Granular bed motion induced by a liquid shear flow is considered as a model of bedload. This flow is characterized by a localization of the granular flow close to the upper surface of the bed. In such a framework, the paper aims at discussing and proposing continuum effective models describing the physical properties, as the rheology, of the two-phase granular suspension at a so-called mesoscale larger than the grain size. The main questions addressed here concern the relevance and the extension of effective rheological models for granular suspensions often extracted from simple-shear flow configurations in a viscous limit, to situations of both granular shear localization as in bedload and weakly inertial flow regime. For this purpose, we consider the steady transport of a granular bed by a laminar Couette fluid flow above it and close to the onset of motion, for a grain-to-fluid density ratio of 2.5, as silica in water, and a range of particle Reynolds numbers Re-p is an element of [0.1, 10] and Shields numbers theta is an element of [0.1, 0.7]. To provide accurate continuum models, the dynamics into the granular shear layer has to be first known down to a scale smaller than each grain, a so-called microscale. Numerical simulations are thus performed at the microscale at which individual grain dynamics is resolved, using an immersed boundary method (IBM) coupled to a discrete element method (DEM) granular solver. Upscaling is then performed to obtain the equivalent momentum balance at the mesoscale, characterized by continuum phases, using a spatial averaging method on volume element larger than the grain diameter. This approach allows us obtaining stresses, strains, and their relationships for the fluid phase, the granular phase, and the equivalent mixture, independently. The main contribution of this work is threefold: (i) we highlight the relevance of mesoscopic rheological continuum law for localized granular shear flow; (ii) we extract rheological models from direct numerical simulations (IBM/DEM) in a weakly inertial regime, going beyond purely viscous situations; and (iii) we extend Coulomb-like model mu(I) of a granular suspension to incorporate fluid/particle inertial effects showing a different dependence of fluid phase and granular phase contributions with dimensionless numbers.

River beds frequently exhibit a lateral variation of roughness. For example, in the case of an overflowing river, the main channel has a smoother topography compared to the adjacent floodplains where vegetation and land occupation yield an important hydraulic roughness. The lateral difference in roughness can induce a high lateral velocity gradient within the river cross- section that gives birth to a mixing layer. This mixing layer leads to fluid and momentum transfers between the two adjacent beds. To understand such mix- ing processes in rivers is important for predicting stage-discharge relationships and the velocity distribution within the cross-section. In order to address these issues in the context of a shallow water flow with a water depth h of the same order as the roughness elements of the bed, experiments were undertaken in a 26 m long and 1.1 m wide glass-walled open-channel flume. One half-side of the bed was covered with an array of cubes of height k arranged in a square configuration, the other side with smooth glass. Three different levels of cube submergence h/k were examined (h/k = 0.8, 1.5 and 2). The experiments and measurements were designed to yield the flow in the complete volume of the interstices across the cube array. To achieve this, 2C-3D linear-scanning PIV measurements with zero-parallax optics were developed and set up. The mea- surements revealed the complexity of the flow structure around the interface between the rough and smooth beds. The results show that the ability of the mixing layer to exchange momentum is highly dependent on the level of the cube submergence h/k.

Multi-plane PIV measurements were performed in an open-channel flume filled with elongated prisms of height k and width l to investigate the effect of the deepening of the canopy on the flow structure. Velocity measurements were performed both inside the canopy and above it. Analysis of the spatial convergence for the double-averaged quantities shows that for canopy flow investigations (z < k), at least 5 measurement planes are required to obtain a relative spatial convergence error below 3% for the dispersive shear stress, the quantity the most sensible to spatial sampling. With only three measurement planes, the spatial convergence is below 1% only in the flow region above the canopy (z > k). Three canopy aspect ratios, k/l = [1, 3, 6] were investigated for a fixed modified-submergence ratio β = (h - k)=l = 3 where h is the water depth. As the canopy deepens, the hydraulic roughness decreases and the velocity near the bottom of the canopy becomes gradually constant, as expected for deep canopies. We show how the highly converged (both in space and time) profiles of double-averaged longitudinal velocity and total shear stress can be used to calculate the vertical distribution of drag in the canopy. With this methodology, values of the drag coefficient CD(z) can be calculated, and are found to be always close to unity, even in the upper part of the canopy.