Investigation on the Kinematics of Entrapped Air Pockets in Stormwater Storage Tunnels by Carmen Chosie A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama December 14, 2013 Keywords: Stormwater tunnels; Air pockets; Laboratory experiments Copyright 2013 by Carmen Chosie Approved by Jose G. Vasconcelos, Chair, Assistant Professor of Civil Engineering Prabhakar Clement, Professor of Civil Engineering Xing Fang, Associate Professor of Civil Engineering Abstract Various mechanisms can lead to the entrapment of air pockets within stormwater storage tunnels when they undergo rapid lling during intense rain events [Vasconcelos and Wright(2006)]. These entrapped air pockets are linked to operational is- sues within systems such as damaging surges, storage capacity loss, and severe geysering upon their release through water- lled ventilation shafts. Therefore, tracking entrapped air pockets and their celerity is important in the context of numerical simulation to assess the risk of the aforementioned operational issues. Previous studies focused on quantifying the magnitude of pressure surges associated with air pocket compression or in obtaining the minimum ow velocities required to expel the entrapped air pockets from water mains (hydraulic clearing). However, the conditions controlling the motion of these nite volume pockets following entrapment require further investigation. A balance between drag and buoyancy forces is expected to control the motion of discrete air pockets in closed conduits, yet there have been limited studies in terms of how factors such as varying pipeline slope, background ow, and air pocket volume a ect air pocket motion. This research aims to explore a link between ambient ow velocity, pipeline slope, and the celerity of entrapped air pockets of various volumes. This work presents experimental results from an investigation on the kinematics of entrapped air pockets in pressurized water ows under various shallow slopes (up to 2% favorable and adverse). Results of pocket trajectories and celerities are systematically compared for various tested slopes, ow rates, and pocket volumes. These experimental results are useful for the future development of numerical models that can include the motion of entrapped air pockets in closed conduits. ii Acknowledgments I would like to thank my professor and advisor, Professor Jose Vasconcelos, for his time and patience spent in the development of my experimental research as well as writing conference papers, monographs, and this thesis. I also would like to thank Amanda Summers, Holly Guest, and Andrew Patrick for all their hands-on help in the laboratory. I would also like to acknowledge the support provided by LimnoTech Inc. and Auburn University which has funded part of this research. I would like to thank my parents, Dr. Kim and Mr. Kern Clark, who have always supported my motivation and drive to keep learning. Without their sacri ces and gifts, I would not have been able to complete both my degrees at Auburn University. My ance, Nick, deserves much thanks for putting up with all my late nights of stress and o ering to buy me ice cream and chocolate. Finally, I am also thankful for all my friends in the civil engineering department who made classes, lab work and writing a great experience. These include: Holly Guest, Kristi and Tyler Mitchell, Jacob Kearley, Andrew Patrick, Kyle Moynihan, Tom Hatcher, Mitchell Moore, Catherine Butler, and Farhad Jazaei. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Mechanisms for air pocket entrapment . . . . . . . . . . . . . . . . . . . . . 1 1.2 Studies on gravity currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Hydraulic clearing of air pockets in water mains . . . . . . . . . . . . . . . . 11 1.4 Slug ows/Plug ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Summary of Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Issues related to air pocket entrapment . . . . . . . . . . . . . . . . . . . . . 17 2.2 Experimental investigations on air pocket kinematics . . . . . . . . . . . . . 24 2.3 Numerical investigations on air pocket kinematics . . . . . . . . . . . . . . . 26 3 Knowledge Gap and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 General description and rationale of the experiments . . . . . . . . . . . . . 33 4.2 Experimental program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Comparison with numerical model . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.1 Air pocket spreading in horizontal slope . . . . . . . . . . . . . . . . . . . . 40 iv 5.2 Air pocket spreading in adverse slopes . . . . . . . . . . . . . . . . . . . . . 44 5.3 Air pocket spreading in favorable slopes . . . . . . . . . . . . . . . . . . . . 47 5.4 Air pocket motion and spreading compared to numerical model prediction . 53 6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A Additional Experimental Results for Trajectory and Celerity . . . . . . . . . . . 64 v List of Figures 1.1 Air entraining vortex at intake structure due to insu cient pump submergence [Pinott and Moller(2011)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Pocket entrapment due to inadequate ventilation [Vasconcelos and Wright(2006)]. 3 1.3 Pocket entrapment due to misplaced ventilation [Vasconcelos and Wright(2006)]. 3 1.4 Pocket entrapment due to interface breakdown [Vasconcelos and Wright(2006)]. 4 1.5 Pocket entrapment due to shear ow instability [Vasconcelos and Wright(2006)]. 4 1.6 Pocket entrapment due to GFRT [Vasconcelos and Wright(2006)]. . . . . . . . . 5 1.7 Air pocket entrapment mechanism related to the re ection of in ow fronts from the system boundary [Vasconcelos and Leite(2012)]. . . . . . . . . . . . . . . . . 6 1.8 Gravity current propagation velocity adapted from [Benjamin(1968)]. . . . . . . 7 1.9 Diagram of ow schematic from [Wilkinson(1982)]. . . . . . . . . . . . . . . . . 9 1.10 Velocity as function of channel slope from [Baines(1991)]. . . . . . . . . . . . . . 10 1.11 Comparison of advance of air intrusion observed in experiments with numerical prediction by [Vasconcelos and Wright(2008)]. . . . . . . . . . . . . . . . . . . . 11 1.12 Three stages of advancing air volume released from one end of a channel of water by [Simpson(1997)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 vi 1.13 Forces acting on air pocket in closed conduit with ow [Pozos et al(2010b)]. . . 13 1.14 Forces acting on air pocket in closed conduit with ow [Pozos et al(2010b)]. . . 14 1.15 Plug and slug ow patterns as distinguished by [Falvey(1980)]. . . . . . . . . . . 15 2.1 Di erent urban geysering incidents: (A) Image from geyser video, reportedly recorded at Chicago, IL on 06/23/2010; (B) geyser lifting a car in Montreal, Quebec on 07/18/2011. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Numerical predictions for normalized free-surface level rise for di ering initial free-surface levels and ventilation tower diameters [Vasconcelos and Wright(2011)]. 19 2.3 Transient air pressure, liquid velocity, and air volume [Martin(1976)]. . . . . . . 20 2.4 Pressure oscillation pattern for water hammer e ect, Type 3 [Zhou et al(2002)]. 20 2.5 Decrease of pressure peak for various slopes, valve obstructions, and air pocket volume by [Vasconcelos and Leite(2012)]. . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Conceptual sketch by [Wright et al(2011)] of geyser during release of large air pocket: (a) pocket migrates toward vertical shaft; (b) momentum of air pocket into shaft due to buoyancy cause rise in water level and (c) high velocity air may entrain liquid due to ooding instability. . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Schematic of experimental apparatus used by [Perron et al.(2006)]. . . . . . . . 25 2.8 Stages of ow transition by [Li and McCorquodale(1999)]. . . . . . . . . . . . . 28 2.9 Measured and predicted pressures at pipe crown for x =0.39, dorif<0.5, and slope 1% by [Trindade and Vasconcelos(2013)]. . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Sketch of apparatus used in horizontal and adverse pipeline slope experiments. . 34 vii 4.2 Sketch of apparatus used in favorable pipeline slope experiments. Changes were only in the location of internal knife gate valves. . . . . . . . . . . . . . . . . . . 35 5.1 Trajectory of the air pocket leading edge and observed celerity for horizontal slope and various air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. . . . . . . . . . . . . . . . . 41 5.2 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with horizontal slope, ambient ow, and Vol =1.2 and Vol =2.2. 43 5.3 Sequence of snapshots illustrating the trajectory of the backward moving pocket trajectory for Q*=0.26, Vol*=4.1 and horizontal slope, illustrating the shearing process. Interface between air and water is enhanced. . . . . . . . . . . . . . . . 45 5.4 Trajectory of the air pocket leading edge and observed celerity for various adverse slopes and air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. . . . . . . . . . . . . . . . . . . . . . 46 5.5 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% adverse slope, ambient ow, and Vol =0.95 and Vol =1.9. 48 5.6 Trajectory of the air pocket leading edge and observed celerity for various fa- vorable slopes and air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. . . . . . . . . . . . . . . . . 49 5.7 Trajectory of the air pocket leading edge for 2% (chart a), 1% (chart b) and 0.5% (chart c) adverse and favorable slopes and various air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. . 50 5.8 Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% favorable slope, ambient ow, and Vol =0.95 and Vol =1.9. 52 viii 5.9 Trajectory of the air pocket leading edge (chart a), observed celerity (chart b) and relative celerity (chart c) for conditions with 1% adverse and 1% favorable slopes and ambient ow for Vol =1.9. . . . . . . . . . . . . . . . . . . . . . . . 53 5.10 Air front trajectory comparison between experiments and both integral models for various background ows and pocket volumes: a) Vol = 1:27, b) Vol = 2:22, c) Vol = 3:18, and d) Vol = 4:13. The solid lines represent the integral model predictions, and the data markers represent experimental values. . . . . . . . . . 55 A.1 Trajectory and celerity for 0.5% adverse slope and Q 0.12. . . . . . . . . . . 65 A.2 Trajectory and celerity for 0.5% adverse slope and Q 0.25. . . . . . . . . . . 66 A.3 Trajectory and celerity for 0.5% adverse slope and Q 0.37. . . . . . . . . . . 67 A.4 Trajectory and celerity for 0.5% favorable slope and Q 0.12. . . . . . . . . . 68 A.5 Trajectory and celerity for 0.5% favorable slope and Q 0.25. . . . . . . . . . 69 A.6 Trajectory and celerity for 0.5% favorable slope and Q 0.37. . . . . . . . . . 70 A.7 Trajectory and celerity for 2% adverse slope and Q 0.12. . . . . . . . . . . . 