We present an experimental investigation aimed at understanding the effects of surface roughness on the time-mean drag coefficient ($\bar {C}_{D}$) of finite-span cylinders ($\text {span/diameter} = \text {aspect ratio}$, $0.51 \le AR \le 6.02$) freely rolling without slip on an inclined plane. While lubrication theory predicts an infinite drag force for ideally smooth cylinders in contact with a smooth surface, experiments yield finite drag coefficients. We propose that surface roughness introduces an effective gap ($G_{eff}$) resulting in a finite drag force while allowing physical contact between the cylinder and the plane. This study combines measurements of surface roughness for both the cylinder and the plane panel to determine a total relative roughness ($\xi$) that can effectively describe $G_{eff}$ at the point of contact. It is observed that the measured $\bar {C}_{D}$ increases as $\xi$ decreases, aligning with predictions of lubrication theory. Furthermore, the measured $\bar {C}_{D}$ approximately matches combined analytical and numerical predictions for a smooth cylinder and plane when the imposed gap is approximately equal to the mean peak roughness ($R_p$) for rough cylinders, and one standard deviation peak roughness ($R_{p, 1\sigma }$) for relatively smooth cylinders. As the time-mean Reynolds number ($\overline {Re}$) increases, the influence of surface roughness on $\bar {C}_{D}$ decreases, indicating that wake drag becomes dominant at higher $\overline {Re}$. The cylinder aspect ratio ($AR$) is found to have only a minor effect on $\bar {C}_{D}$. Flow visualisations are also conducted to identify critical flow transitions and these are compared with visualisations previously obtained numerically. Variations in $\xi$ have little effect on the cylinder wake. Instead, $AR$ was observed to have a more pronounced effect on the flow structures observed. The Strouhal number ($St$) associated with the cylinder wake shedding was observed to increase with $\overline {Re}$, while demonstrating little dependence on $AR$.

Two-dimensional (2-D) quadrant analysis is generally used for investigating flow and sediment dynamics around a rigid structure in open channel flows. Given that particle distribution around rigid obstacles is not spatially uniform and changes in time, while vortices evolve to become three-dimensional (3-D) structures, 2-D quadrant analysis might be unsuitable to completely determine the sediment transport. Hence, 3-D quadrant and 3-D octant analyses should be considered, using the 3-D instantaneous velocity data and relative 3-D bursting process to define sediment transport surrounding the submerged square and circular cylinders. The turbulent kinetic energy (TKE), transition probabilities, occurrence probabilities, stress fraction and angles of inclination of 3-D bursting events are considered to quantify the coherent structures surrounding the cylinders and their interaction with bed particles. Experiments were conducted at the Hydraulics and Water Resources Engineering Laboratory, School of Infrastructure, Indian Institute of Technology, Bhubaneswar, Odisha, India, and velocity data were recorded at different cross-sections around the submerged cylinders using an acoustic Doppler velocimeter. Results show that the TKE is greater for internal ejection, external ejection, and internal sweep, external sweep in the upstream of the circular and square cylindrical structures. On the other part, the TKE is significant for internal ejection, external ejection, and internal sweep, external sweep in the downstream of the aligned square cylindrical structures, which justifies the highest scour depth that occurred upstream of the circular and square cylindrical structures and downstream of the aligned square cylindrical structure. The transition probability of the bursting events was determined using the Markov process from the measured velocity data to investigate the consecutive occurrence of bursting events. Further, the importance of sweeps and ejections on sediment erosion surrounding the cylinders within the scour hole at various stages of its development was investigated via 3-D quadrant analysis of the bursting occurrences. The outcomes show that external sweep and internal ejection events are active mechanisms for bed particle transport surrounding the cylinder. The maximum transition probability values are found around aligned cylindrical structures in comparison with the circular and square cylindrical structures in the transverse direction. This depicts the formation of a trailing vortex on both sides of the aligned square cylindrical object. The results reveal that the effect of inclination angles with respect to the water flow is greater for internal ejection and external sweep from upstream to downstream within the scour hole surrounding the cylindrical structures at various phases of development as horseshoe vortices and downflow develop upstream of the cylindrical structures while trailing vortices and wake vortices form at the top and downstream of the cylindrical structures. Internal and external ejection have a higher stress fraction than an internal and external sweep for square cylinders with alignment angles of 0°, 20° and circular cylinders over underdeveloped and developed scoured beds, respectively. With the higher percentage of fractional contributions for internal sweeps, the external sweep is predicted close to the cylindrical objects in comparison with the internal ejection and external ejection events because of the formation and warping of the horseshoe vortex close to the cylindrical objects, suggesting a significant probability of 3-D bursting occurrences with sediment movement near the cylindrical structures.

