Academic literature on the topic 'Bagnold Rheology'

We have performed numerical studies of dense granular flows on an incline with a rough bottom in two and three dimensions. This flow geometry produces a constant density profile that satisfies scaling relations of the Bagnold, rather than the viscous, kind. No surface-only flows were observed. The bulk and the surface layer differ in their rheology, as evidenced by the change in principal stress directions near the surface; a Mohr–Coulomb type failure criterion is seen only near the surface. In the bulk, normal stress anomalies are observed both in two and in three dimensions. We do not observe isostaticity in static frictional piles obtained by arresting the flow.

In this work we present a multilayer shallow model to approximate the Navier–Stokes equations with the ${\it\mu}(I)$-rheology through an asymptotic analysis. The main advantages of this approximation are (i) the low cost associated with the numerical treatment of the free surface of the modelled flows, (ii) the exact conservation of mass and (iii) the ability to compute two-dimensional profiles of the velocities in the directions along and normal to the slope. The derivation of the model follows Fernández-Nieto et al. (J. Comput. Phys., vol. 60, 2014, pp. 408–437) and introduces a dimensional analysis based on the shallow flow hypothesis. The proposed first-order multilayer model fully satisfies a dissipative energy equation. A comparison with steady uniform Bagnold flow – with and without the sidewall friction effect – and laboratory experiments with a non-constant normal profile of the downslope velocity demonstrates the accuracy of the numerical model. Finally, by comparing the numerical results with experimental data on granular collapses, we show that the proposed multilayer model with the ${\it\mu}(I)$-rheology qualitatively reproduces the effect of the erodible bed on granular flow dynamics and deposits, such as the increase of runout distance with increasing thickness of the erodible bed. We show that the use of a constant friction coefficient in the multilayer model leads to the opposite behaviour. This multilayer model captures the strong change in shape of the velocity profile (from S-shaped to Bagnold-like) observed during the different phases of the highly transient flow, including the presence of static and flowing zones within the granular column.

In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient ${\it\mu}$. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent ${\it\mu}(I)$-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number $I$ in the friction coefficient ${\it\mu}$. In this paper it is shown that the ${\it\mu}(I)$-rheology is well-posed for intermediate values of $I$, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.

Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged ${\it\mu}(I)$-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional ${\it\mu}(I)$-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.

Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the $\unicode[STIX]{x1D707}(I)$-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.

The effect of base roughness on the transition and dynamics of a dense granular flow down an inclined plane is examined using particle based simulations. Different types of base topographies, rough bases made of frozen particles in either random or hexagonally ordered configurations, as well as sinusoidal bases with height modulation in both the flow and the spanwise directions, are examined. The roughness (characteristic length of the base features scaled by the flowing particle diameter) is defined as the ratio of the base amplitude and particle diameter for sinusoidal bases, and the ratio of frozen and moving particle diameters for frozen-particle bases. There is a discontinuous transition from an ordered to a disordered flow at a critical base roughness for all base topographies studied here, indicating that it is a universal phenomenon independent of base topography. The transition roughness does depend on the base configuration and the height of the flow, but is independent of the contact model and is less than 1.5 times the flowing particle diameter for all of the bases considered here. The bulk rheology is independent of the base topography, and follows the Bagnold law for both the ordered and the disordered flows. The base topography does have a dramatic effect on the flow dynamics at the base. For flows over frozen-particle bases, there is ordering down to the base for ordered flows, and the granular temperature is comparable to that in the bulk. There is virtually no velocity slip at the base, and the mean angular velocity is equal to one-half of the vorticity down to the base. For flows over sinusoidal bases, there is significant slip at the base, and the mean angular velocity is approximately an order of magnitude higher than that in the bulk within a region of height approximately one particle diameter at the base. This large particle spin results in a disordered and highly energetic layer of approximately 5–10 particle diameters at the base, where the granular temperature is an order of magnitude higher than that in the bulk. Thus, this study reveals the paradoxical result that gentler base topographies result in large slip and large agitation at the base, whereas rougher topographies such as frozen-particle bases result in virtually no slip and no agitation at the base for both ordered and disordered flows.

Abstract A lahar is a flowing mixture of rock debris and water (other than normal streamflow) from a volcano, which encompasses a continuum from debris flows (sediment concentration > or =60% per volume) to hyperconcentrated streamflows (sediment concentration from 20 to 60% per volume). Debris flow deposits are poorly sorted and massive with abundant clasts. Lahars can be either syn-eruptive, post-eruptive or have a non-eruptive origin. Four types of lahars can be generated during an eruption, based on distinct sources of water (i.e. ice, snow, crater lake, river, and rain) that allow the sediments to be removed and incorporated in the lahar (e.g., Mount St.-Helens in 1980, Nevado del Ruiz in 1985). Post-eruptive lahars, which are rain-triggered, occur during several years after an eruption (e.g., still occurring at Pinatubo). Non-eruptive lahars are flows generated on volcanoes without eruptive activity, particularly in the case of a debris avalanche or a lake outburst (e.g., Kelud or Ruapehu). Lahars flow as pulses, whose velocity and discharge are much higher than those of streamflows, including catchments similar in size. Sediment transport capacity of lahars is exceptional, owing to buoyancy, dispersive pressure, and to the amount of cohesive clay and silt. However, the finding of recent experimental works indicates that even clay-rich lahar mixtures have little true cohesion. Therefore, the typical classification of lahars into "cohesive" and "non cohesive" seems to be inappropriate at present. Besides, past work on lahar mechanics used models based on the Bagnold's or the Bingham's theories. Recent advances in experimentation show that a lahar has specific rheological properties: it moves as a surge or series of surges, driven by gravity, by porosity fluctuation, and by pore fluid pressures, in accordance with the Coulomb grain flow model. Grain size distribution and sorting control pore pressure distribution. Lahar mechanics depend on much more than steady-state rheology, because lahars are highly unsteady and typically heterogeneous flows. Lahar can show a succession of debris flow phases, hyperconcentrated flow phases, and sometimes transient streamflow phases. Therefore, some fluids-mechanics concepts and terminology, such as "viscous", "laminar" or "non-Newtonian" are inappropriate to describe the mechanical properties of lahars. Processes of deposition are complex and poorly known. Interpretation of massive and unsorted lahar deposits commonly ascribe the deposition regime to a freezing en masse process. However, recent laboratory experiments highlight that debris-flow deposits may result from incremental deposition processes.