Transient flows and quasi-dynamic hydraulic coefficients at unpressurized circular pipe confluences

In one-dimensional (1D) transient simulations of unpressurized drainage networks, confluence junctions are commonly represented using static hydraulic coefficients, which cannot adequately capture their dynamic hydrodynamic behavior. To address this limitation, three-dimensional (3D) simulations validated against experiments were conducted to investigate the flow physics and hydraulic characteristics of circular pipe confluences. The results demonstrate that, beyond the junction angle θ and the discharge ratio q*, the filling ratio y* plays an important role in governing the flow structure. As y* increases, it non-linearly modifies the cross-sectional aspect ratio and increases the submergence of the branch inflow, driving a transition from smooth merging to vigorous rolling with intensified turbulent kinetic energy dissipation. Based on these flow physics insights, a dynamic 1D control volume confluence branch junction model was developed, where the momentum correction and localized head-loss coefficients vary with the transient flow. To complete the model, a database of 150 3D simulations was established, and the control volume boundaries were identified at 0.2, 0.9, and 1.4 pipe diameters upstream, branch, and downstream sections, respectively. Formulas for the quasi-dynamic hydraulic coefficients were then derived as functions of θ, q*, and y*. By incorporating these formulas, the 1D model demonstrated high accuracy for transient flow simulation: reducing the maximum dynamic time warping water-depth alignment error by 68.69% and the peak water-depth relative error from 5% to 0.1% in their respective cases. These results demonstrate that incorporating dynamic junction characteristics enables the proposed 1D model to more accurately represent confluence hydrodynamics in unpressurized circular pipe systems.