Experimental Study on the Compression of Entrapped Air Pockets in  
Stormwater Tunnel Systems 
 
by 
 
Gabriel Maciel Leite 
 
 
 
 
A thesis submitted to the Graduate Faculty of  
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
August 4, 2012 
 
 
 
 
Keywords: stormwater hydraulics, air pockets, experimental investigation, numerical modeling 
 
 
Copyright 2012 by Gabriel Maciel Leite 
 
 
Approved by 
 
Jos? Goes Vasconcelos Neto, Chair, Assistant Professor of Civil Engineering 
Prabhakar Clement, Professor of Civil Engineering 
Xing Fang, Associate Professor of Civil Engineering 
 
 
 
 
 
 
 
 
ii 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
To Luana, which she gave strength and protection 
 when I needed the most.     
iii 
 
Abstract 
 
 
Stormwater storage tunnels may undergo rapid filling pipe conditions during extreme rain 
events. Such conditions are relevant as adverse conditions may develop, such as surging due to 
entrapment and compression of entrapped air pockets. Operational issues such as structural 
damage, geysering and return of conveyed water to grade, among others, have been linked to air 
pocket entrapment. This work presents results from experimental investigations on pressure 
surges caused by sudden air pocket entrapment. Steady flow was supplied in a pipeline in such a 
way that pressurized flows existed at the upstream end while the downstream end experienced 
free surface flow due to free discharge conditions at the downstream end. A combination of flow 
rates and slopes resulted in several gradually varied flow profiles at the discharge, with various 
volumes of atmospheric air at the discharge end. By sudden closing the downstream discharge 
valve, an air pocket was entrapped and surges were recorded. To emulate conditions in which 
surge relief is provided during air pocket compression, valve maneuvering also included cases 
with partial valve obstruction. Among the obtained results, one can notice significantly different 
surges for cases with and without pressure relief (e.g. with total or partial obstruction), and that 
the larger is the obstruction degree the larger are the surges. Such findings are useful in the 
development of numerical models to simulate rapid filling of stormwater tunnels incorporating 
these findings. 
 
 
 
 
 
 
 
 
 
 
 
 
iv 
 
Acknowledgement 
 
 I will begin to be intensely grateful to my advisor, Doctor Vasconcelos, for giving the 
time I need for my recuperation and to have so much confidence in my research. Thank you so 
much, professor, for believe that I would pass this obstacle in my life. I would also thank my 
friend, Kyle Moynihan, for helping me to build the apparatus experimental and collecting the 
data. 
 I would like to express my genuine gratitude to my family (Salom?o, Ciene, Osmarina, 
Louise and Lorena) that without your security and support, I will have never achieved and 
succeeded. I will be enormous grateful to the Doctor Clement and to your church for providing a 
place for my parents and my friends to stay and helping so much in times of need. Thank you 
very much to all the people (Bernardo, David, Scoot, Sherry, Matthew, Kyle, Sushban, Nirajan, 
Manoj, Tom Hart, Luana, Lenilza, Salom?o, Ciene, Jos? Goes, Denise, Derong, Xiaolong, Sam 
and if I forgot somebody, please forgive me) who visited in the hospital and the rehabilitation 
center, I really appreciated all your kindness. I would be hugely grateful to everyone that helped 
me get up and continue to live my life. Thank you all very much for my friends and my family 
without you I wouldn't be living this fulfilling moment of my graduation. 
Luana, my dear friend, you showed me that is possible to realize my dreams and 
ambitions. You transformed my life without I even noticed. I already found someone with best 
kindness and immense sympathy in the world. My luck and my destiny is to have known you.  
 
 
 
 
v 
 
Table of Contents 
 
Abstract .......................................................................................................................................... iii 
Acknowledgement ......................................................................................................................... iv 
List of Figures ............................................................................................................................... vii 
List of Tables ................................................................................................................................ xii 
Chapter one - Introduction .............................................................................................................. 1 
1.1. Stormwater systems described in two-phase flow framework ......................................... 2 
1.2 Modeling Framework of rapid filling pipe problems ............................................................ 6 
1.2.1. Air phase modeling ........................................................................................................ 6 
1.2.2. Water phase modeling ................................................................................................... 7 
1.3. Thesis Outline .................................................................................................................... 13 
Chapter two - Literature Review ................................................................................................... 14 
2.1 Air cavity motion and shape in closed conduit flows ......................................................... 14 
2.2. Rapid filling flows in closed conduits and effects of entrapped air ................................... 15 
2.3. Numerical modeling of stormwater drainage systems in extreme flow conditions ........... 22 
Chapter three - Knowledge gap and Objectives ........................................................................... 26 
Chapter four - Methodology ......................................................................................................... 28 
4.1. Experimental apparatus construction ................................................................................. 28 
4.2. Experimental program ........................................................................................................ 32 
4.2.1. Experimental procedure ............................................................................................... 32 
4.2.2. Experimental variables and tested range ..................................................................... 35 
4.3. Numerical Model ? Lumped Inertia Method ..................................................................... 36 
vi 
 
4.3.1. Mathematical model: Water phase momentum equation ............................................ 36 
4.3.2. Mathematical model: Description of the air phase volume ......................................... 39 
4.3.3. Mathematical model: Description of the air phase pressure........................................ 39 
4.3.4. Implementation of the numerical solution ................................................................... 40 
Chapter five - Results.................................................................................................................... 43 
5.1. Experimental Results.......................................................................................................... 43 
5.1.1.Total obstruction experiments - Pressure measurements ............................................. 44 
5.1.2. Total obstruction experiments - Flow velocity measurements .................................... 52 
5.1.3.Partial obstruction experiments - Pressure measurements ........................................... 55 
5.1.4 Partial obstruction. Experiments - Flow velocity measurements ................................. 66 
5.1.5 Discussion of experimental results ............................................................................... 73 
5.2 Numerical modeling results ................................................................................................ 91 
Chapter six ? Conclusions and Recommendations for Future Work ............................................ 98 
References ................................................................................................................................... 101 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
vii 
 
 
 
 
List of Figures 
 
Figure 1.1: Air pocket entrapment at low point shaft predicted by numerical simulation ............. 3 
 
Figure 1.2: Horizontal, two-Phase flow patterns (Falvey, 1980) .................................................... 5 
 
Figure 1.3: The equation of motion for a control volume (Wylie and Streeter, 1993) ................... 8 
 
Figure 1.4: Control volume for continuity equation (Wylie and Streeter, 1993) ......................... 11 
 
Figure 2.1: Two-Phase flow apparatus (Hager, 1999) .................................................................. 15 
 
Figure 2.2: Schematic of key parameters used in Martin (1976) model. ...................................... 16 
 
Figure 2.3: Sketch of key variables used by Liou and Hunt (1996) model for water mains. ....... 17 
 
Figure 2.4: Characteristics of the installation (Fuertes et al., 2000) ............................................. 18 
 
Figure 2.5: Comparison between the model of Izquierdo et al. (1999) and the measured values 
(Fuertes et al., 2000). .................................................................................................................... 19 
 
Figure 2.6: Experimental model layout by Li and McCorquodale (1999). .................................. 20 
 
Figure 2.7: Diagram of experimental apparatus (Zhou et al., 2002). ............................................ 21 
 
Figure 2.8: Scheme of Preissmann Slot concept ........................................................................... 24 
 
Figure 4.1: Illustration of experimental apparatus construction. .................................................. 30 
 
Figure 4.2: Sketch of the adapter used to connect the pressure transducers to the pipeline. ........ 31 
 
Figure 4.3: MicroADV wooden support. ...................................................................................... 32 
 
Figure 4.4: Sketch of the experimental apparatus used in the investigations ............................... 34 
 
Figure 4.5: Force balance in a pipeline flow................................................................................. 37 
 
Figure 4.6: Screen capture of the numerical model interface. ...................................................... 42 
 
viii 
 
Figure 5.1: Sequence of a complete valve obstruction, air pocket formation and motion. 
(Q*=0.44, Vair*=2.3, horizontal slope). Interface between air water phases has been artificially 
enhanced. ...................................................................................................................................... 46 
 
Figure 5.2: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.38 to 0.40, Vair*=3.4, horizontal slope). Subscripts up and dw denotes upstream 
pressure transducer results (X*=0.5) and downstream pressure transducer results (X*=1.0), 
respectively. .................................................................................................................................. 46 
 
Figure 5.3: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.42, Vair*=2.55, horizontal slope). .......................................................................... 47 
 
Figure 5.4: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.48, Vair*=1.08, horizontal slope). .......................................................................... 48 
 
Figure 5.5: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.51 to 0.52, Vair*=0.38 to 0.42, horizontal slope). .................................................. 49 
 
Figure 5.6: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.49 to 0.51, Vair*=0.13 to 0.38, varying pipe slopes) ............................................. 50 
 
Figure 5.7: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.37 to 0.38, Vair*=0.64 to 3.35, varying pipe slopes) ............................................. 51 
 
Figure 5.8: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.51 to 0.53, Vair*=0.317 to 0.381, varying pipe slopes) ......................................... 52 
 
Figure 5.9: Non-dimensional velocity evolution following total obstruction by the knife gate 
valve, horizontal slope results. ...................................................................................................... 53 
 
Figure 5.10: Non-dimensional velocity evolution following total obstruction by the knife gate 
valve, 2.7% adverse slope results. ................................................................................................ 54 
 
Figure 5.11: Non-dimensional velocity evolution following total obstruction by the knife gate 
valve, 1.0% favorable slope results............................................................................................... 55 
 
Figure 5.12: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. (Q*=0.37 to 0.52, horizontal slope) ................................................................................... 56 
 
Figure 5.13: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.38 to 0.50, horizontal slope) ................................................................................... 57 
 
Figure 5.14: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. (Q*=0.20 to 0.49, 2.7% adverse slope) .............................................................................. 58 
 
Figure 5.15: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. (Q*=0.50 to 0.54, 1.0% favorable slope) ........................................................................... 59 
ix 
 
Figure 5.16: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.22 to 0.48, 2.7% adverse slope) .............................................................................. 60 
 
Figure 5.17: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.50 to 0.52, 1.0% favorable slope) ........................................................................... 61 
 
Figure 5.18: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. (Q*=0.42, varying slopes) .................................................................................................. 63 
 
Figure 5.19: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.41-0.43, varying slopes).......................................................................................... 64 
 
Figure 5.20: Non-dimensional pressure evolution following for 81% obstruction by the knife 
gate of all the slopes with minimal air pocket volume (Vair*=0.15 to 0.31) ............................... 65 
 
Figure 5.21: Non-dimensional pressure evolution following for 89% obstruction by the knife 
gate of all the slopes with minimal air pocket volume (Vair*=0.11 to 0.37) ............................... 66 
 
Figure 5.22: Non-dimensional velocity evolution following 81% obstruction by the knife gate 
valve, horizontal slope results. ...................................................................................................... 68 
 
Figure 5.23: Non-dimensional velocity evolution following 81% obstruction by the knife gate 
valve, 2.7% adverse slope results. ................................................................................................ 69 
 
Figure 5.24: Non-dimensional velocity evolution following 81% obstruction by the knife gate 
valve, 1.0% favorable slope results............................................................................................... 70 
 
Figure 5.25: Non-dimensional velocity evolution following 89% obstruction by the knife gate 
valve, horizontal slope results. ...................................................................................................... 71 
 
Figure 5.26: Non-dimensional velocity evolution following 89% obstruction by the knife gate 
valve, 2.7% adverse slope results. ................................................................................................ 72 
 
Figure 5.27: Non-dimensional velocity evolution following 89% obstruction by the knife gate 
valve, 1.0% favorable slope results............................................................................................... 73 
 
Figure 5.28: Non-dimensional peak pressures measured at two locations along the pipeline 
(X/L=X*=1.0 and 0.5) for experiments involving complete valve obstruction, horizontal slope 
and different anchoring schemes .................................................................................................. 74 
 
Figure 5.29: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; horizontal slope results shown............................................................. 75 
 
Figure 5.30: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 1.3% adverse slope results shown ....................................................... 77 
 
x 
 
Figure 5.31: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 2.7% adverse slope results shown ....................................................... 78 
 
Figure 5.32: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 0.5% favorable slope results shown .................................................... 79 
 
Figure 5.33: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 1.0% favorable slope results shown .................................................... 80 
 
Figure 5.34: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. Horizontal slope results shown ..................................................................... 81 
 
Figure 5.35: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 1.3% adverse slope results shown. ................................................................ 82 
 
Figure 5.36: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 2.7% adverse slope results shown ................................................................. 83 
 
Figure 5.37: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 0.5% favorable slope results shown, and are qualitatively similar to the ones 
obtained with the 1.0% favorable slope ........................................................................................ 84 
 
Figure 5.38: Non-dimensional peak pressures as a function of air pocket volume and slope for 
total obstruction experiments. ....................................................................................................... 85 
 
Figure 5.39: Non-dimensional peak pressures as a function of air pocket volume and slope for 
81% obstruction experiments. ....................................................................................................... 86 
 
Figure 5.40: Non-dimensional peak pressures as a function of air pocket volume and slope for 
89% obstruction experiments. ....................................................................................................... 87 
 
Figure 5.41: Non-dimensional peak pressures as a function of initial flow rate and slope for total 
obstruction experiments ................................................................................................................ 88 
 
Figure 5.42: Ratio between peak pressures measured at X*=0.5 and X*=1.0 as a function of the 
entrapped air pocket volume ......................................................................................................... 89 
 