Laboratory experiments and particle-resolved simulations are employed to investigate the settling dynamics of a pair of rigidly connected spherical particles of unequal size. They yield a detailed picture of the transient evolution and the terminal values of the aggregate's orientation angle and its settling and drift velocities as functions of the aspect ratio and the Galileo number $Ga$, which denotes the ratio of buoyancy and viscous forces acting on the aggregate. At low to moderate values of $Ga$, the aggregate's orientation and velocity converge to their terminal values monotonically, whereas for higher $Ga$-values the aggregate tends to undergo a more complex motion. If the aggregate assumes an asymmetric terminal orientation, it displays a non-zero terminal drift velocity. For diameter ratios much larger than one and small $Ga$, the terminal orientation of the aggregate becomes approximately vertical, whereas when $Ga$ is sufficiently large for flow separation to occur, the aggregate orients itself such that the smaller sphere is located at the separation line. Empirical scaling laws are obtained for the terminal settling velocity and orientation angle as functions of the aspect ratio and $Ga$ for diameter ratios from 1 to 4 and particle-to-fluid density ratios from 1.3 to 5. An analysis of the accompanying flow field shows the formation of vortical structures exhibiting complex topologies in the aggregate's wake, and indicates the formation of a horizontal pressure gradient across the larger sphere, which represents the main reason for the emergence of the drift velocity.

We study the behaviour of a particle bed immersed in water when a flow generated by an oscillating plate is induced above it. We first consider a rigid plate submerged and oscillated over a particle bed. During upward motion of the plate, a portion of the bed fails, allowing particle displacement, and the bed surface to deform into a heap. We have already determined the flow of the fluid above and within the bed. This work describes the particle motion within the failed region of the bed: when the particles are mobile, they follow the fluid. We depth average the balance of mass and obtain an evolution equation for the displacement of the bed surface. We solve this equation and compare the predictions with the measurements of surface displacement in earlier experiments on rigid square plates. We carry out new experiments to measure the surface displacements under elongated plates. Elongated rigid plates behave similarly to the rigid square ones. Flexible plates produce multiple heaps. We determine that the peaks of these heaps are correlated with the flexural modes of the plates and occur at points along the bed at which the fluid pressure has its extreme values. Different plate flexural modes, resulting in different numbers of heaps, are produced by driving the plate at different frequencies. The particle motion within the bed and heap evolution under a flexible plate can be roughly described by regarding it as two or more rigid plates. We test the predictions of the theory against experiments.

We employ the methods of statistical mechanics to obtain closures for the balance equations of momentum and fluctuation kinetic energy that govern the ballistic motion of grains rebounding at a rigid, bumpy bed that are driven by turbulent or non-turbulent shearing fluids, in the absence of mid-trajectory collisions and fluid velocity fluctuations. We obtain semi-analytical solutions for steady and fully developed saltation over horizontal beds for the vertical profiles of particle concentration and stresses and fluid and particle velocities. These compare favourably with measurements in discrete-element numerical simulations in the wide range of conditions of Earth and other planetary environments. The predictions of the particle horizontal mass flux and its scaling with the amount of particles in the system, the properties of the carrier fluid and the intensity of the shearing also agree with numerical simulations and wind-tunnel experiments.