71 A.8 Trajectory and celerity for 2% adverse slope and Q 0.25. . . . . . . . . . . . 72 A.9 Trajectory and celerity for 2% adverse slope and Q 0.37. . . . . . . . . . . . 73 A.10 Trajectory and celerity for 2% favorable slope and Q 0.12. . . . . . . . . . . 74 A.11 Trajectory and celerity for 2% favorable slope and Q 0.25. . . . . . . . . . . 75 A.12 Trajectory and celerity for 2% favorable slope and Q 0.37. . . . . . . . . . . 76 ix List of Tables 4.1 Experimental variables considered with respective tested ranges . . . . . . . . . 36 x List of Abbreviations A Pipeline area= 4D2 C Celerity C Normalized celerity= CpgD D Pipeline diameter g Gravitational acceleration L Pipeline length L1 Location along pipeline length=XL Q Flow rate in the pipeline Q Normalized ow rate in the pipeline=Q=pgD5 t Time t Normalized time= tqD g vflow Flow velocity v flow Normalized ow velocity=vflowpgD Vol Air pocket volume Vol Normalized air pocket volume=Vol=D3 X Location along pipeline length X Normalized location along pipeline length= XX0 xi X0 Initial air pocket length=2.1 m xii Chapter 1 Introduction Deep storage tunnels have been utilized in highly urbanized areas as the means to provide relief to the stormwater collection systems during intense rain events as well as treatment to runo . Operational problems such as damaging pressure surges and geysering episodes in these systems led to investigations with the goal of identifying the causes of these problems, which have been observed when these tunnels undergo rapid lling during intense rain events. The role of entrapped air pockets and the problems associated with these pockets is currently being identi ed by recent investigations. Evidence suggests that the entrapped air pockets have major impacts in system behavior relating to pressure surges magnitudes, loss of storage capacity, an geysering. A better understanding of air pocket behavior is required so that better design guidelines can be developed and implemented to avoid these adverse conditions. 1.1 Mechanisms for air pocket entrapment Rapid lling of closed pipes can lead to the entrapment of air pockets through many di erent mechanisms, which depend entirely on the type of hydraulic system under consid- eration. There are various contexts in which air in closed conduits creates an issue, which arise from the fact that these conveyance systems are not designed to operate in two-phase ows. Various causes for air entrainment in transmission mains have been identi ed to date, and a comprehensive summary is presented by [Lauchlan et al(2005)]. These mechanisms include 1) entrainment at in ow (drop chamber, inlet, or intake) and out ow (outfalls in tidal areas placed above sea level) locations; 2) entrainment due to vortices, turbulence and hydraulic jumps; 3) insu cient pump submergence; 4) lling or emptying of pipelines; and 1 Figure 1.1: Air entraining vortex at intake structure due to insu cient pump submergence [Pinott and Moller(2011)]. 5) negative pressures at the pipe inlet. Figure 1.1 displays an air entraining vortex at an intake structure due to insu cient pump submergence. Other mechanisms have been identi ed in the context of the lling of stormwater sys- tems. An early study by [Hamam and McCorquodale(1982)] indicates that interface stability caused by the relative motion between air and water can lead to surface waves in water that can grow and eventually promote an air pocket entrapment. This and other mechanisms for air pocket entrapment in the context of stormwater systems have been studied experimen- tally by [Vasconcelos and Wright(2006)], which also includes mechanisms such as insu cient (see Figure 1.2) or misplaced ventilation (Figure 1.3), breakdown of pressurization air-water interfaces (Figure 1.4), air-water shear ow instabilities (Figure 1.5), and gradual ow regime transition (GRFT) as shown in Figure 1.6. Figure 1.7 displays an air pocket formation mechanism observed during a numerical simulation of an actual rapid lling scenario. The re ection of an in ow front from the 2 Figure 1.2: Pocket entrapment due to inadequate ventilation [Vasconcelos and Wright(2006)]. Figure 1.3: Pocket entrapment due to misplaced ventilation [Vasconcelos and Wright(2006)]. 3 Figure 1.4: Pocket entrapment due to interface breakdown [Vasconcelos and Wright(2006)]. Figure 1.5: Pocket entrapment due to shear ow instability [Vasconcelos and Wright(2006)]. 4 Figure 1.6: Pocket entrapment due to GFRT [Vasconcelos and Wright(2006)]. system boundary entraps air forming a large pocket at this location. This is a poten- tially harmful condition if this pocket is evacuated through a water- lled ventilation pipe, an event that can only be assessed by learning more about the kinematics of entrapped air pockets. More studies related to the formation and motion of air in stormwater sys- tems are illustrated by [Hamam and McCorquodale(1982)], [Li and McCorquodale(1999)], [Zhou et al(2002)] and [Lautenbach et al(2008)]. While relevant, these studies do not pro- vide means to track the location of discrete entrapped air pockets to assess likelihood and attempt prevention of deleterious air-water interactions. 1.2 Studies on gravity currents Other related areas of investigation to those involving air-water ow in closed conduits include the propagation of gravity currents, particularly the case of non-Boussinesq currents 5 Figure 1.7: Air pocket entrapment mechanism related to the re ection of in ow fronts from the system boundary [Vasconcelos and Leite(2012)]. such as the intursion of an air cavity in a pipe initially lled with water. Classical contri- butions in uid mechanics in this area are exempli ed in the works by [Zukoski (1966)], [Benjamin(1968)], [Wilkinson(1982)], [Baines(1991)], and [Simpson(1997)] among others. Such air cavities are referred to as gravity currents in this context. Unlike cases in which air intrusion is caused by water emptying from a pipeline, the motion of air cavities of - nite volume is essentially analogous to air pockets and also studied in the realm of gravity currents. The study by [Zukoski (1966)] aimed to determine in a more precise manner the in uence of surface tension, viscosity, and tube inclination on the propagation rate of air bubble in vertical tubes for the ow regime in which surface tension e ects are important. It was determined that tube material had no in uence on ow when the diameter was larger than 2 cm. Also, for Reynolds numbers greater that 200, propagation rate become substantially independent of viscous e ects. The propagation rate of these bubbles increases to a maximum value as the inclination angle decreases from the vertical position to 45 degrees; a further 6 Figure 1.8: Gravity current propagation velocity adapted from [Benjamin(1968)]. reduction in the inclination angle causes the propagation rate to decrease. It was also noted that uid occupies roughly the lower half of the tube for a horizontal inclination position and the critical speed of the air bubble is roughly pga, where a is the tube radius. The work by [Benjamin(1968)] was one of the pioneers in establishing a that the cav- ity celerity scales with pgH, with g as gravitational acceleration and H the pipe charac- teristic dimension (e.g. diameter). Among various important observations it was found that the gravity current celerity generally decreased with its thickness. Also, this work by [Benjamin(1968)] di erentiated the dissipation-free (loss-free) intrusion and the dissipative intrusion (e.g. having a trailing bore, such as Figure 1.8). 7 The experimental investigation by [Gardner and Crow(1970)] on the movement of large long air bubbles in stationary water in a horizontal channel of rectangular cross-section con- rms the results by [Benjamin(1968)] for very deep channels, but stresses that consideration must be given to the in uence of surface tension for channels as deep as 175 mm, the deepest channel investigated. This investigation also focused on the explanation of the in uence of surface tension on bubble velocity. It was also noted that for deep channels, the ow is similar to that conceived by [Benjamin(1968)] except for the tip of the curved nose front near the top wall of the channel, and bubble velocity is reasonably constant with respect to distance down the channel. Surface tension was found to have a substantial e ect, which is the same conclusion reached by [Zukoski (1966)] with respect to large bubbles in horizontal tubes. A limitation of these studies, similar to those previously mentioned, is the lack of ambient water ow. The study from [Wilkinson(1982)] expanded the work by Benjamin by incorporating surface tension e ects, which can markedly a ect the shape and celerity of the air cavity and is more pronounced for cavities of smaller depth. However, experiments conducted in deeper ducts con rmed that surface tension becomes increasingly less signi cant, and the shape of the cavity approaches that of the idealized model proposed by [Benjamin(1968)]. [Wilkinson(1982)] selected frames of reference that allowed the frontal region and bore region of the cavity to be treated as steady ows, even though the ow associated with the intrusion of an air cavity into a long duct may be of an unsteady nature. Figure 1.9 shows the distinct regions of ow including: upstream region, the region of energy-conserving ow, bore, and downstream region of uniform ow. [Baines(1991)] work followed these studies and studied an interesting air gulping ow feature observed when the pipe was set in a downward slope, in which the outlet was at a lower elevation than the inlet. This work showed that the observed celerity increased slightly with the tested slopes (up to 8 degrees), as seen in Figure 1.10, and that the celerity discrete air pockets formed after gulping also scaled with pgD. A limitation to these studies is 8 Figure 1.9: Diagram of ow schematic from [Wilkinson(1982)]. that there is no ambient water ow prior to the arrival of the air cavity; however, a general expression for the pockets celerity can be expressed as: C(h) = k(h) p gh (1.1) where C is the air pocket celerity, k is a factor that depends on the pocket thickness, h is the pocket thickness and g is gravitational acceleration. [Vasconcelos and Wright(2008)] presented an experimental and numerical work on the advance of an air cavity considering the acceleration of the water column due to varying pressure gradients. Figure 1.11 displays the advance of air intrusion observed in experimental work compared to the authors? numerical prediction. The initial advance of the air cavity is gradually halted and reverted by the water column velocity. For the largest values of H=D (pressure head at the reservoir divided by the pipeline diameter), observed and predicted trajectory agree well, but for decreasing values of H=D the results were not as accurate. The 9 Figure 1.10: Velocity as function of channel slope from [Baines(1991)]. authors explain this discrepancy is due to the assumptions introduced into their calculations, such as assuming constant air cavity intrusion celerity. This work concluded that the loss- free intrusion analyzed by [Benjamin(1968)] does in fact adequately represent ow conditions during the initial phases of air intrusion; however, the application of this criteria to the rapid lling of an initially empty pipe does not appear to be valid. It is also important to note that this study by [Vasconcelos and Wright(2008)] has not considered such e ects in discrete air pockets. [Simpson(1997)] also presents a study on nite-volume cavities. This experimental work involved releasing a xed volume of air into a closed horizontal tank full of water. These experiments show the clear stages of development of a gravity current of air that is released from behind a gate as it progresses above the water, as shown in Figure 1.12. Phase A shows the pocket front occupying almost exactly half the depth of the tank, and the pocket is moving at a constant speed. In phase B, a hydraulic jump (or bore) has formed at the 10 Figure 1.11: Comparison of advance of air intrusion observed in experiments with numerical prediction by [Vasconcelos and Wright(2008)]. tail of the gravity current. Finally, phase C shows that the hydraulic jump has reached the head of the gravity current, and the speed during this phase is no longer constant. 