Asymmetric flux models for particle-size segregation in granular avalanches

Abstract

Debris and pyroclastic flows often have bouldery flow fronts, which act as a natural dam resisting further advance. Counter intuitively, these resistive fronts can lead to enhanced run-out, because they can be shouldered aside to form static levees that self-channelise the flow. At the heart of this behaviour is the inherent process of size segregation, with different sized particles readily separating into distinct vertical layers through a combination of kinetic sieving and squeeze expulsion. The result is an upward coarsening of the size distribution with the largest grains collecting at the top of the flow, where the flow velocity is greatest, allowing them to be preferentially transported to the front. Here, the large grains may be overrun, resegregated towards the surface and recirculated before being shouldered aside into lateral levees. A key element of this recirculation mechanism is the formation of a breaking size-segregation wave, which allows large particles that have been overrun to rise up into the faster moving parts of the flow as small particles are sheared over the top. Observations from experiments and discrete particle simulations in a moving-bed flume indicate that, whilst most large particles recirculate quickly at the front, a few recirculate very slowly through regions of many small particles at the rear. This behaviour is modelled in this paper using asymmetric segregation flux functions. Exact non-diffuse solutions are derived for the steady wave structure using the method of characteristics with a cubic segregation flux. Three different structures emerge, dependent on the degree of asymmetry and the non-convexity of the segregation flux function. In particular, a novel 'lens-tail' solution is found for segregation fluxes that have a large amount of non-convexity, with an additional expansion fan and compression wave forming a 'tail' upstream of the 'lens' region. Analysis of exact solutions for the particle motion shows that the large particle motion through the 'lens-tail' is fundamentally different to the classical 'lens' solutions. A few large particles starting near the bottom of the breaking wave pass through the 'tail', where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate. † Email address for correspondence: parmesh.gajjar@alumni.manchester.ac.uk Asymmetric breaking size-segregation waves 461

Asymmetric breaking size-segregation waves in dense granular free-surface flows

“…They also showed that the large particle velocity displayed a peak at approximately φ = 0.55, proving that the coarse grains rise quickest as a group. Gajjar & Gray (2014) showed that the normal constituent velocities associated with the segregation equation (1.1) are…”

“…In order to model the asymmetric behaviour between large and small grains, Gajjar & Gray (2014) introduced a new class of flux functions with the following properties: (i) F(φ) is skewed towards φ = 0, with a maximum occurring at 0 < φ max < 1/2; (ii) F(φ) is normalised to have the same amplitude as the quadratic flux (1.2); and (iii) F(φ) has at most one inflexion point φ inf in the interval (φ max , 1). Although there are other ways of normalising the class of flux functions, e.g.…”

“…It is this inflexion point which causes the large particle velocity to have a peak at an intermediate concentration with 'jump' brackets f = f + − f − denoting the discontinuity in f across the shock (Gray, Shearer & Thornton 2006). Note that the right-hand side of (1.7) is proportional to the gradient of the chord on flux F(φ) between φ = φ − and φ = φ + (Gajjar & Gray 2014). 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 1986), where characteristics only collide with one side of the shock and are tangential to it on the other.…”

“…Concentration φ 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 φ E is important in the structure described in § 2.3, and determines which of the two non-convex solutions is formed. Tunuguntla, Bokhove & Thornton (2014) showed that asymmetry causes the distance for complete segregation of an initially homogeneous mixture to become dependent on the initial conditions, and Gajjar & Gray (2014) 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.…”

“…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 2014), 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 2014;van der Vaart et al 2015) on both the structure of a two-dimensional breaking size-segregation wave, and the particle recirculation within it.…”