Scale Effects in Free-Flow Nonlinear Weir Head-Discharge Relationships

Using similitude relationships, laboratory-scale models have been used for decades to predict and confirm prototype hydraulic structure performance. For free-flow weir discharge conditions, gravity and inertia typically represent the dominant forces, and the Froude number scales the head-discharge performance between model and prototype. Under low-head flow conditions, other forces (e.g., viscous and surface-tension forces) can become relevant, resulting in differences between the model and prototype performance. In this study, the head-discharge relationships for 15 different nonlinear weirs (labyrinth and piano key) with prototype-to-model length ratios of 2, 3, 6, and 12 (based upon Froude modeling) were evaluated, with the largest weirs ($\sim 1\mathrm{m}$ tall) serving as prototypes. This study found differences in the head-discharge performance between prototype and model that exceeded what could be explained solely by model and measurement effects, confirming the presence of scale effects. The range of small upstream heads influenced by scale effects, in general, increased with decreasing model size. The minimum dimensionless head above which scale effects were negligible increased with decreasing model size. As such, model size and geometric scale appear relevant because no single minimum upstream head limit was found that characterized a scale effects limit for all nonlinear weirs tested herein.