Influence of Entrapped Air on Hydraulic Transients During Rapid Closure of a Valve Located Upstream and Downstream of an Air Pocket in Pressurised Pipes

Abstract

This paper examines the dual impact of trapped air on fluid transients in pressurised conduits, highlighting both its beneficial and detrimental impacts. This research analyses transient pressures caused by rapid valve closure in pipelines that contain air pockets and small bubbles dispersed within the liquid phase, by a hydraulic jump occurring at the downstream edge of the pockets. Experiments and numerical simulations were conducted with the valve positioned at the ends of the test section on both the inflow and outflow sides. A numerical model utilising the four-point centred scheme and method of characteristics was developed to resolve the governing equations of two-phase flow and was experimentally validated. The results indicate that entrapped air significantly influences hydraulic transients. When the valve is positioned downstream, air pockets and bubbles reduce pressure transients, illustrating a favourable effect. Conversely, when the valve is positioned upstream, adverse pressure transients occur, highlighting a detrimental impact. These outcomes underscore the importance of considering trapped air in pipeline systems, as its existence can either mitigate or exacerbate transient pressures depending on the configuration of the pipeline. The research highlights the significance of considering entrapped air in the design and evaluation of pressurised conduits to improve performance and prevent adverse effects.

1. Introduction

Although it is practically impossible to completely eliminate air from pipelines, identifying and addressing the sources of air ingress can significantly reduce its occurrence. For instance, water under standard conditions contains about 2% dissolved air by volume [1]. Additionally, pipelines are initially filled with air, and incomplete expulsion through valves, vents, and standpipes can leave residual air pockets at high points within the system [2]. Similarly, pumps can introduce air through vortex action during suction, contributing approximately 5–10% of the water discharge [3]. Furthermore, if pipeline pressure falls below atmospheric levels, air may infiltrate through faulty seals at joints and valves [4]. These factors collectively highlight the challenges of managing air in pipeline systems and the importance of targeted measures to minimise its presence.

The trapped air within pipelines may result in a variety of operational challenges. Notably, the formation of air pockets at elevated sections within enclosed conduits can substantially reduce flow rates by increasing head loss significantly [5,6]. Additionally, free air can compromise the accuracy of flow measurement devices. The intermittent shift from partially full to full flow conditions caused by these air pockets can induce vibrations within the pipeline. In extreme cases, substantial accumulation of air may trigger considerable pressure surges, resulting in vibrations and potential structural damage [7]. Moreover, large air pockets have the capacity to partially or completely block water flow, thereby reducing the efficiency and capacity of both pump-driven and gravity-fed systems [8]. Furthermore, when distributed along pipelines, these pockets can decrease overall system capacity and consequently increase energy consumption [9].

Air entrainment within water pipelines can result in the formation of pockets at the highest elevations of these systems, particularly when air valves are either absent or malfunctioning. These trapped air pockets can cause the water flowing beneath them to shift into a free-surface flow regime. This phenomenon has been explored by several researchers [10,11,12].

The rapid release of air from pipelines via air valves or orifices can trigger substantial pressure transients. Various researchers have investigated this phenomenon, highlighting crucial factors that influence peak transient pressures. These include the dimensions of orifices, characteristics of air pockets, driving heads, and the length of water columns [13,14,15,16,17,18,19,20]. Recent studies by Aguirre-Mendoza et al. [21] have utilised 2D CFD software to model rapidly filling manoeuvres involving air valves. Their work considers diverse air pocket sizes and pressure variations. Additionally, Aguirre-Mendoza et al. [22] examined how uncontrolled filling processes affect air pocket pressure using similar CFD models. This scenario often involves high velocities in water columns that can cause premature closure of air valves, resulting in additional pressure peaks. In a related study, Hurtado-Misal et al. [23] employed a 2D computational fluid dynamics program to analyse sub-atmospheric pressure patterns during the emptying of irregular pipelines without air valves.

In 2022, Fuertes-Miquel [24] underscored the significance of air valves in water distribution systems. Nevertheless, it is acknowledged that installing air valves does not invariably guarantee optimal system performance. Similarly, Ramos et al. [25] investigated the intricate dynamics of two-phase flows within pipelines, both with and without air vents, thereby highlighting the role and limitations of current assumptions regarding air valve functionality. Additionally, Tasca et al. [26] stressed the importance of precise characterisation of air valves for reliable transient simulations. They identified discrepancies in characteristic curves provided by manufacturers and demonstrated how these inaccuracies can result in substantial errors.

The impact of trapped air on pressure surges in water distribution systems is multifaceted, exhibiting both advantageous and disadvantageous effects. These effects are contingent upon several factors, including the volume of air present, its spatial distribution within the pipeline, the geometry of the system itself, and the nature of the transient event triggering these phenomena [27,28]. In water and wastewater pipelines, entrapped air commonly manifests as either stationary pockets or dispersed bubbles suspended within the liquid phase [29]. For instance, research has demonstrated that small air pockets can substantially amplify transient pressures to levels capable of causing pipe rupture [30,31,32,33,34]. Conversely, large air pockets forming within pipelines can serve to alleviate overpressures generated by hydraulic transients [35,36].