Figure 5.43: Predicted and measured pressure hydrographs for Vair*=3.5, Q*=0.38, horizontal 
slope, and total cross section obstruction condition...................................................................... 92 
 
Figure 5.44: Predicted and measured pressure hydrographs for Vair*=0.40 Q*=0.51-0.52, 
horizontal slope, and total cross section obstruction condition .................................................... 93 
 
Figure 5.45: Predicted and measured pressure hydrographs, for Vair*~3.4, Q*=0.37, horizontal 
slope, and 81% of cross section obstruction condition ................................................................. 94 
 
xi 
 
Figure 5.46: Predicted and measured pressure hydrographs, for Vair*~0.40, Q*=0.515, horizontal 
slope, and 81% of cross section obstruction condition ................................................................. 95 
 
Figure 5.47: Predicted and measured pressure hydrographs, for Vair*~2.97, Q*=0.382, horizontal 
slope, and 89% of cross section obstruction condition ................................................................. 96 
 
Figure 5.48: Predicted and measured pressure hydrographs, for Vair*~0.367, Q*=0.503, 
horizontal slope, and 89% of cross section obstruction condition. ............................................... 97 
 
  
xii 
 
 
 
 
  List of Tables 
 
Table 4-1- Experimental variables and tested range ..................................................................... 35 
 
Table 5-1- Comparison of peak H*=H/D pressures for horizontal slope experiments averaged 
between repetitions ....................................................................................................................... 57 
1 
 
Chapter one - Introduction 
 
Stormwater drainage systems are designed to drain the excess of superficial water derived 
from rain or ground water from the city pavements. Among other components, this system is 
includes main and secondary conduits, also referred as stormwater sewer system, that receive the 
runoff captured by the inlets at dropshafts. The dimension (e.g. diameter of conduits) of 
stormwater sewer systems often reflects flows from rain events with relatively low return periods 
and with the assumption that the system will operate in a free surface regime. However, in the 
case of a rapid filling during an intense rain, the conveyance capacity of the system may be 
exceeded and a pressurized flow regime will occur in the conduit. This is observed from the 
smallest conduits at the upstream end of the urban catchment until the largest conduits such as 
stormwater storage tunnel systems. 
In this process, the air phase initially present in the conduits can also become entrapped 
and pressurized, resulting in a two-phase flow condition. The transition between the two phases 
has been linked to structural problems in drainage systems and the geysering phenomenon. Guo 
and Song (1991) show pictures of geysers that occurred in Minnesota stormwater drainage 
system pointing that these events resulted in undesirable flooding and pollution problems. Zhou 
et al. (2002) pointed out similar issues and infrastructure damage caused by intense rain events in 
the city of Edmonton in 1995. 
The knowledge about this problematic is still very scarce and limited, mainly in the 
interaction of the two phases in an extreme inflow condition.  This study aims to contribute in the 
better understanding of the issues in drainage systems during intense rain events focusing on the 
effects generated by entrapped air pockets. To attain this objective, research was conducted 
involving numerical and experimental investigations in order to represent more precisely the 
effects of sudden air pockets entrapment and associate pressures surges in an unventilated 
system. 
The following sections include a description of flow in stormwater sewers as two-phase 
flows, considering all possible interactions between air and water in rapid filling scenarios. Also, 
2 
 
these flows also are unsteady due to the nature of the rainfall-runoff generation process, and a 
section on unsteady flow formulation for these systems is presented.  
1.1. Stormwater systems described in two-phase flow framework 
 
Air presence in stormwater system flows is of great relevance to the analysis of these in 
unsteady conditions. Air may become an issue in hydraulic systems when the filling rate reaches 
a certain level that leads to the entrainment and/or entrapment of air in the pressurized portions 
of the flow. These rapid inflow conditions are central to the hydraulic analysis of these systems 
since operational issues that have been reported to date in stormwater systems are generally 
associated with intense rain events. The effects of the air presence in the closed conduits are 
mainly three (Pozos, 2007): 
1. High air compressibility can generate higher pressures than the pipeline walls 
resistance, causing the rupture of the conduit (Wylie and Streeter, 1993); 
2. Air volume alter system dynamics, and can lead to change in the flow characteristics, 
velocity and discharge, and have been traced to episodes such as geysering 
(Vasconcelos and Wright, 2011); 
3. The air accumulation in the intermediate or high points with no ventilation can form a 
partial blockage in flows, creating additional energy losses and in cases of stormwater 
tunnels loss of storage capacity, which may lead to premature overflow episodes. 
 
Stormwater flows are said to be single phase if water phase fills the entire conduit 
volume; when air phase is also present, they are classified as biphasic or two-phase flows.  When 
small air bubbles and pockets are traveling in conduits due to drag forces caused by the water 
flow, this volume of air tends to accumulate in the high points of the conduit, a problem that is of 
relevance in the context of water main design; in other instances air pockets may be generated 
from the air volume originally present in the tunnel that is entrapped by the water flow, as shown 
in Vasconcelos and Wright (2006). Figure 1.1 exemplifies a mechanism of air pocket entrapment 
3 
 
based on reflection of inflow fronts off of system boundaries predicted by numerical simulation.  
As it is presented below, classification of two-phase flows generally involves a clear 
identification of the two phases in the conduits, which occupy distinct places in the pipe cross 
section. Adequate representation of these conditions in numerical models requires two-phase 
flow modeling approaches (Vasconcelos et al, 2011). However, for some applications where 
very small air bubbles are well-distributed in the entire pipe cross section, flows may be regarded 
as homogenous, and a valid modeling alternative is to represent such conditions in a single-phase 
flow framework. In these circumstances, modeling can be done considering a single phase fluid 
and adjusting the acoustic wave velocity calculation, among other relevant flow parameters such 
as density and bulk modulus of elasticity (Wylie and Streeter, 1993). 
 
Figure 1.1: Air pocket entrapment at low point shaft predicted by numerical simulation 
 
Air accumulation in the closed conduits affects the behavior of the flow, and may cause 
additional head losses and reduction of acoustic wave propagation speed (Wylie and Streeter, 
1993). This topic has been of research interest since the decade of 1940?s, with the studies by 
Kalinske and Bliss and Kalinske and Roberts on the motion of air bubbles and the air 
4 
 
entrainment in hydraulic jumps. Benjamin (1968) performed studies on the motion of air cavities 
in closed conduits in the context of gravity current flows. Wallis (1969) presents a 
comprehensive study on unsteady two-phase flows proposed a classification based in the 
morphological arrangement of the components of the two-phase flow. This study also indicates 
that the classification can often be revealed through visual or photographic observations but it 
can?t define completely the flow regime due to other independents criteria such as the turbulence 
of the system. The relative proportion of air and water flowing through the pipeline depends on 
the slope and the quantity of air. The presence of air in water flows receives different 
denominations depending on the relative size of the cavity, as illustrated in the Figure 1.2.  
Falvey (1980) described the flow patterns for two-phase horizontal flows: 
? Bubble Flow - Air forms in bubbles at pipe crown that has approximately the same 
velocity as the water flow; it is denominated as ?Front Flow? if the bubbles are dispersed;  
? Plug flow - The air bubbles combine in plugs and rapid alternation between air flow and 
water flow is observed in cross sections; 
? Stratified flow - The air and water flow creates a horizontal interface separation and both 
are flowing downstream; 
? Wave flow- Surface waves appear in stratified flow if air flow is increased; 
? Slug flow - Air pocket are large enough to close half of the pipe cross section. The air 
slug travels with velocity higher than the average water flow velocity. 
? Annular flow - Water flows as a film on the pipeline wall, while air phase flows with 
greater velocity at the conduit center. 
? Spray flow ? For even higher air flow velocity, the water film is broken by shear and 
carried away in water droplets.  
5 
 
 
Figure 1.2: Horizontal, two-Phase flow patterns (Falvey, 1980) 
In stormwater system flows, anticipated two-phase flows are either slugs or bubbles flow, 
depending on inflow conditions, and system geometry characteristics such as pipeline slope. The 
definition of bubbles and slug sizes varies between researchers. Bubbles are identified by Wisner 
et al. (1975) as ellipsoidal shaped air cavities with the size varying from 1 mm to 5 mm. In the 
case of Pozos (2007), bubbles are identified as air cavities with the longitudinal length less than 
or equal to the diameter of the pipe. On the others hand, air slugs and pockets are defined by 
Pozos (2007) as an air cavities with longitudinal length greater than the diameter of the pipe. 
Falvey (1980) describe that the transition from bubble flows to slug flows occurs when the 
bubble diameter is one-half of the conduit diameter. 
 
6 
 
1.2 Modeling Framework of rapid filling pipe problems 
 
Because of potential operational issues in stormwater systems undergoing rapid filling 
events, numerical models have been developed to anticipate the magnitude of pressure surges 
and/or simulate the filling process of these systems, particularly in the context of large 
stormwater tunnel systems (Song, et al., 1983; Cardle and Song, 1988; Vasconcelos, et al. 2006; 
Politano, et al. 2007; among others). While most current models neglect effects of entrapped air 
in these systems, in the scope of the present investigations the inclusion of these are essential. 
This section presents a brief summary of the mathematical framework used in the development 
of the numerical model of this thesis. 
 
1.2.1. Air phase modeling  
 
While there are different alternatives to simulate the behavior of entrapped air pockets that 
includes one-dimensional, discretized formulations as presented by Issa and Kempf (2003), 
Trindade and Vasconcelos (2011), among others, the simplest formulation that has been used to 
date is based on the application of the ideal gas law. This alternative has been used by Martin 
(1976), Li and McCorquodale (1999), Zhou, et al. (2002), among others. 
The ideal gas law was Clapeyron in the XIX century, who demonstrated that there was an 
inversely proportional relationship between pressure and volume of a gas. Kr?nig, in 1856, 
demonstrated that these results can also be derived from kinetic theory of gases. In an isothermal 
process, a gas pressure, volume, and temperature are related by equation (1.1). 
 
g1842g1848 =g1866g1844g1846 
 
(1.1) 
which P is the pressure, V is the volume, n is the amount gas substance (in moles), R is the ideal, 
or universal gas constant, and T is the temperature. 
7 
 
While the ideal gas law is good approximation to the behavior of real gases at any pressure or 
any temperature, it has various limitations that one example was the negation of both molecular 
size and intermolecular attraction. However, for the process involving air entrapment in water 
flows in closed conduits, the ideal gas law has been applied successfully to describe pressure 
changes upon air compression/expansion cycles. This equation, expressed in terms of an ordinary 
differential equation on time, is enforced during the water flow simulation at air-water interfaces. 
1.2.2. Water phase modeling  
 
Time-independent water flows, in which none of its fundamental characteristics such as 
pressure and velocity are varying in time, are classified as steady flows. However, due to the 
nature of rainfall-runoff generation, stormwater flows present significant variation over time, and 
thus are referred to as unsteady flows. The one-dimensional formulation of these unsteady flows 
requires a mathematical model based on the fundamental laws of mass and linear momentum 
conservation, as presented below.  
1.2.2.1. Momentum equation 
 
The theoretical foundation of the momentum equation is in Newton's second law, which 
states that the sum of the forces acting on a control volume must be balanced by varying the 
amount of linear movement amount of it. Essentially, in water flows, three forces are 
predominant: weight; frictional forces; and pressure forces. Figure 1.3 illustrates a typical free-
body diagram over a control volume for application of equation of motion in a closed conduit. 
 
8 
 
 
Figure 1.3: The equation of motion for a control volume (Wylie and Streeter, 1993) 
 
The application of Newton?s 2nd law in the control volume presented in Figure 1.3 results 
in  equation (1.2): 
g1868g1827?g3428g1868g1827+g2034(g1868g1827)g2034g1876 g2012g1876g3432+g3436g1868+g2034g1868g2034g1876g2012g18762g3440g2034g1827g2034g1876g2012g1876?g2028?g2024g1830g2012g1876?g2025g1859g2012g1876g1827g1871g1861g1866(g2009)=g2025g1827g2012g1876g1856g1848g1856g1872 (1.2) 
 
Where p is pressure in the control volume, ?x is the length of the control volume, ? is 
water density, A is the cross-sectional area, D is the pipe diameter, V is the velocity in the 
section, g is the gravity, and H is the height hydraulic grade line, ?o is the shear tension, x is the 
position on the length, z is the elevation of the pipe center and t is the time.   
Provided that ?x is sufficiently small, all terms which in the order (?x)? become 
negligible. This results in equation (1.3). 
9 
 
g2034g1868
g2034g1876g1827+g2028?g2024g1830+g2025g1859g1827g1871g1861g1866(g2009)+g2025g1827
g1856g1848
g1856g1872 =0 (1.3) 
 
Despite shear forces be dependent of local acceleration terms (Wylie and Streeter, 1993), 
particularly in fast transient (waterhammer) applications, for simplicity it is assumed the validity 
of the Darcy-Weisbach equation relating a pressure drop caused by friction, the friction factor 
and the velocity head as defined by equation (1.4):  
?g1868=g2025g1858g1838g1830 g1848
g2870
2  (1.4) 
 
If this pressure drop is caused by shear forces, one may write: 
?g1868g2024g1830
g2870
4 =uni0009g2028?g2024g1830g1838uni0009uni0009 (1.5) 
 
From this relationship, an expression for the shear tension is obtained: 
g2028?=g2025g1858g1848|g1848|8 uni0009uni0009 (1.6) 
 
Since velocity V is a function of both time and space, its total differential can be defined 
as: 
g1856g1848
g1856g1872 =
g2034g1848
g2034g1872 +g1848
g2034g1848
g2034g1876 (1.7) 
 
When equations 1.6 and 1.7 are used in equation 1.3, relation (1.8) is derived: 
g2034g1868
g2034g1876
1
g2025+g1848
g2034g1848
g2034g1876 +
g2034g1848
g2034g1872 +g1859g1871g1861g1866(g2009)+
g1858g1848|g1848|
2g1830 uni0009=0uni0009uni0009uni0009uni0009 (1.8) 
 
As pointed out by Wylie and Streeter (1993), for low Mach flows this expression may be 
expressed by the expression below of the momentum conservation: 
10 
 
g2034g1868
g2034g1876
1
g2025+
g2034g1848
g2034g1872 +g1859g1871g1861g1866(g2009)+
g1858g1848|g1848|
2g1830 uni0009=0uni0009uni0009uni0009uni0009uni0009 (1.9) 
 
Further manipulation includes represent the pressure terms in terms of the piezometric 
head H. The relationship between pressure p and piezometric head H is: 
g1868 =g2025g1859(g1834?g1878)uni0009 uni0009(1.10) 
with z as the elevation of the CV. 
 Therefore, one derives: 
g2034g1868
g2034g1876 =g2025g1859g3436
g2034g1834
g2034g1876 ?
g2034g1878
g2034g1876g3440=uni0009g2025g1859g3436
g2034g1834
g2034g1876 ?g1871g1861g1866(g2009)g3440uni0009 (1.11) 
 
which is substituted into the equation 1.9 to derive an alternative expression for the momentum 
equation: 
g2034g1848
g2034g1872 +g1859
g2034g1834
g2034g1876 +
g1858g1848|g1848|
2g1830 uni0009=0 (1.12) 
 
Equation 1.9 and 1.12 are applicable for any one-dimensional, homogeneous, low Mach 
fluid flows. Further simplification of this expression is possible, yielding an ordinary differential 
equation that forms the basis for the Rigid Column Model. 
 
