IntroductionTwo-phase flow dynamics is important when dealing with conduits that are subjected to transitional gravity filling conditions, such as elements of an urban drainage system (UDS) during intense rainfall. If UDS tunnels are poorly ventilated, trapped air pockets may occur and become pressurized, negatively influencing the operating conditions of the system (Vasconcelos and Wright 2009). When a transition to pressurized flow in UDS sections occurs, an air–water front appears, resulting in a regime with both gravity and pressurized flow or mixed flow (e.g., Vasconcelos et al. 2006; Politano et al. 2007; Leon et al. 2010).Pressure surges in pipelines can be associated with fast pressure fluctuations or prolonged pressure alterations. High pressure peaks in UDS during storm events can be induced by the release of trapped air pockets, which cause flow acceleration that may result in conduit failure or induce flooding. The formation of air pockets in sewers due to the bridging of stratified flow during the filling of the conduit may be accompanied by a mobile hydraulic jump. The flow transients, which are related to changes in flow depth, can act to entrain air into the water column, resulting in prolonged pressure change and deviations of the operating conditions of the system until the air pocket collapses.When trapped air pockets are released, flow acceleration in the undulating pipeline can result in unexpected pressure surges (Liou and Hunt 1996). For example, if a rapidly advancing water column passes a knee point when undulating pipeline profiles are being filled, negative gauge pressures can arise (Liou and Hunt 1996; Malekpour and Karney 2011). This phenomenon can introduce intrusion of contaminants (Funk et al. 1999) or even structural buckling (Chaudhry 2014).The transition from stratified to slug flow in a horizontal pipe is facilitated by the pressurization bore forming a nonstationary flow, which results in a mobile hydraulic jump. This hydraulic jump acts to entrain air from the air pocket until the fully filled condition is reached. Filling experiments on an industrial-scale pipeline carried out at Deltares (formerly Delft Hydraulics) (Hou et al. 2014) confirmed the intrusion of water–air fronts with different speeds along the pipeline, starting from a full vertical water–air front at the pipe bridge. The Deltares experimental apparatus also enabled the researchers to generate stratified flow, which touched the pipe obvert (i.e., bridged) due to the formation of a mobile hydraulic jump and the consequent occurrence of an air pocket during pipeline filling. Measurements of the flow rate, pressure, and water level along the horizontal polyvinyl chloride (PVC) pipeline, which was 261 m long and had an internal diameter of 236 mm (Hou et al. 2014), were used with camcorder recordings to quantify the transitional flow process. When air pockets form during the filling of a pipeline, a volumetric water flux difference can occur between the inflow and outflow. If the inflow rate is significantly larger than the outflow rate, the pressurization bore velocity may be considerable, and unexpected changes in the pressure gradient (i.e., subatmospheric conditions) can occur in the downward sloping pipe (Politano et al. 2007).There are no clear criteria for predicting the formation of hydraulic jumps in closed conduits with concurrent air pocket entrapment during filling. Air pocket formation in the pipe may be related to stratified flow instabilities: shear flow instability (SFI), and long wave instability. Currently, the shear flow instability criterion is used for pipe flow (Kordyban 1990; Li and McCorquodale 1999). Investigation of the advance of air cavities in pipes was introduced by Zukoski (1966) and Benjamin (1968). The Zukoski number was used by Laanearu et al. (2012) to characterize the air-cavity dynamics in the case of large-scale pipeline emptying in which air pockets do not form during stratified flow. The difference between an interfacial Froude-like parameter (Eldayih et al. 2020), which is related to the SFI, and the Zukoski number criterion, which is related to the air–water interaction in the closed conduit, is the definition of celerity. For the SFI criterion, the relative velocity (between air and water in the stratified-flow regime) is normalized by the square root of gravitational acceleration times the hydraulic depth, whereas for the Zukoski number, the velocity of the air cavity is normalized by the square root of gravitational acceleration times the pipe diameter. The validity of the stratified flow instability criteria in closed conduits is unclear with respect to the presence of a shear flow and a long wave, while the air flow is insignificant. In this study, the SFI condition of stratified flow is represented with respect to the Zukoski number. The long wave instability is based on the free-surface-flow Froude number that is defined herein for the conduit geometry.The ability to predict and understand the phenomenon of air–water mixed flow in UDS is of great interest. It can be speculated that in a storm sewer pipe, a series of air pockets may form during filling, changing the operating conditions. The numerical study of the transition from stratified to slug flow via a pressurization bore raises a number of challenges in terms of solving the air–water interaction conditions. The Storm Water Management Model (SWMM) (Rossman 2015), which