1.3 Hydraulic clearing of air pockets in water mains The "hydraulic clearing" of transmission mains consists in a related application involving air pocket motion in a di erent context. Because of the potential issues in closed conduits, various studies have been presented to date attempting to determine a minimum value for the ambient water ow that would ensure that air pockets could be removed. [Kalinske and Bliss(1943)] presented relevant information for a pipeline designer includ- ing the water discharge necessary to maintain air removal from any given pipe size placed at any slope based on experimental work using a 4 inch and 6 inch pipeline with varying water ow rates and pipeline slopes. It was observed that smaller air bubbles could be moved by owing water more easily than the larger bubbles. However, these smaller bubbles gradually combine into larger bubbles, which in turn could not be moved by the water ow and as a 11 Figure 1.12: Three stages of advancing air volume released from one end of a channel of water by [Simpson(1997)]. result traveled upstream and passed through the hydraulic jump, merging with the upstream bubble. The work by [Kalinske and Robertson(1943)] presented experimental data relating the ability of a hydraulic jump to entrain air and have it carried away by owing water by using an apparatus with an outside diameter of 6 inches, an inside diameter of 0.49 feet, a length of 35 feet, and slopes ranging from 0.2% to 30%. It was observed that above a certain critical condition, which depends on the Froude number of the ow before the hydraulic jump, the rate of air removal from an air pocket in a pipeline will depend of the ability of the hydraulic jump which is formed within the pipe to entrain air, and an empirical relationship was presented. These pioneer studies provide useful information, but have not focused in presenting observed displacement and celerity of entrapped air pockets for various combinations of ambient ows and pipe slopes, including adverse slopes conditions. More studies on hydraulic clearing, exempli ed by works of [Falvey(1980)], [Little et al(2008)], [Pothof and Clemens(2008)] and [Pozos et al(2010b)] provide means to 12 Figure 1.13: Forces acting on air pocket in closed conduit with ow [Pozos et al(2010b)]. compute the velocity magnitude at which drag forces overcomes the buoyancy forces and pockets are thus dragged downstream. Figure 1.13 presents conditions for which it is an- ticipated that air pockets (e.g. bubbles) will be dragged with ow (hydraulic clearing) or against ow (buoyancy) according to the pipeline geometric characteristics and background ow rate. Figure 1.14 displays the forces that act on an entrapped air pocket presented in the work by [Pozos et al(2010b)]. The mathematical expressions for air pocket removal depend on pipeline angle and pipeline diameter, as presented in the following equation: vp gD = k p S0 (1.2) where v is the critical removal velocity of water, k is 1.27, g is gravitational acceleration, D is the pipeline diameter, and S0 is the pipeline bottom slope. This equation is similar to Equation 1.1 presented in the work of [Baines(1991)] on gravity currents. 13 Figure 1.14: Forces acting on air pocket in closed conduit with ow [Pozos et al(2010b)]. 1.4 Slug ows/Plug ows Some pipeline systems are characterized by the active pumping/injection of two uid phases (e.g. liquid/gas). These systems are usually analyzed in the realm of two-phase ows, and various types of ow regimes exist in this context. Of particular interest are slug and plug ows, which are visually similar to the air-pocket ows. It is important to note the di erences between plug ow and slug ow in the context of multiphase ows, and Figure 1.15 displays the various types of ow regimes for horizontal two-phase ow. [Falvey(1980)] di erentiates slug ow as having wave amplitudes that are large enough the seal the conduit, and this wave forms a frothy slug where it touches the roof of the conduit; this slug travels with a larger velocity than the average liquid velocity. Plug ow, however, occurs for increased air ow rates. These air bubbles combine with plugs of air and water alternately owing along the top of the conduit. [Lauchlan et al(2005)] de nes plug ow as pockets or plugs that are entrapped in the main water ow, which are transported with ow along the top of the pipeline, and slug ow occurs when surface waves are large enough to seal the conduit causing the slug to travel 14 Figure 1.15: Plug and slug ow patterns as distinguished by [Falvey(1980)]. 15 with a velocity larger than that of the liquid, which are similar to those de nitions presented by [Falvey(1980)]. Two types of slugging mechanisms were identi ed in the work by [Lauchlan et al(2005)]: terrain induced slugging and shear induced slugging. Terrain induced slugging occurs when gas slugs form in ows in sloped terrain, and results from the accumulation of liquid phase in lower points in the pipeline, and the sudden release of gas slugs that are backed up at the upstream side of these low points. Slugs can also be generated at low elbows, dissipate at high elbows, as well as shrink and grow in length as they travel along the pipeline. Shear induced slugging occurs when hydraulic jumps promote air pocket entrainment such as Figure 1.5. The turbulent shear region contributes substantially to the air-water transfer at a hydraulic jump because its large air content and small bubble sizes resulting from large turbulent shear stress creates a very large region of air-water interface. 1.5 Summary of Introduction There are many causes for air pocket formation and motion and di erent analysis frame- works have been developed to study such ow conditions. The frameworks proposed to study air-water ows in various contexts such as air pocket entrapment in hydraulic systems, stud- ies on gravity currents, hydraulic clearing, and slug ow studies. However, some operational issues such as geysering, pressure surges, and ooding require deeper understanding of en- trapped air pocket kinematics. This focus on operational issues and air pocket kinematics is further explored in Chapter 2. 16 Chapter 2 Literature Review Below grade stormwater storage tunnels create a cost-e ective method for relief to stormwater collection systems during intense rain events in densely urbanized areas. These storage tunnels can prevent combined sewer over ow (CSO) events as well as other related environmental issues. In most cases, these tunnel systems are very large structures, ranging in diameter sizes from 15 to 30 ft and spanning over several miles below cities. The appli- cation of these systems can be seen as early as the late 1970?s in areas such as Chicago and San Francisco. Operational problems in these systems, like damaging pressure surges, led to investigations which focused in identifying the causes of these problems which have been observed when tunnel undergo rapid lling during intense rain events. Recent studies have been helpful in identifying the role that entrapped air pockets have in these ows. Evidence suggests that these entrapped air pockets have major impacts in system behavior such as pressure surge magnitude, storage capacity loss, and geysering. An improved understanding of air pocket behavior in these systems can result in the creation of better design guidelines to avoid such adverse conditions, which are sometimes present in other hydraulics systems such as water mains. 2.1 Issues related to air pocket entrapment Other operational issues that have appeared in stormwater storage tunnels during in- tense rain events include: damaging pressure surges, manhole lid blow-o , and return of conveyed water to grade. A survey conducted by [Lautenbach and Klaver(2010)] with tunnel operators in urban areas provides an idea of the extent of these issues. Of the nine stormwa- ter tunnel systems surveyed, seven operators indicated the occurrence of operational issues 17 Figure 2.1: Di erent urban geysering incidents: (A) Image from geyser video, reportedly recorded at Chicago, IL on 06/23/2010; (B) geyser lifting a car in Montreal, Quebec on 07/18/2011. with varying levels of severity from displaced manholes, to structural damage and return of conveyed water to grade. One of the most extreme examples of the adverse interactions between stormwater and entrapped air is the occurrence of urban geysering, which is de ned as the sudden release of a mix of conveyed water and air through ventilation shafts. Figure 2.1 presents images of recent geyser episodes recorded in urban areas resulting from intense rain events. These images suggest a variety of public health and safety hazards including the release of poor quality water to the ground surface and the ooding of major roadways. Evidence of the relationship of geysering and the rapid lling of closed conduits may be found in works by [Guo and Song(1991)], [Nielsen and Davis(2009)] and [Wright et al(2011)]. 