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Experimental and Numerical Investigation on the Motion of Discrete Air
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Pockets in Pressurized Water Flows
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Carmen D. Chosie1 Thomas M. Hatcher2 Jose G. Vasconcelos3 , A.M. ASCE
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ABSTRACT
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Entrapped air pockets are linked to various operational issues in closed pipe flows. Tracking of entrapped
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air pockets is a relevant task in numerical simulation of hydraulic systems subjected to air pocket entrapment.
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However, few studies focused on the kinematics of discrete air pockets and the motion of these at various
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combinations of pipe flow velocities, air pocket volumes and pipe slopes. Such information is relevant for
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the improvement of current numerical models for the simulation of transition between pressurized and free
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surface flows, and related air pocket entrapment. This work explores the link between pipe flow velocity, pipe
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slope, and entrapped air pocket celerities. Systematic experimental studies were performed and provided
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insights on the nature of entrapped air pocket motion in cases when buoyancy forces are either summed or
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opposed to flow drag. Results indicated that when flow drag and buoyancy forces are summed the observed
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air pocket celerity can be approximated by the celerity in quiescent conditions plus the pipe velocity. This
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approximation fails in cases when drag force opposed buoyancy.This work also proposes a novel approach
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for the simulation of the air pocket motion as non-Boussinesq gravity currents. Modeling results compared
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favorably with experimental measurements in horizontal slope and pipe flow velocity cases. Such results
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are an indication of the feasibility of integrating this approach into more complex hydraulic models able to
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simulate flow regime transitions.
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Keywords: Closed pipe flows; Air pockets; Laboratory experiments; Numerical modeling
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INTRODUCTION
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A key problem for closed conduit flows in which two-phase flows may occur is the limited information
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on the kinematics of entrapped air pockets. The ability of describing the motion of entrapped pockets is
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important as it can indicate the likelihood of uncontrolled air releases through ventilation shafts in stormwater
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systems. Such information could also help assess the effectiveness of air pocket removal in transmission 1 Graduate student, Dept. of Civil Engrg., Auburn University, 238 Harbert Engineering Center, Auburn, AL, 36849. E-mail:
[email protected] 2 Ph.D. candidate, Dept. of Civil Engrg., Auburn University, 238 Harbert Engineering Center, Auburn, AL, 36849. E-mail:
[email protected] 3 A.M. ASCE, Assistant Professor, Dept. of Civil Engrg., Auburn University, 238 Harbert Engineering Center, Auburn, AL, 36849. E-mail:
[email protected]
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mains. Finally, the ability to track entrapped air pockets is relevant for the simulation of flow regime
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transition between free surface and pressurized flows, with the goal of detecting a priori deleterious air-
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water interactions. Past investigations dealing with topics related to the motion of air pockets in closed
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conduit flows yielded important insights in this process. Yet, as is discussed, some knowledge gaps remained
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both in terms of experimental and numerical approaches on how to predict this motion.
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Most water conveyance systems based on closed conduits are not designed to operate in two-phase flows,
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yet such undesirable flows have been observed in various transmission pipelines as well in in stormwater sys-
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tems. Various mechanisms for large air pocket entrapment have been identified to date, and a comprehensive
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summary is presented in Lauchlan et al (2005) in the context of transmission mains. Other mechanisms have
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been identified in the context of the rapid filling of stormwater systems. The pioneer study by Hamam
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and McCorquodale (1982) indicated that interface instability caused by the relative motion between air and
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water can lead to surface waves in water that can grow and eventually promote air pocket entrapment. Other
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mechanisms for air pocket entrapment in the context of stormwater systems have been studied experimentally
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by Vasconcelos and Wright (2006).
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The presence of air pockets in closed conduit water flows is generally undesirable because it can cause
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various operational problems. One of the clearest adverse impacts is the increased potential for structural
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damage, caused by the great compressibility of air that leads to increased surging. The pioneer study by
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Martin (1976), which was complemented by Zhou et al (2002) among others, indicated that compressed air
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pressure can greatly exceed the pressure that caused the flow prior to air compression. Zhou et al (2002)
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points out that such compression may be behind an episode in which an entire manhole structure was blown
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off the pipe. Issues related to potential structural damage in rapid filling of closed conduits have also been
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explored by Song et al (1983) and Guo and Song (1991) in the context of pressurization of large stormwater
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tunnels. A more recent work by Vasconcelos and Leite (2012) confirmed these earlier findings, but also
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showed that when relief is provided next to the location where air is compressed, as would be anticipated in
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some real applications, the pressure rise can be significantly mitigated.
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Another adverse impact related to air pocket entrapment is loss of conveyance and storage capacity
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in closed conduits. As shown in Falvey (1980), entrapped air pockets act as flow constrictions generating
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additional turbulence and hence energy losses. Also, large entrapped air pockets will occupy a volume in
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closed conduits that would otherwise be filled with water. This could be a relevant issue in the context of
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stormwater storage tunnels since early pressurization of these systems caused by large entrapped air pockets
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have the potential of increasing frequency of combined sewer overflows (CSO) episodes.
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Release of entrapped air pockets can be a problematic issue as well. In the context of water transmission
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mains one concern is the possibility of air slamming, defined as the waterhammer-type pressures observed
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at the moment when air is completely evacuated through air valves. The study by Lingireddy et al (2004)
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presents a formulation to calculate such peaks in terms of the diameter of the pipeline, the diameter of the
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air valve and the air pocket pressure. In context of large collection sewers and tunnels, release of large air
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pockets through water-filled ventilation shafts have been linked to geysering phenomena. This phenomenon
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was observed in physical modeling studies by Vasconcelos and Wright (2011), as well as field observations of
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large trunk sewers Wright et al (2011) and in penstocks Nielsen and Davis (2009). An interesting aspect in
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Wright et al (2011) studies is that while water may be ejected violently above ground, the hydraulic grade
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line was far below the ground surface elevation.