is used widely to simulate storm water flow conditions, is not able to capture the dynamics of rapid flow changes in hydraulic systems (Vasconcelos et al. 2018). Computational fluid dynamics (CFD) models that use the volume of fluid (VOF) method to solve the two-phase flow have been able to reproduce the pressurization bore and predict the behavior of the air–water mixed flow (Laanearu and Kaur 2018; Eldayih et al. 2020). The Reynolds-averaged Navier–Stokes (RANS) approach, which requires a solution to the Reynolds stress term, is one of the most widely used methods for turbulent-flow modeling. Two-equation models that simulate the eddy viscosity via turbulent kinetic energy production and dissipation (e.g., Launder and Spalding 1972; Shih et al. 1995) commonly are used. Air bubbles aggregate and decompose in the turbulent environment of a hydraulic jump, dissipating the mechanical energy of the flow process (Chanson and Brattberg 2000). However, studies of turbulent kinetic energy profiles in air–water mixed flow are scarce (Pothof 2011), due to the presence of air bubbles, buoyancy, and turbulence.This work explains the bridging conditions of stratified flow in a closed conduit using experimental and numerical model findings. Based on these findings, the pressurization bore in the closed conduit was quantified, and a parameter of the mixing efficiency of strongly coupled immiscible fluids was introduced.MethodologyExperimental MethodFilling experiments were carried out on a horizontal pipeline containing a bridge which was used to accelerate the water flow. Transparent polymethyl methacrylate (PMMA) pipe segments 2 m long and with an inner diameter of 0.08 m were used to assemble a pipeline 18 m in length. During the experiments, this pipeline was fed from an upstream tank, and the pressure head was kept constant. The pipeline discharged into a base reservoir of 150  m3, which was used to fill the upstream tank. The experimental apparatus is illustrated in Fig. 1. The numbered locations on the schematic of the experimental facility are used for reference in the descriptions of the instrumentation and procedure given herein.The operating conditions of the experimental apparatus first were mapped in a series of tests. The intent was to operate the apparatus in a manner that resulted in the formation of a sufficiently large air pocket to be able to investigate the advance of the pressurization bore along the pipeline. We aimed to create a stratified flow interface bridging 5 m downstream of the pipe bridge, and to achieve a stationary inflow rate throughout the series of experiments. The inflow rate was adjusted by changing the initial pressure in the upstream tank (Fig. 1, 1) and the setting of the inflow valve (Fig. 1, 3). A suitable operating frequency was found for the pump supplying the tank (Fig. 1, 1) to keep the water level in the tank constant. The Reynolds number for fully filled flow at the target flow rate was R=56,000.In the initiation stage, the section of pipe (Fig. 1, 3–6) between the upstream tank and bridge was filled with water, i.e., all air was expelled. The outflow valve (Fig. 1, 13) was fully open, and the position of the inflow valve (Fig. 1, 3) was adjusted to achieve the target flow rate. Section 6–13, initially containing only air, was subjected to free-surface water inflow via the pipe bridge (Fig. 1, 6) at a constant inflow rate Q=3.5  l s−1, corresponding to pipe Froude number f=Q/(A(g D)1/2)=0.8 according to Hager (1999), where A = water flow cross-sectional area of partially filled case, yielding the formation of an air cavity. Bridging of the stratified flow occurred 5 m after the pipe bridge (Fig. 1, 6) at Section 10–11, resulting in the formation of an air pocket. The pressurization bore travelled upstream with celerity, cjump as the air was mixed and entrained via the roller near the pipe obvert, thereby gradually decreasing the volume of the air pocket. The velocity distributions (at Location 9), water levels (at Locations 8 and 10), and pressure changes (at Locations 5, 7, and 11) were captured at different stages of development of the transitional flow process, for a constant inflow rate and varying outflow rate. Electromagnetic flow meters (Fig. 1, 4 and 12) were used to determine the water volumetric flux differences (data from the downstream meter were filtered after the measurements to account only for the full cross-section water flow portions). The flow development stages were defined with respect to Location 9, the section at which particle image velocimetry (PIV) measurements were taken, as follows: 1.stratified flow was when the bore was located 8 pipe diameters downstream of Location 9;2.roller was when the bore was located at Location 9; and3.slug flow was when the bore was located 3 pipe diameters upstream of Location 9.After the first series of measurements of Stages 1–3, additional air was supplied to the air pocket through a valve at the crown of the pipe bridge (Fig. 1, 6), thus readjusting the bridging location to 5 m from the pipe bridge and restarting the advance of the pressurization bore along the horizontal pipe section. Fifteen repeat measurements were taken of each stage. Before ensemble averaging was conducted, the individual PIV frame sequences were analyzed in order to discard frames in which the roller was not precisely at the predetermined position. Additional measurements of fully filled flow were conducted for a numerical model sensitivity study (Fig. 2) after the pressurization bore reached the bridge crown, i.e., when all air was expelled from the pipeline. The instrumentation used in the experiments was as follows: •Electromagnetic flow meters (Fig. 1, Locations 4 and 12) were used at the up- and downstream ends of the pipeline. The sampling frequency for both flow meters was set to 100 Hz. The uncertainty in the readings was ±0.6% for the upstream flow meter and ±0.5% for the downstream meter, for water-only conditions.