\displaystyle w^{l}({\it\phi})=w+S_{r}{\displaystyle \frac{F({\it\phi})}{1-{\it\phi}}},\quad w^{s}({\it\phi})=w-S_{r}{\displaystyle \frac{F({\it\phi})}{{\it\phi}}}, & & \displaystyle\end{eqnarray}$$

with both velocities uniquely determined by the geometry of the flux function $F({\it\phi})$ at every concentration ${\it\phi}$ . The velocity of the large particles $w^{l}({\it\phi})$ (1.3a ) is directly proportional to the gradient of the chord, namely the gradient of the straight line segment (Clapham & Nicholson Reference Clapham and Nicholson2009), joining $(1,0)$ with $({\it\phi},F({\it\phi}))$ . Similarly, the velocity of the small particles $w^{s}({\it\phi})$ is directly proportional to the gradient of the chord joining $(0,0)$ with $({\it\phi},F({\it\phi}))$ . A pair of these two chords for ${\it\phi}={\it\phi}_{max}$ are shown in figure 7(b). Since the quadratic segregation flux (1.2) utilised by many segregation models is symmetric about ${\it\phi}=0.5$ (figure 7 a), it gives linear segregation velocities for the large and small grains

(1.4a,b ) $$\begin{eqnarray}\displaystyle w^{l}({\it\phi})=w+S_{r}{\it\phi},\quad w^{s}({\it\phi})=w-S_{r}(1-{\it\phi}). & & \displaystyle\end{eqnarray}$$

The maxima of these velocities are equal in magnitude (figure 7 c), and so (1.2) is unable to capture the asymmetry measured by van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015). In order to model the asymmetric behaviour between large and small grains, Gajjar & Gray (Reference Gajjar and Gray2014) introduced a new class of flux functions with the following properties: (i) $F({\it\phi})$ is skewed towards ${\it\phi}=0$ , with a maximum occurring at $0<{\it\phi}_{max}<1/2$ ; (ii) $F({\it\phi})$ is normalised to have the same amplitude as the quadratic flux (1.2); and (iii) $F({\it\phi})$ has at most one inflexion point ${\it\phi}_{inf}$ in the interval $({\it\phi}_{max},1)$ . Although there are other ways of normalising the class of flux functions, e.g. by the area, there were no qualitative differences between the different methods. The simplest flux function fitting all of the above requirements is the cubic form

(1.5) $$\begin{eqnarray}\displaystyle F({\it\phi})=A_{{\it\gamma}}{\it\phi}(1-{\it\phi})(1-{\it\gamma}{\it\phi}), & & \displaystyle\end{eqnarray}$$

where ${\it\gamma}$ is the asymmetry parameter and $A_{{\it\gamma}}$ is a normalisation constant. Note that the limit ${\it\gamma}\rightarrow 0$ of (1.5) recovers the symmetric quadratic flux (1.2). For small amounts of asymmetry, $0\leqslant {\it\gamma}\leqslant 0.5$ , $F({\it\phi})$ is convex up (Clapham & Nicholson Reference Clapham and Nicholson2009), whilst for greater amounts of asymmetry $0.5<{\it\gamma}\leqslant 1$ , $F({\it\phi})$ is non-convex with a single inflexion point

(1.6) $$\begin{eqnarray}\displaystyle {\it\phi}_{inf}={\displaystyle \frac{1+{\it\gamma}}{3{\it\gamma}}}. & & \displaystyle\end{eqnarray}$$

As shown in figure 7(c), the cubic functions (1.5) are able to reproduce the asymmetric behaviour that a small particle will percolate down more quickly at low ${\it\phi}$ (figure 7 e) than a large particle rises upwards at high ${\it\phi}$ (figure 7 g). In addition, figure 7(b) shows how the presence of an inflexion point (1.6) means that the chord joining $({\it\phi},F({\it\phi}))$ with $(1,0)$ initially has an increasing gradient as ${\it\phi}$ increases from 0 to ${\it\phi}_{\text{M}}$ , and a decreasing gradient thereafter. Thus, the non-convex flux functions display a maximum in the large particle velocity at an intermediate concentration ${\it\phi}_{\text{M}}$ (figure 7 f). This behaviour will be known as the collective motion of the large particles.