1.2.1.2. Continuity Equation 
 
The one-dimensional continuity equation expresses mass conservation in pipeline flows. The 
control volume geometry is represented by Figure 1.4, where the upstream boundary of the 
control volume is located in section x. The mass conservation law establishes that the net inflow 
into the control volume equals to the rate of mass increase, expressed by equation 1.13: 
 
11 
 
 
Figure 1.4: Control volume for continuity equation (Wylie and Streeter, 1993) 
 
? g2034g2034g1876g4670g2025g1827(g1848?g1873)g4671g2012g1876 = g1856g1856g1872(g2025g1827g2012g1876) (1.13) 
 
Where u is the velocity in the conduit wall velocity and the differential operator g1856 g1856g1872uni2044  
that appears in equation in equation 1.13 is expressed as: 
g1856
g1856g1872 =g1873
g2034
g2034g1876+
g2034
g2034g1872 (1.14) 
 
The growth rate of the length of the control volume is given by the formula (1.15). 
g1856
g1856g1872g2012g1876 =
g2034g1873
g2034g1876g2012g1876 (1.15) 
 
Taking these expressions into equation 1.12 results in: 
12 
 
g2034
g2034g1876g4670g2025g1827(g1848?g1873)g4671g2012g1876+g2012g1876
g1856
g1856g1872(g2025g1827)+g2025g1827
g1856
g1856g1872(g2012g1876)=0 (1.16) 
 
Further manipulations leads into 
g2034
g2034g1876(g2025g1827g1848)?
g2034
g2034g1876(g2025g1827g1873)+
g1856
g1856g1872(g2025g1827)+g2025g1827
g2034g1873
g2034g1876 =0 (1.17) 
g2034
g2034g1876(g2025g1827g1848)?
g2034
g2034g1876(g2025g1827g1873)+g1873
g2034(g2025g1827)
g2034g1876 +
g2034(g2025g1827)
g2034g1872 +g2025g1827
g2034g1873
g2034g1876 =0 (1.18) 
 
Assuming that the last term on the left hand side of equation 1.18 is zero, equation 1.19 
or 1.20 are obtained as expressions of the continuity equation for compressible flows. 
g2034(g2025g1827)
g2034g1872 +
g2034(g2025g1827g1848)
g2034g1876 =0 (1.19) 
g2034(g2025g1827)
g2034g1872 +g1848
g2034(g2025g1827)
g2034g1876 +g2025g1827
g2034(g1848)
g2034g1876 =0 (1.20) 
 
Wylie and Streeter (1993) proceed with equation 1.20, and using the concept of acoustic 
wave speed (the celerity of pressure wave propagation in closed conduit flows) defined as 
g1853g2870 = g3012/g3096g2869g2878(g3012/g3002)(g2940g3002/g2940g3043), (with ? as density and K the fluid?s bulk modulus of elasticity) showed that 
for Mach number flows, one can express the unsteady continuity as: 
g2034g1834
g2034g1872 +
g1853g2870
g1859
g2034g1848
g2034g1876 =0 (1.21) 
 
One relevant observation is that the equations (e.g. 1.12 and 1.21) have been derived with 
the assumption of unsteady, pressurized flows. During rapid inflow scenarios is likely that 
portions of the system will be pressurized, but others would be operating in unsteady, free 
surface flows. In the latter case, Saint-Venant would be applicable. Stormwater tunnel models 
may either apply either equation at these flow regions, tracking the position where pressurization 
interface exists, or apply conceptual models such as the Preissmann slot (Cunge and Wegner, 
13 
 
1964) or TPA (Vasconcelos, et al. 2006) and use a single equation in the two flow regime 
domains. The approach in this study involves tracking the location of the pressurization 
interfaces over time. 
 
 
1.3. Thesis Outline 
 
Chapter Two includes the literature review for this research, focusing in recent 
investigation on two-phase flow characterization; air effects in unsteady closed conduit flows; 
experimental investigations on rapid inflow conditions in stormwater systems; and numerical 
modeling approaches to extreme flow conditions in stormwater systems. Chapter three presents 
the perceived knowledge gaps and research objectives. Chapter Four includes the methodology 
of the work, including a description of the experimental apparatus construction, experimental 
program, and numerical modeling strategy. Chapter Five presents research results and a 
discussion of research findings that are of practical relevance for the design of stormwater 
conveyance systems. Finally, Chapter Six presents the conclusions of this research and it gives 
recommendations for subsequent studies. 
 
 
 
 
 
  
14 
 
Chapter two - Literature Review 
 
The literature review is divided in three subsections to facilitate the understanding current 
stage of the research on extreme flows in stormwater systems including air effects. The initial 
section presents two-phase flow investigations in the context of air cavity motion and shape in 
closed conduits. The second section presents the research on rapid filling flows in closed 
conduits and effects of entrapped air, both in the context of sloping water pipelines and 
stormwater systems.  Finally, the last section presents some of the numerical modeling 
alternatives to simulate the flows including the transition between free surface and pressurized 
conditions, which are anticipated in stormwater systems undergoing rapid filling. 
 
2.1 Air cavity motion and shape in closed conduit flows 
 
The advance of an air cavity into a pressurized flow is an example of a gravity current 
problem, and has been studied by a number of researchers as early as the decade of 1960?s. An 
important contribution was the work presented by Benjamin (1968), who considered the advance 
of an air cavity into a horizontal pipeline water flow as it was emptied. Among other important 
contributions, this work demonstrated an important relationship between the thickness of the 
gravity current and the celerity of its advance. A follow-up study presented by Wilkinson (1981) 
demonstrated that no-slip effects and surface tension, which were not accounted for in 
Benjamin?s work had an influence the air cavity shape and celerity. 
Research by Baines (1991) included the advance of an air cavity in an emptying pipeline 
considering slope effects. It was observed that increased slopes resulted in a larger celerity of the 
air pocket for the narrow range of slopes considered. Moreover, it was observed a variation of air 
pocket thickness along its length. It also expressed that the air pocket shape at the air-water 
interface front is very depending on the air quantity that enters into the conduit, and that under 
certain conditions a led to gulping at the water outlet, which broke the integrity of the advancing 
air cavity, generating discrete air pockets.  
15 
 
Goldring (1983) analyzed the air voids caused by the downshaft-tunnel bend water flow 
of middle size diameter. He noticed that there was a range between air and water flows to an air 
void become stable in the bend. A related problem was presented by Hager (1999), who 
investigated air cavity shape formed as water discharged freely from a horizontal pipeline. The 
on systematic experiments assumed the Froude number in the outlet as the controlling variable. 
This studied modified the Boussinesq equation and proposed equations to represent the shape of 
the cavity profile and the downstream end depth ratio for various flow conditions. Figure 2.1 
represents a portion of the experimental two-phase flow apparatus, which has a discharge setup 
that resembles the one used in the present research.  
 
 
Figure 2.1: Two-Phase flow apparatus (Hager, 1999) 
  
2.2. Rapid filling flows in closed conduits and effects of entrapped air 
 
One of the pioneering studies involving unsteady pipeline flows with air was presented 
by Holley (1969), who conducted analytical and experimental studies in order to investigate the 
type and magnitude of surges that can happen in pipe system that contains several check 
structures. The author suggested that that the storage and release of air cavities inside the 
pipeline can cause head surges initiation in the system.  
16 
 
Martin (1976) proposed a model based on the lumped inertia approach that combined air 
modeling based on the ideal gas law. This model was developed to determine the magnitude of 
the pressure surges caused by compression of air cavities at the pipeline discharge. The model 
shows that the presence of entrapped air may lead to peak surge pressures that are many times 
larger than the driving pressure in the water flow due to air compressibility. This model proposed 
ideas that later served as basis for important studies, such as Zhou, et al. (2002) and De Martino 
et al. (2008). A schematic of key parameters of Martin (1976) model is presented in Figure 2.2. 
 
Figure 2.2: Schematic of key parameters used in Martin (1976) model. 
 
Possibly, the first numerical model of water main filling simulation was designed by Liou 
and Hunt (1996). It applied to the filling of long water mains with variations in topographic 
profiles, but the model does not account explicitly for air phase effects. According to the authors, 
the model is applicable as long as the flow velocity is higher than the celerity of an intruding air 
cavity, such as the ones discussed in section 2.1. In these conditions, a rigid column model was 
proposed and implemented, assuming perfect air ventilation and that the inflow front remains 
well defined and vertical, resembling a water piston. As the inflow front advances over an sloped 
terrain, increasing frictional forces slow its advance. The authors compared the model 
predictions with experimental results obtained with a small diameter pipeline submitted to a high 
17 
 
driving head and obtained good agreement. Figure 2.3 presents a sketch of the key variables used 
by this model in an undulating water main.  
 
Figure 2.3: Sketch of key variables used by Liou and Hunt (1996) model for water mains. 
  
Izquierdo et al. (1999) presented a numerical model to simulate the filling of pipelines 
based on the momentum equation, considering the hypothesis of water accumulation in the lower 
points of the water pipeline, which would lead to air pockets entrapment upon flow startup at the 
upstream end of the system. Air pockets were located between the points of water accumulation 
and they were highly compressed by the surrounding water columns as the flow was initiated. 
The authors concluded that the smaller air pockets volumes led to greater the pressure surges 
upon flow initiation, a result in agreement with the earlier work by Martin (1976). The model 
assumes that the transition interface between phases remains well defined and that the entrapped 
air does not intrude on the water advance. While Izquierdo (1999) improved Liou and Hunt 
(1996) model by adding the capability of air pocket compression during a rapid filling event, it 
still has applicability limitations. One is the lack of air ventilation on the formulation, a feature 
that could be present in anticipated in stormwater systems in terms of vertical shafts and/or 
manholes. A follow-up experimental study presented by Fuertes, et al. (2000) assessed the 
validity of Fuertes, et al. (1999) model using an experimental apparatus comprised by a pipe with 
6.90 m of length and a small internal diameter of 18.8 mm, shown in Figure 2.4. Good agreement 
18 
 
between experiments and numerical predictions was observed, as presented in Figure 2.5. An 
important remark is that experimental conditions included only cases with high ratios of driving 
pressure head over pipe diameter, which are not representative for flows in stormwater systems. 
 
Figure 2.4: Characteristics of the installation (Fuertes et al., 2000) 
 
19 
 
 
Figure 2.5: Comparison between the model of Izquierdo et al. (1999) and the measured values 
(Fuertes et al., 2000).  
 
Li and McCorquodale (1999) constructed an experimental model to simulate the pressure 
transients due to the transition from the free-surface to the pressurized flow in the context of 
rapid filling pipeline flows. The study of consisted of the 12.12 m long pipeline and the diameter 
was 152 mm, connecting upstream to open-channel section and downstream to a reservoir tank. 
This research analyzes continuously surges due to the entrapped air pocket and considers the 
movement of the air bubbles, which consists an in important innovation.  The authors also 
proposed a numerical model that extended and refined the previous one proposed by Hamam and 
McCorquodale (1982) that adopted the hypothesis of a stationary bubble. One limitation of this 
study is that it doesn?t consider the effect of a ventilation system acting in the air cavity for storm 
sewers. Figure 2.6 presents schematics of their experimental apparatus. 
 
20 
 
 
Figure 2.6: Experimental model layout by Li and McCorquodale (1999). 
 
