SUPPLEMENTAL MATERIALS

The types of PKW based on crest overhang and associated geometric and hydraulic characteristics are summarized with full specifications are given below.Type-A Piano Key Weir, the upstream and downstream overhangs are kept equal to maintain a constant crest length and identical nappe development conditions. This allows the study to isolate the effect of internal key geometry on flow separation and energy conversion, which governs discharge efficiency in CFD analysis. In Type-B, only the outlet key overhang is present, which is kept constant in the model to maintain a uniform nappe detachment condition at the exit. This allows the study to isolate the effect of inlet-to-outlet flow contraction, shear development, and momentum amplification, which control discharge efficiency in CFD simulations. In Type-C PKW, only the inlet key overhang is provided and retained unchanged to ensure consistent upstream pressure distribution and boundary-layer initiation. This enables assessment of the influence of flow entry alignment, separation suppression, and energy preservation within the key system, which dictate hydraulic efficiency. For Type-D PKW, no overhangs exist on either side, eliminating geometric extension of the crest and preventing nappe aeration enhancement from overhang effects. Therefore, the analysis focuses purely on core-flow acceleration, pressure build-up, and vortex dissipation inside the keys, which govern hydraulic performance under submerged flow regimes.