Figure 7. There is an intrinsic geometric relationship between the segregation flux $F({\it\phi})$ shown in (a), and its segregation velocities $w^{{\it\nu}}$ (1.3) shown in (c). At any concentration ${\it\phi}$ , the gradient of the chords (straight line segment) joining $({\it\phi},F({\it\phi}))$ with $(1,0)$ and $(0,0)$ are proportional to the velocities (1.3) of the large and small particles, respectively. These chords are illustrated in (b) for ${\it\phi}={\it\phi}_{max}={\it\phi}_{\text{R}}$ . The quadratic flux (1.2) is symmetric about ${\it\phi}=0.5$ , and thus gives linear segregation velocities (1.4) that have the same magnitude. The cubic flux is skewed towards ${\it\phi}=0$ with a maximum occurring at $0<{\it\phi}_{max}={\it\phi}_{\text{R}}<1/2$ , and is normalised by (2.8) to have the same amplitude as the quadratic flux. This gives asymmetric segregation velocities, with a single small particle (e) having a greater velocity that a single large particle (g). For higher amounts of asymmetry, measured by the asymmetry parameter ${\it\gamma}$ , the cubic flux has an inflexion point at ${\it\phi}_{inf}=(1+{\it\gamma})/3{\it\gamma}$ . It is this inflexion point which causes the large particle velocity to have a peak at an intermediate concentration ${\it\phi}_{\text{M}}$ , with large particles moving quickest when in close proximity to other large particles (f). (d) The image point ${\it\phi}^{o}$ (1.8) of concentration ${\it\phi}$ is defined as the point at which the gradient of the tangent to the flux function $F^{\prime }({\it\phi}^{o})$ is equal to the gradient of the chord joining ${\it\phi}$ to ${\it\phi}^{o}$ on $F$ . These pairs of concentrations $\{{\it\phi},{\it\phi}^{o}\}$ (filled black circles) cause the formation of semi-shocks, where only the characteristics of concentration ${\it\phi}$ collide with shock on one side, whilst the characteristics of concentration ${\it\phi}^{o}$ lie tangential to the shock on the other side. Two pairs of concentrations $\{1,1^{o}={\it\phi}_{\text{M}}\}$ , and $\{{\it\phi}_{\text{E}},{\it\phi}_{\text{E}}^{o}=1\}$ (open circles) are particularly important in the solutions, with the chords tangential at ${\it\phi}={\it\phi}_{\text{M}}$ and ${\it\phi}=1$ respectively. Note that the segregation flux in (b) and (d) is the cubic flux (1.5) with ${\it\gamma}=0.9$ .

Gajjar & Gray (Reference Gajjar and Gray2014) were able to examine the influence of asymmetry on the segregation process by constructing exact solutions to the non-diffuse ( $D_{r}=0$ ) hyperbolic segregation equation (1.1) using the method of characteristics (e.g. Whitham Reference Whitham1974; Billingham & King Reference Billingham and King2001). Concentration ${\it\phi}$ is constant along characteristic curves, which are also simply known as characteristics. The characteristics combine to form distinct features in the solution, such as rarefaction fans, shocks, semi-shocks and compressions, with physical definitions of these features provided in appendix A. Characteristics may diverge and form an expansion fan, with a smoothly varying concentration field, or converge and form a shock with a sharp jump in concentration from the rearward ( $-$ ) side to the forward ( $+$ ) side. The propagation of the shock surface $z_{s}(t,x,y)$ is governed by