Understanding settling motion of coral grains is important in terms of protection of coral reef systems and resotoration of the associated ecosystems. In this paper, a series of laboratory experiments was conducted to investigate the settling motion, using optical microscopy to measure shape parameters of coral grains and the particle-filtering-based object tracking to reconstruct the three-dimensional trajectory. Three characteristic descent regimes, namely, tumbling, chaotic and fluttering, are classified based on the three-dimensional trajectory, the spiral radius variation and the velocity spectrum. It is demonstrated that if one randomly picks up one coral grain, then the probabilities of occurrence of the three regimes are approximately $26\,\%$, $42\,\%$ and $32\,\%$, respectively. We have shown that first, the dimensionless settling velocity generally increases with the non-dimensional diameter and Corey shape factor and second, the drag coefficient generally decreases with the Reynolds number and Corey shape factor. Based on this, the applicability of existing models on predicting settling velocity and drag coefficient for coral grains is demonstrated further. Finally, we have proposed extended models for predicting the settling velocity. This study contributes to better understanding of settling motion and improves our predictive capacity of settling velocity for coral grains with complex geometry.

New experiments in highly turbulent, steady, subcritical and uniform water open-channel flows have been carried out to measure the mean turbulent kinetic energy (TKE) budget of sediment-laden boundary layer flows with two sizes (dp = 3 mm and 1 mm) of Plexiglas particles $({\rm relative\ density}\ = 1.192)$. The experiments covered energetic sediment transport conditions (Shields number of $0.35 < \theta < 1.2$) ranging from non-capacity to full-capacity flows in bedload-to-suspension-dominated transport modes (suspension number of $0.5 < {w_s}/{u_\ast } < 1.3$ where ${w_s}$ is the settling velocity and $u_*$ is the friction velocity) and for weakly to highly inertial, finite size turbulence-particle conditions (Stokes number of 0.1 < St < 3.5 and dp/η > 10 where $\eta$ is the Kolmogorov length scale). It was shown that the effects of sediments on the TKE budget are very pronounced in all large particle experiments for which a bedload layer of several grain diameter thickness is developed above the channel bed. When compared with the corresponding reference clear-water flows, the TKE shear-production rate for the 3 mm particle flows is strongly reduced in the wall region corresponding to the bedload layer. This turbulence damping is seen to increase with sediment load until full capacity for flows with constant Shields value, as well as with Shields number value. Inside this damped TKE shear-production zone, a distinct peak of maximal turbulence production appears to coincide with the upper edge of the bedload layer delimited by a sharp gradient in mean sediment concentration. This vertically upshifted peak of TKE production is accompanied by an enhanced net downward oriented TKE flux when compared with the reference clear-water flows. The downward diffused TKE is found to act in the bedload layer as a local energy source in reasonable balance with the sediment transport term. The mechanism behind this downward TKE transport was further analysed on the basis of coherent flow structure dynamics controlled by ejection- and sweep-type events. The agreement between the height of downward directed mean TKE flux and the height below which sweep-type events dominate the Reynolds shear-stress contribution over ejections, revealed the leading role played by sweeps in mean TKE transport. This agreement holds for all reference clear-water flows supporting the well-known wall-roughness-induced dominance of the sweep contribution in turbulent, rough clear-water boundary layer flows. Furthermore, for all 3 mm particle flows, the two referred to transition levels were significantly and similarly upshifted to the upper edge of the bedload layer. Only for these sediment-laden flows, the bedload layer thickness is seen to exceed the wall-roughness sublayer of the reference clear-water flows. This supports a strong analogy between wall-roughness effects in clear-water flows and bedload layer effects in sediment-laden flows, on the mean TKE budget induced by a similarly modified coherent flow structure dynamics. The bedload layer-controlled wall roughness is finally confirmed by the good prediction of the wall-roughness parameter ks of the logarithmic velocity distribution. An empirical formulation fitting the presented measurements is presented, valid over the range of Shields number values covered herein.

