scholarly journals Segregation-induced fingering instabilities in granular free-surface flows

2012 ◽  
Vol 709 ◽  
pp. 543-580 ◽  
Author(s):  
M. J. Woodhouse ◽  
A. R. Thornton ◽  
C. G. Johnson ◽  
B. P. Kokelaar ◽  
J. M. N. T. Gray
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Free Surface Flows ◽  
Numerical Results ◽  
Small Scale ◽  
Size Segregation ◽  
Numerical Resolution ◽  
Bulk Motion ◽  
Large Particles ◽  
Ill Posed

AbstractParticle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.

scholarly journals Asymmetric flux models for particle-size segregation in granular avalanches

2014 ◽  
Vol 757 ◽  
pp. 297-329 ◽  
Author(s):  
P. Gajjar ◽  
J. M. N. T. Gray
Keyword(s):  
Particle Size ◽  
Small Particle ◽  
Volume Fraction ◽  
Free Surface Flows ◽  
Two Dimensions ◽  
Combined Processes ◽  
Maximum Point ◽  
Size Segregation ◽  
Flux Function ◽  
Large Particles

AbstractParticle-size segregation commonly occurs in both wet and dry granular free-surface flows through the combined processes of kinetic sieving and squeeze expulsion. As the granular material is sheared downslope, the particle matrix dilates slightly and small grains tend to percolate down through the gaps, because they are more likely than the large grains to fit into the available space. Larger particles are then levered upwards in order to maintain an almost uniform solids volume fraction through the depth. Recent experimental observations suggest that a single small particle can percolate downwards through a matrix of large particles faster than a large particle can be levered upwards through a matrix of fines. In this paper, this effect is modelled by using a flux function that is asymmetric about its maximum point, differing from the symmetric quadratic form used in recent models of particle-size segregation. For illustration, a cubic flux function is examined in this paper, which can be either a convex or a non-convex function of the small-particle concentration. The method of characteristics is used to derive exact steady-state solutions for non-diffuse segregation in two dimensions, with an inflow concentration that is (i) homogeneous and (ii) normally graded, with small particles above the large. As well as generating shocks and expansion fans, the new asymmetric flux function generates semi-shocks, which have characteristics intersecting with the shock just from one side. In the absence of diffusive remixing, these can significantly enhance the distance over which complete segregation occurs.

scholarly journals Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches

2008 ◽  
Vol 19 (1) ◽  
pp. 61-86 ◽  
Author(s):  
M. SHEARER ◽  
J. M. N. T. GRAY ◽  
A. R. THORNTON
Keyword(s):  
Particle Size ◽  
Free Surface ◽  
Shear Flow ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Small Particles ◽  
Surface Flows ◽  
Scalar Conservation Law ◽  
Large Particles ◽  
Scalar Conservation

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

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

2016 ◽  
Vol 794 ◽  
pp. 460-505 ◽  
Author(s):  
P. Gajjar ◽  
K. van der Vaart ◽  
A. R. Thornton ◽  
C. G. Johnson ◽  
C. Ancey ◽  
...  
Keyword(s):  
Particle Motion ◽  
Wave Structure ◽  
Free Surface Flows ◽  
Pyroclastic Flows ◽  
Breaking Wave ◽  
Size Segregation ◽  
Small Particles ◽  
Flux Function ◽  
Large Particles ◽  
Over The Top

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.

scholarly journals A theory for particle size segregation in shallow granular free-surface flows

2005 ◽  
Vol 461 (2057) ◽  
pp. 1447-1473 ◽  
Author(s):  
J.M.N.T Gray ◽  
A.R Thornton
Keyword(s):  
Free Surface ◽  
Plug Flow ◽  
Mixture Theory ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Void Space ◽  
Small Particles ◽  
Surface Flows ◽  
Large Particles ◽  
Steady State Solutions

Abstract Granular materials composed of a mixture of grain sizes are notoriously prone to segregation during shaking or transport. In this paper, a binary mixture theory is used to formulate a model for kinetic sieving of large and small particles in thin, rapidly flowing avalanches, which occur in many industrial and geophysical free-surface flows. The model is based on a simple percolation idea, in which the small particles preferentially fall into underlying void space and lever large particles upwards. Exact steady-state solutions have been constructed for general steady uniform velocity fields, as well as time-dependent solutions for plug-flow, that exploit the decoupling of material columns in the avalanche. All the solutions indicate the development of concentration shocks, which are frequently observed in experiments. A shock-capturing numerical algorithm is formulated to solve general problems and is used to investigate segregation in flows with weak shear.

scholarly journals Bulbous head formation in bidisperse shallow granular flow over an inclined plane

2019 ◽  
Vol 866 ◽  
pp. 263-297 ◽  
Author(s):  
I. F. C. Denissen ◽  
T. Weinhart ◽  
A. Te Voortwis ◽  
S. Luding ◽  
J. M. N. T. Gray ◽  
...  
Keyword(s):  
Numerical Solutions ◽  
Pyroclastic Flows ◽  
Surface Velocity ◽  
Small Scale ◽  
Efficient Computation ◽  
Flow Front ◽  
Mean Velocity ◽  
Discrete Particle ◽  
Large Particles ◽  
Particle Simulations