(1.7) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial z_{s}}{\partial t}}+u{\displaystyle \frac{\partial z_{s}}{\partial x}}+v{\displaystyle \frac{\partial z_{s}}{\partial y}}-w=-S_{r}{\displaystyle \frac{\unicode[STIX]{x27E6}F({\it\phi})\unicode[STIX]{x27E7}}{\unicode[STIX]{x27E6}{\it\phi}\unicode[STIX]{x27E7}}}, & & \displaystyle\end{eqnarray}$$

with ‘jump’ brackets $\unicode[STIX]{x27E6}f\unicode[STIX]{x27E7}=f^{+}-f^{-}$ denoting the discontinuity in $f$ across the shock (Gray, Shearer & Thornton Reference Gray, Shearer and Thornton2006). Note that the right-hand side of (1.7) is proportional to the gradient of the chord on flux $F({\it\phi})$ between ${\it\phi}={\it\phi}^{-}$ and ${\it\phi}={\it\phi}^{+}$ (Gajjar & Gray Reference Gajjar and Gray2014). The characteristics usually collide with both sides of a shock, but the non-convex cubic flux functions give rise to a special type of shock, known as a semi-shock (Rhee, Aris & Amundson Reference Rhee, Aris and Amundson1986), where characteristics only collide with one side of the shock and are tangential to it on the other. The image point ${\it\phi}^{o}$ of concentration ${\it\phi}$ is defined as the point at which the gradient of the tangent to the flux function $F^{\prime }({\it\phi}^{o})$ is equal to the gradient of the chord joining ${\it\phi}$ to ${\it\phi}^{o}$ on $F$ , with the shock condition (1.7) giving the relation

(1.8) $$\begin{eqnarray}\displaystyle F^{\prime }({\it\phi}^{o})={\displaystyle \frac{F({\it\phi})-F({\it\phi}^{o})}{{\it\phi}-{\it\phi}^{o}}}. & & \displaystyle\end{eqnarray}$$

By this definition, the characteristics of concentration ${\it\phi}^{o}$ lie tangential to the shock, whilst the characteristics of concentration ${\it\phi}$ collide with the other side. For the cubic flux function (1.5), the relationship (1.8) between concentrations ${\it\phi}$ and ${\it\phi}^{o}$ simplifies to

(1.9) $$\begin{eqnarray}\displaystyle {\it\phi}^{o}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1+{\it\gamma}}{{\it\gamma}}}-{\it\phi}\right). & & \displaystyle\end{eqnarray}$$

An example pair of concentrations $\{{\it\phi},{\it\phi}^{o}\}$ is shown with closed black circles in figure 7(d). It is possible that the characteristics of concentration ${\it\phi}^{o}$ may collide with another semi-shock; characteristics of concentration $({\it\phi}^{o} )^{o}={\it\phi}^{oo}$ would lie tangential to this semi-shock on the other side. An example of the relationship between ${\it\phi}$ , ${\it\phi}^{o}$ and ${\it\phi}^{oo}$ is illustrated in figure 8. Two pairs of concentrations $\{1,1^{o}={\it\phi}_{\text{M}}\}$ , and $\{{\it\phi}_{\text{E}},{\it\phi}_{\text{E}}^{o}=1\}$ are of particular importance in the exact solutions, with

(1.10a,b ) $$\begin{eqnarray}\displaystyle 1^{o}={\it\phi}_{\text{M}}={\displaystyle \frac{1}{2{\it\gamma}}}\quad \text{and}\quad {\it\phi}_{\text{E}}={\displaystyle \frac{1-{\it\gamma}}{{\it\gamma}}}, & & \displaystyle\end{eqnarray}$$

using the short hand notation $1^{o}={\it\phi}^{o}|_{{\it\phi}=1}$ . As shown by the open circles in figure 7(d), the chord between $({\it\phi}_{\text{M}},F({\it\phi}_{\text{M}}))$ and $(1,0)$ is tangential to the segregation flux $F$ at ${\it\phi}={\it\phi}_{\text{M}}$ , whilst the chord between $({\it\phi}_{\text{E}},F({\it\phi}_{\text{E}}))$ and $(1,0)$ is tangential to $F$ at ${\it\phi}=1$ . Concentration ${\it\phi}_{\text{M}}$ has the physical significance that it is the concentration at which the large particles reach their maximum velocity and is important in the solution structure described in § 2.2, whilst concentration ${\it\phi}_{\text{E}}$ is important in the structure described in § 2.3, and determines which of the two non-convex solutions is formed.