Another relevant work was presented by Zhou et al. (2002), which included an 
experimental study of the rapid filling of a short and horizontal conduit with 10 meter of length 
and 35 mm of diameter. One important goal was to estimate the effects of the air ventilation to 
the observed transient pressures with the goal of obtaining the peak pressure. The authors also 
proposed a numerical model based on the lumped inertia approach with the additional 
consideration of the air pressurization effects. A limitation of Zhou et al. (2002) study is the use 
of a small pipeline diameter and a very large driving pressure head/diameter ratio, which are not 
representative conditions that would be observed in actual stormwater systems. The numerical 
model uses the same assumption as Liou and Hunt (1996) and Izquierdo, et al. (1999) model of a 
well-defined inflow interface; the authors assume a vertical inflow front in the filling process. A 
follow-up work by Zhou et al (2004) introduced a vertical pipe in the apparatus in order to 
investigate the effect of the vertical conduit in the air-water flow condition, but the behavior of 
21 
 
the system and the results did not change considerably. Figure 2.7 exemplified a sketch of 
experimental apparatus used Zhou et al. (2002). 
 
 
Figure 2.7: Diagram of experimental apparatus (Zhou et al., 2002). 
 
Aimable and Zech (2003) conducted experiments that agreed with the experiment 
proposed by Li and McCorquodale (1999) for the formation of air pockets in sewers. Their 
apparatus consisted of pipe with two manholes of 0.2 m and 0.144 m diameter spaced 5.0 m 
along the pipeline. The downstream reach initially flowed in a steady pressurized state and the 
two other parts were in free surface conditions. The study shows that a sudden increase in the 
pressure at the downstream end generated a flow regime transition front, and confirmed the 
mechanism of pocket motion described by Li and McCorquodale (1999). 
Arai and Yamamoto (2003) performed experiments in apparatus 1/50 scale of actual 
underground drainage system. The experimental apparatus have the pipeline of 0.198 m diameter 
and a length of 122.08 m to investigate the effects of the air pressurization when there is flow 
already on the system. Their results indicate that the air pocket presence accelerates the transition 
from free surface to pressurized flow, suppress the tube oscillation and increases the maximum 
pressure and increases air releases. 
22 
 
Despite many of these previous studies resulted in numerical models that are constructed 
with the assumption of well-defined air-water interfaces, in conditions when the driving pressure 
head/diameter relationship is not high this assumption of may not be accurate. Guizani, et al. 
(2005) conducted numerous experiments on the filling of water pipelines with different inflow 
conditions and slopes. In all cases the filling front shape resembled a wedge, similar to a dam-
break front propagating within the pipe. In this research, it was point out that a vertical interface 
would be an applicable supposition only if the flow rate within the conduit is very large.  
More recently, De Martino, et al. (2008) reported an experimental installation to analyses 
the water flow transients that was caused by the expulsion of the air pocket. The De Martino, et 
al. (2008) utilized a galvanized steel pipe was around 70 m long, a diameter of 53 mm and the 
downstream end an air venting orifice. In this research, the pressure oscillation through an orifice 
was studied and it was closely related to the ones observed by Zhou et al. (2002). This study 
avoided the difficulty of assuming a vertical air-water interface by orienting the pipe flow and 
direction of air compression vertically. De Martino, et al. (2008) showed that the driving 
pressure head and orifice diameter are more important factors toward the pressure surge 
magnitude than the length of the air pocket. 
However relevant, one argues that these extreme conditions used in most of past 
investigations are too extreme and not representative of the situations anticipated in stormwater 
systems in rapid filling scenarios where air pockets may become entrapped and compressed. In 
addition to this, important air pocket formation mechanisms, such as the one generated by the 
reflection of water flow fronts within the system downstream boundaries has not been 
sufficiently executed and tested to this date to assess the magnitude of surges that are anticipated 
following these entrapment mechanisms. These are outstanding research questions that are 
addressed by the present work. 
 
2.3. Numerical modeling of stormwater drainage systems in extreme flow conditions  
 
23 
 
As mentioned previously, numerical modeling has been an important tool to analyze the 
behavior of stormwater systems, particularly large below-grade stormwater tunnels, under rapid 
inflow conditions that are anticipated in intense rain events. These models provide information as 
to what are the anticipated pressure surges, elevations of HGL at selected location, flow rates and 
other relevant results for designers. A common assumption is that flows are single-phase, and 
unsteady. These models? ability to incorporate air phase interactions is limited partly because of 
the few experimental studies conducted to date on these interactions in closed conduit flows. 
Because filling processes will be characterized by the transition between free surface 
flow conditions into a pressurized flow condition, models that need to handle both flow 
unsteadiness and the transition between these two flow regimes. These models are referred to in 
literature as Mixed Flow models or Flow Regime Transition models, and the first model was 
proposed by Cunge and Wegner (1964) to study the hydraulics of a tail-race tunnel in the context 
of a hydroelectric dam. The model used the Saint-Venant equation to perform the simulation, 
which is expressed below in conservative, vectorial format: 
 
g2034g1847g4652g4652g1318
g2034g1872 +
g2034g1832g1318
g2034g1876 =g1845g1318 
(2.1) 
g1847g4652g4652g1318 =g3428g1827g1843g3432uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009g1832g1318 =g4686
g1843
g1843g2870
g1827 +g1859g1827?g3030
g4687uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009g1845g1318=g3428 0g1859g1827g3435g1845
g3042 ?g1845g3033g3439
g3432 
(2.2) 
Where U is the vector of conserved variables, F is the vector of conserved fluxes, S is the 
vector of source terms, A is the flow cross sectional area, Q is the flow rate, g is gravity 
acceleration, hc is the distance from the cross-section centroid and the free surface, So is the 
conduit slope and Sf is the friction head.  
Because this equation becomes non-defined if the free surface width reaches zero (e.g. at 
pipe crown), the authors used the conceptual model proposed by Preissmann (1961). He 
hypothesized that if a virtual narrow slot would be added at the top of a closed conduit both 
pressurized and free surface flow could be analyzed by the St. Venant equations since the 
24 
 
resulting cross section would still have a free surface interface. This slot would allow for the 
surcharge pressure that would be anticipated in pressurized flow conditions. A schematic of the 
Preissmann slot is represented by Figure 2.8, with g1872g3033 as the slot width, ?g3046g3039g3042g3047 as the water level 
inside the slot, corresponding to a surcharge pressure head, and D as the conduit diameter.  
 
Figure 2.8: Scheme of Preissmann Slot concept 
 
The Preissmann slot idea is simple and allows for a single set of equations to represent 
two flow regimes simultaneously. Moreover, these models may be implemented using 
sophisticated flow solvers based on non-linear schemes, with advantages in the simulation of 
flow discontinuities (e.g. bores) when these solvers are constructed with shock-capturing 
capabilities (Toro, 2001). The Preissmann slot was included in various numerical models, 
including versions of EPA SWMM 5 (Rossman, 2004) and HEC RAS 4.0 (Brunner, 2008). A 
limitation of the conceptual model is that it cannot simulate negative water pressures, as it would 
regenerate free surface flow conditions even in the absence of ventilation, a non-physical 
outcome. 
Song, et al. (1983), Cardle and Song (1988), Fuamba (2002), Politano, et al. (2007) and 
others have proposed an alternative way to simulate unsteady flows involving the transition 
25 
 
between pressurized and free surface flows. These models are referred to as shock-fitting mixed 
flow models, and apply the set of unsteady pressurized flow equations (eq. 1.12 and 1.21) 
alongside with unsteady flow equations for free surface flows based (eq. 2.1 and 2.2). Explicit 
tracking of the pressurization interface is needed to determine the boundaries of these two flow 
regimes so that the appropriate set of equations may be applied. Despite of the added complexity 
of these models that comes from the tracking of multiple pressurization fronts and the use of two 
sets of unsteady flow equations, they were extensively applied in the simulation of extreme flows 
in stormwater tunnels since Preissmann slot models are unable to simulate sub-atmospheric 
flows. 
The limitation that the Preissmann slot posed was overcome by a new conceptual model 
proposed by Vasconcelos, et al. (2006). Rather than consider a virtual slot, the authors assumed 
an elastic behavior for the conduit walls so that there would be a direct relationship between the 
pressure experienced in pressurized flows (either positive or negative/sub-atmospheric) and 
changes in the full-pipe cross-sectional area. This conceptual model, referred to as TPA model, 
has been applied in water main filling modeling (Vasconcelos, 2007; Vasconcelos, et al. 2009), 
stormwater tunnel systems (Lautenbach, et al. 2008), stormwater collection networks (Sanders 
and Bradford, 2011), and has been expanded to include effects of pressurized air phase 
(Vasconcelos and Wright, 2009; Trindade and Vasconcelos, 2011), as well effects of distributed 
cavitation (Vasconcelos and Marwell, 2011). The TPA model introduces changes to the Saint-
Venant equation, resulting in the following system of partial differential equations: 
 
g2034g1847g4652g4652g1318
g2034g1872 +
g2034g1832g1318
g2034g1876 =g1845g1318 
(2.3) 
 
Where 
g1847g4652g4652g1318 =g3428g1827g1843g3432uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009g1832g1318 =g4686
g1843
g1843g2870
g1827 +g1859g1827(?g3030 +?g3046)
g4687uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009uni0009g1845g1318=g3428 0g1859g1827g3435g1845
g3042 ?g1845g3033g3439
g3432 
(2.4) 
26 
 
The term hs represent the pressure head anticipated only in pressurized flow conditions, and is 
calculated as: 
 
uni0009?g3046 =g1853
g2870
g1859
(g1827?g1827g3030)
g1827g3030  
(2.5) 
With a as the acoustic wavespeed and Ac the cross-sectional area of the conduit. 
In a completely different context, multiphase models have been developed to simulate the 
slug flow in oil or gas transport through pipelines with an example the work by Issa and Kempf 
(2003). However it is uncertain that these models consider all the relevant physics of stormwater 
hydraulics into the continuity and momentum equations, since, for instance, there is no specific 
term that handles pressure surcharge in the conduits. 
Chapter three - Knowledge gap and Objectives 
      
The research proposed here aims to provide insights on the interactions between 
entrapped air cavities inside a stormwater tunnel without air ventilation by means of 
experimental and numerical investigations. Based on previous related research in stormwater 
tunnels systems, the pertinent knowledge gaps related to air pocket entrapment and compression 
include: 
i. The pipeline flow within system boundaries is one of the causes of air pocket formation 
and pressurization. However, the air pocket by the reflection of water flow fronts within the 
system downstream boundaries has not been sufficiently executed and tested to this date. Such 
cases are not anticipated to be as extreme as the conditions that have been considered in previous 
studies (Martin, 1976 and Zhou, et al. 2002) because of the pressure relief that shafts may 
provide. Yet, they are very relevant as they may result in loss of tunnel storage volumes. Of 
particular interest is the magnitude of the surges upon pocket entrapment and whether in such 
cases negative pressures will be observed; 
ii. It is probable that rapid flows will produce large pressure upon the valve or blockage. If 
there is air pockets, this large pressure will make them smaller and compressed (Martin, 1976). 
27 
 
In most of situations when there is a blockage, the flow is going to elevate pressure and the water 
will passes even in smaller quantities. So far, no research has considered this factor, principally 
in the context of air pocket entrapment in stormwater tunnels. This may difficult the 
development and calibration of numerical models to simulate tunnel filling events accounting for 
air pressurization effects. 
This study presents the numerical and experimental results from a constant investigation 
on surges caused by air pocket entrapment. The idea is to measure the magnitude of those 
pressure surges for varying air pocket volumes, system slopes, initial flow rates and obstruction 
levels caused by a discharge valve. These researchers intend to follow the conditions in which 
water removal causes surge decrease by varying the valve barriers. It?s assumed that the knife 
gate valve would create conditions similar to rising water at shafts, decreasing the outflow into 
the shafts during the air pocket compression process. By investigating this particular flow 
condition, the main goal is to gain better insight of the underlying physics and with this improve 
current modeling approaches to this problem by better incorporating air pressurization effects. 
Although, this discussion of numerical and experimental models for the simulation of stormwater 
drainage tunnels filling indicates that there are still opportunities for improvement. Future 
models should include more realistic assumptions, when considering effects of air pocket 
pressure.  
 
  
28 
 
Chapter four - Methodology 
           
This methodology chapter presents how the proposed investigation on air pockets 
entrapment was conducted in three main sections. The first section presents how the required 
experimental apparatus was designed and constructed. The second section discussion covers how 
the experimental program was conducted: procedures, experimental variables, conditions tested, 
etc. The third and last section focuses on the numerical modeling development, which aimed to 
replicate the conditions tested in experiments. 
The fundamentals of the experimental and numerical investigation of air pocket in water 
flow applied similar approach to the works of Martin (1976) and Zhou et al. (2002), but with 
some important distinctions. In those earlier works, the flow rate was zero in the pipeline, going 
all the water to compress the air pocket. However, in this study, the flow rate was steady and 
non-zero value with a gradual transition between free surface and pressurized flows. Upon 
rapidly valve maneuver, a bore was generate at the valve and started to move upstream  of the 
conduit and initiates the air cavity entrapment. As a result of the air pocket entrapment 
mechanism, it?s assumed to be describing the actual characterized of air entrapments in authentic 
stormwater tunnels in which air pockets are trapped between water columns, particularly in the 
cases where compression occurred in parallel to pressure relief provided by partial openings at 
the discharge valve. 
 