•Strain-gauge pressure probes were used on the pipe (Fig. 1, Locations 5, 7, and 11), and on the upstream tank (Fig. 1, 2). The sampling frequency for all pressure probes was set to 100 Hz, and the uncertainty in the pressure measurement readings was ±0.3%.•A PIV system was used (Fig. 1, Location 9), consisting of a high-speed camera and a continuous-wave laser. The frequency of measurement varied from 1,155 to 2,982  images/s, depending on the Reynolds numbers for fully filled and stratified flow. The uncertainties in the captured PIV velocity field components and the derived values of turbulent kinetic energy were quantified using the uncertainty propagation techniques presented by Sciacchitano and Wieneke (2016). The uncertainties of the horizontal along-pipe and vertical cross-pipe velocity components, hereinafter corresponding to axial and vertical velocity components, and the turbulent kinetic energy at the different stages of flow development are presented in Table 1. The larger uncertainty in the vertical velocity component arose from the settings for the PIV sampling frequency; this depended on the mean axial velocity, which was more than 1 order of magnitude larger than the vertical velocity component for the measurements of stratified, roller, and slug flow.•Capacitive sensors for water level metering (Fig. 1, 8 and 10) were placed at each side of Section 9. The sampling frequency for both was set to 100 Hz, and the accuracy of the measurement was 1 mm.•Two synchronized data acquisition modules were used to collect flow rate, pressure, and water level data.Table 1. Uncertainties in PIV measurements of axial and vertical velocities and turbulent kinetic energy at different stages of flow development (%)Table 1. Uncertainties in PIV measurements of axial and vertical velocities and turbulent kinetic energy at different stages of flow development (%)QuanitityStratified flowRollerSlug flowAxial velocity0.6752.451.98Vertical velocity12.411.69.16Turbulent kinetic energy0.9877.433.48Numerical ProcedureTo interpret the different stages of the pressurization bore development, CFD modeling was applied. A three-dimensional computational domain geometry was created based on the physical dimensions of the experimental apparatus. Simulations of the two-phase flow were conducted using the interFoam solver of OpenFOAM v1906 (OpenCFD Ltd.) software. The flow regime was turbulent, and the air and water phases were considered to be incompressible, isothermal, Newtonian, and immiscible fluids (Hou et al. 2014). Interface capturing for the air–water flow was resolved by applying the VOF phase fraction–based approach (Hirt and Nichols 1981; Bombardelli et al. 2001). The transport equations for continuity [Eq. (1)] and momentum [Eq. (2)], in which the two immiscible fluids are considered as a single joint fluid through the domain, are (1) (2) ∂(ρU)∂t+∇·(ρUU)=−∇p′−g·x∇ρ+σκ∇γ+∇·(μ∇U)+(∇U)·∇μwhere U=γUl+(1−γ)Ug = velocity vector of two-phase fluid; ρ=γρl+(1−γ)ρg = average density of the fluid within a cell; t = time; μ=γμl+(1−γ)μg = average dynamic molecular viscosity; p′ = pressure, modified to exclude hydrostatic contribution; g=(0,0,−g), where g = acceleration due to gravity; x = position vector; σ = surface tension; κ=∇·n = interface curvature, where n=−∇γ/|∇γ|; and subscripts l and g correspond to liquid and gas, respectively.To define the portion of the cell occupied by the fluid, the phase fraction γ is used as an indicator function. The transport of γ is expressed by an advection function ∂γ/∂t+∇·(γU)=0. The phase fraction, γ, representing the excess density, can range within 0≤γ≤1, with values 0 or 1 corresponding to regions entirely of air or water, respectively. To provide a sharper interface resolution, especially for large density difference two-phase flows, Weller (2002) introduced an extra term of artificial compression in the phase fraction equation, so the transport equation is (3) ∂γ∂t+∇·(γU)+∇·(γ(1−γ)Ur)=0where Ur=Ul−Ug = velocity field to compress interface.Because the water in our experiments was circulated from an indoor reservoir, the physical properties in the numerical model were chosen based on the laboratory air temperature of 20°C. The density and dynamic molecular viscosity of water were set to 998.2  kg m−3 and 1.002×10−3  kg m−1 s−1, and those for air were set to 1.204  kg m−3 and 1.82×10−3  kg m−1 s−1, respectively. The interfacial tension plays an important role in the two-phase flow of immiscible fluids that are strongly coupled due to the formation of the air pocket. The surface tension coefficient was set to σ=0.0728  N m−1 for the air–water interfaces.The performance of the RANS model for the transitional flow was analyzed by comparing the measured and modeled velocities, the turbulent kinetic energy distributions, the water levels and the location of formation and dynamics of the hydraulic jump. Based on the performance