Craters form as the lander's exhaust interacts with the planetary surfaces. Understanding this phenomenon is imperative to ensuring safe landings. We investigate the crater morphology, where a turbulent air jet impinges on granular surfaces. To reveal the fundamental aspect of this phenomenon, systematic experiments are performed with various air-jet velocities, nozzle positions and grain properties. The resultant crater morphology is characterized by an aspect ratio. We find a universal scaling law in which the aspect ratio is scaled by a dimensionless variable consisting of the air velocity at the nozzle, the speed of sound in air, the nozzle diameter, the nozzle-tip distance from the surface, the grain diameter, the density of the grains and the density of air. The obtained scaling reveals the cross-over of the length scales governing the crater aspect ratio, providing a useful guideline for ensuring safe landings. Moreover, we report a novel drop-shaped sub-surface cratering phenomenon.

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$-rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$. In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$-rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$-rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

The behaviour of granular column collapses is associated with the dynamics of geohazards, such as debris flows, landslides and pyroclastic flows, yet their underlying physics is still not well understood. In this paper, we explore granular column collapses using the sphero-polyhedral discrete element method, where the system contains two types of particles with different frictional properties. We impose three different mixing ratios and multiple different particle frictional coefficients, which lead to different run-out distances and deposition heights. Based on our previous work and a simple mixture theory, we propose a new effective initial aspect ratio for the bi-frictional granular mixture, which helps unify the description of the relative run-out distances. We analyse the kinematics of bi-frictional granular column collapses and find that deviations from classical power-law scaling in both the dimensionless terminal time and the dimensionless time when the system reaches the maximum kinetic energy may result from differences in the initial solid fraction and initial structures. To clarify the influence of initial states, we further decrease the initial solid fraction of granular column collapses, and propose a trial function to quantitatively describe its influence. Due to the utilization of a simple mixture theory of contact occurrence probability, this study can be associated with the friction-dependent rheology of granular systems and friction-induced granular segregations, and further generalized to applications with multiple species of particles in various natural and engineering mixtures.

We consider steady, fully-developed flows of deformable, inelastic grains driven by gravity between identical bumpy walls. Using constitutive relations from extended kinetic theory (EKT) for the erodible bed near the centreline and the collisional flow between the surfaces of the bed and the walls, we calculate the fields of mean velocity, fluctuation velocity and solid volume fraction across the chute. We consider both situations in which the solid volume fraction at and near the centreline is high enough to form a bed and when it is not. We compare results predicted by EKT with recent discrete element simulations results, and obtain very good agreement.

Numerical simulation based on the discrete element method (DEM) is used to investigate the flow field generated when a cylindrical obstacle is placed in a supersonic granular stream. Robust validation of the simulation model is performed by comparing numerical results with experiments. Experiments are performed using a two-dimensional set-up generating rapid granular flow owing to gravity. DEM simulations demonstrate that a rapid gas-like stream of grains suddenly decelerates across the shock wave and finally collapses into a slow-moving heap at the cylinder. The volume fraction suddenly increases across the shock layer and remains constant thereafter. The flow physics of the shock wave and the granular heap is elucidated through fundamental fluid dynamic quantities such as the velocity, volume fraction, pressure and granular temperature. It is shown that the interaction of grains with a cylindrical obstacle results in the generation of pressure, which is responsible for sustaining static granular heaps on the cylinder. The total pressure is resolved into collisional and streaming components. A streaming pressure is generated owing to velocity fluctuations, and is found to be significant only in the shock wave region. The observations show that the rheological complexity offered by granular shock waves is a direct manifestation of the dissipative and frictional nature of granular collisions. The new insight into the granular heaps could be relevant to a variety of applications involving granular-fluid–solid interactions.

We report on the discovery, using numerical simulations, of a segregation pattern in high-speed granular flows, in which size segregation is primarily driven by two-dimensional granular temperature gradients, rather than by gravity. In contrast to slower flows on gentle slopes, in high-speed flows on steep slopes, large particles no longer accumulate in the upper layers of the flow, but are trapped in the interior. The strong temperature gradients that develop between the interior of the flow and the surrounding dilute periphery appear to govern the segregation mechanism. Interestingly, these new segregated flows run at a much faster speed than similar monodisperse flows. This opens up promising perspectives for transporting granular material with enhanced efficiency. Importantly, we show that the kinetic theory for dense, inclined flows of binary mixtures can provide a relevant theoretical framework to explain the segregation patterns observed in the numerical simulations.