An experimental investigation identifying the effects of surface roughness on the drag coefficient ($C_{D}$) of freely rolling spheres is reported. Although lubrication theory predicts an infinite drag force for an ideally smooth sphere in contact with a smooth wall, finite drag coefficients are obtained in experiments. It is proposed that surface roughness provides a finite effective gap ($G$) between the sphere and panel, resulting in a finite drag force while also allowing physical contact between the sphere and plane. The measured surface roughnesses of both the sphere and panel are combined to give a total relative roughness ($\xi$). The measured $C_{D}$ increases with decreasing $\xi$, in agreement with analytical predictions. Furthermore, the measured $C_{D}$ is also in good agreement with the combined analytical and numerical predictions for a smooth sphere and wall, with a gap approximately equal to the root-mean-square roughness ($R_q$). The accuracy of these predictions decreases for low mean Reynolds numbers ($\overline {Re}$), due to the existence of multiple scales of surface roughness that are not effectively captured by $R_{q}$. Experimental flow visualisations have been used to identify critical flow transitions that have been previously predicted numerically. Path tracking of spheres rolling on two panels with different surface roughnesses indicates that surface roughness does not significantly affect the sphere path or oscillations. Analysis of sphere Strouhal number ($St$) highlights that wake shedding and sphere oscillations are coupled at low $\overline {Re}$ but with increasing $\overline {Re}$, the influence of wake shedding on the sphere path diminishes.

We present the results from a series of experiments investigating the dynamics of gravity currents which form when a dense saline or particle-laden plume issuing from a moving source interacts with a horizontal surface. We define the dimensionless parameter $P$ as the ratio of the source speed, $u_a$, to the buoyancy speed, $(B_0/z_0)^{1/3}$, where $B_0$ and $z_0$ are the source buoyancy flux and height above the horizontal surface, respectively. Using our experimental data, we determine that the limiting case in which $P=P_c$ the gravity current only spreads downstream of the initial impact point occurs when $P_c=0.83\pm 0.02$. For $P< P_c$, from our experiments we observe that the plume forms a gravity current that spreads out in all directions from the point of impact and the propagation of the gravity current is analogous to a classical constant-flux gravity current. For $P>P_c$, we observe that the descending plume is bent over and develops a pair of counter-rotating line vortices along the axis of the plume. The ensuing gravity current spreads out downstream of the source, normal to the motion of the source. Analogous processes occur with particle-laden plumes, but there is a second dimensionless parameter $S$, the ratio of the particle fall speed, $v_s$, to the vertical speed of a plume in a crossflow, $(B_0/u_a z_0)^{1/2}$. For $S\ll 1$, particles remain well mixed in the plume and a particle-driven gravity current develops. For $S\gg 1$, particles separate from the plume prior to impacting the boundary which leads to a fall deposit and no gravity current. We discuss these results in the context of deep-sea mining.

Multigrain/polydispersity has a significant impact on turbidity current (TC). Despite the fact that several researches have looked into this effect, the impact of the fluid–particle interactions is not fully understood. Motivated by this, we employ the Eulerian–Lagrangian computational fluid dynamics–discrete element method model to investigate the dynamics of the bidisperse lock-exchange TC. Results show that, because the coarse particles will settle faster and stop moving forward, the two phases of bidisperse transport and fine component transport can be distinguished in the evolution of the bidisperse TC. During the bidisperse transport stage, the upper interface of each component is primarily determined by their own settling and transport characteristics and does not strongly depend on the relative fine particle volume fraction $\phi _F$. Fine particles are primarily responsible for the vortical structures near the upper interface of the TC head, and the increase of $\phi _F$ promotes their streamwise development. In comparison, fragmented vortical coherent structures are closely related to the presence of coarse particles, which can be seen in the lower layers. Bidisperse segregation alters the collision process between dispersed phases, which differs from monodisperse TC. The collisions and segregation-induced flow establish interconnections between the two dispersed phases. In the latter stage, the transport of fine particles is inhibited by both the lift force and the contact force produced by the collision with the deposited materials. As $\phi _F$ rises, the negative contact force weakens, and its change is essentially balanced by the rise in negative lift force.