analysis, the renormalization group (RNG) k-ε model was used in the numerical experiments.The RNG k-ε turbulence model was derived using a statistical technique called renormalization group theory (Yakhot and Orszag 1986; Smith and Reynolds 1992; Smith and Woodruff 1998). The RNG model contains an additional term in the equation for ε that improves the accuracy for rapidly strained flows. The effect of swirl on turbulence is included in the RNG model, which enhances the accuracy of swirling flows. In addition, the RNG theory provides an analytical formula for turbulent Prandtl numbers, whereas the standard k-ε model uses empirical, constant values (Launder and Sharma 1974). The values of the constants in the equations for k and ε stemming from the RNG analysis are Cμ=0.0845, Cε1=1.42, Cε2=1.68, and σk=σε=0.72 (Orszag et al. 1996; Pope 2000). The advantage of the RNG k-ε turbulence model is that it also accounts for small scales of fluid motion, and therefore it is more suitable for flows with a low Reynolds number.In the present CFD solver, the wall functions approach is based on universal flow profiles in the boundary layer along a wall, which is divided into three regions based on the viscous, buffer, and inertial sublayers. The production of turbulence occurs at varying normalized distances from the wall, and depends on the Reynolds number. The turbulence model in the CFD solver used in this study places the first internal grid point in the vicinity of the pipe wall in the inertial sublayer and approximates the viscous sublayer by the wall function boundary condition.At the domain inlet, initial values of the turbulent kinetic energy and turbulence dissipation rate are required. The initial values of k and ε were determined from the measurements and used in the setup of the model. These values were set to k=0.00124  m2 s−2 and ε=0.00128  m2 s−2. Concurrently, a constant inflow rate boundary condition was set for the inlet, and a zero gradient boundary condition was set for the outlet, in accordance with the experiments.The computational domain was discretized using a hybrid grid containing a blend of structured and unstructured grid areas. The snappyHexMesh utility from OpenFOAM, which creates the grid using triangulated surface geometries, was used to generate the mesh. The domain was discretized more or less uniformly over the pipe cross-section, considering that the mixing processes of particular interest occur near the centerline, whereas the boundary layer was resolved by applying wall functions. A mesh convergence analysis was performed on the full-scale computational domain. Fig. 2 compares the measured axial velocity profile and the modeled profile obtained with three different grid resolutions for fully filled flow. It is apparent from the graph that the finer grid resolutions with 0.54 million and 0.77 million elements gave almost identical results for the velocity distribution, and therefore we used 0.54 million finite volumes in this study.Parameterization of Stratified FlowEldayih et al. (2020) found that SFI during the regulated emptying of a pipeline was characterized by an interfacial Froude-like parameter (FI), which depends on the relative velocity between the air and water, and on the hydraulic water depth (area/free-surface width) of the flow region in free-surface flow mode. If this value exceeded a critical value, defined as FC, then SFI occurred during the emptying of the pipeline. The expressions for both FI and FC were adopted from Li and McCorquodale (1999) and Kordyban (1990). In the present study, a modified SFI criterion was tested for the formation of an air pocket during rapid filling of a horizontal pipe via a pipe bridge to clarify the conditions under which it occurs.The onset of shear-flow-type instability can be characterized by the Zukoski number, which is defined as the ratio of the air cavity speed to the celerity of a shallow water wave, based on the pipe diameter. The Zukoski number represents a dimensionless speed that is related to the rate of change of the length of the water column due to the intrusion of air above the water layer. In the case of pipe filling, the air cavity speed is defined as the difference between the water inflow velocity of the fully filled section, UVF, and the velocity of the upper water–air front at the upstream end of the cavity UVF1 (Fig. 3).The Zukoski number then is defined as (4) where D = internal diameter of pipe. In the case of a frozen water–air upstream front in the pipe, the upper-front velocity UVF1 is zero; the Zukoski number therefore is determined by the fully filled inflow velocity, UVF, and is equal to Zu=0.8. In the present experiment, the instability appearing at the downward sloping section of the pipe bridge was thought to be of the Helmholtz type, modified by the influence of the gravitational field, which is not perpendicular to the air–water interface. Zukoski (1966) found that the amplitude of Helmholtz instability in the interface also increases with increasing cavity speed. While the upper water–air front is frozen, i.e., the cavity moves with a velocity equal and opposite with respect to the water column, the instability of the lower water–air front advancing into the horizontal pipe section results in a mobile hydraulic jump. The formation of the cavity, with respect to the pipe Froude number value, is in accordance with values mapped by Hager (1999) because the