A blunt obstacle in the path of a rapid granular avalanche generates a bow shock (a jump in the avalanche thickness and velocity), a region of static grains upstream of the obstacle, and a grain-free region downstream. Here, it is shown that this interaction is qualitatively altered if the incline on which the avalanche is flowing is changed from smooth to rough. On a rough incline, the friction between the grains and the incline depends on the flow thickness and speed, which allows both rapid (supercritical) and slow (subcritical) steady uniform avalanches. For supercritical experimental flows, the material is diverted around a blunt obstacle by the formation of a bow shock and a static dead zone upstream of the obstacle. Downslope, a grain-free vacuum region forms, but, in contrast to flows on smooth beds, static levees form at the boundary between the vacuum region and the flow. In slower, subcritical, flows the flow is diverted smoothly around the dead zone and the obstacle without forming a bow shock. After the avalanche stops, signatures of the dead zone, levees and (in subcritical flows) a deeper region upslope of the obstacle are frozen into the deposit. To capture this behaviour, numerical simulations are performed with a depth-averaged avalanche model that includes frictional hysteresis and depth-averaged viscous terms, which are needed to accurately model the flowing and deposited regions. These results may be directly relevant to geophysical mass flows and snow avalanches, which flow over rough terrain and may impact barriers or other infrastructure.

Vertical chutes and pipes are a common component of many industrial apparatus used in the transport and processing of powders and grains. Here, a typical arrangement is considered first in which a hopper at the top feeds the chute and a converging outlet at the bottom controls the mass flux. Discrete element method (DEM) simulations reveal that steady uniform flow is only observed for intermediate flow rates, with jamming and unsteady waves dominating slow flows and non-uniform wall detachment in fast flow. Focusing on the steady uniform regimes, a progressive idealisation is carried out by matching with equivalent DEM simulations in periodic cells. These investigations justify a one-dimensional continuum modelling of the problem and provide key test data. Novel exact solutions are derived here for vertical flow using a linear version of the ‘$\mu(I),\varPhi(I)$-rheology’, for which the bulk friction $\mu$ and steady solid volume fraction $\varPhi$ depend on the inertial number I. Despite not capturing the full nonlinear complexities, the solutions match important aspects of the DEM flow fields and reveal simple scaling laws linking many quantities of interest. In particular, this study clearly demonstrates a linear relation between the chute width and the size of the shear zones at the walls. This finding contrasts with previous works on purely quasi-static flow, which instead predict a roughly constant shear zone width, a difference which implies that finite-size effects are minimal for the inertial flows studied here.

Starting from the Boltzmann–Enskog kinetic equations, the charge transport equation for bidisperse granular flows with contact electrification is derived with separate mean velocities, total kinetic energies, charges and charge variances for each solid phase. To close locally averaged transport equations, a Maxwellian distribution is presumed for both particle velocity and charge. The hydrodynamic equations for bidisperse solid mixtures are first revisited and the resulting model consisting of the transport equations of mass, momentum, total kinetic energy, which is the sum of the granular temperature and the trace of fluctuating kinetic tensor, and charge is then presented. The charge transfer between phases and the charge build-up within a phase are modelled with local charge and effective work function differences between phases and the local electric field. The revisited hydrodynamic equations and the derived charge transport equation with constitutive relations are assessed through hard-sphere simulations of three-dimensional spatially homogeneous, quasi-one-dimensional spatially inhomogeneous bidisperse granular gases and a three-dimensional segregating bidisperse granular flow with conducting walls.

Flowing granular materials segregate due to differences in particle size (driven by percolation) and density (driven by buoyancy). Modelling the segregation of mixtures of large/heavy particles and small/light particles is challenging due to the opposing effects of the two segregation mechanisms. Using discrete element method (DEM) simulations of combined size and density segregation we show that the segregation velocity is well described by a model that depends linearly on the local shear rate and quadratically on the species concentration for free surface flows. Concentration profiles predicted by incorporating this segregation velocity model into a continuum advection–diffusion–segregation transport model match DEM simulation results well for a wide range of particle size and density ratios. Most surprisingly, the DEM simulations and the segregation velocity model both show that the segregation direction for a range of size and density ratios depends on the local species concentration. This leads to a methodology to determine the combination of particle size ratio, density ratio and particle concentration for which a bidisperse mixture will not segregate.