18 Figure 2.2: Numerical predictions for normalized free-surface level rise for di ering initial free-surface levels and ventilation tower diameters [Vasconcelos and Wright(2011)]. One of the clearest adverse impacts of air pocket entrapment is the increased potential for structural damage, caused by the great compressibility of air that leads to increased surging. The pioneer study by [Martin(1976)], which was followed by [Zhou et al(2002)] among others, indicates that compressed air pressure can greatly exceed the pressure that drives the ow prior to air compression, as displayed in Figure 2.3. [Zhou et al(2002)] points that such compression may be behind an episode in which an entire manhole structure was blown o the pipe in Edmonton, Alberta. This study also separates pressure oscillations patterns in to three types of behavior: Type 1 - Negligible Water Hammer E ect, Type 2 - Mitigated Water Hammer E ect, and Type 3 - Water Hammer Dominated. These patterns were separated by observing the pressure oscillations associated with changing the size of the ventilation ori ce located at the downstream end of the experimental apparatus. Figure 2.4 shows the pressure peaks associated with the most extreme of these behaviors, Type 3, which creates a great potential for structural damage. 19 Figure 2.3: Transient air pressure, liquid velocity, and air volume [Martin(1976)]. Figure 2.4: Pressure oscillation pattern for water hammer e ect, Type 3 [Zhou et al(2002)]. 20 Issues related to potential structural damage in rapid lling of closed conduits have also been explored by [Song et al(1983)], [Guo and Song(1991)], in the context of pressurization of large stormwater tunnels. Such earlier works, however, focused in a single phase framework for this medium description neglecting the in uences of air phase in the problem. A more recent work by [Vasconcelos and Leite(2012)] involving air-water interaction con rms earlier ndings by [Martin(1976)], but also shows that when relief is provided next to the location where air is compressed, as would be anticipated in actual tunnel geometries, the pressure rise can be signi cantly mitigated as presented in Figure 2.5. Another adverse impact related to air pocket entrapment is loss of conveyance and stor- age capacity in closed conduits. As shown as early as [Falvey(1980)], entrapped air pockets act as ow constrictions generating additional turbulence and hence energy losses. Also, large entrapped air pockets will occupy a volume in closed conduits that would otherwise be lled with water. This is a relevant issue in the context of stormwater storage tunnels since early pressurization of these systems can lead to more frequent episodes of combined sewer over ow (CSO). Release of entrapped air pockets can be a problematic issue as well. In the context of force mains one concern is the possibility of air slamming, de ned as the waterhammer- type pressures observed at the moment when entrapped air pockets are completely evacuated through air valves. The study by [Lingireddy et al(2004)] presents a formulation to calculate such peaks in terms of the diameter of the pipeline, the diameter of the air valve and the air pocket pressure. Another issue related to air release is geysering episodes, as sketch in Figure 2.6. The ini- tial theoretical framework for the investigation of geysers presented by [Guo and Song(1991)] was based on inertial oscillations of the water mass within closed con- duits. This single-phase approach assumed that when the hydraulic grade line (HGL) is above 21 Figure 2.5: Decrease of pressure peak for various slopes, valve obstructions, and air pocket volume by [Vasconcelos and Leite(2012)]. 22 Figure 2.6: Conceptual sketch by [Wright et al(2011)] of geyser during release of large air pocket: (a) pocket migrates toward vertical shaft; (b) momentum of air pocket into shaft due to buoyancy cause rise in water level and (c) high velocity air may entrain liquid due to ooding instability. grade geysers will occur. While this is correct, such an approach could not explain the vio- lence and intensity of the episodes observed in penstocks [Nielsen and Davis(2009)], in physi- cal modeling studies [Vasconcelos and Wright(2011)], and in eld observations of large trunk sewers [Wright et al(2011)]. Evidence of eld pressure measurements during an actual geyser- ing episode presented by [Wright et al(2011)] show that these strong geyser episodes occurred even when the HGL was far below grade. Experimental work by [Vasconcelos and Wright(2011)] has shown that the release of large air pockets through water- lled ventilation towers may lead to geysering, and the likelihood of these events increases signi cantly for smaller diam- eter ventilation towers. Experimental results shown in Figure 2.2 indicated that air pocket release through water lled shafts of smaller diameters increased geyser likelihood. 23 While these previous investigations are very relevant, a key problem that is less under- stood is the kinematics of these entrapped air pockets. The ability of describing the motion of entrapped pockets would be useful as it would help to assess the e ectiveness of air pocket removal in transmission mains. It would also assess the likelihood of uncontrolled air release through ventilation shafts in stormwater systems. 2.2 Experimental investigations on air pocket kinematics The experimental investigation presented by [Aimable and Zech(2003)] depicts the re- sults on the formation of an air cavity and intermittent ows in a small-scale laboratory sewer model (D = 0:144 m and S = 1:5% to 1:45%) that is similar to a reach of a real-world sewer system. The authors intended to mimic shear force mechanisms for pocket generation such as in the work by [Li and McCorquodale(1999)]. This work concludes that the behav- ior of an air cavity is in uenced by discharges and the rate of variation in the downstream boundary of the system. An important observation is that the velocities of the two ends of a mixed ow section are rather di erent during the air release process. [Perron et al.(2006)] presented a work investigating the in uence of surface inclination and bubble volume on the terminal velocity of relatively large air bubbles with volumes ranging from 0.3 to 0.9 cubic centimeters and a surface inclination varying from 2 to 10 degrees. An important note is that the apparatus used in these experiments was not a pipe, but an inclined plate as seen in Figure 2.7. The authors observed that the terminal velocity of a given bubble volume increased with inclination angle, and this increase was important for both low bubble volumes and low inclination angles. Both surface tension and viscous forces played a role for bubble of small volume at low inclination angles. For higher bubble volumes, the increase in terminal velocity followed a more linear relationship. These studies have a limitation in their ability to evaluate e ects such as air pocket spreading observed in horizontal slopes. 24 Figure 2.7: Schematic of experimental apparatus used by [Perron et al.(2006)]. [Glauser and Wickenhauser(2009)] provided an experimental study on the dynamics of an air cavity advancing in a pressurized pipe. The focus was on the shape and movement of single air bubbles in continuous air-water ow with the intention to determine the bubble volume that represents the stagnation velocity based on a balance of buoyancy and drag forces. It was noted that in favorably sloped pipes, volumes larger than this critical volume move against this ow due to buoyancy and volumes smaller than this critical volume would be dragged by the ow against buoyancy. Similarly to studies in hydraulic clearing of water mains, it was determined that pipeline slope, background water velocity, and bubble volume controls the observed pocket celerity. This investigation involved the use of slopes ranging from 1.7% to 8.7%, thus always involving conditions where buoyancy forces opposed ow drag forces. 25 2.3 Numerical investigations on air pocket kinematics Numerical studies have been presented in attempting to describe patterns related to the motion of entrapped air pockets in closed conduit ow. The majority of these stud- ies are in the realm of multi-phase ow applications, and exempli ed by the works of [Barnea and Taitel(1993)], [DeHenau and Raithby(1995)] and [Issa and Kempf(2003)]. In these applications there is a continuous injection of gas and liquid in closed conduits, and the mechanisms for pocket formation are diverse (e.g. terrain induced slugging) from the ones that are generally observed in water, wastewater and stormwater systems. [Issa and Kempf(2003)] presented the following equations to describe one-dimensional strat- i ed and slug ow, which are solved for the conservation of mass and momentum for both gas and liquid phases: @( g g) @t + @( g g g) @x = _ mb (2.1) @( l l) @t + @( l l l) @x = _mb (2.2) @( g g g) @t + @( g g 2g) @x = g @p @x + g ggsin +Fgw +Fi (2.3) @( l l l) @t + @( l l 2l ) @x = l @p @x l lg @h @x cos + l lgsin +Flw Fi (2.4) where g + l = 1 (2.5) The subscripts g, l, and i refer to the gas phase, liquid phase, and interface, respectively. Also, x is the axial coordinate, is the density, is the phase fraction, is the velocity, 26 _mb is the mass transfer per unit volume between the phases, p is the interface (and gas) pressure, is the pipeline angle, h is the height of the liquid surface (assumed to be at) above the pipeline bottom, and g is the gravitational acceleration. It is assumed that the liquid phase is incompressible, while the gas phase is considered compressible obeying the ideal gas law, and the ow is also assumed to be isothermal for simplicity. The second term on the right-hand side of equation 2.3 relates to the hydrostatic pressure in the liquid phase and is speci c to the strati ed and slug ow regimes. The F terms represent the frictional forces per unit volume between each phase and the pipeline wall, and between the phases themselves (at the interface). The study by [Li and McCorquodale(1999)] present mathematical framework for the simulation of ow regime transition (also referred to as mixed ows) using lumped inertia analysis and the ideal gas law, and consider the mechanism for air pocket formation based on shear ow instabilities presented by [Hamam and McCorquodale(1982)]. Figure 2.8 displays the stages in transition of free-surface to pressurized ow by [Li and McCorquodale(1999)]. In the formulation the location of the air-water interface is calculated explicitly, providing then means to compute the advance of an entrapped pocket. Yet, the velocity of the air bubble, rather than be calculated by the model, appears as a calibration parameter in the simulations. More recently [Trindade and Vasconcelos(2013)] presented a study that included a nu- merical framework to simulate pipeline priming operations considering the possibility of air pressurization using the Two-component Pressure Approach (TPA) following the work by [Vasconcelos and Wright(2009)]. With regards to air pressurization, this work included the possibility of air pressure variation along the length of the air pocket, which was computed using the isothermal Euler