pipe bridge acts to trap the upper water–air front and mimics the pipe outlet conditions. In the present study, the Zukoski number value that corresponds to the frozen interface case at the pipe bridge was found.Based on the assumption that air is forcing the interface motion to create water waves, forming instability, Li and McCorquodale (1999) proposed a normalized condition for the transition of free-surface to pressurized flow by discontinuity. They built on the work of Milne-Thomson (1938), who proposed an equation for the instability condition based on the small-amplitude waves between water and air. A modification is proposed here in which the condition is normalized based on the square root of the gravitational acceleration and the pipe diameter (5) |UVF1−UVF2|g·D≥Kf(1−ρaρw)(ρwρatanh(2πhaλ)+tanh(2πhwλ))λ2πDwhere hw and ha = hydraulic depths of water and air, respectively; and ρw and ρa = densities of water and air, respectively. If the air velocity is zero, corresponding to a frozen water–air upstream front in the pipe, then the instability is determined by the lower-front water velocity in the stratified-flow regime UVF2=1.15  m s−1. The water–air front velocity UVF1, the fully filled inflow velocity UVF, and the water velocity in the stratified-flow regime UVF2 are related through the Laanearu et al. (2015) formula, UVF1=UVF/α−UVF2(1−α)/α, where α is the void fraction, representing the effective cross-sectional area of air. The void fraction is proportional to the cross-section average of the parameter (1−γ), represents the effective cross-sectional area of the air, and is equal to unity when the vertical water–air front exists over the cross-sectional area. Hence, UVF2 on the left-hand side of Eq. (5) can be expressed in terms of UVF and UVF1. Because UVF1 is assumed to be practically zero, the SFI criterion can be expressed as (6) UVFg·D≥Kf(1−α)(1−ρaρw)(ρwρatanh(2πhaλ)+tanh(2πhwλ))λ2πDby making use of the definition of the Zukoski number. The Zukoski number is fundamentally related to the stratified flow, similar to an interfacial Froude-like parameter, and unlike the Froude number, which is related only to the water-layer mechanical energy. In this modified SFI criterion, λ is the wavelength of the free-surface water wave, which may vary from 20 mm (for capillary waves) to a few times the water depth. The correction factor Kf is due to the reduction in the air flow area, and the entrance and alignment conditions. According to this criterion, Kf should be on the order of magnitude of 1/10 for the instability to arise, because the criterion value varies in the range 2.8–5.7, whereas λ varies from the capillary wave length to a wavelength equal to the pipe diameter (Li and McCorquodale 1999). The SFI criterion given here would be applicable to air pocket formation if the setup produced a significantly higher air velocity. Therefore, a criterion for the onset of instability that is governed by the Froude number, i.e., long-wave-type instability, is considered. Pratt (1986) described long-wave instability criteria for the sill flows resulting from supercritical solutions in which the propagation of the perturbed-flow interfacial waves is possible only in the downstream direction of flow. Based on the similarity criterion, long-wave instability was applied in this study.The onset of long-wave-type instability during free-surface inflow that results in the formation of a pressurization bore is indicated by the supercritical Froude number. The equation form [Eq. (7)] was derived by solving for the cross-section minimum specific energy, making use of the partially filled circular cross-section of flow area, and is expressed as follows: (7) In the experiments, the water level above the pipe invert was h=0.045  m, D=0.08  m, and Q=3.5  l s−1. The corresponding Froude number for the stratified flows stage then was Fr=2. This corresponds to supercritical flow, but is sufficiently small to facilitate the formation of a long air pocket. The value of the Froude number was calculated based on experimental data captured when the bore in the pipe had already formed.MixingIn the transitional flow process, air–water mixing and the formation of the mobile hydraulic jump are responsible for the entrainment of air into the water (the formation of air bubbles) and water into the air (formation of water droplets in air) inside the pipeline, which is quantified in the numerical model by the phase fraction γ. In the air–water mixing flow process, such as the hydraulic jump, the kinetic energy can be converted to potential energy due to increase of the water level and pressure. The concept of mixing efficiency can be used to relate the amount of mixing in the stratified flow to the amount of energy available to support mixing (Davies Wykes et al. 2015). Mixing efficiency is used to characterize the internal hydraulic jumps of two-layer flows with a small density difference (Ogden and Helfrich 2016). However, the density variations inside the air–water roller are present due to external forcing, and the mixing efficiency therefore is applicable in a general way. As per its definition, the mixing efficiency for the air–water roller can be formulated based on the flux Richardson number, representing the ratio of the buoyant destruction to shear production. In the case of temperature-stratified single-phase flow, because the shear production is positive with the minus sign displayed, the