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Another context where very important contributions in this field originates are investigations on the
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motion of air cavities as gravity currents. In such studies water typically discharges in one end of an initially
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filled pipe. An exchange flow condition is established at the discharge point as an air cavity is admitted
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at the top of the pipe and moves toward the opposite end of the pipe. The work by Benjamin (1968) was √ one of the pioneers in establishing that the celerity of the air cavity leading edge scales with gH, with
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g the gravitational acceleration and H the pipe characteristic dimension (e.g. diameter, D). It was also
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determined that the gravity current celerity generally decreased with its thickness for a given pipe cross-
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sectional dimension. The study from Wilkinson (1982) expanded the work by Benjamin by incorporating
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surface tension effects, which have an effect on the location of the stagnation point located at the leading
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edge of the air-water interface at the pipe crown. Baines (1991) work followed these studies and studied an
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interesting air gulping flow feature observed when the pipe was set in a downward slope, in which the outlet
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was at a lower elevation. This work showed that the observed celerity increased with slope (tested up to 8 √ degrees) and that the discrete air pocket celerity formed after gulping also scaled with gH. A limitation
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to these studies was that there was no pipe flow prior to the arrival of the air cavity. A related work by
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Vasconcelos and Wright (2008) presented the advance of an air cavity against an accelerating pressurized
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flow column and indicated that the increasing water flow velocity could eventually reverse the advance of the
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cavity. At this point in time, however, the authors determined that the air-water shape is no longer strongly
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curved but was strongly affected by the brink depth discharge conditions at the far-downstream end.
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The “hydraulic clearing” of transmission mains presents the problem of air pocket motion in a different
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prism with the very practical objective of removing air from mains. Because of the potential issues in closed
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conduits, various studies have been presented to date attempting to determine a minimum value for the
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pipe flow velocity that would ensure that drag is sufficiently large enough to overcome anticipated air pocket
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buoyancy forces and clear pipelines of pockets. Works of Falvey (1980), Little et al (2008), and Pozos et
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al (2010) provide examples of formulas to compute the velocity magnitude, which depends on pipe angle √ and gD. While very useful, these studies have not focused on predicting the displacement and celerity of
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discrete air pockets for various combinations of pipe flow velocities and pipe slopes.
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Studies performed by Pothof and Clemens (2010) focused on the motion of elongated air pockets in
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favorable slope pipelines. These studies were useful in providing insights as to the controlling aspects of the
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pocket motion, with potential application in mitigating block back occurrences. A separate work presented
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by Pothof and Clemens (2011) focused on the the co-current flow of air and water in downward sloping pipes.
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These studies determined the water discharge rate required to prevent air accumulation as a function of air
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flow rate, pipeline diameter, and pipeline slope. These experimental results pointed various dimensionless
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parameters that are useful in characterizing air-water flow regimes. Such studies, however, have not focused
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in predicting the displacement and celerity of discrete air pockets for various combinations of pipe flows
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velocities and slopes, including cases when buoyancy and drag forces are not in opposition.
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Glauser and Wickenhauser (2009) provided a relevant experimental study on the dynamics of an air
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cavity advancing in a pressurized pipe. The focus of that work was on the shape and movement of single air
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bubbles in continuous air-water flow with the intention to determine the bubble volume that represents the
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stagnation velocity based on a balance of buoyancy and drag forces. It was noted that in favorably sloped
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pipes, volumes larger than this critical volume move against this flow due to buoyancy and volumes smaller
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than this critical volume would be dragged by the flow against buoyancy. Similar to studies in hydraulic
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clearing of water mains, it was determined that pipeline slope, background water velocity, and bubble volume
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controlled the observed pocket celerity. This investigation involved the use of slopes ranging from 1.7% to
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8.7%, so effects such as air pocket spreading observed in horizontal slopes could not be evaluated.
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Numerical studies have been presented that attempt to describe the motion of entrapped air pockets in
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closed conduit flow. The majority of these studies are in the realm of multi-phase flow applications, and
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exemplified by the works of Barnea and Taitel (1993); DeHenau and Raithby (1995) and Issa and Kempf
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(2003). In these applications there is a continuous injection of gas and liquid in closed conduits, and the
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mechanisms for pocket formation are diverse (e.g. terrain induced slugging) from the ones that are more
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frequently observed in water, wastewater and stormwater systems. Two studies were to numerically describe
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the motion of entrapped air pockets in water systems. The study by Li and McCorquodale (1999) presents
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the mathematical framework for the simulation of flow regime transition (also referred to as mixed flows)
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using lumped inertia analysis and the ideal gas law. This works also considered the mechanism for air
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pocket formation based on shear flow instabilities presented in Hamam and McCorquodale (1982). In the
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formulation the location of the air-water interface is calculated explicitly, providing a means to compute the
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advance of an entrapped pocket. Yet, the velocity of the air bubble, rather than being calculated by the
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model, appears as a calibration parameter in the simulations.
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More recently Trindade and Vasconcelos (2013) presented a study that included a numerical framework
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to simulate pipeline priming operations considering the possibility of air pressurization using the Two-
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component pressure approach following the work by Vasconcelos and Wright (2009). With regards to air
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pressurization, this work included the possibility of air pressure variation along the length of the air pocket,
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which was computed using the isothermal Euler equations. The model updates the air-water interface using
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a source term for the Euler equations presented in Toro (2009). The trajectory of air-water interface was
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accurately predicted including complex flow interactions such as interface breakdown of the pressurization
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interface. A model limitation was that the only mechanism that accounted for the motion of the air-water
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interface was the displacement of the air phase caused by the water inflow during the priming event. Also,
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it was assumed that one of the boundaries of the air pocket was fixed at the air release valve. An important
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difficulty in the simulation of entrapped pocket motion is that such one-dimensional models are generally
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constructed with the hypothesis of hydrostatic pressure distribution at the flow cross-sections, which will
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not hold at strongly curved air-water interfaces.
