Connecting Diverse Hydraulic Regime Transitions with the p-Norm Function

Hydraulic processes are frequently characterized by transitions between distinct physical regimes, such as viscous-dominated to inertial-dominated turbulent flow or small-scale to large-scale roughness effects. Traditional models for these transitions often rely on piecewise formulas or complex implicit equations. This note demonstrates that the p-norm function, expressed as Fp=F1p+F2p (where F denotes the target variable, F1 and F2 represent the limiting values for each regime, and p is a case-specific exponent), serves as a robust and unifying framework for such transitions, resolving the fragmentation of traditional piecewise or implicit models. Through analysis of five examples of classical hydraulic phenomena, grain settling velocity, pipe friction factor, velocity distribution in open channel flows, flow resistance with large-scale roughness, and bed-material sediment discharge, it is shown that the p-norm seamlessly connects disparate regimes, while retaining explicit form and physical significance. This framework offers superior simplicity and an explicit analytical form compared to conventional models. The consistency of this approach across diverse phenomena highlights a fundamental mathematical structure underlying regime transition in hydraulics and provides a unified methodology for their prediction.