Experimental physical model studies of hydraulic structures are often conducted to replicate complex flow patterns and intricate transport situations that may occur at the prototype scale. Froude scaling is most often used for open channel model studies (e.g., weir flow) as they are gravity driven and constancy of gravity to inertial forces between model and prototype is met, while other fluid forces (e.g., viscosity, surface tension, elastic) are assumed negligible. As the prototype-to-model characteristic length ratio (i.e., length-scale ratio) increases or the total head decreases, forces such as viscous and surface tension forces at the model scale may influence flow behavior and result in discrepancies between the model and prototype, a phenomenon referred to as size-scale effects. That is, when size-scale effects are present the hydraulic behaviors of the model do not accurately represent the behaviors of the geometrically similar prototype structure. The purpose of this research was to increase understanding of size-scale effects with respect to nonlinear weirs, specifically labyrinth weirs. Laboratory tests were conducted with trapezoidal, single-cycle, 15° sidewall angle labyrinth weir models with two different crest shapes (half- and quarter-round) at five different length-scale ratios. The largest model, which featured a weir height of 3 ft (about 0.3 m) and a cycle width of 8 ft (about 2.4 m), served as the prototype for comparative purposes. The smaller scale models featured length-scale ratios of 2, 3, 6, and 12. Head-discharge data and nappe behavior for vented and non-vented conditions were collected for each model, for dimensionless total head ratios (i.e., total head normalized by the weir height) ranging from 0.01 to 1.00. However, available flow rates limited data collection up to a dimensionless total head ratio of 0.35 at the prototype scale. Discharge coefficients were calculated to investigate size-scale effects in conjunction with an uncertainty analysis to quantify the confidence levels of calculated discharge coefficients. In this study, size-scale effects were determined to be negligible for dimensionless total head ratios greater than ～0.30 for half-round crest shapes and dimensionless total head ratios greater than ～0.35 for quarter-round crest shapes. Below these limits, size-scale effects influenced nappe behavior and the discharge coefficient. For half-round crest shapes, nappe self-aeration began near a dimensionless total head ratio of 0.05 at the prototype scale whereas for the smallest scale model (length-scale ratio = 12), nappe self-aeration began near 0.30. Similarly, for quarter-round crest shapes, nappe self-aeration began near a dimensionless total head ratio of 0.02 at the prototype scale whereas for the smallest scale model, nappe self-aeration began near 0.25. Discharge coefficients were under-estimated by as much as 70% and over-estimated up to 4% of the prototype data for half-round crest shapes. For quarter-round crest shapes, discharge coefficients were under-estimated by as much as 87% of the prototype data while no over-estimation occurred. The most error for both crest shapes occurred at very small total heads. Furthermore, the low-head performance predictive errors increased with decreasing model size (i.e., increasing length-scale ratio). The following limiting total heads are recommended based on the results of this research to avoid size-scale effects: 0.008-0.016 m for half-round crest shapes and 0.007-0.010 m for quarter-round crest shapes. However, the limiting total head is dependent on model size and is based upon an allowable error of the discharge coefficient of ±5%. If additional error can be tolerated in predicting the prototype head-discharge relationship then limiting total heads may be less. Additionally, the application of these limiting heads to other nonlinear weir configurations with half- or quarter-round crest shapes should be conducted with engineering judgement as weir geometry affects flow behavior.
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