sign of the flux Richardson number depends on the sign of the heat flux (Kundu et al. 2012).The mixing efficiency parameter was introduced in this study to estimate the turbulence energy consumed in the air–water mixing process in the pressurization bore. In the case of horizontal flow with a vertical jump, the flux Richardson number can be given as (8) Rif≈−gγU3−u1u3¯dU1dx3where U1 and U3 = axial and vertical velocity components, respectively; and u1 and u3 = respective fluctuating velocity components. In the numerator of Eq. (8), the buoyant destruction is replaced with the loss to potential energy, the parameter that appears in the equation for the mean flow’s kinetic energy per unit mass (Kundu et al. 2012).The mixing efficiency of the air–water interface roller (i.e., at the bridging location) is determined based on the results of full-scale numerical modeling using the RANS equations. The maximum of the profile is located in the region in which the volume fraction gradients occur, i.e., near the interface (Fig. 4). If measurements indicate the presence of turbulent fluctuations, but at the same time the value of Rif is positive, then it can be concluded that the turbulence is decaying. The water level is increased due to long-wave instability forming a hydraulic jump that mixes the air and water phases, virtually mimicking a single-phase-like state in the roller. The experiments indicate that the fluid phases are not well mixed near the pipe obvert (Rif<0) (Fig. 4). In accordance with the positive flux Richardson number, the buoyancy fluxes are suppressing the turbulence through the hydraulic jump development stage, facilitating the separation of the mixed phases. The hydraulic jump roller is trailed by slug flow with mobile air pockets at the pipe obvert. The mixing efficiency parameter can be used to quantitatively assess the mixing regimes of stable and unstable two-phase flow.Control-Volume ModelThe stratified flow transition to slug flow is modeled within the framework of a control volume (CV) using the principles of conservation of mass and momentum. The aim of the CV analysis is to establish a relationship for the speed of the hydraulic jump. Based on a straightforward relationship between the CV mass and momentum equations, the speed of the mobile hydraulic jump can be expressed as: (9) cjump=(Uu−Ud)−Auρu(Aupu−Adpd+Rx)Adρd(Auρu−Adρd)where (pu, Uu, Au, ρu) and (pd, Ud, Ad, ρd) = average pressure and water velocity, water flow cross-section area, and density immediately upstream and downstream of jump, respectively; and Rx = friction force.Therefore, the pressurization bore that occurs during the filling process of the pipe can be characterized by the surge wave speed, which essentially is determined by the pressure, the cross-sectional area of water flow, and the density jumps. The friction force at the hydraulic jump section can be omitted (Rx→0) as the mixing region length is on the order of magnitude of the pipe diameter, and the mixing-efficiency parameter value indicated that the buoyancy fluxes are suppressing the turbulence (Rif>0). Calculations based on experimental data captured during the measurements at the roller stage revealed the value of the hydraulic jump speed cjump≈0.04  m s−1.Results and DiscussionFormation of the Air PocketThe filling process under investigation has three stages: air cavity formation, air pocket formation, and air entrainment. This three-stage mixed flow process lasts about half an hour, until the trapped air is entrained out from the pipeline.The operating conditions of the experimental apparatus were chosen to produce an air pocket. The target flow rate resulted in the cavity formation, as two water–air fronts appeared, the upper water–air front becoming frozen at the pipe bridge crown and the lower front advancing into the horizontal pipe section. The prerequisite of the cavity formation corresponds well to the pipe Froude number that is associated with cavity formation at the horizontal pipe outlet (Hager 1999). In the present study, the pipe bridge mimicked the pipe outlet conditions. In terms of the cavity formation in the undulating pipe, the pipe bridge acts as a cavity trap. The pipe Froude number value corresponds to the range resulting in a trapped air cavity at the pipe bridge crown. The formation of the air cavity caused the stratified flow that led to the formation of a hydraulic jump in the horizontal pipe section.As the mobile hydraulic jump advanced, two distinctive regions were observed behind the jump. The roller acted to entrain air, resulting in bubbles being propelled toward the pipe centerline, forming a bubbly flow [Fig. 5(c)], before aggregating near the obvert and resulting in slug flow. The region of bubbly flow spanned approximately 2 pipe diameters behind the roller.The Deltares experiments, conducted on a pipe with a 235-mm diameter and used here for comparison, also manifested the three stages of the mixed flow process (Fig. 6). The scales of these two experiments were different; in the present experiment, for fully filled flow, the Reynolds number was 56,000, whereas in the Deltares experiments the Reynolds number was 950,000. The length of the air pocket, the end of which was the bridging location, and the region of bubbly flow were dependent on the cross-sectional filling ratio of the pipe, h/D, and consequently on the Froude number. Stronger supercritical flow conditions correspond to