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Understanding of air-water interactions is key to a more accurate description of unsteady closed conduit
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flows. Previous studies have provided significant insights for air-water interactions in closed conduit flows,
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yet significant knowledge gaps remain with respect to air pocket kinematics. These gaps exist both in terms
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of tested experimental conditions and in terms of numerical modeling approaches to describe motion of
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entrapped air pockets. The proposed research attempts to address some of these gaps, as is outlined below
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in the Objectives.
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OBJECTIVES
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This research program has two main objectives. The first is to further explore the link between pipe flow
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velocity, pipeline slope and the celerity of entrapped air pockets of various volumes, including effects of air
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pocket spreading. This work presents results from an experimental program on the kinematics of entrapped
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air pockets in steady pressurized water flows and shallow pipe slopes.
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The second objective is to propose and assess the accuracy of an innovative framework to describe
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the motion of air pockets in horizontal slopes based on gravity current theory. In particular, this proposed
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framework intends to provide a simpler method to model entrapped air pocket motion that can be embedded
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in unsteady flow models in the future. Numerical model and experimental results are compared and discussed
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along with conclusions and recommendations for future work.
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METHODOLOGY
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Experimental methods
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Figure 1 presents a schematic of the apparatus used in the experimental procedure. These experiments
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were performed with a variable slope, clear PVC pipeline supplied upstream by a fixed head reservoir
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with a throttled discharge downstream to ensure pressurized flow throughout the pipeline. The pipeline
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has a length L = 11.1 m and a diameter D = 101.6 mm. The upstream supply reservoir is attached to
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the pipeline by a 50 mm diameter ball valve. Two knife gate valves with the same diameter as that of
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the pipeline were positioned 3 m downstream of the upstream supply reservoir with a 2.1 m separation for
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adverse and horizontal sloped conditions. For positive sloped experiments, the intermediate knife gate valves
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were relocated to a position 3 m upstream of the downstream end of the pipe. Pre-determined volumes of
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air at atmospheric pressure were injected into the region between the two knife gate valves forming an air
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pocket. These partially closed knife gate valves prevented the motion of the air pockets while still allowing
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flow underneath.
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Air pockets of different volumes (V ol) were injected via a graduated acrylic tank located at the bottom
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of the clear PVC pipeline, which is separated from the water flow by a control valve. This supply tank was
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equipped with valves that ensured the air injected into the water filled pipeline was at atmospheric pressure.
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The measurement devices used in the experimental procedure include: • Up to four high definition camcorders (1080 pixels, 30 FPS), providing a total view of 7.0 m upstream and downstream of the intermediate knife gate valves;
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• Two piezoresistive pressure transducers, MEGGIT Endevco 8510B-5, located at the upstream end
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(L1 = Lcoord /L = 0.0) and at approximately 50% of the pipeline length (L1 = 0.5), recording with a
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frequency of 100 Hz for each channel;
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• National Instruments data acquisition board NI-USB 6210 with SignalExpress logging software;
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• Digital manometers located at both ends of the apparatus and at 50% of the pipeline length (L1 = 0.5);
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and • Paddle-wheel flow meter to gage pipe flow rate.
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Conditions that merited testing included those that represented a wide range of air pocket volumes,
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pipeline slopes, and background flow rates that stormwater storage tunnels could encounter during rapid
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filling after an intense rain event.
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The experimental procedure was performed as follows:
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1. Set the selected slope for the pipeline apparatus;
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2. Create steady flow conditions in the PVC pipeline by turning on submersible pumps at desired flow
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rate and opening the intermediate knife gate valves to desired opening percentage; 3. Inject air volume at atmospheric pressure by opening the valve connecting the graduated acrylic cylinder and the water filled PVC pipeline;
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4. Allow system to achieve a new steady state condition due to the increase in energy losses created by the addition of air into the system;
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5. Initiate pressure measurements and collect readings on calibration manometers;
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6. Open the intermediate knife gate valves simultaneously and rapidly (t < 0.6 s) to allow sudden release
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of air pocket; 7. Record motion and spreading of the air pocket(s) with camcorders until moving outside field of view; and 8. Close valves and stop submersible pumps after considerable time and make final readings on the calibration manometers.
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The results presented in this paper include 99 differing conditions. Seven slopes have been tested:
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horizontal slope, 0.5%, 1.0%, and 2.0% adverse slope(downstream end at higher elevation than inlet) and
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0.5%, 1.0%, and 2.0% positive slope (downstream end at lower elevation than inlet). For each of these
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slopes, three different flow rates ranging from 1.4 L/s to 4.0 L/s and up to four air pocket volumes were
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tested. These pocket volumes ranged between 1.3 L and 4.3 L for horizontal slopes and between 0.5 L and
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4.0 L for adverse and positive slopes.
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It was noted visually that a pronounced drag created by larger flow rates caused bits of some air pockets
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to be dragged underneath the knife gate valves intended to keep the pocket in place as a whole until the start
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of the experiment. In order to keep this phenomenon from happening, certain runs with larger air pocket
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volumes and larger flow rates were not able to be performed. Along with the flow rates and air pockets
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volumes mentioned above, experiments were run to study the thinning of the air pocket when a single knife
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gate valve was opened with no background flow.