Cohesive particulate flows play an important role in environmental fluid dynamics, as well as in a wide variety of civil and process engineering applications. However, the scaling laws, constitutive equations and continuum field descriptions governing such flows are currently less well understood than for their non-cohesive counterparts. Grain-resolved simulations represent an essential tool for addressing this shortcoming, along with theoretical investigations, laboratory experiments and field studies. Here we provide a tutorial introduction to simulations of fine-grained sediments in viscous fluids, along with an overview of some representative insights that this approach has yielded to date. After a brief review of the key physical concepts governing van der Waals forces as the main cohesive effect for subaqueous sediment suspensions, we discuss their incorporation into particle-resolved simulations based on the immersed boundary method. We subsequently describe simulations of cohesive particles in several model turbulent flows, which demonstrate the emergence of a statistical equilibrium between the growth and break-up of aggregates. As a next step, we review the influence of cohesive forces on the settling behaviour of dense suspensions, before moving on to submerged granular collapses. Throughout the article, we highlight open research questions in the field of cohesive particulate flows whose investigation may benefit from grain-resolved simulations.

We interpret the Taylor–Green cellular vortex model in terms of the Kolmogorov length and velocity scales, in order to study the balance between aggregation and breakup of cohesive sediment in fine-scale turbulence. One-way coupled numerical simulations, which capture the effects of cohesive, lubrication and direct contact forces on the flocculation process, reproduce the non-monotonic relationship between the equilibrium floc size and shear rate observed in previous experiments. The one-way coupled results are confirmed by select two-way coupled simulations. Intermediate shear gives rise to the largest flocs, as it promotes preferential concentration of the primary particles without generating sufficiently strong turbulent stresses to break up the emerging aggregates. We find that the optimal intermediate shear rate increases for stronger cohesion and smaller particle-to-fluid density ratios, and we propose a simple model for the equilibrium floc size that agrees well with experimental data reported in the literature.

In this work, the modelling of collisional bedload transport is investigated with a focus on the continuum modelling of the granular flow. For this purpose, a frictional–collisional approach, combining a Coulomb model with the kinetic theory of granular flows, is developed. The methodology is based on a comparison with coupled fluid–discrete simulations that the classical kinetic theory model fails to reproduce. This inaccuracy may be explained by the assumptions of negligible interparticle friction and the absence of a saltation model in the continuum approach. In order to provide guidelines for the modelling, the fluctuating energy balance is computed in the discrete simulations and systematically compared with the kinetic theory laws. Interparticle friction is shown to affect the radial distribution function and to increase the energy dissipation, in agreement with previous observations. In addition, a saltation regime is identified, leading to departure from the viscosity and pseudo-thermal diffusivity laws of the kinetic theory in the dilute regime. Based on these observations, modifications to account for interparticle friction are included in the two-fluid model, and the kinetic theory is coupled with a saltation model. The results show that for inelastic frictional particles, interparticle friction controls energy dissipation, and the macroscopic behaviour of the granular flow does not depend on the microscopic particle properties. The proposed model reproduces the $\mu (I)$ rheology in the dense regime of the granular flow. Finally, the model is evaluated with experiments, showing significant improvements concerning the granular flow modelling.