equations. The model updates the air-water interface using a source term for the Euler equations presented in [Toro(2009)]. The equations applied by the TPA model, exempli ed in the work of [Trindade and Vasconcelos(2013)], modify the 27 Figure 2.8: Stages of ow transition by [Li and McCorquodale(1999)]. Saint-Venant equations enabling them to simulate pressurized and free-surface ow regimes and are shown below: @U @t + @F(U) @x = S(U) (2.6) where U = 2 64A Q 3 75 F(U) = 2 66 4 Q (Q)2 A +gA(hc +hs) +gApipehair 3 77 5 S(U) = 2 64 0 gA(S0 Sf) 3 75 (2.7) 28 hair = 8 >>< >>: = 0 ! Free-surface ow without entrapped air pocket or pressurized ow 6= 0 ! Free-surface ow with entrapped air pocket (2.8) hs = 8> >< >>: 0 ! Free-surface ow a2 g (A Apipe) Apipe ! Pressurized ow (2.9) hc = 8> >>> >>< >>> >>> : D 3 3 sin sin3 3 cos 2 sin 2 ! Free-surface ow where = arccos [(y D=2)(D=2)] D 2 ! Pressurized ow (2.10) where U = [A;Q]T is conserved variables vector, A is the cross-sectional area of ow, Q is the ow rate, F(U) is the ux of conserved variables vector, g is gravitational acceleration, hc is the distance between the free surface and the centroid of the ow cross-section (limited to D 2 ), hs is the surcharge head, hair is the extra head due to entrapped air pocket pressurization, is the angle formed by free surface ow width and the pipe centerline, D is the pipeline diameter, Apipe is the pipeline cross-sectional area, and a is the celerity of acoustic waves in pressurized ow. Two modeling approaches were used for the air phase description. The rst one applies the isothermal Euler equation, whereas the second alternative assumes uniform air pressurization (UAPH model). Figure 2.9 indicates the model accurately predicted the trajectory of air-water interface including complex ow interactions, including interface breakdown of the pressurization in- terface. A limitation of that model is that the only mechanism that is accounted for the motion of the air-water interface is the displacement of air that is caused by the advancing water interface during the priming event. Also, it was assumed that one of the boundaries 29 Figure 2.9: Measured and predicted pressures at pipe crown for x =0.39, dorif<0.5, and slope 1% by [Trindade and Vasconcelos(2013)]. of the air pocket was xed at the air release valve. Another di culty for the simulation of entrapped pockets motion is that one dimensional models generally are constructed with the hypothesis of hydrostatic pressure distribution at the ow cross sections, which will not hold at the strongly curved air-water interfaces. 30 Chapter 3 Knowledge Gap and Objectives Although the previous studies in areas such as air pocket velocity, clearing velocity, air pocket movement and air pressure estimation have led to signi cant developments in the area of air-water interaction in stormwater storage tunnels, there still remains a signi cant knowledge gap in this eld. This knowledge gap may be summarized as follows: Gravity current investigations, such as those by [Benjamin(1968)], [Wilkinson(1982)], [Baines(1991)], [Zukoski (1966)], and [Gardner and Crow(1970)], describe motion of air-water interfaces but disregard e ects of ambient water ow. Air cavity intrusion into unsteady ow considered by [Vasconcelos and Wright(2008)] does not consider the case of discrete air pockets. Hydraulic clearing studies, exempli ed in the works of [Falvey(1980)], [Little et al(2008)], [Pothof and Clemens(2008)], [Pozos et al(2010b)], etc., have not focused on the kine- matics of entrapped air pockets for varying conditions of ambient ow and pipeline slope. Studies focused on the motion of entrapped air pockets presented by [Aimable and Zech(2003)], [Perron et al.(2006)], and [Glauser and Wickenhauser(2009)] do not provide information on air pocket spreading in horizontally sloped pipelines or images when drag and buoyancy forces are added. A better understanding of the kinematics of these air pockets in stormwater storage tunnels is needed so that better design guidelines can be proposed to avoid adverse conditions such as pressure surges, storage capacity loss, and geysering. 31 The main objective of this research is to explore a link between ambient ow velocity, pipeline slope and the celerity of entrapped air pockets of various volumes. This research explores these links through previously unexplored processes that include the following: Study the motion of entrapped air pockets with and without ambient ow velocity. Physically contain a discrete entrapped air pocket at a determined location with the use of knife-gate valves while allowing background ow to circulate. Focus on the kinematics of entrapped air pockets for varying adverse, horizontal, and favorable pipeline slopes as well as varying conditions of ambient ow. Explore entrapped air pocket motion when drag and buoyancy forces are opposing one another in very shallow and horizontal pipes. This work presents experimental results from a physical model investigation on the kine- matics of entrapped air pockets in pressurized water ows under shallow slopes. Discussion of the experimental results is presented along with conclusions and recommendations for future work. 32 Chapter 4 Methodology 4.1 General description and rationale of the experiments The apparatus for this experimental investigation was created from 102 mm clear PVC pipe supported by a steel frame which could have its slope adjusted manually and is pre- sented schematically in Figure 4.1. Two reservoirs of approximately equal volume, 0:63 m3, were secured upstream and downstream of the PVC pipeline. The upstream reservoir was connected to the pipeline by a control valve which could stop ow if necessary. Water was allowed to ow steadily underneath partially opened knife gate valves into the downstream reservoir, ensuring pressurized ow regime in the pipeline. A clear, acrylic cylinder was attached to the bottom of the PVC pipeline and connected to it through a ball valve in order to allow for air introduction into the system. Another valve was placed on this acrylic cylinder allowing atmospheric air into it. The experimental protocol was developed with the idea of allowing an air pocket of known volume into a closed conduit system while still allowing a known background ow to circulate. It was determined that once the air pocket was injected into the system, a new steady-state needed to be reached due to the extra head loss throughout the system caused by the addition of this pocket. A ruler was attached along the length of the pipeline in order to track the location of the air pocket front edge during experiments. Figure 4.1 presents a schematic of the apparatus used in the experimental procedure for horizontal and adverse (pipeline downstream end at a higher elevation than upstream end) pipeline slopes. These experiments were performed with a variable slope, clear PVC pipeline supplied upstream by a xed head reservoir with a throttled discharge downstream to ensure pressurized ow throughout the pipeline. The pipeline has a length L = 11:1 m 33 Figure 4.1: Sketch of apparatus used in horizontal and adverse pipeline slope experiments. and a diameter D = 101:6 mm. The upstream supply reservoir is attached to the pipeline by a 50 mm diameter ball valve. Two knife gate valves with the same diameter as that of the pipeline were positioned 3 m downstream of the upstream supply reservoir with a 2:1 m separation for adverse and horizontal sloped conditions. For favorable (pipeline downstream end at a lower elevation than upstream end) slopes, upstream and downstream reaches have lengths of 6 m and 3 m respectively, as shown in Figure 4.2. The apparatus was adjusted to allow video cameras to have a longer viewing range of air pocket pocket motion propagating upstream due to buoyancy forces. 34 Figure 4.2: Sketch of apparatus used in favorable pipeline slope experiments. Changes were only in the location of internal knife gate valves. Pre-determined volumes of air at atmospheric pressure were injected into the region between the two knife gate valves forming an air pocket. These partially closed knife gate valves prevented the motion of the air pockets while still allowing ow underneath. The speci c air volumes were determined by back-calculating the amount of water that lled the graduated acrylic tank located below the clear PVC pipeline once the valve connecting these pieces was opened. A valve connecting the graduated acrylic tank to the atmosphere ensured that the air was at atmospheric pressure when injected into the water- lled pipeline. The measurement devices used in the experimental procedure include: Up to four high de nition camcorders (1080 pixels, 30 FPS), providing a total view of 7:0 m upstream and downstream of the intermediate knife gate valves; 35 Two piezoresistive pressure transducers, MEGGIT Endevco 8510B-5 ( 1% accuracy), located at the upstream end (L1 = X=L = 0:0) and at approximately 50% of the pipeline length (L1 = 0:5), recording with a frequency of 100 Hz for each channel; National Instruments data acquisition board NI-USB 6210 with SignalExpress logging software; EXTECH digital manometers ( 0:3% accuracy) located at both ends of the apparatus and at 50% of the pipeline length (L1 = 0:5); and Cole-Palmer paddle wheel ow meter ( 1% accuracy) to gage ambient ow rate. 4.2 Experimental program The results presented in this paper include 99 di ering conditions. Seven slopes have been tested: horizontal slope, 0:5%, 1:0%, and 2:0% adverse slope(downstream end at higher elevation than inlet) and 0:5%, 1:0%, and 2:0% favorable slope (downstream end at lower elevation than inlet). For each of these slopes, three di erent ow rates ranging from 1:4 L/s to 4:0 L/s and up to four air pocket volumes were tested. These pocket volumes ranged between 1:3 L and 4:3 L for horizontal slopes and between 0:5 L and 4:0 L for adverse and favorable slopes. It is important to note that each case was repeated at least once in order to ensure that data was consistent. Table 4.1 presents a summary of the parameters tested in this investigation. Table 4.1: Experimental variables considered with respective tested ranges Experimental variable Range Flow rate (normalized by Q=pgD5) Three values ranging from Q =0 up to 0.388 Pipeline slope Horizontal slope 0.5%, 1.0%, and 2.0% adverse slopes 0.5%, 1.0%, and 2.0% favorable slopes Air pocket volume (normalized by D3) Horizontal: up to 4 values in the range Vol =1.2-4.1 Adverse: up to 4 values in the range Vol =0.48-3.8 Favorable: up to 4 values in the range Vol =0.48-3.8 36 Conditions that merited testing included those that represented a wide range of air pocket volumes, pipeline slopes, and background ow rates that stormwater storage tunnels could encounter during rapid lling after an intense rain event. Air pocket volume was