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Pressure monitoring throughout the system allowed the following objectives to be completed: gain insight
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into the pressure variation in the system during motion of released air pockets, assess amount of energy loss
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caused by the partially closed knife gate valves holding the pocket in its initial position, and measure
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thickness of air pockets in selected cases since there is evidence in literature of the relationship between
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pocket thickness and celerity during motion. Each case was repeated at least once in order to ensure that
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data was consistent. Table 1 presents a summary of the parameters tested in this investigation.
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Numerical modeling
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According to modeling objectives, simulation of gravity current flows can be performed with three primary
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approaches: 1) accurate CFD simulations solving Navier-Stokes equations; 2) approximations based on
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shallow water equation models; and 3) simple integral models. As this work is not focused on air-water
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flow details and interactions but instead aims to track the endpoint coordinates of air pockets, a simple
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integral model was formulated (this approach is outlined in Ungarish (2009)). For this numerical model, a
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relationship between the gravity current thickness and its propagation speed is utilized to update the location
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of the gravity current fronts/leading edges. Integral models assume uniform flow depth at each time step
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while providing the average gravity current depth and celerity. Because of this uniform depth assumption,
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integral models are also referred to as “box models” Ungarish (2009).
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For the experiments presented in this work, the integral model is developed for horizontal (no slope) flow
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conditions. Figure 2 presents a schematic diagram of this modeling approach where x0 and y0 are respectively
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the initial length and thickness (measured from the pipe crown) of the air pocket. After the knife gate valves
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are fully opened, the air pocket spreads in both directions at unique velocities according to the leading edge
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celerity and relative pipe flow velocity. Front conditions relate the air pocket leading edge celerity and its
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thickness and offer a closure to the integral model along with the air phase continuity equation. The front
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conditions used in integral model framework are adapted to account for pipe flow velocities following the
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rationale by Hallworth et al (1998). The resulting front condition is:
uf =
p dxf = F gD + Vf low dt
(1)
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where xf is the distance traveled by the respective air pocket front, F is the local Froude number at the
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upstream and/or downstream air pocket front, D is the pipe diameter and Vf low is the background flow
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velocity.
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Each front condition uses a different formulation to determine the value of F at the leading edge of the
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air pocket. The first front condition applies the theory by Benjamin (1968) in which there is a momentum
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balance solved at the air pocket leading edge. The control volume (CV) encloses the leading edge so that the
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flow is uniform and hydrostatic at the CV boundaries. This formulation considers surface tension following
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Wilkinson (1982). After integrating Equation 1 and solving for xf , the result is the first integral model
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considered in this work (referred to as M 1): √ uf = (F gD + Vf low )(tnew − told ) 1/2 2πDA2 σ 1 2 3 πD A2 − ρgrc − 4A2 A2 cos α + 6 D sin α F = 1 2 2 πD πD − A2 2
2
(2)
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where tnew is the updated time step; told is the previous time step; α = is the pipe half-angle, with 2α as
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the angle subtended from the center of the pipe to the free surface; σ is the air-water surface tension; rc is
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the leading edge radius of curvature; and A2 is the cross-sectional flow area beneath the air pocket equal to
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π−α+
1 2
sin α
D2 4
.
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The second integral model approach (M 2) utilizes the front condition presented in Vasconcelos and
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Wright (2008), which in turn was based on the work by Townson (1991). This front condition accounts
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for the curvature at the air pocket front and has the advantage of accounting for the depth change at
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the hydraulic jump that trails the leading edge of the gravity current. A limitation is that while this
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front condition formulation does not assume uniform air pocket thickness, the integral model makes this
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assumption. This has impacts in the accuracy of model predictions, as is shown in the Results section. The
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expression for the 2nd proposed integral model (M 2) is: √ uf = (F gD + Vf low )(tnew − told ) 1/2 1 3 1− 3 sin α + 3(π − α) cos α − sin α π − α − sin α cos α 3π F = A2 A2 π 2− π π
(3)
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Since there are initially two air intrusion fronts for the cases studied in this work, Equation 2 or 3 are
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solved twice at each time step. With the front velocities known, the locations of each front are updated
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(xf U and xf D ) according to Figure 2. The methodology used to solve each leading edge location is identical
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except that the sign of U changes depending on the orientation of the background flow with respect to the
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front motion. One of the difficulties in this procedure is that xf (xf U or xf D ) is a function of α and vice
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versa. For the implementation utilized in this work, xf was first computed from parameters determined at
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the previous time step. Then, α was updated using the following volume/continuity expression: Lp D2 V ol = 4
1 α − sin 2α 2
(4)
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in which V ol and Lp = x0 + xf U + xf D are the air pocket volume and length, respectively. This nonlinear
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expression was solved at each time step using the iterative Newton-Raphson method with a tolerance for α
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of 10−5 . A schematic diagram for this integral model formulation is displayed in Figure 2.
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Albeit relatively simple, the integral models formulated in this work account for buoyancy, drag, and
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background flows, but there are important limitations. The model formulation utilizes steady-state theory
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for the air cavity motion in which zero displacement is assumed at the stagnation point. The steady-
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state assumption is valid during the initial slumping stage, but the air pocket motion will diverge from
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steady-state theory in later stages of the propagation Simpson (1997). In addition, as previously stated, the
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spatial variation in depth is neglected. Finally, the adopted radius of curvature at the front leading edge
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(rc = 0.0034 m) followed the work of Wilkinson (1982) in rectangular cross-sections, while results in this
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work were obtained in circular pipes.