Granular column collapse involves complicated granular flow bridging initial and final solid-like states, with intermediate multi-regime spatial and temporal flow patterns and transitions, which include large free-surface variation. In this study, a non-local mesh-free numerical method is proposed to model these flows and capture the entire process from flowing to arresting. Free surface evolution is tracked by the mesh-free method while the non-local theory of peridynamics is used to capture the arresting for the flow. The constitutive relation to calculate the effective viscosity and stress is based on the μ(I) rheology. The non-local mesh-free method is validated to simulate a granular column collapse in which effects from the peridynamic horizon and particle distance are both examined. Non-local modelling is then used to simulate more types of granular column collapses, by comparison with other numerical results and experimental observations in terms of the free surface and velocity variations in both the fluid-like and solid-like states. The collision between two adjacent collapsing granular columns is also simulated, and the interface variations between material from each collapsing column are compared with experimental observations. The non-local modelling is shown to reflect the internal flow characteristics in the granular flow. Throughout these simulations, the non-local mesh-free method is able to calculate the free surface, velocity and interface variation in the granular flows.

Granular systems are of a discrete nature. Nevertheless, it can be advantageous to describe their dynamics by means of continuum mechanical methods. The numerical solution of the corresponding hydrodynamic equations is, however, difficult. Therefore, previous numerical simulations are typically geared towards highly specific systems and frequently restricted to two dimensions or mild driving conditions. Here, we present the first robust general simulation scheme for granular hydrodynamics in three dimensions which is not bound to the above limitations. The performance of the simulation scheme is demonstrated by means of three applications which have been proven as notoriously difficult for numerical hydrodynamic description. Although, by construction, our numerical method covers grain-inertia flows, the presented examples demonstrate that it produces reliable results even in the jammed or high density limit.

Shear dilatation/contraction of granular materials has long been recognized as an important process in granular flows but a comprehensive theoretical description of this process for a wide range of shear rates is not yet available. In this paper, a theoretical formulation of dilatation/contraction is proposed for continuum modelling of granular flows, in which the dilatation/contraction effects consist of a frictional component, which results from the rearrangement of enduring-contact force chains among particles, and a collisional component, which arises from inter-grain collisions. In this formulation, a frictional solid pressure, which considers the rearrangement of contact force chains under shear deformation, is proposed for the frictional dilatation/contraction, while well-established rheological laws are adopted for the collisional inter-grain pressure to account for the collisional dilatancy effect. The proposed formulation is first verified analytically by describing the shear-weakening behaviour of granular samples in a torsional shear rheometer and by capturing the incipient failure of both dry and immersed granular slopes. The proposed dilatation/contraction formulation is then further validated numerically by integrating it into a two-fluid continuum model and applying the model to study the collapse of submerged granular columns, in which the dilatation/contraction plays a critical role.

Particles of differing sizes are notoriously prone to segregation in shear driven flows under the action of gravity. This has important implications in many industrial processes, where particle-size segregation can lead to flow problems and reduced product quality, as well as longer product development and start-up times. Particle-size segregation also readily occurs in many hazardous geophysical mass flows (such as snow avalanches, debris flows and volcanic pyroclastic flows) and can lead to the formation of destructive bouldery flow fronts and significantly longer runouts. Although general theories exist to model particle-size segregation, the detailed functional dependence of the segregation flux on the shear rate, gravity, pressure, particle concentration, grain size and grain-size ratio is still not known. This paper describes refractive-index matched oscillatory shear-cell experiments that shed light on the segregation velocity in the two extreme cases of (i) a single large intruder rising up through a matrix of smaller grains, and (ii) a single small intruder percolating down through a matrix of large particles. Despite the sometimes markedly different time scales for segregation in these two situations, a unifying scaling law has been found that is able to collapse all the experimental data over a wide range of shear rates and grain-size ratios in the range $[1.17,4.17]$. The resulting functional form is easily generalizable to intermediate concentrations and can quantitatively capture laboratory experiments and numerical simulations with a $50\ {:}\ 50$ mix of large and small grains.

Modelling the flow of dry granular materials entering water is crucial for optimizing blending processes in industry and for natural hazard assessment when describing tsunami waves induced by landslides. In this study, we experimentally investigate the case of a jet of grains entering from the air into a water bath. After an initial transient state, a stationary impregnation front appears between the dry and the wet grains. The wet grains are then dispersed in the liquid. To describe this dry-to-wet transition, we focus on the first step of the process, when the liquid invades the dense granular medium. In this regime, the granular jet is modelled as a translating porous material, and we systematically characterize the impregnation process using a combination of experiments, analytical modelling, and numerical tools. We then compare this process to the situation of a confined granular jet entering a water bath. Our approach is a first step toward describing the interplay between dry grains and a liquid and the resulting dispersion of particles.