4.1. Experimental apparatus construction 
 
The experimental apparatus was thought and designed to be a recirculation system, which 
the upstream reservoir would supply water flow for the pipeline until it passes the last valve and 
fall in downstream reservoir. Thus, there are two pumps in the downstream reservoir, which are 
responsible to create the system recirculation. For that reason, it was constructed the wooden 
material to operate such as a recirculation apparatus.  
29 
 
First, it was necessary to construct a wood support to elevate the upper reservoir. 
However, it was required to know that the wood stands the weight of a filled upper reservoir. It 
constructs the project design of the wood support, which it was calculated. Then, it was build the 
pipeline wood holders to control the slope. Since the conduit is little elastic, the wood holders 
also served to maintain the inflexibility of pipeline, making sure that the air pockets are in the 
right position. This was verified and calculated, as a result there were six pipeline holders 
necessary.  
Next, it was required to fit the experimental apparatus pipeline, which is compound to a 
clear PVC pipe of 101.6-mm diameter with a length of 12 meters for horizontal and adverse 
slopes. Later, the experimental apparatus have to change location to another off-campus 
experimental laboratory, thus the new laboratory was a little smaller that the before and the 
experiments with favorable slopes where performed in a 10.6 meter. However, for experiments 
of this thesis, it isn?t altered, significantly, the features of the flow. The next step was the knife 
gate valve that was installed in the pipeline downstream end. The knife gate valve had the 
function of closing almost instantaneous the water flow and entrapping the air cavity. And the 
last piece set up in the pipeline apparatus was the ventilation orifice that was in upstream end. 
The ventilation orifice was able to let air escape to allow the permanent flow and in the next step, 
the valve was close. 
In order to establish the initial flow conditions, the upper and downstream reservoirs 
have, respectively, 0.66 m3 and 0.62 m3 of total volume. Hence, the two pumps were putted in 
downstream reservoir and a white pipeline was built to connect the upper reservoir to the pumps. 
Both of the pumps had a valve to regulate how much flow passes in the pipeline. Another valve 
was putted in white pipeline in order there flow couldn?t come back in opposite direction. This 
valve also served to control the flow that reaches the upper reservoir. All of this were constructed 
and connected to a recirculation apparatus experimental. The Figure 4.1 demonstrates the 
construction of experimental apparatus. 
30 
 
 
Figure 4.1: Illustration of experimental apparatus construction.  
After that, it required that flow and pressure measurements were placed and connecting 
with the computer. The type of the two piezoresistive pressure transducers was MEGGIT-
ENDEVCO 8510-50C, the maximum pressure was 345 kPa and 0.4% of accuracy. The pressure 
transducers were collocated on the wall inside pipeline, one in the middle and another in the 
downstream end. Consequently, it was needed to finish the two holes in the conduit in order to 
collocated the two plastic supports, which the Figure 4.2 presents a sketch of the adapter used to 
connect the pressure transducers to the pipeline. 
31 
 
 
Figure 4.2: Sketch of the adapter used to connect the pressure transducers to the pipeline. 
 Subsequently, the two manometers were placed in the same position and opposite site 
that the pressure transducer. There were also required two holes that were smaller than the rubber 
hose linking to the manometer because it need to make pressure to haven?t any sort of leakage. 
The paddlewheel flow was fitted in the white pipeline to observe how much the flow change in 
the rotation the valve. 
And at last and more complicated was the MicroADV probe that has function to measure 
the velocity. Primary, it was thinking and constructed a wood support for MicroADV, which 
touch the ground, although it doesn?t work. When it modified the slope, the MicroADV had little 
movement and it messes with the connection between the MicroADV and the conduit. The most 
recent idea was to support the MicroADV into the pipeline and a hole in rubber connection to fit 
inside the conduit where there is flow velocity. It demonstrates the wood support into the 
pipeline in Figure 2.3.    
32 
 
 
Figure 4.3: MicroADV wooden support.  
 
 
4.2. Experimental program 
4.2.1. Experimental procedure 
 
 Following the experimental construction there was a period of few weeks of experimental 
troubleshooting, after which the system was ready for experimental runs and for data collection. 
The conditions were designed to create pressurized flow in the pipeline connection with the 
upstream reservoir. Brink depth discharge conditions at the downstream end were designed to 
create conditions of free surface flows at the downstream end, and consequently free discharge 
conditions were anticipated with the presence of an atmospheric air cavity at the pipeline crown. 
The location of the air pocket intrusion would depend on a combination of flow rate and pipe 
33 
 
slope, and conditions were adjusted to avoid air pocket lengths exceeding more than 50% of the 
pipeline length. 
The pipeline slopes and variable discharge rates were meant to represent a wide variety of 
air cavity volumes and when the knife gate valve was fully or partially closed (about 0.3 second) 
it formed an air pocket. Upon obstruction, water flow continued to compress the air phase due to 
water phase inertia, and air pocket pressure increases significantly before water flow reversed.  
 The pressures conditions were captured by the pressure transducers and recorded in the 
computer. The pressures transducers were placed in two positions: at the pipe crown in the 
pipeline downstream end and at half of the apparatus length at the pipe invert. These two points 
were carefully chosen to represent the pressure variations. First one, at pipe crown, measured the 
pressures variations between the air and water flow and the second one measures only water 
pressure to observe if there are some differences between the values or not. However, the 
pressures transducers yielded an analog electrical signal that is directly correlated to pressure 
changes, so it was necessary to place two manometers in the same point as the transducer to 
permit an accurate pressure conversion. The flow velocities were measured by a MicroADV 
probe that was located at 3.0 m from the upstream end, position that always flow in pressurized 
conditions. All the essential elements of the experimental apparatus are presented in Figure 4.1. 
34 
 
  
Figure 4.4: Sketch of the experimental apparatus used in the investigations 
 
For the experimental process, it is described as following: 
1. A required slope was set in the pipeline  
2. Pumps were started and flow valves were opened and regulated to that a predetermined 
steady flow would was supplied to the upstream reservoir and into the pipeline. The 
choice of the water flow depended on the required volume of the air cavity that was 
formed at the downstream end of the pipeline; 
3. Air pocket volume was measured by performing measurements of the air pocket width at 
the clear PVC pipe crown at regular intervals, so that a numerical integration of the 
resulting volume could be performed. The spacing between the stations where the air 
cavity width was measured varied from 0.05 m for the smallest pockets (less than 0.50 
m), 0.10 m for the intermediate size pockets (up to 1.2 m) and 0.30 m for the larger 
pocket; 
4. Three digital camcorders (1920 x 1080 resolution, 30 frames per second) were turned on 
to record the pocket compression process. All the important parameters like date, hour, 
35 
 
discharge, repetition among others relevant experimental parameters, were registered in 
the movies.  
5. The initial pressure readings at the two manometers were recorded, as well the head at the 
upstream reservoir was read;  
6. The two piezoresistive pressure transducers were started at a frequency of 100 Hz. 
7. Velocity acquisition was provided by a MicroADV probe, sampling at a frequency of 25 
Hz; 
8. The knife gate valve was rapidly closed (partially or completely) obstructing the flow at 
the downstream end of the pipeline, and visual observations and comments were made to 
aid the subsequent data analysis; 
9. Next, the pump flow was stopped and downstream reservoir valves were closed, 
separating the upper from downstream reservoir. 
10. After the pressure pulses disappeared, the pressures reached an equilibrium level. The 
final pressure levels at the upper reservoir and at the final manometers reading were 
collected to the pressure transducers. 
 
4.2.2. Experimental variables and tested range 
 
The experimental apparatus provided focus on variables such as the slope, the flow rate 
and the valve obstruction. Table 1 illustrates the range of tested variables. 
Table 4-1- Experimental variables and tested range 
Variables Range tested so far Normalized range 
Flow rate (Q) 
 2.08 to 5.45 L/s Q*=Q/(gD
5)0.5=0.202 to 0.532
Slope (So) 
 
Horizontal; 0.5% and 1.0% favorable; 
1.3% and 2.7% adverse n/a 
36 
 
Valve obstruction 
 
0 mm, 8 mm and 13 mm residual 
opening at the knife gate valve 
100%, 89% and 81% of cross 
section obstruction 
 
The initial idea was to conduct 75 different initial conditions in this experimental system, 
but this number was re-adjusted to 60 considering that some of the conditions were not possible 
to create or were not relevant to the experimental program objectives. A total of 136 
experimental runs were conducted considering repetitions to ensure consistency. Usually two to 
three such repetitions were made and results compared to verify whether pressure and/or ADV 
signals were comparable. When the duplicates had significant difference in values (~10%), a 
more repetitions were introduced. 
 
4.3. Numerical Model ? Lumped Inertia Method 
 
A predictive numerical model based in lumped inertia was developed and its results 
compared to experimental results, using for the code development CodeGear Turbo Delphi that 
is based on the object-oriented computational language Object Pascal. The Object Pascal 
language is a windows-based Pascal platform which allows for a quick development of 
programs, and in its Turbo Delphi Express version the license was free. The idea for the 
numerical model development was to check whether results obtained in experiments would be 
reproducible in a two-phase, one-dimensional modeling framework constructed with the water 
phase momentum continuity equation, and the ideal gas law. The following subsections presents 
detail of the selected mathematical model (based on the lumped inertia approach as described in 
Wylie and Streeter, 1993), and the strategy used in its numerical implementation.   
4.3.1. Mathematical model: Water phase momentum equation  
 
37 
 
The momentum equation for the lumped inertia approach is the simplification of the 
partial differential equations related to pressurized flow that leads a system of two partial 
differential equations (describing mass and linear momentum conservation) into only a single 
one ordinary differential equation for the momentum equation, allowing for a significant 
simplification in the solution procedure. The approach is based on the assumptions of the water 
incompressibility and absolute inelasticity of the conduit walls. Thus, the approach ignores the 
fluid accumulation in the pipeline in function of water pressure changes. Alongside with the pipe 
inelasticity, this implies in a linear pressure gradient at all times, and that changes to the flow 
conditions do propagate instantaneously through the system (infinitely large acoustic wave 
speed).    
  
Figure 4.5: Force balance in a pipeline flow 
 
38 
 
Figure 4.2 sketches terms used in the derivation of the lumped inertia method (Wylie and 
Streeter, 1993). The application of the lumped inertia analysis (Wylie and Streeter, 1993) to a 
control volume leads to equation (4.1). 
1
g2025
g2034g1868
g2034g1876+
g2034g1848
g2034g1872 +g1859sing2009+
g1858g1848uni007Cg1848uni007C
2g1830 = 0 (4.1) 
 
where p is the control volume pressure, ? is the water density, D is the pipeline diameter, V is the 
velocity, g is the gravity acceleration, ? is the acoustic wave celerity, x is the position of conduit 
length,  f is the friction loss coefficient and t is the time. 
As explained, the acoustic wave celerity has an infinitely large value considering that 
instantaneous propagation of flow conditions. Thus, the flow velocity remains constant during 
the whole extension of the conduit and the equation (4.2) can be written. 
g2034g1834
g2034g1876 =
g1834g3045g3032g3046 ?g1834g3028g3047g3040
g1838  
(4.2) 
 
In which g1834g3045g3032g3046 is the reservoir head, g1834g3028g3047g3040 is the atmospheric pressure head and L is the pipeline 
length between the reservoir and the location at which pressure is at atmospheric levels. 
 After this increment and some manipulation on slope effects, the momentum formula for 
the water column is transform in the ordinary differential equation (4.3)  
g1856g1843
g1856g1872 =
g1859g1827
g1838 g3428g1834g3045g3032g3046 ?(g1834g3028g3036g3045 ?g1834g3028g3047g3040)?uni0009g1845g2868g1838?g3436g1858
g1838
g1830+g1837g3039g3042g3030g3028g3039g3440
g1843|g1843|
2g1859g1827g2870g3432 (4.3) 
 
In which, Q is the water flow rate, H air is the air phase pressure head, Klocal is the localized loss 
coefficients. 
 
39 
 
4.3.2. Mathematical model: Description of the air phase volume 
 
A lumped inertia air continuity equation may be formulated in terms of the amount of 
water flow that is admitted upstream end of the pipeline and the water discharged through the 
knife gate valve, computed with an orifice-like equation. It assumed that no air mass is lost or 
gained during the valve maneuvering process. The resulting continuity equation for the air phase 
is expressed in equation (4.4). 
g1856g1848g3028g3036g3045
g1856g1872 =?g1843+g1829g3031g1827g3042g3045g3036g3033g34932g1859(g1834g3028g3036g3045 ?g1834g3028g3047g3040) (4.4) 
 
In which, V air is the air phase volume, C d is the discharge coefficient at the downstream end as 
the valve is being closed and Aorif is the discharge area as the gate valve is being closed. 
 