Rapid shallow granular flows over inclined planes are often seen in nature in the form of avalanches, landslides and pyroclastic flows. In these situations the flow develops an inversely graded (large at the top) particle-size distribution perpendicular to the plane. As the surface velocity of such flows is larger than the mean velocity, the larger material is transported to the flow front. This causes size segregation in the downstream direction, resulting in a flow front composed of large particles. Since the large particles are often more frictional than the small, the mobility of the flow front is reduced, resulting in a so-called bulbous head. This study focuses on the formation and evolution of this bulbous head, which we show to emerge in both a depth-averaged continuum framework and discrete particle simulations. Furthermore, our numerical solutions of the continuum model converge to a travelling wave solution, which allows for a very efficient computation of the long-time behaviour of the flow. We use small-scale periodic discrete particle simulations to calibrate (close) our continuum framework, and validate the simple one-dimensional (1-D) model with full-scale 3-D discrete particle simulations. The comparison shows that there are conditions under which the model works surprisingly well given the strong approximations made; for example, instantaneous vertical segregation.

scholarly journals The kinematics of bidisperse granular roll waves

2018 ◽  
Vol 848 ◽  
pp. 836-875 ◽  
Author(s):  
S. Viroulet ◽  
J. L. Baker ◽  
F. M. Rocha ◽  
C. G. Johnson ◽  
B. P. Kokelaar ◽  
...  
Keyword(s):  
Particle Concentration ◽  
Wave Crest ◽  
Small Scale ◽  
Size Segregation ◽  
Chute Flow ◽  
Glass Spheres ◽  
Roll Waves ◽  
Low Concentrations ◽  
Large Particles ◽  
The Waves

Small perturbations to a steady uniform granular chute flow can grow as the material moves downslope and develop into a series of surface waves that travel faster than the bulk flow. This roll wave instability has important implications for the mitigation of hazards due to geophysical mass flows, such as snow avalanches, debris flows and landslides, because the resulting waves tend to merge and become much deeper and more destructive than the uniform flow from which they form. Natural flows are usually highly polydisperse and their dynamics is significantly complicated by the particle size segregation that occurs within them. This study investigates the kinematics of such flows theoretically and through small-scale experiments that use a mixture of large and small glass spheres. It is shown that large particles, which segregate to the surface of the flow, are always concentrated near the crests of roll waves. There are different mechanisms for this depending on the relative speed of the waves, compared to the speed of particles at the free surface, as well as on the particle concentration. If all particles at the surface travel more slowly than the waves, the large particles become concentrated as the shock-like wavefronts pass them. This is due to a concertina-like effect in the frame of the moving wave, in which large particles move slowly backwards through the crest, but travel quickly in the troughs between the crests. If, instead, some particles on the surface travel more quickly than the wave and some move slower, then, at low concentrations, large particles can move towards the wave crest from both the forward and rearward sides. This results in isolated regions of large particles that are trapped at the crest of each wave, separated by regions where the flow is thinner and free of large particles. There is also a third regime arising when all surface particles travel faster than the waves, which has large particles present everywhere but with a sharp increase in their concentration towards the wave fronts. In all cases, the significantly enhanced large particle concentration at wave crests means that such flows in nature can be especially destructive and thus particularly hazardous.

scholarly journals Visualization of dominant stress-transfer mechanisms in experimental debris flows of different particle-size distribution

2017 ◽  
Vol 54 (2) ◽  
pp. 258-269 ◽  
Author(s):  
Nicoletta Sanvitale ◽  
Elisabeth T. Bowman
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Debris Flows ◽  
Stress Transfer ◽  
Flow Regimes ◽  
Free Surface Flows ◽  
Small Scale ◽  
Size Segregation ◽  
Transfer Mechanisms

Physical modelling of debris flow in a small-scale flume has been carried out to investigate the internal stress-transfer mechanisms within unsteady, saturated, and segregating granular free-surface flows. Measurements of the internal velocity fields within model flows were obtained via planar laser–induced fluorescence and particle image velocimetry. Normalized velocity profiles taken at a section over the flow duration were found to essentially collapse onto a single curve, the shape of which was dependent on the particle-size distribution. While all flows exhibited internal basal slip and shear, for tests on well-graded materials that are most representative of debris flows, the shear rate was found to reduce towards the surface to near-zero, exhibiting near plug-flow. Dimensional analysis shows that particles of different size within these flows experienced different dominant stress-transfer mechanisms — frictional, collisional or viscous. Rapid grain-size segregation therefore is both due to and results in different modes of stress transfer within a single flow. This means that in a segregating and hence, stratified system, different flow regimes will act concurrently at microscale and mesoscale. Results highlight the complexity of debris flows, so that it may be undesirable to ascribe a single microscale constitutive behaviour throughout, and further calls into question the concept of flow regimes for debris flows based on bulk measurements.

Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell

1969 ◽  
Vol 39 (3) ◽  
pp. 477-495 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Saturated Porous Medium ◽  
Mean Amplitude ◽  
Wave Number ◽  
Two Fluids ◽  
Averaged Equations ◽  
The Mean ◽  
Hele Shaw Cell

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

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