The flow structure around limited-size vegetation patches is crucial for understanding sediment transport and vegetation succession trends. While the influence of vegetation density has been extensively explored, the impact of the relative diameter of vegetation stems remains relatively unclear. After validating the reliability of the numerical model with experimental data, this study conducted 2D-URANS simulations (SST k-ω) to investigate the impact of varying relative diameters d/D under different vegetation densities λ on the hydrodynamic characteristics and drag force of vegetation patches. The results show that increasing d/D and decreasing λ are equivalent, both contributing to increased spacing between cylinder elements, allowing for the formation of element-scale Kármán vortices. Compared to vegetation density λ, the non-dimensional frontal area aD is a better predictor for the presence of array-scale Kármán vortex streets. Within the parameter range covered in this study, array-scale Kármán vortex streets appear when aD ≥ 1.4, which will significantly alter sediment transport patterns. For the same vegetation density, increasing the relative diameter d/D leads to a decrease in the array drag coefficient ${\overline{C}}_D$ and an increase in the average element drag coefficient ${\overline{C}}_d$. When parameterizing vegetation resistance using aD, all data points collapse onto a single curve, following the relationships ${\overline{C}}_D=0.34\ln \left(aD\right)+0.78$ and ${\overline{C}}_d=-0.42\ln \left(aD\right)+0.82$.

Large-eddy simulation (LES) is utilized to elucidate the flow characteristics and overall time-averaged drag coefficients of finite-length tandem cylinders. This study focuses explicitly on the three-dimensional effects induced by the free end, a feature absent in classical studies of infinite (two-dimensional) tandem cylinders. By varying the cylinder spacing ratio L/D from 1.5 to 5, the evolution of wake regimes and their variations along the vertical direction of the cylinders are systematically examined. The results reveal a distinct vertical transition of wake patterns: at the mid-height plane, the wake falls into the extended-body regime for L/D = 1.5 and 2, where vortex shedding occurs downstream of the downstream cylinder. When L/D = 3–5, the flow enters the reattachment regime, characterized by the separated shear layers from the upstream cylinder reattaching onto the windward face of the downstream cylinder, while a Kármán vortex street persists in its wake. In contrast, at planes near the free end, the flow characteristics shift towards the co-shedding regime for L/D ≥ 2, though strong downwash suppresses organized vortex shedding. This vertical transition of wake regimes, driven by free-end downwash, clarifies a significant gap in applying two-dimensional regime classifications to finite-length bodies. The overall time-averaged drag coefficients of the upstream and downstream cylinders show opposite trends with increasing L/D: the former decreases, whereas the latter increases. The force on the downstream cylinder changes from an upstream-directed drag to a downstream-directed thrust at L/D = 2. Overall, the results indicate that for L/D = 3–5, the overall drag coefficient of the cylinder is dominated by the co-shedding regime. These findings advance the understanding of flow interference in finite-length tandem configurations and offer refined insights for modeling analogous systems such as adjacent vegetation stems in aquatic environments.

High-speed water flow conditions can cause erosion of the bedrock in engineering areas. Due to the lack of accurate evaluation of bedrock scour and erosion rates, there has been a consumption of manpower and resources without achieving satisfactory engineering outcomes. Therefore, studying the scouring and erosion effects of water flow on bedrock is of significant importance for maintaining the sustainable development and safety of engineering projects. Using the bedrock prototype from the Xiaonanhai site in the upper reaches of the Yangtze River, a model test device was developed to conduct anti-scour tests on the bedrock. The study quantitatively examined the basic physical properties, incipient erosion velocity, and erosion rates of different types of bedrock. The study found that the prototype bedrock under natural exposure, submerged immersion, and alternating wet and dry conditions showed a trend of decreased tensile strength, with the alternating wet and dry conditions being the most detrimental to maintaining the physical properties of the rock mass. The anti-scour velocity of silty claystone and clayey siltstone samples increased with the increase in tensile strength, and the erosion rate increased with the increase in shear stress. If the shear stress is kept constant, the erosion rate decreases with the increase in tensile strength. The erosion rate is inversely proportional to the ratio of the bedrock’s tensile strength to the riverbed shear stress, with the fitting relationship showing a piecewise linear distribution. The research results can provide guidance for the safe production of engineering involving bedrock erosion in engineering reservoir areas that are conducive to sustainable development.