We study experimentally the critical conditions for incipient motion of a single spherical particle deposited on a regular substrate under laminar flow conditions. The substrates are triangular and quadratic arrangements of identical glass spheres. For the latter configuration, the distance between the substrate spheres is varied, resulting in different partial shielding of the deposited particle to the shear flow. For the studied particle Reynolds numbers range between 3 × 10−4 and 3, the critical Shields number is independent from the particle density and from the particle Reynolds number but it depends significantly on the geometry of the substrate. Depending on the spacing between the substrate beads and thus on the exposure of the particle to the flow, we have observed an increase of about 50 percent in the critical Shields number. Studying the onset of particle motion as a function of the orientation of the substrate to the flow direction we find that the critical Shields number changes by up to a factor of 2, which is mainly due to the fact that the particle travels through the troughs of the substrate and hence the shear force in travel direction diminishes if not in line with the flow direction. Besides the critical Shields number we study the initial stage of particle motion by detecting the minimum time that is necessary for maintaining a certain Shields number to change the position of a single particle on the regular substrates. In the range studied, the initial stage of motion on the scale of the substrate's periodicity is mainly governed by the equilibrium particle motion.

We study the incipient motion of single spheres in steady shear flow on regular substrates at low particle Reynolds numbers. The substrate consists of a monolayer of regularly arranged fixed beads, in which the spacing between beads is varied resulting in different angles of repose and exposures of the particle to the main flow. The flow-induced forces and the level of flow penetration into the substrate are determined numerically. Since experiments in this range had shown that the critical Shields number is independent of inertia but strongly dependent on the substrate geometry, the particle Reynolds number was fixed to 0.01 in the numerical study. Numerics indicates that rolling motion is always preferred to sliding and that the flow penetration is linearly dependent on the spacing between the substrate particles. Besides, we propose an analytical model for the incipient motion. The model is an extension of Goldman’s classical result for a single sphere near a plain surface taking into account the angle of repose, flow orientation with respect to substrate topography and shielding of the sphere to the linear shear flow. The effective level of flow penetration is the only external parameter. The model, applied to triangular and quadratic arrangements with different spacings, is able to predict the dependence of the critical Shields number on the geometry and on the orientation of the substrate. The model is in very good agreement with numerical results. For well-exposed particles, we observed that the minimum critical Shields number for a certain angle of repose does not depend sensitively on the considered arrangement. At large angles of repose, as expected in fully armoured beds, the model is consistent with experimental results for erodible beds at saturated conditions.

We experimentally study how neighboring particles affect the incipient motion of particles on regular substrates and exposed to a laminar shear flow. To this end, we determine the critical Shields number and determine whether the particle rolls or slides. The substrates consist of a monolayer of fixed spheres of uniform size that are regularly arranged in triangular and quadratic configurations. Neighboring particles influence the incipient motion by shielding to the shear flow and may inhibit continuous motion once they are in direct contact with the particle. At the low particle Reynolds numbers studied, neighboring spheres on the monolayer only affect the incipient particle motion if they are closer than about 3 particle diameters. Direct contact inhibits continuous motion and results in a strong increase of the critical Shields number. For identical beads, we found two different regimes for the onset of continuous motion. Depending on the substrate geometry, the upstream particle may start to roll like a single particle passing the downstream neighbor or it may push its downstream neighbor forward. In the latter case, the downstream sphere rolls while the upstream bead slides in contact with the downstream neighbor. Both regimes yield about the same critical Shields number although the critical Shields number for single particle motion differs by about 50%. If particle contact is avoided by a sudden jump in the Shields number, the critical Shields number for onset of continuous particle motion can be reduced considerably. Finally, the lowest critical Shields numbers for dislodging buried beads in the configurations studied coincides with the critical Shields number for incipient motion of irregular granular beds.

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We study incipient motion of single beads on regular substrates made of spherical particles of a different size in steady shear flow at small particle Reynolds numbers. We cover a large range of sizes: from small beads that are highly shielded from the shear flow by the substrate spheres, and hence, are susceptible to the flow through the interstices of the substrate, to beads fully exposed to the flow, where the substrate rather acts like roughness of an otherwise flat surface. Numerical and experimental studies agree within measurement uncertainty. To describe the findings, we extend a recently derived model for particles of equal size which was validated over a wide range of substrates [Agudo et al., “Shear-induced incipient motion of a single sphere on uniform substrates at low particle Reynolds numbers,” J. Fluid Mech. 825, 284–314 (2017)]. The extended model covers the entire spectrum of size ratios, where the critical Shields number varies from about zero to infinity. The model properly describes the numerical and experimental data. For well exposed beads, we find a scaling law between the critical Shields number and the size ratio between mobile bead and substrate spheres with an exponent of −1.