Figure 8. A sketch showing the relationship between ${\it\phi}_{\text{R}}$ , ${\it\phi}_{\text{R}}^{o}$ and $({\it\phi}_{\text{R}}^{o})^{o}={\it\phi}_{\text{R}}^{oo}$ for the cubic flux with ${\it\gamma}=0.9$ (see (1.5)). The dash-dotted line shows that the chord joining ${\it\phi}_{\text{R}}$ to ${\it\phi}_{\text{R}}^{o}$ is tangential to the flux function at ${\it\phi}_{\text{R}}^{o}$ , whilst the dashed line shows that the chord joining ${\it\phi}_{\text{R}}^{o}$ with ${\it\phi}_{\text{R}}^{oo}$ is tangential to the flux function at ${\it\phi}_{\text{R}}^{oo}$ . These points are important in the construction of the ‘lens-tail’ structure in § 2.3.

Tunuguntla, Bokhove & Thornton (Reference Tunuguntla, Bokhove and Thornton2014) showed that asymmetry causes the distance for complete segregation of an initially homogeneous mixture to become dependent on the initial conditions, and Gajjar & Gray (Reference Gajjar and Gray2014) specifically found the distance to be dependent on the inflow concentration, with a higher proportion of fines increasing the final segregation distance. In addition, the decreasing large particle velocity at higher concentrations causes semi-shocks to form, where large particles take longer to rise to the upper layer. This creates a stronger dependence of the final segregation distance on the inflow concentration for both homogeneous and normally graded inflow profiles, similar to the linear relationship reported by both Staron & Phillips (Reference Staron and Phillips2014) and van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015). In particular, van der Vaart et al. (Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) were able to fit their data to a non-convex cubic flux with ${\it\gamma}=0.89$ , which also matched their experimental observation of a peak in the large particle velocity around ${\it\phi}=0.55$ . It is also interesting that asymmetric segregation flux functions arise naturally in the work of Gray & Ancey (Reference Gray and Ancey2015), which extends the model of Gray & Chugunov (Reference Gray and Chugunov2006) to account for differences in both particle size and particle density.

1.4 Breaking size-segregation waves

Figure 9. Numerical solutions of the segregation equation (1.12) in a steady uniform flow with a quadratic flux (1.2) show that a monotonically decreasing interface between large and small grains (a) continually steepens in time ( $t=0.0$ ) (b) as small particles are sheared over the top of large particles ( $t=0.5$ ). This interface breaks in finite time ( $t=1.0$ ) (c) and forms a recirculation zone ( $t=1.5$ ) (d), in which the large particles rise upwards towards the surface as they are resegregated before being sheared back towards the front. The recirculating zone has a complex ‘breaking-wave’ structure that oscillates in time, however the oscillations exponentially decay and the structure tends towards a steady state. (e) The steady breaking wave (Thornton & Gray Reference Thornton and Gray2008) for the quadratic flux function (1.2) exists between the vertical heights $H_{down}=0.1$ and $H_{up}=0.9$ , and consists of two expansion fans and two concentration shocks arranged in a ‘lens’-like structure. The two expansion fans are $\text{A}\text{B}\text{C}\text{A}$ centred at point $\text{A}$ and $\text{C}\text{D}\text{A}\text{C}$ centred at point $\text{C}$ , with individual characteristic curves shown with thin solid lines. The edge of the expansion fans are the ${\it\phi}=1$ and ${\it\phi}=0$ characteristics, which lie along $\text{A}\text{B}$ and $\text{C}\text{D}$ , respectively, and are shown with thick dashed lines. The two shocks are $\text{B}\text{C}$ and $\text{D}\text{A}$ , and are shown with thick solid lines. However, this structure is unable to replicate the slow movement of large particles upstream of the main recirculation region that was seen in figure 6.