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RESULTS AND DISCUSSION
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Measurements of air pocket spreading with no pipe flows
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Figure 3 presents the path of the air pocket leading edge location over time and corresponding celerity
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for four different air pocket volumes for horizontal slopes without pipe flows. Such paths are defined in
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this work as air pocket front trajectories, and one may notice the curved nature of the trajectories. This
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curvature indicates a reduction in the leading edge celerity, caused by the air pocket spreading and reducing
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of the air pocket thickness. The slope of the air pocket’s leading edge trajectory decreased for larger air
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pocket volumes, indicating an increase in celerity, an expected result. It can be noticed that the trajectory
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of the back-propagating front is symmetrical to the forward propagating front in the absence of flow in the
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pipe, also as anticipated. In all results, air pocket volumes are presented in a normalized format, with V ol∗
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defined as V ol/D3 .
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Results for adverse slopes (downstream end at a higher elevation than the upstream end) and no pipe
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flows are presented Figure 4. Trajectories and celerity values of the leading edge of the moving air pockets
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are shown for various volumes as well as various adverse pipeline slopes. The results do not present the
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curved trajectory from the air pocket leading edge that was present in horizontal cases even for the smallest
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slope tested of 0.5%. This indicates that the effect of air pocket spreading at the pocket leading edge was
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not significant in the experiments. Another observation of this figure indicates the relative celerity difference
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between pocket of various volumes decreases for larger pipeline slopes. That may indicate that larger slopes
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lead to stronger concentration of air at the leading edge of the air pocket as it propagates, contributing to
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the reduction of these celerity values. Results for the smallest air pocket volume of V ol∗ = 0.48 and 2% slope
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show a much smaller celerity than corresponding cases with shallower slopes, a result which could not be √ explained at this point. However, for most tested cases, the relative celerity C ∗ = C/ gD is limited by 0.47,
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which is consistent with results presented by Baines (1991). Results obtained when slopes were reversed to
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positive and no flow are consistent with the ones with the cases with adverse slope, and are not shown here
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for brevity.
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Measurements of air pocket spreading with pipe flows
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Results are significantly affected by the presence of pipe flows. Figure 5 presents the trajectories for
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the cases with air pocket volumes of V ol∗ =1.2 and V ol∗ =2.2 and horizontal slope for various pipe flows. √ Celerity values C ∗ are complemented by celerity values relative to the pipe flow flow (C - Vf low / gD. The
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symmetry between the two air pocket fronts previously observed for horizontal case is no longer observed due
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to drag forces caused by the introduction of pipe flow. The results in Figure 5 indicate that larger flow rates
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result in an increase in the celerity of the forward moving air pocket front, and that this gain in velocity is
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proportional to the flow velocity. Results also indicate that in the initial stages following the pocket release
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there is an increase in the celerity that is generally finished at the instant t∗ ≈ 2, with the normalized time p defined as t∗ = (t/X0 )/( D/g) . The previously observed air pocket front spreading in the horizontal slope
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and no flow, which led to decreasing celerity values, is not observed is when there is pipe flows. Results in
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Figure 5 support the assumption that for horizontal pipes the pocket celerity within flowing pipes can be
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estimated as the summation of its value in quiescent conditions plus the pipe flow velocity.
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An interesting result is that is even for the smallest pipe flow, the backward propagating air pocket leading
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edge is eventually sheared and no longer observed for values of L1 < 0.3 and t∗ ≈ 1. This indicate that in
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the absence of buoyancy forces the propagation of discrete air pockets against an non-zero pipe flow velocity
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is short lived. Figure 6 presents seven snapshots of the pocket propagation with 1 second time lapse between
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the images, for a condition corresponding to a pocket with V ol∗ =4.1 and flow rate Q∗ =0.27. Air-water
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interfaces in the figure are artificially enhanced for clarity. The initial shape of the front resembles a typical,
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dissipative gravity current with a curved nose with a trailing a hydraulic jump, as described in Benjamin
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(1968). As the front moved upstream, the trailing hydraulic jump entrained air at the leading edge of the
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pocket, and these air bubbles were carried with the pipe flow. The pocket volume thus decreased over time
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and within 7-8 seconds later it effectively vanished. In the process, the observed pocket celerity decreased as
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the thickness of the backward propagating front decreased, a result consistent with gravity current theory.
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The other extremity of the front did propagate as a curved gravity current front. One speculates that this
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type of pocket shearing may also be observed in deep stormwater storage tunnels laid in shallow slopes. This
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behavior was not observed for downward slopes, when buoyancy forces oppose drag forces, as is explained
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below.
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√ Figure 7 presents the trajectories, observed celerity C normalized by gD, and the relative celerity √ accounting for pipe flow velocity (C - Vf low ) also normalized by gD, for the cases with V ol∗ =0.95 and
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V ol∗ =1.9 and 1% adverse slope. As anticipated, the observed air pocket front celerity increases with the
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flow rate, but there is no significant changes in celerity between the two tested pocket volumes. In such
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slopes buoyancy and drag forces are summed and the air pocket celerity can also be approximated by the
329
summation of the pipe flow velocity and the celerity observed in quiescent conditions. An exception to this
330
observation was the case with V ol∗ =0.95 and the highest pipe flow rate Q∗ =0.37. An explanation of this
331
observation is not available at this point.
324
332
The most complex cases studied in this work involved positive slopes, when buoyancy forces opposed drag
333
forces. A sample of these results is presented in Figure 8 that corresponds to the case where pocket volumes
334
V ol∗ =0.95 and V ol∗ =1.9 and 1% positive slope. Negative celerity values are due to the propagation direction
11
335
of the pocket toward the upstream end of the apparatus. Visually, the leading edge of the pockets advanced
336
upstream against the pipe flow due to buoyancy in all tested slopes. Strong turbulence was observed at
337
the trailing hydraulic jump behind the air pocket front. Air was sheared from the air pocket leading edge,
338
but consistently with observations are consistent with those by Pozos (2007) these smaller bubbles gathered
339
downstream, and eventually grew in size and also moved against the flow and rejoin with the main pocket.
