COUPLED AIR-WATER UNIFORM-FLOW MODEL FOR PARTIALLY FILLED CIRCULAR CONDUIT

Abstract

Partially filled circular conduits convey water beneath a confined air headspace in many hydraulic systems, including sewers, stormwater networks, and drainage pipes. While the discharge- flow depth relationship of single-phase flow in circular conduits is well established, the role of air motion and air-induced pressure gradients in modifying steady uniform-flow resistance remains poorly understood. Existing air-water flow studies in circular pipes primarily focus on pressurized two-phase transport with imposed phase flow rates, or on transient phenomena such as air pocket compression during rapid filling, leaving a gap in the description of steady, gravity-driven, partially filled flow where air motion is induced rather than prescribed. In such configurations, interfacial shear and headspace confinement can generate longitudinal air pressure gradients that feedback on the water momentum balance and potentially alter the hydraulic capacity.

This study develops a mechanistic, steady uniform-flow model for air-water coupling in partially filled circular conduits. Separate one-dimensional momentum balances are formulated for the water and air phases, explicitly accounting for wall friction, air-water interfacial shear, and the pressure gradient required to sustain air motion in a confined headspace. By eliminating the air pressure gradient through the air momentum equation and substituting it into the water balance, a fully coupled uniform-flow equation is derived. The formulation retains exact circular geometry and naturally reorganizes into a generalized hydraulic resistance kernel expressed in discharge form. This kernel extends the classical Darcy-Weisbach framework by incorporating interfacial momentum exchange and air-pressure-mediated feedback, without relying on empirical correction factors. An air-activation closure and an air-induction parameter are introduced to represent the onset and intensity of air motion in a physically consistent manner.

The resulting model yields an explicit, dimensionless correction function that quantifies the relative contribution of air dynamics to hydraulic resistance as a function of filling rate. Analytical and numerical results demonstrate that air effects are generally weak but finite, localized near high filling rates, strongly dependent on headspace confinement, and vanish smoothly as the conduit approaches full flow. Depending on the intensity of air induction, the air-water interaction function may exhibit a clear maximum or decay monotonically, illustrating how geometry and induced air motion jointly govern coupling behavior.

The novelty of this work lies in providing the first geometry-explicit, steady uniform-flow formulation that rigorously couples air and water momentum balances in a partially filled circular conduit and expresses the result as a generalized hydraulic resistance kernel in discharge form. Unlike existing two-phase pipe-flow models or open-channel formulations, the present approach captures air-pressure feedback induced by confinement under gravity-driven conditions and clarifies how a low-density air phase can influence hydraulic resistance near closure. This framework bridges the gap between classical open-channel hydraulics and two-phase pipeline theory and offers a new mechanistic basis for assessing air effects in partially filled circular conduits.