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1. Finding the Bed Shear Stress on a Rough Bed Using the Log Law

   https://doi.org/10.1061/(ASCE)WW.1943-5460.0000707  

   Ting, Francis C.; Kern, Gunnar S. (July 2022, Journal of Waterway, Port, Coastal, and Ocean Engineering)
2. Flow and Drag in a Seagrass Bed

   https://doi.org/10.1029/2018JC014862  

   Monismith, Stephen G.; Hirsh, Heidi; Batista, Natasha; Francis, Holly; Egan, Galen; Dunbar, Robert B. (March 2019, Journal of Geophysical Research: Oceans)

   Abstract We report direct measurement of drag forces due to tidal flow over a submerged seagrass bed in Ngeseksau Reef, Koror State, Republic of Palau. In our study, drag is computed using an array of high‐resolution pressure measurements, from which values of the drag coefficients,C_D, referenced to measured depth‐averaged velocities,V, were inferred. Reflecting the fact that seagrass blades deflect in the presence of flow, we find thatC_Dis O(1) when flows are weak and tends toward a value of 0.03 at the highest velocities, behavior that is consistent with existing theory for canopy flows with flexible canopy elements. A limited subset of velocity profiles obey the law of the wall, producing values of shear velocity that, while noisy, broadly agree with values inferred from the pressure measurements. 

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3. Debris-bed friction during glacier sliding with ice–bed separation

   https://doi.org/10.1017/aog.2019.46  

   Iverson, Neal R.; Helanow, Christian; Zoet, Lucas K. (December 2019, Annals of Glaciology)

   Abstract Theory and experiments indicate that ice–bed separation during glacier slip over 2-D hard beds causes basal shear stress to reach a maximum at a particular slip velocity and decrease at higher velocities. We use the sliding theory of Lliboutry (1968) to explore how friction between debris particles in sliding ice and a rock bed affects this relationship between shear stress and slip velocity. Particle–bed contact forces and associated debris friction increase with increasing slip velocity, owing to increased rates of ice convergence with up-glacier facing surfaces. However, debris friction on diminished areas of the bed counteracts this effect as cavities grow. Thus, friction from debris alone increases only slightly with slip velocity, and for sediment particles larger than ~60 mm in diameter, debris friction peaks and decreases with increasing slip velocity. The effect on the sliding relationship is to steepen its rising limb and shift its shear stress peak to a slightly higher velocity. These results, which exclude the effect of debris friction on cavity size and debris concentrations above ~15%, indicate that the effect of debris in ice is to increase basal shear stress but not significantly change the form of the sliding relationship. 

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4. Laboratory study of bed shear stress in gradually varied flow over a sudden change in bed roughness

   https://doi.org/10.1007/s10652-025-10024-6  

   Jamil, Muhammad Farrukh; Ting, Francis_C K; Kafle, Monika (April 2025, Environmental Fluid Mechanics)

   Abstract The evolution of bed shear stress in open-channel flow due to a sudden change in bed roughness was investigated experimentally for rough-to-smooth (RTS) and smooth-to-rough (STR) transitions. The velocity field was measured in the longitudinal-vertical plane from upstream to downstream using a Particle Image Velocimetry system. The bed shear stress was determined from the measured velocity profile and water depth using various methods. It was found that the variation of bed shear stress in gradually varied flow through a roughness transition was influenced by both flow depth and bottom roughness. In both RTS and STR transitions, the bed shear stress adjusted to the new bed condition almost immediately even though the velocity profile away from the bed was still evolving, but unlike external and close-conduit flows the bed shear stress in open-channel flows continued to evolve until the flow depth was uniform. It is shown that the evolution of bed shear stress in a STR transition is dependent on the choice of the displacement height on the rough bed, which affects the mixing length used to derive the logarithmic velocity profile and equivalent roughness. Bed shear stress variation consistent with published data was obtained when the$${k}_{s}/{d}_{90}$$ $k_s/d_{90}$ratio was determined as a function of the$$h/{d}_{90}$$ $h/d_{90}$ratio, where$${k}_{s}$$ $k_s$is the equivalent roughness height,$$h$$ $h$is the flow depth, and$${d}_{90}$$ $d_{90}$is the grain diameter with 90% of finer particles. 

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5. Vertical velocity of a small sphere in a sheared granular bed

   https://doi.org/10.1103/PhysRevResearch.6.L022015  

   Gao, Song; Ottino, Julio M; Lueptow, Richard M; Umbanhowar, Paul B (April 2024, Physical Review Research)

   Small particles fall through sheared beds of larger particles in settings ranging from geophysics to industry, but the study of large-to-small size ratios $R$, spanning the trapping threshold $R_t,$has been neglected. In simulations of noncohesive spheres for $R<R_t,$the small-sphere vertical velocity $v_p$first increases with shear rate $\dot{\gamma }$as trapping time decreases, but $v_p$then decreases as velocity fluctuations frustrate downward mobility. For $R>R_t, v_p$is constant at low $\dot{\gamma },$but again decreases at high $\dot{\gamma }$. We model these behaviors and discuss analogies with electron transport in solids. Published by the American Physical Society2024 

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