We investigate two-dimensional shallow granular flows on a rotating and gravitating elliptical body. This is motivated by regolith flow on small planetary bodies – also called minor planets – which is influenced by the rotation of the body, as well as its irregular topography and complex gravity field. Governing equations are obtained in an elliptic coordinate system attached to the body by extending the framework employed for terrestrial avalanches to incorporate effects of rotation, varying gravity and a curvilinear surface. Additionally, we introduce criteria to monitor grain shedding and to track flow initiation and cessation. We delineate different types of regolith motion that are governed by the rotation rate and surface roughness of the body. We find that grains migrate towards the minor and major axis of the body at low and high rotation rates, respectively. Grains are shed when the basal pressure vanishes, and shedding is encouraged by Coriolis effects during prograde flow. We observe the coexistence of regions of static and mobile regolith and their reorganization owing to the merging or division of flows. We also probe the formation and destruction of dunes – bulges arising from local grain accumulation – and find several aspects of their motion to be different from terrestrial situations. We then perform discrete element simulations that display a good match with theoretical predictions. Finally, we consider the evolution of a bi-disperse regolith. We find that big and small grains occupy, respectively, the top and bottom of the dunes formed on the surface, which is reminiscent of observations on asteroids like Itokawa.

Gravity-driven size segregation is important in mountain streams where a wide range of grain sizes are transported as bedload. More particularly, vertical size segregation is a multi-scale process that originates in interactions at the scale of particles with important morphological consequences for the river bed. To address this issue, a volume-averaged multi-phase flow model for immersed bi-disperse granular flows was developed based on an interparticle segregation force (Guillard et al., J. Fluid Mech., vol. 807, 2016, R1) and a granular Stokesian drag force (Tripathi & Khakhar, J. Fluid Mech., vol. 717, 2013, pp. 643–669). An advection–diffusion model was derived from this model yielding parametrisations for the advection and diffusion coefficients based on the interparticle interactions. This approach makes it possible to bridge the gap between grain-scale physics and continuum modelling. Both models were successfully tested against existing discrete element model (DEM) simulations of size segregation in bedload transport (Chassagne et al., J. Fluid Mech., vol. 895, 2020, A30). Through a detailed investigation of the granular forces, it is demonstrated that the observed scaling of the advection and diffusion coefficients with the inertial number can be explained by the granular drag force dependency on the viscosity. The drag coefficient is shown to be linearly dependent on the small particle concentration. A new scaling relationship for the segregation force including the small particle concentration and the pressure is proposed. Lastly, adding a size-ratio dependency in the segregation force fairly reproduces the DEM results for a large range of small particle concentrations and size ratios.

The incompressible $\mu (I)$-rheology has been used to study subaerial granular flows with remarkable success. For subaquatic granular flows, drag between grains and the pore fluid is substantially higher and the physical behaviour is more complex. High drag forces constrain the rearrangement of grains and dilatancy, leading to a considerable buildup of pore pressure. Its transient and dynamic description is the key to modelling subaquatic granular flows but out of the scope of incompressible models. In this work, we advance from the incompressible $\mu (I)$-rheology to the compressible $\mu (J)$, $\phi (J)$-rheology to account for pore pressure, dilatancy and the scaling laws under subaquatic conditions. The model is supplemented with critical state theory to yield the correct properties in the quasi-static limit. The pore fluid is described by an additional set of conservation equations and the interaction with grains is described by a drag model. This new implementation enables us to include most of the physical processes relevant for submerged granular flows in a highly transparent manner. Both the incompressible and compressible rheologies are implemented into OpenFOAM, and various simulations at low and high Stokes numbers are conducted with both frameworks. We found a good agreement of the $\mu (J)$, $\phi (J)$-rheology with low-Stokes-number experiments, which incompressible models fail to describe. The combination of granular rheology, pore pressure and drag model leads to complex phenomena such as apparent cohesion, remoulding, hydroplaning and turbidity currents. The simulations give remarkable insights into these phenomena and increase our understanding of subaquatic mass transports.

This paper presents an analytical study of the Riemann problem for dilute granular gas using initial conditions that result in a shock–contact–shock wave structure. The Euler equations for molecular gas were perturbed with small granular energy dissipation, resulting in an approximate analytical solution that is valid for early evolution time. This approximate analytical solution shows good agreement with the numerical solution of the full problem obtained using a shock-capturing scheme. It is shown that the wave structure of the Riemann problem for dilute granular gas follows that of molecular gas. However, the variables in regions between the discontinuities are functions of both space and time. Our solution shows that the ‘density overshoot’ – reported by Reddy & Alam (J. Fluid Mech., vol. 779, 2015, R2) – is not part of the shock layer but a signature of the density variation in the star region between the left and right shocks, with the maximum density occurring at the contact discontinuity.