340
The observed celerity C ∗ values for these positive slopes, as anticipated, are smaller than the case with
341
no pipe flow and ranged from -0.2 to -0.4 for all tested conditions. However, the assumption that these
342
celerity values corresponded to the summation of celerity in quiescent conditions and the pipe flow velocity
343
conditions is no longer valid, a result that is likely linked to these complex air-water interactions observed
344
in positive slopes.
345
Comparison with numerical model predictions
346
The proposed gravity current integral model was tested for four different air pocket volumes and all
347
background flow rates, all involving horizontal pipe. The limitation of slope was due to the numerical model
348
inability to incorporate slope effects. Two alternatives (M 1 and M 2) discussed in the Methodology were
349
compared to these experimental results to better understand the role of front conditions in the predictions of
350
air cavity motion. Both alternatives were simple to implement, involved relatively inexpensive calculations,
351
and thus simulations were concluded in a fraction of a second. Another limitation of the numerical modeling
352
is that comparisons only involved downstream moving fronts when pipe flows were present. This is mostly
353
due to the fact that the upstream moving fronts lasted briefly as they were quickly sheared by pipe flows.
354
Such conditions yielded a limited amount of experimental data points and prevented a comparison between
355
experimental and model results.
356
The air cavity front trajectories for horizontal flow experiments are compared to both numerical model
358
alternatives (M1 and M2) in Figure 9. The pocket front coordinate (X) was normalized by the initial √ air pocket length (X0 ) and the time by (X0 / gD). The experimental results are clearly affected by the
359
background flow rate in which the air cavity front velocity is governed by Equation 1 as described in Hallworth
360
et al (1998). The effect of air pocket volume on front velocity in the tested conditions was not as important
361
as the pipe flow velocity.
357
362
Figure 9 indicates that in the absence of background flow, M 1 consistently compared better to experi-
363
mental results, and this agreement was considered good. On the other hand, M 2 consistently underestimated
364
the air pocket front velocity, and this error increased for larger air pocket sizes. M 2 predictions were more
365
accurate or comparable to M 1 predictions only for cases involving smaller air pocket volumes (V ol∗ = 1.2
366
and V ol∗ = 2.2) in the presence of higher background flows. The accuracy of M 1 appears unaffected by
12
367
different background flows and pocket sizes. These results indicate that the surface tension parameters
368
presented in Wilkinson (1982) for rectangular pipe cross sections were adequate for circular pipes.
369
While experimental observations indicated that the initial stages (t∗ 2, results from M 1 are fairly close to the observations from tested conditions.
375
Results yielded by the M 2 model compared well only with the the smallest air pocket V ol∗ = 1.2.
377
The accuracy of air pocket velocity predictions are reflected in a comparison between measured and √ predicted cavity Froude numbers (F = uf / gD), which are provided in Table 2. The smallest (V ol∗ = 1.2)
378
and largest (V ol∗ = 4.1) air pocket volumes from experiments were selected for the comparison. The front
379
velocity (uf ) was computed from the average velocity between t∗ = 0 and t∗ ≈ 2.5. M 1 was more accurate of
380
the two integral models for each one of the experiments tested in this comparison. This model underpredicted
381
F values by only 0.1% for V ol∗ = 1.2 experiments while overpredicting results in V ol∗ = 4.1 experiments by
382
4.2%. M 2 underpredicted F values by an average of 11.5% with larger discrepancies occuring for larger air
383
pockets and smaller background flows.
384
CONCLUSIONS AND FUTURE WORK
376
385
This work presented results on an investigation of the motion of entrapped air pockets in water pipes.
386
The focus was on the kinematic aspects of air pocket motion and air-water interactions related to the motion
387
of these discrete, finite volume pockets. A long term goal of this research is to incorporate in flow regime
388
transition models the ability to track entrapped pockets and prevent negative impacts related to these
389
events. One relevant context in which the findings of this study can be of practical used is in the rapid
390
filling of stormwater storage tunnels. Entrapped air is a source of various operational issues which range
391
from surging, loss in storage, and uncontrolled air release through water-filled shafts (e.g. geysering). The
392
ability of tracking the motion of entrapped air pockets can be useful in determining the likelihood of such
393
operational problems, and adjust designs that would avoid such adverse issues.
394
An experimental program was proposed where different conditions of pipe slope and velocities rates were
395
tested. Use of clear PVC pipe enabled tracking the coordinates of entrapped pockets over time for each
396
tested condition. In addition to the experimental studies, a non-Boussinesq, integral model was proposed to
397
simulate some of the conditions tested in the experiments. The purpose of this gravity current model was the
398
possibility of such a simple, computationally inexpensive modeling approach being included as a sub-model
13
399
400
in a more complex flow regime transition model. The findings of this research may be summarized as follows:
401
• The celerity of discrete air pockets in stagnant flow conditions depended on air pocket volume, par-
402
ticularly for shallower slopes (below 0.5%), in agreement with Benjamin (1968) and Wilkinson (1982)
403
observations. For stronger slopes, celerity values were not as dependent on volumes as air accumula-
404
tion at the leading edge of the air pocket reduced differences on the pocket thickness;
405
• For horizontal slope and stagnant conditions, the air pocket celerity values slightly decreased over
406
time, indicating that the air pocket spreading decreased the leading edge thickness, resulting in
407
smaller celerity values;
408
• It was observed that the propagation of air pockets in horizontal slope pipes against pipe flows is
409
short lived due to the shearing of the air pockets in the hydraulic jumps that trailed the leading edge
410
of the air pocket;
411
• The celerity of the air pocket leading edge in horizontal and adverse slopes was well approximated
412
by the summation of values observed in quiescent conditions and the pipe flow velocity. Such an
413
approximation did not hold for air pocket propagation in positive slopes where buoyancy forces oppose
414
drag forces;
415
• Air pocket propagation in positive slopes was slowed down by pipe flows, and shearing of the leading
416
edge of the main air pocket by the trailing hydraulic jump was also observed. However, these sheared
417
bubbles regrouped in larger bubbles/pockets downstream from the main air pocket. At a certain
418
size, they would have enough velocity to overcome the drag forces and eventually rejoin the main air
419
pocket.
