Mixed Flows with Depressurizing Wavefront in Circular Pipe

Rapid pipe filling or emptying of water mains and sewer systems is accompanied by transitions between free surface and pressurized flow regimes and subatmospheric unsteady flows, leading to the propagation of pressurized/depressurized wavefronts along the pipe. The occurrence of these phenomena is often more common when the flow becomes transient. The modeling of such flows is complex, and it requires the inclusion of several types of propagating wavefront to better figure out the effective transient phenomena and to apply the appropriate equations. When simulating mixed flows in closed conduits such as tunnels, sewers, and water mains with shock fitting models, the wavefront is determined by continuity, momentum, and energy equations. To better manage the application of these equations, experimental investigation was conducted on different depressurizing wavefront shapes. The behaviors, velocities, and effects of included air pockets are analyzed according to a series of pipe slopes and reservoir water levels. The flow behavior presents particular differences according to the pipe slope and the pipe upstream end status. When the upstream end of a pressurized pipe is closed, water remains immobile in the full sectional area at any location along the pipe length. Flow occurs in the pressurized zone only after the passage of the depressurized wavefront (DWF). Results show that the transition from pressurized flow to free-surface flow is accompanied by a negative wavefront illustrated by three physical shapes. First, the wavefront takes a pronounced concave shape followed by a 0.8D flow depth zone for low pipe slope values (less than 1.0%). Second, as the pipe slope value becomes larger (1.8–5.5%), the wavefront head becomes sharper. A third and more complex shape, which is an intermediate between the previous two, is observed for pipe slope values of around 1.0–1.8%. Comparison between experimental and numerical results also shows some limitations in the simplified form of actual numerical solutions of the continuity, energy, and momentum equations due to the negation of variables such as energy loss and forces due to friction, gravity, viscosity, and surface tension. Balancing these equations indicates that only the continuity equation applied around the propagating wavefront matches well with the experimental results. The difference between measured and calculated values by the used energy and momentum equations can reach 10.4% of the diameter and 18.4% of the hydrostatic force in pressurized zones, respectively. Finally, the results show that the combination of the continuity and the energy equations gives the best findings, but with an overestimation of 14.4%. Incorporating the continuity and momentum equations, the wavefront celerity is overestimated by as much as 24.3%. The combination of the momentum and the energy equations presents the worst findings, with a celerity value overestimated at by much as 69.4% and a water velocity value in the free surface underestimated by as little as $-8.6\%$. These results show the limits in applying the simplified form of the momentum and energy equations when wavefronts are examined. Taking into account parameters such as surface tension can balance the equations to obtain better results. Also, the obtained analytical forms may be applied easily by engineers who need simple formulas to estimate hydraulic parameters instead of using complex modeling.