determined by back-calculating the amount of water that owed from the PVC pipeline into the acrylic cylinder when the control valve was opened to inject air. The experimental procedure was performed as follows: 1. Set the selected slope for the pipeline apparatus; 2. Create steady ow conditions in the PVC pipeline by turning on submersible pumps at desired ow 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 gradu- ated acrylic cylinder and the water lled PVC pipeline; 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; 5. Initiate pressure measurements and collect readings on calibration manometers; 6. Open the intermediate knife gate valves simultaneously and rapidly (t< 0:6s) to allow sudden release of air pocket; 7. Record motion and spreading of the air pocket(s) with camcorders until moving outside eld of view; and 8. Close valves and stop submersible pumps after considerable time and make nal read- ings on the calibration manometers. It was noted visually that a pronounced drag created by larger ow rates caused bits of some air pockets to be dragged underneath the knife gate valves intended to keep the pocket in place as a whole until the start of the experiment. In order to keep this phenomenon 37 from happening, certain runs with larger air pocket volumes and larger ow rates were not performed. Along with the ow rates and air pockets volumes mentioned above, experiments were run to study the thinning of the air pocket when a single knife gate valve was opened with no background ow. 4.3 Data analysis Data processing was completed by watching the video recordings of each experimental run frame by frame. The time that the pocket front reached every one foot marker on the scale attached to the pipeline apparatus was recorded. Using these time and pocket front location values, the trajectory was plotted for each experimental run. The celerity, C, and relative celerity, C vflow, values were also plotted for analysis. Piezoresistive pressure transducer data was converted into useful information (e.g. pressure oscillations) by using the linear relationship between force/pressure, strain, and return voltage gaged at the transducer membrane. Using the calibration manometers, this linear relationship was used to predict the varying pressures throughout the system during experimental runs based on the return voltage recorded by the transducers. 4.4 Comparison with numerical model Results from a numerical model based on internal gravity current modeling theory, de- veloped by [Hatcher et al(2013)], were compared to the horizontal (no slope) results acquired during the experimental investigation. This model utilizes an integral model approach as well as a relationship between the gravity current thickness (in this case, air pocket thick- ness) and its propagation speed to update the location of the leading edge of the pocket. This model accounts for surface tension using the approach outlined in [Wilkinson(1982)]. The e ects of ambient cross ows are incorporated into this integral model using the front condition provided in [Hallworth et al(1998)]. For the following integral model formulation, this expression has been adapted to circular cross-sections: 38 uf = dxfdt = FrpgD +U (4.1) where xf is the distance traveled by the respective cavity front, Fr is the local Froude number at the upstream and/or downstream air pocket front, D is the pipe diameter and U is the background ow velocity. Two separate integral models, M1 and M2, that di er in front condition selection were analyzed. These models account for buoyancy, drag, background ows, and surface tension (M1). Further model details are not presented here for brevity. 39 Chapter 5 Results and Discussion As stated earlier, a main research objective for this investigation was to further explore a link between ambient ow velocity, pipeline slope and the celerity of entrapped air pockets of various volumes, including e ects of air pocket spreading. The following sections present the results of this experimental investigation, rst for cases with no ambient ow and then for cases with ambient ow. The small discrepancies in air pocket volumes for cases involving horizontal slope and favorable/adverse slopes does not a ect the overall analysis presented below. 5.1 Air pocket spreading in horizontal slope Figure 5.1 presents the trajectories and celerity values of the air pocket front for four di erent air pocket volumes for horizontal slopes without ambient ow. All velocity results (celerity, relative celerity) values on gures are normalized by pgD 1.0 m/s; trajectory coordinates are normalized by the original pocket length; and time is normalized by pD=g 0.102 s. These results show a slightly curved trajectory of the air pocket leading edge for all the tested volumes. This curvature indicates a reduction in the leading edge celerity, caused by the air pocket spreading and reducing of the air pocket thickness consistent to results by [Benjamin(1968)]. The slope of the air pocket?s leading edge trajectory decreased for larger air pocket volumes, indicating an increase in celerity, an expected result. It can be noticed that the trajectory of the back-propagating front is symmetrical to the forward propagating front in the absence of ow in the pipe, also as anticipated. 40 Figure 5.1: Trajectory of the air pocket leading edge and observed celerity for horizontal slope and various air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. 41 Results are signi cantly a ected by the presence of ambient ows. Figure 5.2 presents the trajectories for the cases with air pocket volumes of Vol =1.2 and Vol =2.2 and hor- izontal slope for various ambient ows. Celerity values C =C=pgD are complemented by celerity values relative to the ambient ow (C - vflow=pgD). The symmetry between the two air pocket fronts previously observed for horizontal case is no longer observed due to drag forces caused by the introduction of ambient ow. The results in Figure 5.2 indicate that larger ow rates result in an increase in the celerity of the forward moving air pocket front, and that this gain in velocity is proportional to the ambient ow velocity. Results also indi- cate that in the initial stages following the pocket release there is an increase in the celerity that is generally over by time t 2. The previously observed air pocket front spreading e ect in the horizontal slope and no ambient ow, which led to decreasing celerity values, is not observed is when there is ambient ows. Results in Figure 5.2 support the assumption that for horizontal pipes the pocket celerity in ambient ow conditions can be estimated as the summation of its value in quiescent conditions plus the ambient ow velocity; however, this estimation is only valid after a certain time where the pocket accelerates from zero to a steady velocity. An interesting result is that is even for the smallest tested ambient ow, the backward propagating air pocket leading edge is eventually sheared and no longer observed for values of X1 < 0:3 and t 1. Also, it is important to note that the v flow from ambient ow was smaller than the initial C of the backward moving air pocket. This indicates that in the absence of buoyancy forces the propagation of discrete air pockets against an ambient ow is short lived. Figure 5.3 presents seven snapshots of the pocket propagation with 1 second time lapse between the images, for a condition corresponding to a pocket with Vol =4.1, ow rate Q =0.27, ow velocity v flow=0.34, and C (initial) 0:68. Air-water interfaces in the gure are arti cially enhanced for clarity. The initial shape of the front resembles a typical, dissipative gravity current with a curved nose with a trailing a hydraulic jump, as described in [Benjamin(1968)]. As the front moved upstream, the trailing hydraulic jump entrained 42 Figure 5.2: Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with horizontal slope, ambient ow, and Vol =1.2 and Vol =2.2. 43 air at the leading edge of the pocket, and these air bubbles were carried with the ambient ow ow. The pocket volume thus decreased over time and within 7-8 seconds it e ectively vanished. In the process, the observed pocket celerity decreased as the thickness of the backward propagating front decreased, a result consistent with gravity current theory. The other extremity of the front did propagate as a curved gravity current front. One speculates that this type of pocket shearing may also be observed in deep stormwater storage tunnels laid in shallow slopes. This behavior was not observed for favorable slopes, when buoyancy forces oppose drag forces, as is explained below. 5.2 Air pocket spreading in adverse slopes Results for adverse slopes (downstream end at a higher elevation than the upstream end) and no ambient ows are presented Figure 5.4. Trajectories and celerity values of the leading edge of the moving air pockets are shown for various volumes as well as various adverse pipeline slopes. The results do not present the curved trajectory from the air pocket leading edge that was present in horizontal cases even for the smallest slope tested of 0.5%. This indicates that air pocket spreading wasn?t as pronounced and that the pocket leading edge was not signi cantly a ected in the experiments. Another observation of this gure indicates the relative celerity di erence between pocket of various volumes becomes less pronounced for larger pipeline slopes. That may indicate that larger slopes lead to more prominent concentration of air at the leading edge of the air pocket as it propagates, contributing to the reduction of these celerity values. Conversely, the pocket overall length for these smaller volume conditions was visibly shorter. Results for the smallest air pocket (Vol = Vol=D3) and 2% show a much smaller celerity than corresponding cases with shallower slopes; this could be due to friction with pipeline walls which was more important for these small pocket volumes. However, for most tested cases, the relative celerity C = C=pgD is limited by 0.47, which is consistent with results presented by [Baines(1991)]. 44 Figure 5.3: Sequence of snapshots illustrating the trajectory of the backward moving pocket trajectory for Q*=0.26, Vol*=4.1 and horizontal slope, illustrating the shearing process. Interface between air and water is enhanced. 45 Figure 5.4: Tra jectory of the air po cket lea ding ed ge and observ ed celerit yfor various adv erse slop es and air po cket volumes, no am bien t o w. Negativ eco ordinates indicate po ckets are propagating tow ard upstream. 