420
• Predictions of the air pockets leading edge coordinate yielded by proposed numerical models compared
421
well with experiments involving horizontal slopes and pipe flows. The M 1 model alternative (based on
422
the work of Benjamin (1968) and Wilkinson (1982)) yielded better results for large air pocket volumes
423
and smaller pipe flows. The M 2 model alternative results were more accurate for the smallest air
424
pockets volumes (V ol∗ = 1.2) and higher pipe flows.
425
This work also indicates various points that should be addressed in future investigations. One is the
426
development of similar studies using other pipe diameters to assess scale effects on these findings. Exper-
427
iments with 50 mm and 200 mm pipes are planned in the future to complement findings presented in this
428
work. Another important area for future work is to assess the performance of the proposed gravity current
429
integral models when they are embedded in hydraulic models able to simulate both flow regime transition
430
and entrapment of air pockets. Experimental studies that involve both occurrences are planned to help
14
431
assess the effectiveness of such a combination in controlled laboratory conditions.
432
ACKNOWLEDGMENTS
433
434
The authors would like to acknowledge the support provided by LimnoTech Inc., and Auburn University which has funded part of this research.
15
435
APPENDIX I. NOTATION The following symbols are used in this paper.
436
A =
1 2
2
sin α) D4
A2
=
Cross-sectional area below air pocket=(π − α +
C
=
Celerity
C∗
=
Normalized celerity= √CgD
CSO
=
Combined sewer overflow
D
=
Pipeline diameter
F
=
Local Froude number at upstream and/or downstream air pocket front
g
=
Gravitational acceleration
L =
Pipeline length
L1
=
Relative pipeline length coordinate= X L
Lp
=
Air pocket length
Q =
437
Pipeline
area= π4 D2
Flow rate in the pipeline
Q∗
=
Normalized flow rate in the pipeline= √Q
rc
=
Radius of curvature
t =
gD 5
Time
tnew
=
Updated time parameter
told
=
Previous time parameter
t∗
=
q 0 Normalized time= t/X D g
uf
=
Front velocity
Vf low
=
Average flow velocity
Vf∗low
=
low Normalized flow velocity= √fgD
V ol
=
Air pocket volume
V ol∗
=
Normalized air pocket volume=V ol/D3
X
=
Location along pipeline length
X0
=
Initial air pocket length=2.1 m
xf
=
Distance traveled by respective cavity front
X∗
=
Normalized location along pipeline length= XX0
α
=
Pipe half-angle: 2α is the angle subtended from the center of the pipe to the free surface
σ
=
Surface tension
v
16
438
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439
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440
Barnea, D. and Taitel, Y. (1993). Model for slug length distribution in gas-liquid slug flow. Int. J. Multiphase
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Benjamin, T.B. (1968). Gravity currents and related phenomena. J. Fluid Mech., 31: 209–248, 1968.
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DeHenau, V. and Raithby, G.D. (1995). Transient two-fluid model for the simulation of slug flow in pipelines.
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Int. J. Multiphase Flow,21(3): 335–349, 1995.
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Falvey, H.T. (1980). Air-water flow in hydraulic structures. USBR Engineering Monograph, n. 41
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Glauser, S. and Wickenhauser, M. (2009). Bubble movement in downward-inclined pipes. J. Hydr. Eng.,
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Martin, C.S. (1976). Entrapped air in pipelines. Proc., 2nd Int. Conf. on Pressure Surges, BHR, Bedford, England, 15–28, 1976. Nielsen, K.D., and Davis, A.L. (2009). Air migration analysis of the Terror Lake tunnel. Proc., 33rd IAHR Congress, IAHR, Vancouver, BC, 262–268, 2009. Pothof, I. and Clemens, F.H.L.R. (2010). On elongated air pockets in downward sloping pipes. J. Hydr. Res., 48(4): 499–503, 2010. Pothof, I. and Clemens, F..H.L.R. (2011). Experimental study of air-water flow in downward sloping pipes. Int. J. Multiphase Flow, 37: 278–292, 2011. Pozos, O.E. (2007). Investigation on the effects of entrained air in pipelines. PhD Thesis, University of Stuttgart, 2007. Pozos, O., Gonzalez, C., Giesecke, J., Marx, W. and Rodal, E. (2010). Air entrapped in gravity pipeline systems. J. Hydr. Res., 49(3): 394–397, 2010.
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Townson, J.M.(1991) Free-surface hydraulics. Cambridge University Press.
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Ungarish, M.(2009) An introduction to gravity currents and intrusions. CRC Press, 2009.
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Vasconcelos, J.G. and Leite, G.M. (2012). Pressure surges following sudden air pocket entrapment in
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stormwater tunnels. J. Hydr. Eng., 138(12): 1080–1089, 2012. Vasconcelos, J.G. and Wright, S.J. (2006). Mechanisms for air pocket entrapment in stormwater storage tunnels. Proc., 2006 ASCE EWRI Congress, Omaha, NE, 2006. Vasconcelos, J.G. and Wright, S.J. (2008). Rapid flow startup in filled horizontal pipelines. J. Hydr. Eng., 134(7): 984–992, 2008.
