Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Incipient particle motion is encountered in a wide range of industrial and natural processes. Environmental examples include the initial process of sediment transport in river and oceans, bed erosion or dune formation among others ^1,2,3. Pneumatic conveying⁴, removal of pollutants or cleaning of surfaces^5,6 are typical industrial applications involving the onset of particle motion.

Due to the broad range of applications, the onset of particle motion has been extensively studied over a century, mostly under turbulent conditions^{7,8,9,10,11,12,13,14,15}. Many experimental approaches have been applied to determine the threshold for the onset of motion. The studies include parameters such as the particle Reynolds number^{13,16,17,18,19,20}, the relative flow submergence^21,22,23,24 or geometrical factors as the angle of repose^16,18,25, exposure to the flow^26,27,28,29, relative grain protrusion²⁹ or streamwise bed slope³⁰.

The current data for the threshold including turbulent conditions are broadly scattered^12,31 and the results often seem inconsistent²⁴. This is mostly due to the inherent complexity of controlling or determining flow parameters under turbulent conditions^13,14. Besides, the threshold for sediment motion strongly depends on the mode of motion, i.e. sliding, rolling or lifting¹⁷ and the criterion to characterize the incipient motion³¹. The latter may be ambiguous in an erodible sediment bed.

During the last decade, experimental researchers have studied incipient particle motion in laminar flows^{32,33,34,35,36,37,38,39,40,41,42,43,44}, where the wide spectrum of length scales interacting with the bed is avoided⁴⁵. In many practical scenarios implying sedimentation, the particles are quite small and the particle Reynolds number remains lower than about 5⁴⁶. On the other hand, laminar flows are able to generate geometric patterns as ripples and dunes as turbulent flows do^42,47. Similitudes in both regimens have been shown to reflect analogies in the underlying physics⁴⁷ so important insight for the particle transport can be obtained from a better controlled experimental system⁴⁸.

In laminar flow, Charru et al. noticed that the local rearrangement of a granular bed of uniformly sized beads, so-called bed armouring, resulted in a progressive increase of the threshold for the onset of motion until saturated conditions were achieved³². Literature, however, reveals different thresholds for saturated conditions in irregularly arranged sediment beds depending on the experimental set-up^36,44. This scattering may be due to the difficulty of controlling particle parameters such as orientation, protrusion level and compactness of the sediments.

The main goal of this manuscript is to describe in detail how to characterize the incipient motion of single spheres as a function of geometrical properties of the horizontal sediment bed. For that purpose, we use regular geometries, consisting of monolayers of fixed beads regularly arranged according to triangular or quadratic configurations. Regular substrates similar to that we use are found in applications such as for the template-assembly of particles in microfluidic assays⁴⁹, self-assembly of microdevices in confined structured geometries⁵⁰ or intrinsic particle-induced transport in microchannels⁵¹. More importantly, using regular substrates allows us to highlight the impact of local geometry and orientation and to avoid any dubiety about the role of the neighborhood.

In laminar flow, we observed that the critical Shields number increased by 50% only depending on the spacing between the substrate spheres and thus on the exposure of the bead to the flow³⁸. Similarly, we found that the critical Shields number changed by up to a factor of two depending on the orientation of the substrate to the flow direction³⁸. We noticed that immobile neighbors only affect the onset of the mobile bead if they were closer than about three particle diameters⁴¹. Triggered by the experiment findings, we have recently presented a rigorous analytical model that predicts the critical Shields number in the creeping flow limit⁴⁰. The model covers the onset of motion from highly exposed to hidden beads.

The first part of this manuscript deals with the description of the experimental procedure used in previous studies at shear Reynolds number, Re*, lower than 1. The laminar flow is induced with a rotational rheometer with a parallel configuration. In this low Reynolds number limit, the particle is not supposed to experience any velocity fluctuation²⁰ and the system matches the so-called hydraulically smooth flow where the particle is submerged within the viscous sublayer.

Once incipient motion at laminar flow is established, the role of turbulence can become clearer. Motivated by this idea, we introduce a novel experimental procedure in the second part of the protocol. Using a Göttingen low-speed wind tunnel with open jet test section, the critical Shields number can be determined in a wide range of Re* including the hydraulically transitional flow and the turbulent regime. The experimental results can provide important insight about how forces and torques act on a particle due to the turbulent flow depending on the substrate geometry. Besides, these results can be used as a benchmark for more sophisticated models at high Re* in a similar way that past work in laminar flow has been used to feed semi probabilistic models⁵² or to validate recent numerical models⁵³. We present some representative examples of applications at Re* ranging from 40 to 150.

The incipient criterion is established as the motion of the single particle from its initial equilibrium position to the next one. Image processing is used to determine the mode of onset of motion, i.e. rolling, sliding, lifting^39,41. For that purpose, the angle of rotation of mobile spheres that were manually marked is detected. The algorithm tracks the position of the marks and compares it with the center of the sphere. A preliminary set of experiments was conducted in both experimental set-ups to clarify that the critical Shields number remains independent of finite size effects of the set-up and relative flow submergence. The experimental methods are thus designed to exclude any other parameter dependent on the critical Shields number beyond Re* and geometrical properties of the sediment bed. The Re* is varied using different fluid-particle combinations. The critical Shields number is characterized as a function of the burial degree, , defined by Martino et al.³⁷ as where is the angle of repose, i.e. the critical angle at which motion occurs⁵⁴, and is the exposure degree, defined as the ratio between the cross-sectional area effectively exposed to the flow to the total cross-sectional area of the mobile bead.