46 Figure 5.5 presents the trajectories, observed celerity C normalized by pgD, and the relative celerity accounting for ambient ow (C - vflow) also normalized by pgD, for the cases with Vol =0.95 and Vol =1.9 and 1% adverse slope. As anticipated, the observed air pocket front celerity increases with the ow rate, but changes in celerity between the two tested pocket volumes was comparatively smaller. In such slopes buoyancy and drag forces are summed and the air pocket celerity can also be approximated by the summation of the ambient ow velocity and the celerity observed in quiescent conditions. An exception to this observation was the case with Vol =0.95 and the highest ambient ow Q =0.37. An explanation of this observation is not available at this point. 5.3 Air pocket spreading in favorable slopes This condition corresponded in the presence of ambient ow to the most complex case studied in this investigation due to the opposition between buoyancy and drag forces. Is it important to note that the conditions tested in this investigation did not involve ow rates that would promote the clearing of the pipeline (hydraulic clearing). The trajectories and celerities of the leading edge of the moving air pockets were plotted for various air pocket volumes and various favorable pipeline slopes without ambient ow. The results displayed in Figure 5.6 for all favorable sloped cases indicate little dependence on pipeline slope toward the trajectory for the largest pocket volume tested. As slope increased, the trajectory of the smaller pockets? leading edge approached that of the largest pocket; however, this was not the case for the smallest tested pocket of Vol =0.48, which was consistently slower. These ndings are consistent with those presented by [Benjamin(1968)], indicating that for smaller pocket thickness celerity values should decrease signi cantly. Another notable observation is that for shallower slopes, air pockets spread over larger reaches of the pipeline. These results are qualitatively similar to those presented in Figure 5.4 for adverse slopes and no ambient ow. It should be noted that the detection of the back extremity of the air pocket was complicated by air pocket fragmentation into smaller pockets that was sometimes observed 47 Figure 5.5: Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% adverse slope, ambient ow, and Vol =0.95 and Vol =1.9. 48 at these locations. Figure 5.7 shows a comparison between the front trajectories for adverse and favorable sloped conditions. Figure 5.6: Tra jectory of the air po cket leading edge and observ ed cele rit y for various fav orable slop es and air po cket volumes, no am bien t o w. Negativ eco ord inates indicate po ckets are propagating tow ard upstream. 49 Figure 5.7: Trajectory of the air pocket leading edge for 2% (chart a), 1% (chart b) and 0.5% (chart c) adverse and favorable slopes and various air pocket volumes, no ambient ow. Negative coordinates indicate pockets are propagating toward upstream. 50 As mentioned, the most complex cases studied in this work involved favorable slopes with background ow, when buoyancy forces opposed drag forces. It was observed visually that the leading edge of the pockets advanced upstream against the ambient ow due to buoyancy in all tested slopes. Strong turbulence was observed at the trailing hydraulic jump behind the air pocket front. Air was sheared from the air pocket leading edge, but consistently with observations by [Pozos(2007)] and others, these smaller bubbles gathered downstream from the main pocket, and eventually grew in size and moved against the ow rejoining with the main pocket. This phenomenon is referred to as "blowback" by [Falvey(1980)] and others and has been linked to structural damage. A sample of these results is presented in Figure 5.8 that corresponds to the case where pocket volumes Vol =0.95 and Vol =1.9 and 1% favorable slope. Negative celerity values are due to the propagation direction of the pocket toward the upstream end of the apparatus. The observed celerity C values for these favorable slopes, as anticipated, are smaller than the case with no ambient ow and ranged from -0.2 to -0.4 for all tested conditions. However, the assumption that these celerity values corresponded to the summation of celerity in quiescent conditions and the ambient ow conditions is no longer valid, a result that is likely linked to these complex air-water interactions observed in favorable slopes (e.g. shearing) that prevent pocket sizes to remain steady. These ndings were consistent for all of the pocket volumes tested. Figure 5.9 helps demonstrate this di erence comparing the trajectories, observed celer- ity, and relative celerity accounting for ambient ow for the case when Vol =1.9 and slope is either 1% adverse or 1% favorable. Appendix A presents example sets of experimental results for 0.5% favorable and adverse slopes with Q 0.12, 0.25, and 0.37 as well as 2% favorable and adverse slopes with Q 0.12, 0.25, and 0.37, which are very similar to those discussed in this previous section. In summary, air pocket kinematics in favorable slopes and ambient ow conditions still demand further investigation. Due to the complexity of air-water interaction in such condi- tions, pocket volume varies and so does celerity. While the general assumption that observed 51 Figure 5.8: Trajectory of the air pocket leading edge, observed celerity and relative celerity for conditions with 1% favorable slope, ambient ow, and Vol =0.95 and Vol =1.9. 52 Figure 5.9: Trajectory of the air pocket leading edge (chart a), observed celerity (chart b) and relative celerity (chart c) for conditions with 1% adverse and 1% favorable slopes and ambient ow for Vol =1.9. celerity is not the summation of ambient ow velocity and pocket celerity in quiescent con- ditions, this relative celerity is in the range 0:4 and 0:6 for all tested cases. Larger scale studies should attempt to validate this estimate for relative celerity values. 5.4 Air pocket motion and spreading compared to numerical model prediction A relevant question is to what extent can these experimental results be replicated by nu- merical models based on gravity current theory framework. Figure 5.10 shows the air pocket front trajectories simulated with both numerical models developed by Thomas Hatcher at 53 Auburn University (please see 4) for all horizontal ow experiments. The experimental re- sults are greatly a ected by background ow, and the e ect of air pocket volumes on front velocity is also signi cant (boundary e ects and surface tension are more important for smaller pocket volumes). Without background ow, M1 performs better in comparison with experimental results. M2 consistently underestimates the air pocket leading edge celerity, and this underestimation increases for larger air pocket volumes. M2 is the more accurate model for the smallest air pocket volumes with background ow, although both models perform well. The accuracy of the M1 model seems to be una ected by di ering background ows and pocket volumes, which suggests that the surface tension parameters used by [Wilkinson(1982)] perform well for circular as well as rectangular cross-sections. Both integral models over-predict the air pocket front velocity during the initial stages of simulation. This can be attributed to that fact that the formulations neglect the initial shear stresses caused and the anticipated local acceleration of water upon the complete opening of the knife gate valves that were keeping the air pocket in place. Once these shear stresses are no longer a ecting the air pocket motion (at about t 20 30), M1 slightly under-predicts the air pocket leading edge celerity, but M2 keeps providing a larger under-prediction. 54 Figure 5.10: Air front trajectory comparison between experiments and both integral models for various background ows and pocket volumes: a) Vol = 1:27, b) Vol = 2:22, c) Vol = 3:18, and d) Vol = 4:13. The solid lines represent the integral model predictions, and the data markers represent experimental values. 55 Chapter 6 Conclusions and Future Work This work presented results on an investigation of the motion of entrapped air pockets in water pipes. The focus was on the kinematic aspects of air pocket motion and air-water interactions related to the motion of these discrete, nite volume pockets. A long term goal of this research is to incorporate into ow regime transition models the ability to track entrapped pockets and prevent negative impacts related to these events in stormwater storage tunnels and transmission mains. An experimental program was proposed where di erent conditions of pipe slope and ambient ow rates were tested. Use of clear PVC pipe enabled tracking the coordinates of entrapped pockets over time for each tested condition. In addition to the experimental studies, a non-Boussinesq, integral model was used to simulate some of the conditions tested in the experiments. The purpose of this gravity current model was the possibility of such a simple, computationally inexpensive modeling approach being included as a sub-model in a more complex ow regime transition model. The ndings of this research may be summarized as follows: The celerity of discrete air pockets in stagnant ow conditions depended on air pocket volume, particularly for shallower slopes (below 0.5%), in agreement with [Benjamin(1968)] and [Wilkinson(1982)] observations. For stronger slopes, celerity values were not as dependent on volumes as air accumulation at the leading edge of the air pocket reduced di erences on the pocket thickness; For horizontal slope and stagnant conditions, the air pocket celerity values slightly decreased over time, indicating that the air pocket spreading decreased the leading edge thickness, resulting in smaller celerity values; 56 It was observed that the propagation of air pockets in horizontal slope pipes against ambient ows is short lived due to the shearing of the air pockets in the hydraulic jumps that trailed the leading edge of the air pocket even when ambient velocities are smaller than air pocket celerity; In ambient ow conditions, the celerity of the air pocket leading edge in horizontal and adverse slopes was well approximated by the summation of celerity values in quiescent conditions and the ambient ow velocity. Such an approximation did not hold for air pocket propagation in favorable slopes where buoyancy forces opposed drag forces; Air pocket propagation in favorable slopes was slowed down by ambient ows, and shearing of the leading edge of the main air pocket by the trailing hydraulic jump was observed. However, these sheared bubbles regrouped in larger bubbles/pockets downstream from the main air pocket. As these bubbles grew to a certain size, they would have enough velocity to overcome the ambient drag and eventually rejoin the main air pocket. Predictions of the air pockets leading edge coordinate yielded by proposed numeri- cal models compared fairly well with experiments with horizontal slopes, with and without ambient ow. The M1 model (based on the work of [Benjamin(1968)] and [Wilkinson(1982)]) yielded