4.3.3. Mathematical model: Description of the air phase pressure 
 
It was considered that the ideal gas law is applicable for this experimental apparatus, so 
the energy equation for the air phase is constructed. The water flow can?t escape all through the 
knife gate valve cavity, so the air pocket is suffered pressure and its air volume has decrease.  
The equation (4.5) is presented as following: 
g1856g1834g3028g3036g3045
g1856g1872 =?g1863
g1834g3028g3036g3045
g1848g3028g3036g3045
g1856g1848g3028g3036g3045
g1856g1872  
 
(4.5) 
in which, k is the polytropic coefficient (assumed as 1.2 in this research).  
 An initial calibration of the experimental values was necessary to approximate the 
numerical model results. The water flow utilized in the experiments contained a certain amount 
of fine sediments, and eventually these sediments became attached to the pipe walls. Although 
40 
 
PVC pipes are very smooth and sometimes it roughness has been ignored, in this particular case 
this was not the case due to sediment accumulation. Therefore, instead of friction factors around 
0.021-0.022 anticipated for smooth PVC pipes at such flow rates, the friction factor number used 
in the computations that best matched the experimental results was 0.025. The local loss 
coefficient was considerable, mostly for the junctions, valve and a small pipe segment using two 
inches diameter at the upstream reservoir outlet.  
The initial strategy of calculating the pressure changes assumed a totally or partially 
instantaneous valve closure procedure, effectively changing Aorif into zero right at the simulation 
start. However, this approach led to pressure peaks that were between 50% and 100% larger than 
what was observed the experimental measurements. A close examination of the movies records 
indicates that rather than instantaneous, the valve closure lasted between 0.18 s to 0.30 seconds 
(6 to 9 recording frames in a 30 FPS camera). A parabolic closure relationship was proposed to 
express the decrease of Aorif over the initial 0.20 second of the simulation based on movies.  
A second adjustment was on the discharge coefficient Cd utilized in the calculations 
since the predicted discharges after the knife gate maneuver left a clearance for water discharge. 
The commonly adopted value of Cd=0.60 was consistently yielding flow rates that were much 
above the measurements based on the MicroADV probe velocity. This discrepancy was larger 
for the experiments that used a smaller discharge gap at the valve (8 mm). After matching the 
discharge values yielded by the model with the observations, the discharge coefficient Cd varied 
from a maximum of 0.40 (for some 13 mm opening down) to 0.30 (for 8 mm opening 
experiments). 
4.3.4. Implementation of the numerical solution 
 
The 4th Runge-Kutta method (Press et al., 1989) is a popular approach to solve one or 
more ordinary differential equations. This method was selected to solve equations 4.3 and 4.5, 
with errors per time step (?t) is in the order of ?t5 and the accumulated error is in the order of 
41 
 
?t4. It uses four estimates (where defined as k1, k2, k3 and k4) for the slope (rate of change) of 
the dependent variable and the resulting change is a weight average of these values. The 4th 
Runge-Kutta method is exemplified for the equation of momentum for water column. 
g1858(g1872g3041,g1843g3041)=g1856g1843g1856g1872 =g1859g1827g1838 g3428g1834g3045g3032g3046 ?(g1834g3028g3036g3045 ?g1834g3028g3047g3040)?uni0009g1845g2868g1838?g3436g1858g1838g1830+g1837g3039g3042g3030g3028g3039g3440g1843g3041|g1843g3041|2g1859g1827g2870 g3432uni0009 (4.6) 
 
The Runge-Kutta considers the derivative in four intermediate points as the equations 
(4.7) and (4.8). 
g1843g3041g2878g2869 =g1843(g1872+?g1872)=g1843g3041 +?g1872.g3436g1863g2869 +2g1863g2870 +2g1863g2871 +g1863g28726 g3440uni0009 (4.7) 
g1872g3041g2878g2869 =g1872g3041 +?g1872uni0009uni0009 (4.8) 
Where the coefficients of the intermediate points are calculated as follows: 
g1863g2869 =g1858(g1872g3041,g1843g3041)uni0009uni0009 (4.9) 
g1863g2870 =g1858g3436g1872g3041 +?g18722 ,g1843g3041 +?g18722 g1863g2869g3440uni0009uni0009uni0009uni0009 (4.10) 
g1863g2871 =g1858g3436g1872g3041 +?g1872uni00092 ,g1843g3041 +?g1872uni00092 g1863g2870g3440uni0009uni0009uni0009uni0009uni0009 (4.11) 
g1863g2872 =g1858(g1872g3041 +?g1872,g1843g3041 +?g1872g1863g2871)uni0009uni0009uni0009uni0009uni0009 (4.12) 
 
 Figure 4.6 illustrates an interface of numerical program that was for entering the data and 
showing the results. 
42 
 
 
Figure 4.6: Screen capture of the numerical model interface. 
 
 
 
 
43 
 
 
 
 
Chapter five - Results 
 
 The experimental apparatus and the numerical code on the compression of entrapped air 
pockets in stormwater tunnels generate many results that were divided into experimental and 
numerical. First, it will be demonstrate the experimental data and, after that, it will be presented 
the numerical data, always comparing with the experimental results. Initially, it was 
demonstrated and discussed the results of total valve obstruction and then, the partial valve 
obstruction.  
 
5.1. Experimental Results 
 
The experiments that involved complete valve obstruction represent an extreme case of a 
bore reflection against a system boundary without any storage capability, thus with no pressure 
relief for entrapped air pockets present at stormwater tunnels. Yet, they were included because 
they are similar to previous experimental studies (Martin, 1976; Zhou, et al. 2002; De Martino et 
al., 2008) and to make comparison between the total and the partial valve obstruction.  
Unlike experiments with partial valve obstruction, presented ahead, these experimental 
cases yielded significant down surges and sub-atmospheric pressures. An oscillatory-like pattern 
for the pressures and flow velocity measurements was noticed, with an oscillation period that 
depended on the entrapped air pocket volume. The discussion of experimental results separates 
the pressure and the flow velocity in different sections. A sequence of still images following a 
total obstruction maneuver is presented in Figure 5.1 for a horizontal slope case. The figure 
presents a bore that is formed at the moment of the valve total closure and propagates upstream. 
44 
 
After some time a curved pocket front moves back towards the valve and eventually an air 
pocket is spread at that portion of the pipe. 
 
 
 
 
5.1.1.Total obstruction experiments - Pressure measurements  
 
Pressure measurements at the upstream transducer (X*=X/Lpipe=0.5) and at the 
downstream transducer (X*=X/Lpipe=1.0) are presented in Figures 5.2, 5.3, 5.4 and 5.5, grouped 
by the flow rate/initial air pocket volume for the horizontal slope, and including all repetitions. In 
these charts, one notices a sudden rise in pressure was followed by an oscillation pattern in the 
pressure over time as the incoming flow decelerates due to the air pressure build-up caused by 
valve obstruction. Time is normalized by the factor L/(g. D)0.5. One may also notice that the 
pressure peaks increase for smaller air pockets, while pressure oscillation period increased with 
entrapped pocket volume as anticipated by Martin (1976). The magnitude of the oscillations was 
strongly damped at each oscillation cycle, a feature that is further discussed in ahead in the 
numerical modeling results section. 
The air pocket shape upon valve closure indicated a very complex configuration, with the 
initial air intrusion breaking into several smaller pockets, and moving within the pressurized 
region. Discrepancies among experimental repetitions are generally caused by the inability to 
enforce exactly equal initial conditions prior to valve maneuver, especially with respect to the 
initial air pocket volume. Figure 5.1 illustrates the sequence of pipeline still images following a 
total closing of last valve. 
45 
 
 
 
Figure 5.1: Sequence of a complete valve obstruction, air pocket formation and motion. 
(Q*=0.44, Vair*=2.3, horizontal slope). Interface between air water phases has been artificially 
Figures 5.2, 5.3, 5.4 and 5.5 
repetitions of the same experimental condition
pressure for the tested repetitions for various air pockets as the inflow rate values varied between 
the cases. All the conditions presented in th
experiments, and these different flow rate values resulted
downstream end, as predicted by Hager (1999)
Figure 5.2: Non-dimensional pressur
valve. (Q*=0.38 to 0.40, Vair*=3.4, horizontal slope). Subscripts up and dw denotes upstream 
pressure transducer results (X*=0.5) and downstream pressure transducer results (X*=1.0), 
46 
enhanced. 
present pressure hydrographs with either 
. These charts present variations on the resulting 
ese figures correspond to horizontal slope 
 in different sizes of air cavities at the 
 
e evolution following complete obstruction of the knife gate 
respectively. 
two or three 
 
 
Figure 5.3: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.42, Vair*=2.55, horizontal slope).
47 
 
 
 
Figure 5.4: Non-dimensional pressure evolution following complete obstruction of the knife gate 
valve. (Q*=0.48, Vair*=1.08, horizontal slope).
48 
 
 
 
Figure 5.5: Non-dimensional pressure evolution following complete obstruction of the knife
valve. (Q*=0.51 to 0.52, Vair*=0.38 to 0.42, horizontal slope).
 
Figure 5.6 and 5.7 present a comparison between the total obstruction experiments 
performed at various slopes while keeping the flow rate approximately the same
considered in Figure 5.6 were the largest tested for sloped conditions, and ranged between 0.49 
and 0.51. In such conditions the resulting differences in pocket volumes are very large between 
horizontal/adverse slope experiments and favora
associated with a large portion of the flow that could not be sustained in pressurized mode 
considering that large conveyance associated with the favorable slope and the brink depth 
discharge conditions. Thus, one notices that
conditions are much smaller than correspondent cases for horizontal and adverse slopes. 
49 
 
ble slope experiments. The latter cases were 
 peak pressures associated with these favorable slope 
 
 gate 
. The flow rates 
 
 
Results in Figure 5.7 are similar to the ones presented in Figure 
experimental conditions were such that the flow rates were much smaller (Q*=0.37
However, favorable slopes results are not presented since there 
entire pipeline for these lower flows. The results presented for adverse and horizontal 
indicate that larger peak flow rates are associated with smaller air pocket volumes, as anticipated.
Figure 5.6: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.49 to 0.
50 
was no pressurized flow
51, Vair*=0.13 to 0.38, varying pipe slopes)
5.6, except that 
-0.38). 
 in the 
slope 
 
 
 
 
Figure 5.7: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.37 to 0.38, Vair*=0.64 to 3.35, varying pipe slopes)
 
 A slight increase in the flow rate for favorable slopes (Q*~ 0.51
reduction in the air pocket cavity observed at the downstream end. A comparison between the 
horizontal (in this same flow rate range) and favorable slope results pre
indicate that pressure peak results and oscillaiton periods from these cases are comparable, but 
there seems to be a more rapid energy damping in the cases associated with favorable slope 
conditions. No clear explanation for that is av
of the entrapped air pocket following complete valve closure could be associated with added 
energy losses as the resulting contraction from the pocket presence would be larger than the one 
anticipated for a thinned air pocket 
51 
-0.53) results in a major 
sented in Figure 5.8 
ailable, but one speculates that the upward motion 
evenly spread in horizontal slope conditions.  
 
 
 
 
Figure 5.8: Non-dimensional pressure evolution following complete obstruction by the knife gate 
valve. (Q*=0.51 to 0.53, Vair*=0.317 to 0.381, varying pipe
5.1.2. Total obstruction experiments 
 
The MicroADV probe was located 
prior to the valve maneuver velocity measurements were characterized by a steady 
anticipated fluctuations associated with 
drop in the flow velocities, followed by the same damped oscillation pattern observed with the 
pressure. Results in Figure 5.9 are presented for one of eac
52 
 slopes)
 
- Flow velocity measurements 
3 m downstream from the reservoir
turbulence. As the valve was closed there was a 
h different sizes of air pockets tested. 
 
 
(X*=0.25), and 
value with 
rapid 
 
 
Figure 5.9: Non-dimensional velocity evolution following total obstruction by the knife gate 
The velocity measurements are similar to the pressure 
also present a damped oscillatory pattern. Oscillation period was smaller for larger flow rates 
considering that for such conditions the entrapped air pocket volumes were smaller. 
was also observed with other slo
the cases of maximum adverse and favorable slopes, respectively. Velocity results obtained for 
the other tested slopes present the same trend and are not shown.
 
53 
valve, horizontal slope results. 
measurement results in that they 
pes tested, as shown in Figures 5.10 and 5.11 measurements for 
 
 
 
 
This trend 
 
Figure 5.10: Non-dimensional velocity evolution following total obstruction by the knife gate 
valve, 2.7% adverse slope results.
54 
 
 
 
Figure 5.11: Non-dimensional velocity evolution following total obstructio
valve, 1.0% favorable slope results.
 A comparison between the velocity measurements between the different slopes indicate 
that the case with the smallest pocket volume and favorable slope presented a more rapid 
velocity damping then the other cases analyzed. One speculates that this is caused by the motion 
of this small pocket towards the upstream end of the pipeline following the rapid gate closure. 
This pocket motion could possibly generate additional energy losses as it would constit
contraction in the water flow. Such a condition would not be anticipated either in horizontal or 
adverse slope conditions. 
 