One of the strengths of the continuum theory is its ability to reveal the structure and development of the recirculation zone that plays a vital role in the formation of bouldery fronts (Thornton & Gray Reference Thornton and Gray2008; Gray & Ancey Reference Gray and Ancey2009; Johnson et al. Reference Johnson, Kokelaar, Iverson, Logan, LaHusen and Gray2012). The simplest recirculation structure arises in the case of steady uniform flow (Pouliquen Reference Pouliquen1999b ; Rognon et al. Reference Rognon, Roux, Naaim and Chevoir2007; Forterre & Pouliquen Reference Forterre and Pouliquen2008), in which the flow thickness $h$ is constant. The combination of the propensity of the avalanche to form an upward coarsening size distribution through particle size segregation and the shear profile

(1.11) $$\begin{eqnarray}\displaystyle \boldsymbol{u}=(u(z),0,0), & & \displaystyle\end{eqnarray}$$

means that a monotonically decreasing interface separating large particles above from small particles below (figure 9 a) will continually steepen as fine grains are sheared over the top of coarse grains (figure 9 b). The interface eventually breaks in finite time (figure 9c, Gray et al. Reference Gray, Shearer and Thornton2006), forming a recirculation zone (figure 9 d) in which the large grains lying immediately below small grains are resegregated back towards the surface, and then swept downstream by the shear velocity (Thornton & Gray Reference Thornton and Gray2008; Gray & Kokelaar Reference Gray and Kokelaar2010a ,Reference Gray and Kokelaar b ). The similarity with classical breaking waves formed when an air–water interface steepens and breaks (Basco Reference Basco1985; Shand Reference Shand2009) led Thornton & Gray (Reference Thornton and Gray2008) to refer to the propagating recirculation zone as a breaking size-segregation wave.

The bulk velocity field (1.11) implies that the segregation equation (1.1) reduces to

(1.12) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial {\it\phi}}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}({\it\phi}u(z))-{\displaystyle \frac{\partial }{\partial z}}(S_{r}F({\it\phi}))=0. & & \displaystyle\end{eqnarray}$$

Numerical solutions to (1.12) using a simple TVD Lax–Friedrichs shock-capturing finite volume scheme (Yee Reference Yee1989; Tóth & Odstrčil Reference Tóth and Odstrčil1996; LeVeque Reference LeVeque2002) show that the breaking size-segregation wave initially has a complex structure (figure 9 d) that oscillates back and forth in time like a spinning rugby ball (Thornton & Gray Reference Thornton and Gray2008). Exact solutions for the structure have only been derived for the early stages of wave breaking (McIntyre et al. Reference McIntyre, Rowe, Shearer, Gray and Thornton2007), however, the simulations show that oscillations are transient and exponentially decay, with the structure tending towards a steady state. Thornton & Gray (Reference Thornton and Gray2008) generated an exact solution for the steady wave with the quadratic flux (1.2). As shown in figure 9(e), it consists of two expansion fans and two concentration shocks arranged in a ‘lens’-like structure. In general, the breaking wave forms between the two vertical heights $z=H_{down}$ and $z=H_{up}$ , and propagates at a speed $u_{wave}$ that is equal to the mean speed between these heights,

(1.13) $$\begin{eqnarray}\displaystyle u_{wave}={\displaystyle \frac{1}{H_{up}-H_{down}}}\int _{H_{down}}^{H_{up}}u(z)\,\text{d}z. & & \displaystyle\end{eqnarray}$$

