A Favre-Averaging Shallow Water Framework for Aerated Flows with Friction Factor Decomposition

Accurate prediction of flow resistance in high-Froude-number aerated flows remains challenging due to air entrainment, which causes strong spatial variability in mixture density. Here, we introduce for the first time a density-weighted (Favre) averaging approach within a Shallow Water Equation framework specifically tailored to account for this strong mixture density variability. Within this framework, we present a novel Darcy-Weisbach friction factor formulation that decomposes contributions associated with uniform flow, spatially varying flow, and temporally evolving flow, and incorporates momentum and pressure correction factors reflecting the vertical structure of the mixture. Application to experimental data demonstrates that spatial flow development systematically reduces the effective friction factor relative to the uniform-flow estimate, and that momentum-based and energy-based formulations yield nearly identical results. The framework recovers classical uniform-flow predictions in the quasi-uniform downstream region and reduces to standard single-phase formulations in the absence of aeration. Overall, it provides a physically consistent tool for resistance prediction in high-Froude-number spillways, chutes, and open-channel systems, with a structure compatible with depth-averaged numerical solvers.