smaller volumes of captured air. The size of the air pocket also is related to the volumetric water flux difference between the inflow and outflow of the pipeline.Table 2 presents experimentally determined values of the normalized water–flux differences, air pocket volumes, hydraulic jump speeds, and Froude numbers. The hydraulic jump speed cjump was calculated in the reference frame of the pipe, considering that the bore moves on top of the stratified inflow. Two groups based on the filling ratio, at which point the air pocket formed, are apparent in Table 2. Experiments that resulted in mobile hydraulic jump formation from a filling ratio h/D>0.5 corresponded to the comparatively less-supercritical Froude number and thus the higher hydraulic jump speed. The more supercritical Froude-number experiments with h/D<0.2 had a lower air–water front speed. Because the two experimental facilities were of a different scale, the water flux differences between the inlet and outlet were normalized by the volumetric inflow rate, (Qinflow−Qoutflow)/Qinflow, and the air pocket volumes were normalized based on the cube of the pipe diameter (V/D3).Table 2. Average normalized water flux differences, normalized air pocket volumes, hydraulic jump speeds, and Froude numbers for air pocket formation induced by weaker and stronger supercritical flow conditionsTable 2. Average normalized water flux differences, normalized air pocket volumes, hydraulic jump speeds, and Froude numbers for air pocket formation induced by weaker and stronger supercritical flow conditionsPipe diameter (m)Filling ratioNormalized water flux differenceNormalized air pocket volumecjumpFr0.080.630.00211.6——0.080.630.0120.90.0420.080.630.01222.6——0.080.630.0225.6——0.2350.530.06450.0673.50.2350. the present experiments, the air entrained through the roller, and the jump approached the bridge with speed cjump, which decreased during the two-phase process. The numerical modeling suggested that the speed of the hydraulic jump changes during the process. During the first 5 s after the bridging event cjump≈0.07  m s−1, which decreased to 0.01  m s−1 over the next 1.5 min. Formula-based calculations using Eq. (9), which included experimental data captured during the roller measurements, revealed that the value of the hydraulic jump speed was cjump≈0.04  m s−1. This relatively slow process investigated here had an approximate air entrainment rate of 7×10−5  m3 s−1. Pressure changes also were apparent, with a small vacuum forming in the air pocket. For the laboratory-scale system, the pressure gradient through the jump was about 200 Pa. Subatmospheric conditions also were detected in measurements of the Deltares experiments during the mobile hydraulic jump’s motion towards upstream.Velocity and Turbulent Kinetic Energy ProfilesFigs. 7–9 show the measured and modeled velocities and turbulent kinetic energy profiles in normalized form. The PIV measurements were facilitated by flow visualization, which involved seeding the water flow with oxygen and hydrogen bubbles generated via electrolysis. However, for two-phase mixed flow, visualization of the air flow was not straightforward, and the air flow therefore was not captured with PIV, although in the comparative figures, modeling results are shown for both water and air.Normalization consistently was done based on the maxima of the water flow profiles. The measured profiles were used as the basis, and the axial and vertical velocity profile maxima were found. The modeled axial and vertical velocities then were normalized based on the cell value corresponding to the coordinates of the experimentally determined maxima. Turbulent kinetic energies were normalized based on the cell value corresponding to the experimentally determined maxima of the axial velocity. The values of the normalization constants for the measured and modeled flow profiles and their differences are presented in Table 3.Table 3. Values of normalization constants for measured and modeled flow profiles and their differencesTable 3. Values of normalization constants for measured and modeled flow profiles and their differencesFlow stageQuanitityDifference (%)MeasuredModeledStratified flowAxial velocity1.3641.18513Vertical velocity0.0150.01310Turbulent kinetic energy0.0120.01414RollerAxial velocity1.3361.16213Vertical velocity0.1070.11810Turbulent kinetic energy0.0120.01423Slug flowAxial velocity0.9821.0052Vertical velocity0.0350.02432Turbulent kinetic energy0.0170.01132The axial velocity component had better qualitative agreement than the vertical component between the measured and modeled transitional flow stage of stratified flow (Fig. 7). Comparison of the modeling and experiments of the stratified flow stage revealed slight differences in the interface behavior. In the experiments, it was observed that the vertical air–water mixing already was present in the stratified flow portion, and the mixing intensity increased gradually toward the roller. The numerical model had behavior more similar to that of a gradually varied flow, resulting in the same bridging location. Concurrently, the experiment revealed faster vertical component growth in this stage.The measured and modeled axial and vertical velocity components in the roller flow stage had the overall best qualitative (Fig. 8) and quantitative (Table 3) agreement. Due to the model’s ability to reproduce the roller’s behavior consistently, it was possible to