better results for large air pocket volumes and smaller am- bient ows. The M2 model results were more accurate only for the smallest air pocket volumes (Vol = 1:2) and higher ambient ows. This work also highlighted points that should be addressed in future investigations. One is the development of similar studies using other pipe diameters to assess scale e ects on these ndings. Experiments with 50 mm and 200 mm pipes are planned in the future to complement ndings presented in this work. Experimental studies that involve both ow regime transition and entrapment of air pockets are also planned to help assess the e ectiveness of such a combination in controlled laboratory conditions. Further numerical 57 investigations, including the use of computational uid dynamics (CFD) packages, would also be useful. Operational di culties with the experimental setup that could be addressed in the future include: 1)development of uniform mechanism/procedure to close knife gate valves to keep the air pocket from moving before the beginning of each experimental run; 2) better system for opening both knife gate valves simultaneously to release the pocket in both the upstream and downstream directions; and 3) developing a mechanism that would allow small bubbles of air occasionally admitted from the upstream reservoir to be expelled prior to experimental runs. It would also be helpful in the future to use all data collected during experimental runs. Pressure monitoring that occurred throughout the system can be used to assess the amount of energy loss caused by the partially closed knife gate valves holding the pocket in its initial position. Also, measurements of the air pocket thickness in selected cases would be valuable information since there is evidence in literature of the relationship between pocket thickness and celerity during motion. 58 Bibliography [Aimable and Zech(2003)] Aimable, R. and Zech, Y. (2003). Experimental results on tran- sient and intermittent ows in a sewer pipe model. Proc. of 30th IAHR Congress, Vol. B, Thessaloniki, Greece: 377{384, 2003. [Baines(1991)] Baines, W.F. (1991). Air cavities as gravity currents on slope. J. Hydr. Eng.,117(12): 1600{1615, 1991. [Barnea and Taitel(1993)] Barnea, D. and Taitel, Y. (1993). Model for slug length distribu- tion in gas-liquid slug ow. Int. J. Multiphase Flow,19(5): 829{838, 1993. [Benjamin(1968)] Benjamin, T.B. (1968). Gravity currents and related phenomena. J. Fluid Mech., 31: 209{248, 1968. [DeHenau and Raithby(1995)] DeHenau, V. and Raithby, G.D. (1995). Transient two- uid model for the simulation of slug ow in pipelines. Int. J. Multiphase Flow,21(3): 335{ 349, 1995. [DeMartino et al (2008)] De Martino, G., Fontana, N. and Giugni, M. (2008). Transient ow caused by air expulsion through an ori ce. J. Hydr. Eng.,134(9): 1395{1399, 2008. [Falvey(1980)] Falvey, H.T. (1980). Air-water ow in hydraulic structures. USBR Engineer- ing Monograph, n. 41 [Gardner and Crow(1970)] Gardner, G.C. and Crow, I.G. (1970). The motion of large bub- bles in horizontal channels. J. Fluid Mech., 43: 247{255, 1970. [Glauser and Wickenhauser(2009)] Glauser, S. and Wickenhauser, M. (2009). Bubble move- ment in downward-inclined pipes. J. Hydr. Eng., 135(11): 1012{1015, 2009. [Guo and Song(1991)] Guo, Q. and Song, C.S.S. (1991). Dropshaft hydrodynamics under transient conditions. J. Hydr. Eng., 117(8): 1042{1055, 1991. [Hallworth et al(1998)] Hallworth, M.A., Hogg, A.J., and Huppert, H.E. (1998). E ects of external ow on compositional and particle gravity currents. J. Fluid Mech., 359: 109{142, 1998. [Harris et al(2001)] Harris, T.C., Hogg, A.J. and Huppert, H.E. (2001). A mathematical framework for the analysis of particle-driven gravity currents. Proc. R. Soc. Lond., A(457): 1241{1272, 2001. 59 [Hamam and McCorquodale(1982)] Hamam, M.A. and McCorquodale, A. (1982). Transient conditions in the transition from gravity to surcharged sewer ow. Can. J. Civ. Engrg, (9): 189{196, 1982. [Hatcher et al(2013)] Hatcher, T.M., Chosie, C.D., and Vasconcelos, J.G. (2013). Modeling the motion and spread of air pockets within stormwater sewers. CHI, 2013. [Issa and Kempf(2003)] Issa, R.I. and Kempf, M.H.W. (2003). Simulation of slug ow in horizontal and nearly horizontal pipes with the two- uid model. Int. J. Multiphase Flow, 29(1): 69{95, 2003. [Kalinske and Bliss(1943)] Kalinske, A.A. and Bliss, P.H. (1943). Removal of air from pipe lines by owing water. Proc. Am. Soc. Civ. Eng. (ASCE), 13(10), 1943. [Kalinske and Robertson(1943)] Kalinske, A.A. and Robertson, J.M. (1943). Closed conduit ow. Proc. Am. Soc. Civ. Eng. (ASCE), 108: 1435{1447, 1943. [Lauchlan et al(2005)] Lauchlan, C.S., Escarameia, M., May, R.W.P., Burrows, R. and Ga- han, C.(2005). Air in pipelines - A literature review. HR Wallingford Technical Report SR., 649. [Lautenbach et al(2008)] Lautenbach, D. J., Vasconcelos, J. G., Wright, S. J., Wolfe, J. R., Cassidy, J. F. and Klaver, P. R. (2008). Analysis of transient surge in the proposed District of Columbia Water and Sewer Authority deep tunnel system. Proc., 2008 WEF Collection Systems Conference, Pittsburgh, PA., 2008. [Lautenbach and Klaver(2010)] Lautenbach, D. J. and Klaver, P. (2010). Experience of built tunnel systems with surges, transients and geysering. Memorandum from LimnoTech Inc. to London Tideway Tunnels Delivery Team, 2010. [Li and McCorquodale(1999)] Li, J. and McCorquodale, A. (1999). Modeling Mixed Flow in Storm Sewers. J. Hydr. Eng., 125(11): 1170{1180, 1999. [Lingireddy et al(2004)] Lingireddy, S., Wood, D.J. and Zloczower, N. (2004). Pressure surges in pipeline systems resulting from air releases. J. AWWA, 96(7): 88{94, 2004. [Little et al(2008)] Little, M.J., Powell, J.C. and Clark, P.B. (2008). Air movement in water pipelines - some new developments. Proc., 10th Int. Conf. on Pressure Surges, BHR, Edinburgh, UK, 111{122, 2008. [Martin(1976)] Martin, C.S. (1976). Entrapped air in pipelines. Proc., 2nd Int. Conf. on Pressure Surges, BHR, Bedford, England, 15{28, 1976. [Nielsen and Davis(2009)] 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. [Perron et al.(2006)] Perron, A., Kiss, L.I. and Poncsak, S. (2006). An experimental in- vestigation of the motion of single bubbles under a slightly inclined surface. Int. J. Multiphase Flow, 32: 606{622, 2006. 60 [Pinott and Moller(2011)] Pinott, M. and Moller, G. (2011). Intake structure of the Handeck 2 power plant (Switzerland), physical model investigation. Swiss Federal Institute of Technology, Zurich, 2011. [Pothof and Clemens(2008)] Pothof, I. and Clemens, F. (2008). Air migration analysis of the Terror Lake tunnel. Proc. 11th Int. Conf. Urban Drainage, Edinburgh, Scotland, 2008. [Pothof and Clemens(2010a)] Pothof, I. and Clemens, F. (2010). Interface drag on plugs in downward sloping pipes. Proc. 7th Int. Conf. Multiphase Flows, Tampa, Florida: 1{6, 2010. [Pothof and Clemens(2010b)] Pothof, I. and Clemens, F. (2010). On elongated air pockets in downward sloping pipes. J. Hydr. Res., 48(4): 499{503, 2010. [Pothof and Clemens(2012)] Pothof, I. and Clemens, F. (2012). Air pocket removal from downward sloping pipes. Proc. 9th Int. Conf. Urban Drainage Modeling, Belgrade, Serbia, 2012. [Pozos(2007)] Pozos, O.E. (2007). Investigation on the e ects of entrained air in pipelines. PhD Thesis, University of Stuttgart, 2007. [Pozos et al(2010a)] Pozos, O., Giesecke, J., Marx, W., Rodal, E.A. and Sanchez, A. (2010). Experimental investigation of air pockets in pumping pipeline systems. J. Hydr. Res., 48(2): 269{273, 2010. [Pozos et al(2010b)] 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. [Simpson(1997)] Simpson, J.E. (1997). Gravity currents in the environment and the labora- tory. Cambridge University Press, 1997. [Song et al(1983)] Song, C.S.S., Cardle, J.A. and Leung, K.S. (1983). Transient mixed- ow models for storm sewers. J. Hydr. Eng., 109(11): 1487{1504, 1983. [Toro(2009)] Toro, E. (2009). Riemann solvers and numerical methods for uid dynamics: a practical introduction. Springer Verlag. [Townson(1991)] Townson, J.M.(1991) Free-surface hydraulics. Cambridge University Press. [Trindade and Vasconcelos(2013)] Trindade, B.C. and Vasconcelos, J.G. (2013) Modeling of water pipelines lling events accounting for air phase e ects. J. Hydr. Eng., (): 1943{7900, 2013. [Vasconcelos and Leite(2012)] Vasconcelos, J.G. and Leite, G.M. (2012). Pressure surges following sudden air pocket entrapment in stormwater tunnels. J. Hydr. Eng., 138(12): 1080{1089, 2012. 61 [Vasconcelos and Wright(2006)] 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 and Wright(2008)] Vasconcelos, J.G. and Wright, S.J. (2008). Rapid ow startup in lled horizontal pipelines. J. Hydr. Eng., 134(7): 984{992, 2008. [Vasconcelos and Wright(2009)] Vasconcelos, J.G. and Wright, S.J. (2009). Rapid lling of poorly ventilated stormwater storage tunnels. J. Hydr. Res., 47(5): 547{558, 2009. [Vasconcelos and Wright(2011)] Vasconcelos, J.G. and Wright, S.J. (2011). Geysering gen- erated by large air pockets released through water- lled ventilation shafts. J. Hydr. Eng., 135(5): 543{555, 2011. [Wickenhauser and Kriewitz(2009)] Wickenhauser, M. and Kriewitz, C.R. (2009). Air-water ow in downward inclined large pipes. Proc., 33rd IAHR Congress, IAHR, Vancouver, BC, 5354{5361, 2009. [Wilkinson(1982)] Wilkinson, D.L. (1982). Motion of air cavities in long horizontal ducts. J. Fluid Mech., 118: 109{122, 1982. [Wright et al(2011)] Wright, S.J., Lewis, J.W. and Vasconcelos, J.G. (2011). Geysering in rapidly lling storm-water tunnels. J. Hydr. Eng., 137(5): 543{555, 2011. [Zhou et al(2002)] Zhou, F., Hicks, F.E. and Ste er, P.M. (2002). Transient ow in a rapidly lling horizontal pipe containing trapped air. J. Hydr. Eng., 128(6): 625{634, 2002. [Zukoski (1966)] Zukoski, E.E. (1966). In uence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech., 25(4): 821{837, 1966. 62 Appendices 63 Appendix A Additional Experimental Results for Trajectory and Celerity 64 Figure A.1: Trajectory and celerity for 0.5% adverse slope and Q 0.12. 65 Figure A.2: Trajectory and celerity for 0.5% adverse slope and Q 0.25. 66 Figure A.3: Trajectory and celerity for 0.5% adverse slope and Q 0.37. 67 Figure A.4: Trajectory and celerity for 0.5% favorable slope and Q 0.12. 68 Figure A.5: Trajectory and celerity for 0.5% favorable slope and Q 0.25. 69 Figure A.6: Trajectory and celerity for 0.5% favorable slope and Q 0.37. 70 Figure A.7: Trajectory and celerity for 2% adverse slope and Q 0.12. 71 Figure A.8: Trajectory and celerity for 2% adverse slope and Q 0.25. 72 Figure A.9: Trajectory and celerity for 2% adverse slope and Q 0.37. 73 Figure A.10: Trajectory and celerity for 2% favorable slope and Q 0.12. 74 Figure A.11: Trajectory and celerity for 2% favorable slope and Q 0.25. 75 Figure A.12: Trajectory and celerity for 2% favorable slope and Q 0.37. 76