Equivalent Roughness Height for Nondensely Distributed Submerged Obstacles in Open Channel Flows

Abstract The estimation of equivalent sand-grain roughness height ( k s ) for open channel flows is well-established for densely packed roughness but remains ambiguous for nondensely distributed submerged obstacles. This study presents a practical model to predict k s for such configurations. The flow physics around these obstacles is governed by complex separation phenomena and vortex dynamics; however, direct simulation of these details is impractical for routine engineering prediction. Instead, we adopt a pragmatic bulk-resistance approach: a validated friction law, originally derived for densely packed gravel beds, is extended to characterize total resistance in flows over nondensely distributed obstacles. Starting from this extended friction law, we inversely derive k s from experimental data encompassing three representative roughness elements, namely, cylindrical rods, cubes, and triangular prisms, which were selected to span slender vegetation models and engineered stocky bodies with distinct geometric and flow-separation behaviors. Analysis revealed that k s is proportional to obstacle height ( k ) and dependent on the frontal roughness density ( λ f ), yielding a generalized empirical scaling relation: k s = 4 λ f 1 / 3 k . This formula integrates resistance characterization across all types, providing a practical tool for predicting resistance in flows with nondense roughness. The model’s success suggests that the derived k s effectively characterizes the net hydraulic effect of the underlying complexities, a link that merits future study.