use the modeled velocities in mixing efficiency calculations.The measured and modeled vertical velocity and turbulent kinetic energy of the slug flow stage (Fig. 9) had the lowest quantitative agreement (Table 3). This primarily was due to the frequency of PIV measurements, and secondly because the measurement results were affected by the mixed-in air releasing due to buoyancy. This also was apparent in the case of axial velocity in Fig. 9, in which the measured and modeled profiles coincide merely for the water phase.The data in the figures and in Table 3 reveal that the qualitative similarity between the measured and modeled profiles is apparent. In addition, the distribution of turbulence energy is qualitatively similar to that of other studies. Turbulence intensity and Reynolds stress distributions measured in and behind the hydraulic jump in open channel rectangular flow by Resch and Leutheusser (1972) revealed turbulence energy maxima near the centerline of the flow behind the jump, as also was the case in the present study.ConclusionThis study provided evidence of the formation of an air pocket under conditions of nearly-zero velocity of the air phase in a poorly ventilated conduit. Experiments, numerical modeling and parameterization were used to gain insight into the bridging conditions of stratified flow.The upstream boundary condition in the experiments was regulated by pressure, measured by a flowmeter, and controlled with the water–air front at the pipe bridge. The experiments made it possible to validate the applicability of the VOF method of numerical modeling of the mobile hydraulic jump in a conduit with known geometry and filling conditions. However, the PIV velocity field measurements of the air phase in the two-phase flow process remain a challenge. Therefore, the numerical model was validated based on water-phase measurements and bridging dynamics, and the validated model was used to analyze the air behavior.In general, the CFD results agreed with the observations of the experiments. Nevertheless, the modeling and experimental results revealed some differences in the interface behavior. The numerical modeling resulted in less-turbulent structures in the stratified flow region leading up to the hydraulic jump roller. Even though the sensitivity study conducted based on the experimental stage in which all air had been entrained had good agreement, higher resolutions might capture the air and water mixing more accurately. In terms of the bridging location of the stratified flow and the dynamics of the resulting mobile hydraulic jump in experiments, comparison with the standard k-ε turbulence model and RNG k-ε turbulence model indicated that the latter predicts the behavior more consistently. The transitional flow process produces rapid strain and streamline curvature. Due to the additional term in the transport equation of turbulent dissipation, the RNG model yields lower turbulent viscosity in rapidly strained flows than the standard k-ε turbulence model. It is assumed that, because the bridging location is set by the aggregation of the perturbation signals, the standard model results in an inaccurate location due to higher rates of turbulent kinetic energy.The stratified flow was parameterized to explain the instability that results in air pocket formation. The results revealed that the SFI condition would hold if the velocity in the air cavity were significantly higher. Therefore it was concluded that the hydraulic jump formed in the closed conduit due to long-wave instability, supported by the Froude number value (Fr>1). The experimental conditions that led to the air cavity and subsequent air pocket formation with a sufficient length were characterized by a relatively low supercritical Froude number. Comparison of the measurement results from pipelines with 80- and 235-mm internal diameters confirmed that larger filling ratios with smaller Froude number values resulted in a comparatively higher mobile hydraulic jump speed, due to the lower water layer velocities.The positive value of the mixing efficiency parameter showed that the buoyancy fluxes suppressed the shear production in the hydraulic jump roller stage. This in turn made it possible to neglect the friction force effects when determining the speed of the surge wave in the horizontal pipe. Thus, the surge wave speed is determined by the pressure, the cross-sectional area of water flow, and the density jumps. The surge-wave-speed formula [Eq. (9)] may be useful to assess the duration of the prolonged pressure alteration in a closed conduit when the hydraulic conditions are known.It can be concluded that the VOF-based approach of two-phase flow makes it possible to simulate the transitional flow dynamics in the pipe under varied initial and boundary conditions. CFD modeling can be applied at different pipeline design and construction stages to estimate the conditions of surge wave formation. However, the turbulence modeling of the two-phase process is challenging in the numerical modeling of surge wave characteristics. Therefore, the experimental investigations support the understanding of the transitional flow process in the pipeline. The propagation speed of the surge wave in UDS pipes determines the pressure changes due to the transitional flow and the concurrent pressure head that may be related to the water column height in manholes. 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