A common way to transport solids in large quantities is by using a carrier fluid to transport the solids as a concentrated solid/liquid mixture or slurry through a pipeline. Typical examples are found in dredging, mining and drilling applications. Dependent on the slurry properties and flow conditions, horizontal slurry pipe flow is either in the fixed-bed, sliding-bed or fully-suspended regime. In terms of non-dimensional numbers, the flow is fully characterized by the bulk liquid Reynolds number (Re), the Galileo number (Ga, a measure for the tendency of particles to settle under gravity), the solid bulk concentration (ϕ_b), the particle/fluid density ratio (ρ_p/ρ_f), the particle/pipe diameter ratio (D_p/D_pipe), and parameters related to direct particle interactions such as the Coulomb coefficient of sliding friction (μ_c). To further our fundamental understanding of the flow dynamics, we performed experiments and interface-resolved Direct Numerical Simulations (DNS) of slurry flow in a horizontal pipe. The experiments were performed in a transparent flow loop with D_pipe=4 cm. We measured the pressure drop along the pipeline, the spatial solid concentration distribution in the cross-flow plane through Electrical Resistance Tomography (ERT), and used a high-speed camera for flow visualization. The slurry consisted of polystyrene beads in water with D_p=2mm, ρ_p/ρ_f=1.02, Ga between 40–45 and ϕ_b between 0.26–0.33. The different flow regimes were studied by varying the flow rate, with Re varying from 3272 till 13830. The simulations were performed for the same flow parameters as in the experiments. Taking the experimental uncertainty into account, the results from the DNS and the experiments are in reasonably good agreement. The results for the pressure drop agree also fairly well with popular empirical models from literature. In addition, we performed a parametric DNS study in which we solely varied Re and Ga. In all flow regimes, a secondary flow of Prandtl's second kind is present, ascribed to the presence of internal flow corners and a ridge of densely packed particles at the pipe bottom during transition towards the fully-suspended regime. In the bulk of the turbulent flow above the bed, secondary flow transport of streamwise momentum dominates over turbulent diffusion in regions where the secondary flow is strong and vice versa where it is weak. The transition between flow regimes appears to be governed by the competition between the net gravity force on the particles and shear-induced particle migration from particle–particle interactions. This competition can be expressed by the Shields number, θ. For θ≲0.75, gravity is dominant and the flow is in the fixed-bed regime. For θ≳0.75, shear-induced migration becomes progressively more important for increasing θ. Low-concentration zones flanking the sliding bed start to form at the top corners of the bed, and gradually expand downwards along the pipe wall till the pipe bottom is reached. For θ≳1.5, shear-induced migration is responsible for lifting the particle bed away from the wall, associated with the onset of the suspended regime. For θ≫1, gravity is of minor importance and the mean flow eventually reaches axi-symmetry with a high-concentration particle core at the pipe center and negligible secondary flow.