Note that the recirculation zone within the moving-belt flume in § 1.1 occupies the entire height, hence $H_{down}=0$ and $H_{up}=1$ . Since the velocity $u(z)$ is monotonically increasing, the breaking wave propagates faster than the basal velocity but slower than both the surface velocity and the front velocity $u_{front}$ (Gray & Ancey Reference Gray and Ancey2009). At a height $z=z_{\text{R}}$ , the bulk velocity is equal to $u_{wave}$ . Above $z_{\text{R}}$ , $u(z)>u_{wave}$ , and so material is swept towards the breaking wave from the left, whilst for $z<z_{\text{R}}$ , $u(z)<u_{wave}$ and so material flows towards the breaking wave from the right. The change in flow direction relative to the ‘lens’ at $z=z_{\text{R}}$ is crucial, and thus both expansion fans are initiated at this height, centred at points $\text{A}$ and $\text{C}$ . The ${\it\phi}=1$ characteristic lies between points $\text{A}$ and $\text{B}$ , whilst the ${\it\phi}=0$ characteristic lies between points $\text{C}$ and $\text{D}$ . Two concentration shocks join point $\text{B}$ with $\text{C}$ and point $\text{D}$ with $\text{A}$ , respectively. Although the upper portion of the ‘lens’ $\text{A}\text{B}\text{C}\text{A}$ contains lower concentrations than the lower portion of the ‘lens’ $\text{C}\text{D}\text{A}\text{C}$ , the positions of the characteristics, expansion fans and shocks are rotationally invariant about the centre of the lens. This is a direct result of the symmetry of the quadratic flux (1.2) about ${\it\phi}=0.5$ .

Gray & Ancey (Reference Gray and Ancey2009) derived the structure of the steady-state recirculation zone in a non-uniform depositing flow that was reconstructed from a travelling wave solution to the depth-averaged avalanche equations (Savage & Hutter Reference Savage and Hutter1989; Pouliquen Reference Pouliquen1999a ,Reference Pouliquen b ; Wieland, Gray & Hutter Reference Wieland, Gray and Hutter1999; Gray, Tai & Noelle Reference Gray, Tai and Noelle2003). They found that the breaking wave also consisted of two expansion fans and two shocks arranged in a ‘lens’, but surrounding a central ‘eye’ of constant concentration. The wave is located at a unique position behind the flow front and determines the concentration deposited within the basal layer. The model was able to qualitatively describe the features of their experimental two-dimensional depositing flow constrained by lateral side walls, namely the coarse-grained flow front, the rapidly moving large particles on the surface and the static layer of coarse grains at the base sandwiching an intermediate layer of fine grains. The experiments, were, however, too grainy to resolve the finer structure of the breaking wave.

In the absence of the two-dimensional side-wall restrictions, Johnson et al. (Reference Johnson, Kokelaar, Iverson, Logan, LaHusen and Gray2012) numerically solved for the structure of the recirculation zone on the centreline of a three-dimensional front, which has a more elaborate ‘breaking-wave structure’. Numerical solutions suggest that both the characteristic curves and the particle paths continually spiral inwards, because of the sidewards advection of mass into the lateral levees. The exact analytic structure of the three-dimensional recirculation zone is still proving illusive.

Figure 6(e) shows a breaking-wave structure using the simple asymmetric cubic model (1.5) from § 1.3. As with the structure of the symmetric flux in figure 9, the asymmetric wave also has a ‘lens’-like structure towards the downstream end. The asymmetry also causes a new upstream ‘tail’ to be produced, through which a few large particles recirculate slowly. Although this behaviour is very similar to the individual particle motion observed in experiments and simulations in § 1.1, the shape and structure of the ‘tail’ region are qualitatively different. There are a number of other factors present within the moving-bed flume set-up used in both the experiments and simulations that are unaccounted for by the simple theory. Streamwise spatial variations in the velocity field, diffusive remixing and the differential friction of the two particles on the moving base may all have an influence on the ‘tail’ shape. The size of the system and finite-size effects may also contribute to the discrepancy in the ‘tail’ structure. Further experimental work and extensive simulations are currently being conducted in order to understand more about the slow particle movement through the ‘tail’. Nevertheless, without any knowledge of the exact shape of the segregation flux function in this environment (Gajjar & Gray Reference Gajjar and Gray2014), the fact that a simple asymmetric cubic flux produces a ‘tail’ means that it is of interest to understand the derivation and particle paths. This paper examines the effect of an asymmetric segregation flux function (Gajjar & Gray Reference Gajjar and Gray2014; van der Vaart et al. Reference van der Vaart, Gajjar, Epely-Chauvin, Andreini, Gray and Ancey2015) on both the structure of a two-dimensional breaking size-segregation wave, and the particle recirculation within it.