5.1.3.Partial obstruction experiments 
 
Experimental conditions involving partial obstruction of the downstream valve enforced 
obstructions of 81% and 89% by means of a partial obstruction of the knife gate valve. The valve 
55 
n by the knife gate 
 
- Pressure measurements  
 
ute a 
 
was quickly closed, but rather than interrupt the flow entirely, wooden 
of the valve shaft either 13 mm or 8 mm from the valve seat. 
19% and 11% of the valve cross sectional area, respectively.
Figures 5.12 and 5.13 present pressure measurement results at the downstre
for the partial valve obstruction for three flow rates, horizontal slope, and valve obstruction of 
81% and 89% respectively. In general the results were significantly different from the ones 
obtained with the full valve obstruction conditions
not noticed in these experiments, and no oscillations were noticed for the 81% flow interruption. 
As Figure 5.12 indicates, few oscillations were noticed in some cases when 89% obstruction 
occurred, yet the magnitude of those were greatly damped and rapidly dissipated. This behavior 
is dissimilar from the one observed in the total obstruction experiments.
Figure 5.12: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. 
56 
pieces blocked the motion 
The resulting gap was measured as 
 
. The significant negative pressure surges were 
 
(Q*=0.37 to 0.52, horizontal slope) 
am transducer 
 
 
Figure 5.13: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (
The three flow rates represented in both Figures 
minimum, average, and maximum flow rates tested in the full obstruction experiments under 
horizontal slope. Other results are omitted for br
much smaller than the ones observed for the complete valve obstruction, as 
5.1 below: 
Table 5-1- Comparison of peak H*=H/D pressures for horizontal slope 
averaged between repetitions 
Initial flow Total obstruction
Q*?0.51 H*?125
Q*?0.45 H*?57 
Q*?0.38 H*?24 
57 
Q*=0.38 to 0.50, horizontal slope) 
5.12 and 5.13 correspond to the 
evity. It?s noticeable that the peak flows are 
indicated 
experiments 
 89% obstruction 81% obstruction
 H*?57 H*?21
H*?21 H*?13
H*?14 H*?6
 
in Table 
 
 
 
 
 
 
As it would be anticipated
upon valve maneuver. The maximum surge peak drop 
of maximum flow (Q*~0.51). On the other hand,
horizontal cases was 42% for the case of minimum flow r
Table 5.1 one notices that even though the flow rates may be very close between repetitions, the 
associate variation in air pocket volumes may be more significant due to the 
of cavity volumes to flow rates.
also observed for other tested slopes. 
obstruction data for the 2.7% adverse and 1.0% favorable slopes
Figure 5.14: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. (Q*=0.20 to 0.49, 2.7% adverse slope)
58 
, the smaller obstruction of 81% yielded smaller pressure peaks
for horizontal cases was 83% for the case 
 the minimum surge peak drop 
ate (Q*~0.38) and 89% obstruction. In 
large 
 The same trend presented in Table 5.1 for horizontal slope is 
Figures 5.14 and 5.15 illustrate 81%
 respectively. 
 
 
observed for 
sensitiveness 
 partial valve 
 
 
Figure 5.15: Non-dimensional pressure evolution following 81% obst
valve. (Q*=0.50 to 0.54, 1.0% favorable slope) 
Maximum peaks surges for both cases (V
magnitude despite the difference of 8% in the flow rate and 5% in the pocket volume. 
interesting feature that is shown in 
time, regardless of the flow rate and associated air pocket volume. This is not observed in the 
81% obstruction, 1% favorable slope case (Figure 5.15), when
occur slightly sooner than the ones associated with larger smaller flow rates and larger air pocket 
volumes. The explanation for this behavior
the results in Figure 5.14 clearly indicate that there?s a single pressure 
with the favorable slope show a slightly oscillation 
89% obstruction results, as presented below.
59 
ruction by the knife gate 
 
air*~0.3) have approximately the same 
Figure 5.14 is that the peak surges occur at about the same 
 the larger peak surges tend to 
 is related to the associated volume of the air
pulse, results obtained 
pattern, which is more pronounced for the 
 
 
An 
. While 
 
Figure 5.16: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. 
60 
(Q*=0.22 to 0.48, 2.7% adverse slope)  
 
 
Figure 5.17: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.50 to 0.52, 1.0% favorable slope) 
 
Figures 5.16 shows pressure hydrographs for 89%, various flow rates and the largest 
adverse slope, while Figure 5.17 presents comparable results with the maximum favorable 
slopes. Unlike the comparison between Figures 5.14 and 5.15, maximum peaks surges for 
favorable slope and Vair*~0.3 is almost twice as large as the one for the adverse slope case (flow 
rate difference is 7% and 5% in the pocket volume). There is no clear explanati
Remarks that were made for Figures 5.14 and 5.15 with regards to the timing of the pressure 
oscillations are also applicable here. 
moderate oscillation pattern, with quick damping. This
intermediate case between the full pipe obstruction surge oscillatory pattern and the single
surge pattern observed in most other partial obstruction experiments. 
61 
 
Finally, results obtained with the favorable slope show a 
 oscillation patter is considered as an 
 
 
on for that. 
-pulse 
62 
 
 Entrapped air pocket volumes for the adverse slope were much smaller than the 
comparable ones at horizontal and favorable slopes, and as result the peak flows were larger for 
the cases with adverse slopes. Figure 5.18 presents the result for the 81% flow obstruction for 
adverse and horizontal slopes when the flow rate was about Q*~0.4., and even though there was 
a difference in the air pocket volumes between the two adverse slope conditions, the pressure 
peak in both cases were fairly similar. At this point we still could not find an explanation for this 
result, but one speculates that this may be in part due to differences in the knife gate valve 
maneuvering velocity between conditions. Favorable slope results aren?t shown since at these 
flow rates there were no pressurized flow conditions. 
  Figure 5.19 results are similar to the Figure 5.18, except that it presents the result for the 
89% flow obstruction for different adverse slopes. The same trend is also noticed, with the 
horizontal slope yielding smaller pressure peaks because of the larger air pocket. The significant 
differences are that the discrepancy between the horizontal case and the adverse slope is now 
larger than the case presented in Figure 5.18, and that the pressure peaks, as anticipated, are 
significantly larger. Likewise, there is no favorable slope results because no pressurized flow 
could be sustained at flow rates around Q*~0.4. 
 
Figure 5.18: Non-dimensional pressure evolution following 81% obstruction by the knife gate 
valve. 
63 
(Q*=0.42, varying slopes) 
 
 
 
Figure 5.19: Non-dimensional pressure evolution following 89% obstruction by the knife gate 
valve. (Q*=0.41
 In order to obtain a comparison of all tested 
consider a large flow rate (Q*~0.5) so that pressurized flows can be sustained even in favorable 
slope conditions. Figure 5.20 and 5.21 present such results for all tested slopes using the 81% 
and 89% flow obstruction respectively. In all tested cas
small, ranging from Vair*=0.15 up to 0.37. All peak surges observed for the 81% obstruction are 
very similar (H*~23-27), and all presented the single peak surge pattern. Results for 89% 
obstruction were consistent except for the favorable 
(H*~60 vs. ~85), and the hybrid pattern between single pulse and oscillatory surges.
64 
-0.43, varying slopes) 
slopes for partial obstruction, one ne
es, the air pocket volume was generally 
slopes, which presented larger peak surges 
 
eds to 
 
 
Figure 5.20: Non-dimensional pressure evolution following for 81% 
gate of all the slopes with minimal air pocket volume (Vair*=0.15 to 0.31) 
65 
obstruction by the knife 
 
 
 
Figure 5.21: Non-dimensional pressure evolution following for 89% obstruction by the knife 
gate of all the slopes with mini
 It also maybe noticed in Figures 5.20 and 5.21 that the steady pressure levels obtained for 
favorable slopes differ slightly from the ones obtained for horizontal and adverse slopes. 
no clear explanation for that but it is speculated that this may be caused by the disassembly
reassembly of the experimental apparatus between the horizontal/adverse and the favorable slope 
experiments when the apparatus was moved to its current position in the off
Research Laboratory at Skyway Drive, Auburn, AL.
 
5.1.4 Partial obstruction. Experiments 
 
Flow velocity measurements for the partial obstruction experiments reflected the pressure 
measurements. Upon valve maneuver, flow decayed rapidly and readjusted to a new steady state 
66 
mal air pocket volume (Vair*=0.11 to 0.37)
-campus
 
- Flow velocity measurements 
 
 
There is 
 and 
 Hydraulic 
67 
 
in under 1.2 seconds (T*<0.1). Generally the drop in flow velocity measured by the ADV, and 
hence flow rate in the pipe, ranged between 50%-64% of the initial flow rate for the case with 
81% obstruction; the experiments with 89% obstruction indicated a drop of 72%-80% of the 
initial flow rate.  
Figure 5.22 and 5.25 present velocity measurements for 81% and 89% obstructions and 
horizontal slope, respectively, with three flow rate ranges represented. It is interesting to notice 
that, regardless of the initial flow rate, the flow measured after the valve maneuver was very 
similar between the different cases tested. One may also notice that for the cases with 89% 
obstruction there was some oscillations on the predicted velocity, which also lasted briefly. 
There were no noticeable oscillations on the results measured with the 81% obstruction.  
Figures 5.23 and 5.26 are analogous to 5.22 and 5.25, but instead present results for the 
2.7% adverse slope. Favorable slope (1.0%) results are presented in Figures 5.24 and 5.27 for 
81% and 89% obstruction respectively. Velocity results for these slopes are present two 
interesting features. Adverse slope velocities are significantly different following valve 
maneuver, which is possibly explained by the larger flow difference between the tested cases 
when compared with horizontal or favorable slope. Also, velocity measurements in these partial 
for favorable results present oscillations that are consistent with the pressure measurements 
presented in Figures 5.15 and 5.17. 
 
Figure 5.22: Non-dimensional velocity evol
68 
ution following 81% obstruction by the knife gate 
valve, horizontal slope results. 
 
 
Figure 5.23: Non-dimensional velocity evolution following 81% obstruction by the knife gate 
valve, 2.7% adverse slope results.
69 
 
 
 
Figure 5.24: Non-dimensional velocity evolution following 81% obstruction by the knife gate 
valve, 1.0% favorable slope results.
70 
 
 
 
Figure 5.25: Non-dimensional velocity evolution following 89% obstruction 
71 
by the knife gate 
valve, horizontal slope results. 
 
 
Figure 5.26: Non-dimensional velocity evolution following 89% obstruction by the knife gate 
valve, 2.7% adverse slope results.
72 
 
 
 
Figure 5.27: Non-dimensional velocity evolution following 89% obstruction by the knife gate 
valve, 1.0% favorable slope results.
5.1.5 Discussion of experimental results
 
The initial setup of the experimental apparatus assumed that the upstream reservoir, 
supported by a wooden platform, was a sufficient anchor to perform the proposed experiments. 
At the downstream end there was no anchor, but the initial assumption is that 
deflection on the experimental runs would be negligible. This assumption turned out to be 
inaccurate as such deflections were noticeable in the experiments involving the smaller pockets, 
where even the wooden tower would be pushed by the p
and total obstruction of the knife gate valve. 
To assess the extent that this displacement affected the experimental results, experiments 
were performed with a ?single anchor? configuration (only the reservoir weight) and a ?dual 
73 
 
 
ressure forces generated upon sudden 
 
 
any longitudinal 
 
anchor? configuration, in which both the extremities of the apparatus wer
opposing walls by wooden beams. Peak pressures following total obstruction and horizontal 
pipeline diameter are presented in Figure 
downstream end of the pipeline (X*=1.0) and for a 
(X*=0.5). 
There is a tendency of the dual anchor results to be slightly above the single anchor 
configurations, particularly for smaller air pocket volumes. Because of these findings, all 
subsequent experiments adopted dual anchor configuration to minimize any longitudinal 
displacement, as a means to achieve better comparison with previous investigations and 
numerical modeling predictions. 
Figure 5.28: Non-dimensional peak pressures m
(X/L=X*=1.0 and 0.5) for experiments involving complete valve obstruction, horizontal slope 
and different anchoring schemes
74 
e supported against the 
5.28. Pressure measurements are presented for the 
point halfway through the pipeline length 
easured at two locations along the pipeline 
 
 
75 
 
 
Figures 5.29, 5.30 and 5.31 present all measured peak pressures grouped by slope and 
obstruction degree; Figure 5.29 (horizontal slope results) also presents the minimum negative 
pressure observed in the full obstruction conditions. This graph facilitates the comparison 
between different air pocket volumes and obstruction degrees. In agreement with the data 
presented in Table 5.1, the pressure peak results with the 89% obstruction are slightly less than 
half than the ones observed in total valve obstruction for similar air pocket volumes; results from 
81% obstruction were about half of the obtained with the 89% obstruction. The scatter in the 
results is attributed either to small flow rate variations between experimental repetitions and/or to 
the inability to shut the knife gate valve at the exact same time between the repetitions.  
 
Figure 5.29: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; horizontal slope results shown 
 
76 
 
Figures 5.30 and 5.31 present measured peak pressures for adverse slopes of 1.3% and 
2.7%, respectively. Even though the general trend from Figure 5.29 is also represented, one 
notices that the pressure drops much more rapidly with larger air pockets than what is observed 
for horizontal slope results.  This result is linked to the geometric characteristics of the upward 
slope free discharge and the initial flow rate. For steeper adverse slopes conditions, the flow rate 
required to sustain a certain cavity volume at the downstream end is comparatively smaller than 
the flow required for shallower or horizontal slopes. Thus, with smaller flow rates, peak surges 
are also reduced.  
This also explains why this trend is reversed for the case of favorable slopes, as indicated 
in Figures 5.32 and 5.33 for slopes 0.5% and 1.0% respectively. One notices that the rate of the 
pressure decrease with the size of the air pocket diminishes for larger positive slopes. In such 
favorable slopes the flow rate required to sustain a certain air cavity at the downstream end of the 
pipe for favorable slopes is comparatively much larger. Even though the pocket volume is in 
some cases much larger than the correspondent ones for horizontal and adverse slope cases, the 
high flow rate still create conditions for large H*=H/D values, above 25 in some conditions. 
 