Particle-resolved Direct Numerical Simulations have been performed on the gravitational settling of mono-disperse solid spheres in a viscous fluid and triply periodic domain. In a comprehensive study, the bulk solid volume concentration was varied from ϕ=0.5 to 30%. To study the effect of inertia, three different Galileo numbers were considered in the inertial regime, Ga=144, 178 and 210, for which a single settling sphere exhibits distinctly different wake and path characteristics. The particle/fluid mass density ratio was fixed at 1.5. We find that for ϕ=2−30% the suspension microstructure and dynamics depend predominantly on the bulk concentration. In qualitative agreement with previous studies in literature, three different sedimentation regimes can be distinguished: (1) the dilute concentration regime for ϕ≲2% with preferential settling of particles in vertical trains, (2) the moderate concentration regime for 2%≲ϕ≲10% with preferential settling of particles in horizontal pairs with an interparticle distance of ∼ 1.5 particle diameters, and (3) the dense concentration regime for ϕ≳10% with a nearly random (“hard-sphere”) distribution of the particles in space. The clustering of particles is dictated by, respectively, trapping of particles in the wake of other particles, a drafting–kissing–tumbling (DKT) instability by which two vertically aligned particles quickly reorient themselves into a horizontally aligned particle pair, and short-range multiparticle interactions through viscous lubrication and to a lesser extent collisions between particles. In all cases, hindered settling at a reduced speed is observed as compared to a single settling sphere. The well-known Richardson–Zaki relation for the mean sedimentation velocity appears valid only for the dense concentration regime. We provide ample evidence that in the dense regime the characteristic velocity and time scales of particle motion are proportional to gD_p and D_p/g, respectively, with g the gravitational acceleration and D_p the particle diameter. We also observe an ω⁻³ scaling of the particle velocity spectra for ωD_p/g≳0.4 and we propose a model to explain this scaling behavior, based on the inertial response of the particles to small-scale flow perturbations. Kinematic waves, i.e., vertically propagating plane waves in the local concentration field, are observed in all cases, though unrelated particle motions are responsible for significant loss of the spatio-temporal coherence of the waves. The wave speed was determined from repeated space–time autocorrelations of the local concentration field and appears in reasonable agreement with Kynch sedimentation theory using the Richardson–Zaki relation. The passage of kinematic waves causes perturbations in the particle velocity at a frequency that matches well with peak frequencies in the particle velocity spectra for concentrations up to ϕ≈10%. The time-lagged cross-correlation of the vertical and horizontal particle velocity suggests that kinematic waves may trigger DKT instabilities, while conversely DKT instabilities may be responsible for the onset of kinematic waves. Finally, we suggest that obstruction and perturbation of the particle wake by neighboring particles could offer an explanation for the small influence of the Galileo number on the suspension behavior for ϕ=2−30%.

When two spherical particles submerged in a viscous fluid are subjected to an oscillatory flow, they align themselves perpendicular to the direction of the flow leaving a small gap between them. The formation of this compact structure is attributed to a nonzero residual flow known as steady streaming. We have performed direct numerical simulations of a fully resolved, oscillating flow in which the pair of particles is modeled using an immersed boundary method. Our simulations show that the particles oscillate both parallel and perpendicular to the oscillating flow in elongated figure-8 trajectories. In absence of bottom friction, the mean gap between the particles depends only on the normalized Stokes boundary layer thickness δ∗, and on the normalized, streamwise excursion length of the particles relative to the fluid Ar∗ (equivalent to the Keulegan-Carpenter number). For Ar∗≲1, viscous effects dominate and the mean particle separation only depends on δ∗. For larger Ar∗ values, advection becomes important and the gap widens. Overall, the normalized mean gap between the particles scales as L∗≈3.0δ∗1.5+0.03Ar∗3, which also agrees well with previous experimental results. The two regimes are also observed in the magnitude of the oscillations of the gap perpendicular to the flow, which increases in the viscous regime and decreases in the advective regime. When bottom friction is considered, particle rotation increases and the gap widens. Our results stress the importance of simulating the particle motion with all its degrees of freedom to accurately model the system and reproduce experimental results. The insights of the particle pairs provide an important step towards understanding denser and more complex systems.

Macroscopic properties of sedimenting suspensions have been studied extensively and can be characterized using the Galileo number (Ga), solid-to-fluid density ratio (π_p) and mean solid volume concentration (ϕ¯). However, the particle–particle and particle–fluid interactions that dictate these macroscopic trends have been challenging to study. We examine the effect of concentration on the structure and dynamics of sedimenting suspensions by performing direct numerical simulation based on an Immersed Boundary Method of monodisperse sedimenting suspensions of spherical particles at fixed Ga= 144 , π_p= 1.5 , and concentrations ranging from ϕ¯ = 0.5 to ϕ¯ = 30 %. The corresponding particle terminal Reynolds number for a single settling particle is Re_T= 186. Our simulations reproduce the macroscopic trends observed in experiments and are in good agreement with semi-empirical correlations in literature. From our studies, we observe, first, a change in trend in the mean settling velocities, the dispersive time scales and the structural arrangement of particles in the sedimenting suspension at different concentrations, indicating a gradual transition from a dilute regime (ϕ¯ ≲ 2 %) to a dense regime (ϕ¯ ≳ 10 %). Second, we observe the vertical propagation of kinematic waves as fluctuations in the local horizontally-averaged concentration of the sedimenting suspension in the dense regime.