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500 501 502 503
List of Tables 1 Experimental variables considered √ with respective tested ranges . . . . . . . . . . . . . . . . . 2 Cavity Froude number (F = uf / gD) comparison for the largest and smallest air pocket volumes tested with various background flows. . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
21 22
TABLE 1.
Experimental variables considered with respective tested ranges
Experimental variable p Flow rate (normalized by Q/ gD5 ) Pipeline slope
Air pocket volume (normalized by D3 )
Range Three values ranging from Q∗ =0 up to 0.388 Horizontal slope 0.5%, 1.0%, and 2.0% adverse slopes 0.5%, 1.0%, and 2.0% positive slopes Horizontal: up to 4 values in the range V ol∗ =1.2-4.1 Adverse: up to 4 values in the range V ol∗ =0.48-3.8 Positive: up to 4 values in the range V ol∗ =0.48-3.8
21
√ TABLE 2. Cavity Froude number (F = uf / gD) comparison for the largest and smallest air pocket volumes tested with various background flows. p Q∗ = Q/ gD5 0 0.13 0.27 0.39 V ol∗ = 1.2 Experiments M1 prediction M2 prediction
0.284 0.268 0.253
Experiments M1 prediction M2 prediction
0.419 0.427 0.323
0.451 0.467 0.431 V ol∗ 0.594 0.612 0.496
22
0.672 0.664 0.617 = 4.1 0.751 0.809 0.683
0.804 0.826 0.773 N/A 0.972 0.840
504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532
List of Figures 1 Sketch of apparatus used in experiments. For positive slopes, the length of the reaches upstream and downstream from the valve were adjusted to 6.0 m and 3.0 m respectively. . . . . 2 Schematic Diagram of an air pocket propagating in a circular pipe using the integral model assumption. The direction of uf D is always positive while the sign of uf U will change depending on the magnitude of U and y/D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Trajectory of the air pocket leading edge and observed celerity for horizontal slope and various air pocket volumes, no pipe flow. Negative coordinates indicate pockets are propagating toward upstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Trajectory of the air pocket leading edge and observed celerity for various adverse slopes and air pocket volumes, no pipe flows. Negative coordinates indicate pockets are propagating toward upstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with horizontal slope and pipe flows for V ol∗ =1.2 and V ol∗ =2.2. . . . . . . . . . . . . . . . . 6 Sequence of snapshots illustrating the trajectory of the backward moving pocket trajectory for Q∗ = 0.26, V ol∗ =4.1 and horizontal slope, illustrating the shearing process. Interface between air and water is enhanced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% adverse slope and pipe flows for V ol∗ =0.95 and V ol∗ =1.9. . . . . . . . . . . . . . . . 8 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% positive slope and pipe flows for V ol∗ =0.95 and V ol∗ =1.9. . . . . . . . . . . . . . . . 9 Air pocket front trajectory comparison between experiments and both integral models for various background flows and pocket volumes: a) V ol∗ = 1.2, b) V ol∗ = 2.2, c) V ol∗ = 3.1, and d) V ol∗ = 4.1. The solid lines represent the integral model predictions, and the data markers represent experimental values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Air pocket front celerity comparison between experiments and both integral models for various background flows and pocket volumes: a) V ol∗ = 1.2, b) V ol∗ = 2.2, c) V ol∗ = 3.1, and d) V ol∗ = 4.1. The solid lines represent the integral model predictions, and the data markers represent experimental values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
24
25
26
27 28
29 30 31
32
33
FIG. 1. Sketch of apparatus used in experiments. For positive slopes, the length of the reaches upstream and downstream from the valve were adjusted to 6.0 m and 3.0 m respectively.
24
FIG. 2. Schematic Diagram of an air pocket propagating in a circular pipe using the integral model assumption. The direction of uf D is always positive while the sign of uf U will change depending on the magnitude of U and y/D.
25
FIG. 3. Trajectory of the air pocket leading edge and observed celerity for horizontal slope and various air pocket volumes, no pipe flow. Negative coordinates indicate pockets are propagating toward upstream.
26
FIG. 4. Trajectory of the air pocket leading edge and observed celerity for various adverse slopes and air pocket volumes, no pipe flows. Negative coordinates indicate pockets are propagating toward upstream.
27
FIG. 5. Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with horizontal slope and pipe flows for V ol∗ =1.2 and V ol∗ =2.2.
28
FIG. 6. Sequence of snapshots illustrating the trajectory of the backward moving pocket trajectory for Q∗ = 0.26, V ol∗ =4.1 and horizontal slope, illustrating the shearing process. Interface between air and water is enhanced.
29
FIG. 7. Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% adverse slope and pipe flows for V ol∗ =0.95 and V ol∗ =1.9.
30
FIG. 8. Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% positive slope and pipe flows for V ol∗ =0.95 and V ol∗ =1.9.
31
FIG. 9. Air pocket front trajectory comparison between experiments and both integral models for various background flows and pocket volumes: a) V ol∗ = 1.2, b) V ol∗ = 2.2, c) V ol∗ = 3.1, and d) V ol∗ = 4.1. The solid lines represent the integral model predictions, and the data markers represent experimental values.
32
FIG. 10. Air pocket front celerity comparison between experiments and both integral models for various background flows and pocket volumes: a) V ol∗ = 1.2, b) V ol∗ = 2.2, c) V ol∗ = 3.1, and d) V ol∗ = 4.1. The solid lines represent the integral model predictions, and the data markers represent experimental values.
33