Figure 5.30: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 1.3% adverse slope results shown
77 
 
 
 
Figure 5.31: Non-dimensional peak pressures a
valve obstruction degree; 2.7% adverse slope results shown
78 
s a function of entrapped air pocket volumes, and 
 
 
 
Figure 5.32: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction 
79 
degree; 0.5% favorable slope results shown 
 
 
Figure 5.33: Non-dimensional peak pressures as a function of entrapped air pocket volumes, and 
valve obstruction degree; 1.0% favorable slope results shown
 
The relationship between
presented in Figures 5.34, 5.35
obstruction types are plotted in terms of the normalized inflow rate Q*=Q/(gD
while Figure 5.34 indicates that for the horizontal slope a total obstruction will cause a peak 
pressure H*=75 with a Q*=0.5, this flow rate is smaller for the 1.3% adverse slope results 
(Q*=0.425) and 2.7% adverse slope (Q*
comparison can be drawn from partial obstruction experiments. For instance, for the 89% 
obstruction results at horizontal slope, the peak pressure H*=25 is attained with Q*=0.45, while 
for 1.3% adverse slope this flow rate is smaller 
 
80 
 
 surge intensity and initial flow rate is supported by the results 
 and 5.36, where the peak pressures for different slopes and 
5)
?0.4) as indicated in Figures 5.35 and 
at Q*=0.37. 
 
0.5. For instance, 
5.36. Similar 
 
Figure 5.34: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 
81 
Horizontal slope results shown 
 
 
Figure 5.35: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 
82 
1.3% adverse slope results shown. 
 
83 
 
 
Figure 5.36: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 2.7% adverse slope results shown 
 
Figure 5.37: Non-dimensional peak pressures as a function of initial flow rate, and valve 
obstruction degree. 0.5% favorable slope results shown, and are qualitatively simi
obtained with the 1.0% favorable slope
 
When the relation between the non
considered for favorable slopes, as presented in Figure 
these variables is noticed. The main reason is that for favorable slopes the required flow rate to 
maintain cavities, even large ones, was generally large. The sensitivity of the cavity volume with 
the flow rate, as noticed in Figure 
range, some conditions also yielded small initial air cavities, associated with large surges.
Figures 5.38, 5.39 and 5.40
as a function of the air pocket volume, but this time results are grouped by the obstruction type. 
Figure 5.38 results include all the maximum and minimum pressures for the total obstruction 
experiments. Despite the data scatter, one clearly sees the
smaller air pockets.  Scatter is more pronounced for the partial obstruction experimental results, 
84 
 
-dimensional flow rate and peak surge head is 
5.37, a much weaker relationship between 
5.37, then becomes much smaller. However, in the tested flow 
 present results of the peak pressures for different pipe s
 trend of largest pressure peaks for 
 
lar to the ones 
 
lopes 
 
and one may notice the tendency of the 1.3% adverse slope peak results to stay between the 
horizontal results and 2.7% adverse slope results. Favorable slope experiments led to results that 
were generally larger than the correspondent ones obtained with horizontal or adverse slopes. 
This separation of the results obtained with different slopes becomes more pronounced for la
air pocket volumes, a result also linked to the differences in the inertia to create air cavities with 
different flow rates and pipe slopes.
 
 
Figure 5.38: Non-dimensional peak pressures as a function of air pocket volu
85 
 
me and slope for 
total obstruction experiments.  
rger 
 
 
Figure 5.39: Non-dimensional peak pressures as a function of air pocket volume and slope for 
86 
81% obstruction experiments. 
 
 
Figure 5.40: Non-dimensional peak pressures as a function of air pocket volume and slope for 
87 
89% obstruction experiments.  
 
 
Figure 5.41: Non-dimensional peak pressures as a function of initial flow rate and slope for total 
Figure 5.41 present measured H* for different slopes as a function of Q*, only for total 
obstruction experiments. While the points are somewhat scattered for larger flow rates 
(Q*>0.53), the series representing the pipe slopes become clearly separ
increase. One notices clearly that peak surges results for adverse slopes have much more 
dependency on the initial flow rate when compared to the favorable slope results
The ratio between the peak pressures measured at the two selected pipe stations (halfway 
through the pipe length X*=0.5 and at downstream end X*=1.0) is presented in Figure 
all tested cases, as a function of the air pocket volume. Despite of the d
that for horizontal and adverse pipe slopes and smaller air pocket volumes the pressure peak ratio 
ranged between 0.55 to 0.75, whereas this ratio had a tendency to increase for larger air pocket 
volumes to value above 0.85. For 
88 
obstruction experiments 
ated as the pocket volume 
ata scatter, it noticeable 
favorable pipe slopes the ratio between the pressure peaks was 
 
. 
5.42 for 
 
consistently at or above 0.8, even with few cases when the pressure at the upstream transducer 
was larger than the one measured at the downstream end.
Figure 5.42: Ratio between peak pressures measured at X*=0.5 and X*=1.0 as a function of the 
With regards to the velocity measurements, an interesting remark is that the flow rates are 
fairly similar right after the valve maneuver within 
the initial head at the upstream reservoir. There was also a tendency of the measured velocity to 
increase after valve maneuvering, as result of water level accumulation at the upstream tank. A 
possible explanation for these similar velocity results, even with different reservoir heads, is that 
the most of the energy losses upon partial valve obstruction occur at the valve itself. This strong 
energy loss may by such that differences in reservoir heads (in the or
may not have an important impact in the final discharge velocity. 
89 
 
entrapped air pocket volume 
the same obstruction degree, independently of 
der of tens of centimeters) 
 
 
90 
 
  
91 
 
5.2 Numerical modeling results 
 
The pressure head predictions resulting from the numerical model represented by 
equations 4.3, 4.4 and 4.5 is presented in this section. Figures 5.43 to 5.48 present a comparison 
between the predicted pressures within the air phase and correspondent measurements. The 
simulations presented in this section are for the horizontal slope experiments only. 
Figures 5.43 and 5.44 presents the comparison between the predicted pressures for the 
total obstruction and the experimental measurements, with Figure 5.43 presenting the minimum 
flow rate conditions (largest air pocket entrapped) and Figure 5.44 presenting the maximum 
pressure condition. It was considered that the valve shut the flow entirely in 0.20 seconds, based 
on observations of the recorded experiment movies. General trend of the measurements is 
captured by the numerical model, with a damped pressure oscillation pattern. However, the 
predicted pressure peaks were larger than the measurements. This may be attributed to some 
extent to the assumption of the valve closure expression, but in fact the results seem to indicate 
that the system has more elasticity than actually accounted for in the numerical model. This 
elasticity in the air-water that mitigates the surges could be due to unaccounted for air volumes, 
rubber joints used in some pipe connections, larger energy dissipation, among others. 
92 
 
 
Figure 5.43: Predicted and measured pressure hydrographs for Vair*=3.5, Q*=0.38, horizontal 
slope, and total cross section obstruction condition. 
 
Which is also unclear is the source for the additional pressure damping in the results 
which is clearly noticed both in Figures 5.43 and 5.44. This behavior was also reported in similar 
studies, such as Li and McCorquodale (1999). Considering that the cause of this discrepancy 
could be linked to thermodynamic effects, a modified version of the numerical model based on 
the work presented by Lee (2005) was developed. This model proposes a thermal damping 
component in the air phase energy equation, modified from the ideal gas law: 
 
g1856g1834g3028g3036g3045
g1856g1872 =g1863
g1834g3028g3036g3045
g1848g3028g3036g3045
g1856g1848g3028g3036g3045
g1856g1872 +
(g1863?1)g1834g1827g3044g1834g3028g3036g3045,g2868g1848g3028g3036g3045,g2868
g1848g3028g3036g3045g1839g3034g1844g3034 g46781?
g1834g3028g3036g3045g1848g3028g3036g3045
g1834g3028g3036g3045,g2868g1848g3028g3036g3045,g2868g4679 
 
(5.1) 
93 
 
In which: A q is the contact area between air and water phases, H is the head transfer coefficient, 
Mg the mass of gas and R g the gas constant.  
However, this modified model has not yielded results that were significantly different 
from the original version proposed with equations 4.3, 4.4 and 4.5. This discrepancy in the 
pressure damping thus remains an open question. 
 
Figure 5.44: Predicted and measured pressure hydrographs for Vair*=0.40 Q*=0.51-0.52, 
horizontal slope, and total cross section obstruction condition 
 
Figures 5.45 and 5.46 present the results of the proposed model in simulating the 81% 
partial obstruction experimental results for the maximum and minimum tested flow rates. The 
discharge coefficient of the valve varied from Cd=0.30 to 0.40, calibrated based on observed 
flow rate after the valve maneuver. The proposed model predicts fairly well the single peak 
behavior of the system as well the pressure variation following the valve maneuver. The same 
observation also applies to the comparison with 89% obstruction presented in Figures 5.47 and 
94 
 
5.48. Particularly interesting for the 89% obstruction predictions is the model?s ability to capture 
to some extent the much damped oscillations that are observed following the pressure peak. 
 
Figure 5.45: Predicted and measured pressure hydrographs, for Vair*~3.4, Q*=0.37, horizontal 
slope, and 81% of cross section obstruction condition 
95 
 
 
Figure 5.46: Predicted and measured pressure hydrographs, for Vair*~0.40, Q*=0.515, horizontal 
slope, and 81% of cross section obstruction condition 
96 
 
 
Figure 5.47: Predicted and measured pressure hydrographs, for Vair*~2.97, Q*=0.382, horizontal 
slope, and 89% of cross section obstruction condition 
97 
 
 
Figure 5.48: Predicted and measured pressure hydrographs, for Vair*~0.367, Q*=0.503, 
horizontal slope, and 89% of cross section obstruction condition.  
98 
 
Chapter six ? Conclusions and Recommendations for Future Work 
 
 The purpose of this research was to provide further insights in the interactions between 
air and water in rapid filling pipe flows in the context of large, below grade stormwater tunnels. 
The focus was in the interactions between entrapped air pockets of various sizes suddenly 
entrapped in pressurized water flows. Unlike previous investigations, surges that are generated 
by the compression of air by water are performed both in cases when there is pressure relief, in 
this instance provided by an orifice, and without pressure relief resembling most previous related 
investigations. The idea of having this setup was to approximate experimental conditions the 
anticipated conditions in actual stormwater tunnels. A numerical modeling study was also 
developed to assess the ability of simplified, one-dimensional models to replicate the results 
obtained in the experiments.  
 As results of the research, there are three essential ideas. At the beginning, the collection 
of valuable experimental data and results on the problem of air pocket entrapment and 
compression in a fairly wide range of conditions. Then, it was made the numerical modeling of 
lumped inertia approach to describe this type of interaction in the pipeline. And, finally, with the 
two first model included it can be arrived a better idea on the physics of the air-water flow 
interactions and how the experimental and numerical are comparable. 
The experimental included essential features present in stormwater storage tunnels. In 
actual conditions, pressure relief would be provided by structures such as surge shafts and 
dropshafts that are frequently tens of meters deep. Even in reduced scale experiments replication 
of these geometric conditions would be difficult, thus the pressure relief was provided by means 
of an orifice that provides escape for water during the surge process. This orifice was created by 
the gap of a partially closed knife gate valve. It is thus possible that pressure surges obtained here 
may be regarded as conservative estimates of air-water surges that would occur in stormwater 
tunnel systems. This assessment would require more detailed investigations, which would 
include larger scale experimental apparatus. 
99 
 
The experimental program had a very important task of highlighting the difference 
between flow characteristics for total and partial flow obstructions. As mentioned, the majority 
of previous investigations focused in situations involving air entrapped in water flow, in which 
the flow would compress with little or no air ventilation. The experiments performed with 
complete obstruction conditions yielded general pressure oscillations with characteristics similar 
to a spring-mass system. Similarly to observations by previous investigations, these experiments 
yielded surge oscillations with higher magnitude and frequencies for smaller air pocket volumes. 
Frequently the pressure oscillations reached sub-atmospheric levels. 
The experiments with air pocket compression while allowing for pressure relief presented 
a pattern that was not reported in previous investigations. To horizontal and adverse, there was 
observed a single pressure pulse, and in most cases no residual oscillations was detected, an 
outcome significantly different from the total obstruction systems. To favorable, there was notice 
that it is a connection between the case with too much opening and the total obstruction by the 
valve. Moreover, the pressure magnitudes are significantly smaller while negative pressures were 
not observed in almost all cases. The identification of this distinctive behavior is important so 
that numerical tools may be constructed to incorporate those observations. 
This research also has been useful in isolating the role of the initial flow rate in the surge 
magnitudes separately from air pocket volumes. Because experiments with comparable pocket 
volumes were performed with significantly different initial flow rates, it was verified that surge 
peaks increased with the initial inflow rates, and could be significant even in cases when air 
pocket volumes were high. Previous comparable studies have not varied systematically the flow 
rates while performing surge analysis, so these findings are also very useful in the calibration of 
future numerical models.   
While all research objectives were successfully attained, it is recognized that this study is 
just an initial step to the path of improving our understanding of two-phase flow in stormwater 
tunnels systems. Clearly there are other relevant topics that should be addressed in future 
100 
 
investigations. Among them is the study of air pocket motion and pocket spreading following 
entrapment. The study of air pockets migration is very relevant in the design of ventilation points 
and tunnel filling dynamics. Another relevant topic to be addressed in future investigations is the 
air entrainment caused by sweeping pipe-filling bores. Depending on the characteristics of the 
moving bore, the volume of air entrained may be significant, affecting the continuity in the 
pressurized flow zone. This is study is also relevant because entrained air reduces the acoustic 
wavespeed, which has impacts in pressure predictions.   
Another area that warrants further research is the creation of an improved numerical 
model able to simulate the filling of stormwater tunnels considering effects of entrapment, 
compression, motion, and spreading of entrapped air pockets. In order to incorporate in future 
numerical model, three fundamental gaps will need to be addressed: 1) the air entrainment 
mechanisms into pipe filling bores; 2) motion of entrapped air pocket due to drag and shear 
forces; and 3) motion and ventilation of entrapped air pockets.  
  
101 
 
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