We use an established discrete element method (DEM) Reynolds-averaged Navier–Stokes (RANS)-based numerical model to simulate non-suspended sediment transport across conditions encompassing almost seven orders of magnitude in the particle–fluid density ratio $s$, ranging from subaqueous transport ($s=2.65$) to aeolian transport in the highly rarefied atmosphere of Pluto ($s=10^7$), whereas previous DEM-based sediment transport studies did not exceed terrestrial aeolian conditions ($s\approx 2000$). Guided by these simulations and by experiments, we semi-empirically derive simple scaling laws for the cessation threshold and rate of equilibrium aeolian transport, both exhibiting a rather unusual $s^{1/3}$-dependence. They constitute a simple means to make predictions of aeolian processes across a large range of planetary conditions. The derivation consists of a first-principle-based proof of the statement that, under relatively mild assumptions, the cessation threshold physics is controlled by only one dimensionless control parameter, rather than two expected from dimensional analysis. Crucially, unlike existing models, this proof does not resort to coarse-graining the particle phase of the aeolian transport layer above the bed surface. From the pool of existing models, only that by Pähtz et al. (J. Geophys. Res.: Earth, vol. 126, 2021, e2020JF005859) is somewhat consistent with the combined numerical and experimental data. It captures the scaling of the cessation threshold and the $s^{1/3}$-dependence of the transport rate, but fails to capture the latter's superimposed grain size dependence. This hints at a lack of understanding of the transport rate physics and calls for future studies on this issue.

In this study, we assess charcoal records from eolian deposits within the Cooloola Sand Mass, a subtropical coastal dune system in eastern Australia, to determine whether they can be used as a proxy for Holocene fire history. We excavate four profiles in depositional wedges at the base of dune slipfaces (footslope deposits) and calculate charcoal concentrations for three size classes (180–250 μm, 250–355 μm, and 355 μm–2 mm) at predetermined depth intervals. Age–depth models are constructed for each profile using radiocarbon measurements (n = 46) and basal optically stimulated luminescence ages (n = 4). All records appear intact with little evidence of postdepositional mixing as demonstrated by minimal age reversals and consistent trends in charcoal concentration and accumulation rates (CHAR) among size classes. Combining all four records, we generate a ca. 7 cal ka BP terrestrial fire history that depicts distinct peaks representing periods of increased local fire activity at <0.3, 1.1–0.4, 2.2–1.6, 3.4–2.6, and 6.7–5.3 cal ka BP. Our findings parallel regional records and highlight the utility of dune footslopes as ecological and sedimentary archives. As dune fields are much more common than wetlands and lakes in semiarid and arid areas, these deposits have the potential to increase the spatial resolution of fire records globally.

The interaction of supercritical turbulent flows with granular sediment beds is challenging to study both experimentally and numerically. This challenging task has hampered advances in understanding antidunes, the most characteristic bedform of supercritical flows. This article presents the first numerical attempt to simulate upstream-migrating antidunes with geometrically resolved particles and a liquid–gas interface. Our simulations provide data at a resolution higher than laboratory experiments, and they can therefore provide new insights into the mechanisms of antidune migration and contribute to a deeper understanding of the underlying physics. To manage the simulations’ computational costs and physical complexity, we employ the cumulant lattice Boltzmann method in conjunction with a discrete element method for particle interactions, as well as a volume-of-fluid scheme to track the deformable free surface of the fluid. By reproducing two flow configurations of previous experiments (Pascal et al., Earth Surf. Process. Landf., vol. 46, issue 9, 2021, pp. 1750–1765), we demonstrate that our approach is robust and accurately predicts the antidunes’ amplitude, wavelength and celerity. Furthermore, the simulated wall shear stress, a key parameter governing sediment transport, is in excellent agreement with the experimental measurements. The highly resolved data of fluid and particle motion from our simulation approach open new perspectives for detailed studies of morphodynamics in shallow supercritical flows.

