Non-unique solutions for flow depth and velocity estimations in free-surface annular-channel for single-phase viscous flows

Abstract

Flow depth and mean velocity estimations for partially filled annular ducts are performed using a one-dimensional free-surface flow formulation. The formulation considers a Newtonian fluid flowing in a closed channel with a circular annular cross-section, in steady-state and uniform regimes. The Darcy–Weisbach friction factor, combined with the Churchill equation and hydraulic-diameter approach, is used. Expressions for the flow cross-section area and the wetted perimeter of the partially full annular channel flow, both terms of hydraulic-diameter definition, are derived from trigonometric rules as a function of the flow depth. In contrast to partially filled pipe flow where a bi-univocal and monotonic trend solution between flow depth and flow rate occurs, for annular-channel, multiple values for the flow depth satisfy the formulation for a single flow rate value. Multiple solutions existence is accounted for the abrupt changes in the flow geometry, mainly related to the wetted perimeter, when (i) the flow height enters in contact with the internal duct, (ii) the internal duct becomes completely submerged, and (iii) the flow completely fills the duct cross-section. Calculations for a broad range of geometries and flow conditions (inclination angles: 15°–85°, viscosity: 0.001–0.576 Pa s., and Reynolds numbers: 101–8×105) show errors up to 47% can be reached for flow velocity and flow depth estimations. Specifically, for turbulent flows, discrepancies are lower, being less than 10%. Thus, the higher the inclination angle and viscosity, the more significant errors. The paper outlines the implication of the flow depth and velocity estimations using a well-accepted formulation that impacts on industrial systems’ design and operation stages. We draw attention to the existence of non-unique solutions for free-surface flows in annulus and the need for experiments to elucidate which of these solutions is physically possible and consequently improve predictions for the flow depth and mean velocity in this kind of flow.