The dynamics of sedimenting particles under gravity are surprisingly complex due to the presence of effective long-ranged forces. When the particles are polar with a well-defined symmetry axis and non-uniform density, recent theoretical predictions suggest that prolate objects will repel and oblate ones will weakly attract. We tested these predictions using mass polar prolate spheroids, which are composed of 2 mm spheres glued together. We probed different aspect ratios ($\kappa$) and centre of mass variations ($\chi$) by combining spheres of different densities. Experiments were done in both quasi-two-dimensional (2-D) and three-dimensional (3-D) chambers. By optically tracking the motion of single particles, we found that the dynamics were well described by a reduced mobility matrix model that could be solved analytically. Pairs of particles exhibited an effective repulsion, and their separation roughly scaled as $(\kappa - 1)/\chi ^{0.39}$, i.e. particles that were more prolate or had smaller mass asymmetry had stronger repulsion effects. In three dimensions, particles with $\chi >0$ were distributed more uniformly than $\chi =0$ particles, and the degree of uniformity increased with $\kappa$, indicating that the effective 2-body repulsion manifests for a large number of particles.

Reliable prediction of the erosion rate of sediment beds is important for many applications in coastal and river engineering. Theoretical understanding of empirically derived scaling relations is still lacking. This applies in particular for the scaling anomaly between low and high Shields number conditions. In this work, the erosion process is studied from the perspective of the phase-averaged turbulent kinetic energy (TKE) equations. The multi-phase TKE equations are written in a form that allows for a direct comparison with the TKE equation that appears for a stratified single-phase flow under the Boussinesq approximation. This reveals that next to buoyancy destruction, several other TKE modulation mechanisms become important at high Shields numbers and concentrations. Two scaling laws are derived for both moderate and high Shields numbers, and are tested against a wide range of experimental data.

The turbidity current (TC), a ubiquitous fluid–particle coupled phenomenon in the natural environment and engineering, can transport over long distances on an inclined terrain due to the suspension mechanism. A large-eddy simulation and discrete element method coupled model is employed to simulate the particle-laden gravity currents over the inclined slope in order to investigate the auto-suspension mechanism from a Lagrangian perspective. The particle Reynolds number in our TC simulation is $0.01\sim 0.1$ and the slope angle is $1/20 \sim 1/5$. The influences of initial particle concentration and terrain slope on the particle flow regimes, particle movement patterns, fluid–particle interactions, energy budget and auto-suspension index are explored. The results indicate that the auto-suspension particles predominantly appear near the current head and their number increases and then decreases during the current evolution, which is positively correlated with the coherent structures around the head. When the turbidity current propagates downstream, the average particle Reynolds number of the auto-suspension particles remains basically unchanged, and is higher than that of other transported particles. The average particle Reynolds number of the transported particles exhibits a negative correlation with the Reynolds number of the current. Furthermore, the increase in particle concentration will enhance the particle velocity, which allows the turbidity current to advance faster and improves the perpendicular support, thereby increasing the turbidity current auto-suspension capacity. Increasing slope angle will result in a slightly larger front velocity, while the effect of that on the total force is insignificant.

Understanding the statistics of bedload particle motions is of great importance. To model the hop events which are defined as trajectories of particles moving successively from the start to the end of their motions, recently, Wu et al. (Water Resour. Res., vol. 56, 2020, p. e2019WR025116) have successfully performed individual-based simulations according to the Fokker–Planck equation for particle velocities. However, analytical solutions are still not available due to (i) difficulties in treating the velocity-dependent diffusivity, and (ii) a knowledge gap in incorporating the termination of particle motions for the equation. To tackle the above-mentioned challenges, we first specify a Robin boundary condition representing the deposition of particles. Second, for analytical solutions of hop statistics, a variable transformation is devised to deal with the velocity-dependent diffusivity. The original bedload transport problem is thus found to be governed by the classic equation for the solute transport in tube flows with a constant diffusivity after the transformation. Finally, through solving the spatial and temporal moments of the governing equation, we investigate the influence of the deposition rate on three key characteristics of particle hops. Importantly, we have related the deposition rate to the mean travel times and hop distances, enabling a direct determination of this physical parameter based on measured particle motion statistics. The analytical solutions are validated by experimental observations with different bedload particle diameters and transport conditions. Based on the limited experimental datasets, the deposition frequency is shown to decrease as the shear stress increases when the flow rate is not small.