The occurrence of small air pockets generating significant overpressures is a consequence of its low density, which leads to minimal inertia and allows for rapid compression. In contrast, larger air pockets act as cushions or dampers, playing a key role in mitigating transient pressures. Gahan [37], following an extensive review of studies on hydraulic transients involving entrapped air, highlights that categorising an air pocket as small or large depends on its influence on transient behaviour. This distinction underscores the importance of understanding how air pocket size affects system dynamics in pipeline design and analysis.

Similarly, the uniform distribution of air in the form of small bubbles adds complexity to predicting its impact on hydraulic systems. A significant consequence is the considerable reduction in wave propagation speed, even when only a small amount of air is present. When a homogeneous mixture of water and air flows through a pressurised conduit, transient pressures are typically lowered due to the internal reflection of pressure surges within water–air mixtures [38]. However, the magnitude of transient pressures is also influenced by the proportion of air in the mixture. In some cases, these pressures can exceed those observed in scenarios where air is not considered, a phenomenon linked to the expansion and compression cycles of air bubbles [39]. This dual behaviour highlights the intricate role of air bubbles in modifying transient responses in pipeline systems. This study builds upon earlier work by Pozos [40] to further investigate numerically and experimentally how trapped air pockets and two-phase mixtures collectively influence hydraulic transients in pipelines.

This paper introduces a numerical model that integrates the four-point centred scheme and the method of characteristics (MOC). The four-point centred scheme is extensively used for simulating unsteady flows in both pressurised and open-channel systems, owing to its computational robustness and reliability [41,42,43]. Furthermore, numerical models based on the MOC are recognised for yielding accurate results and have been shown to be effective [44]. These models are now actively employed in the development and analysis of water pipelines, particularly in scenarios characterised by cavitation phenomena, the presence of entrapped air, and the identification of leaks in pipelines [45,46,47]. The combination of these methodologies ensures a robust framework for addressing complex hydraulic transient phenomena.

The numerical model investigates how the influence of valve position—whether situated upstream or downstream relative to an entrapped gas cavity (air pocket)—affects fluid transient behaviour in systems with gas cavities terminating in a rapid flow transition (hydraulic jump). This jump produces gas bubbles, resulting in a dispersed gas–liquid mixture throughout the liquid medium. Hydraulic transients were induced via the abrupt closure of a butterfly valve. The research demonstrates, both experimentally and numerically, that entrapped pockets with their corresponding void fractions substantially reduce pressure transients when the valve is located downstream of the air pocket. This finding is particularly pertinent for designing systems that are more resilient to pressure surges, thereby minimising the potential for costly repairs and operational downtime. Conversely, less favourable pressure transient characteristics were observed when the valve was situated at the upstream boundary of the pipeline.

Previous research has acknowledged the impact of entrapped air on simplified scenarios, such as single horizontal pipes [19,20,48,49,50,51]. However, real-world pipeline systems for water distribution and wastewater management are often more complex, with undulating profiles and varying pipe gradients. This research addresses a gap in the existing literature by examining hydraulic transients in more realistic, undulating pipelines where air pockets and hydraulic jumps are common. This prompted the author to design and construct an experimental apparatus to verify the proposed model. The setup incorporates a dedicated segment designed to replicate an elevated section of a pipeline within a topographically varied configuration. Notably, numerical predictions and experimental results demonstrated strong agreement, underscoring the robustness and precision of the proposed model in capturing the dynamics of hydraulic transients in complex pipeline systems.

This research offers experimental validation for a numerical model designed to simulate hydraulic transients in pipelines containing entrapped air. This validation is of paramount importance, as it ensures that the model can be reliably employed for application in the development and evaluation of real-world piping networks. A comprehensive understanding of air behaviour within pipelines is essential for the effective design, evaluation, and operation of pressurised conduits. By accurately accounting for entrapped air, engineers can enhance system performance and mitigate potential adverse effects, such as pipeline damage or operational inefficiencies. The data presented in this study demonstrates the model’s capacity to predict the impact of air pockets on transient pressures under varying operating conditions, thereby providing valuable insights for pipeline design and management.

2. Numerical Model

A numerical model was developed to investigate both the adverse and advantageous impacts of trapped air on hydraulic transients resulting from the sudden closure of a valve in conduits incorporating an entrapped gas cavity and dispersed gas bubbles in the bulk fluid medium. The fundamental partial differential equations governing time-dependent gas–liquid flow regimes were solved employing the four-point centred scheme and the MOC.

This research studies a static gas cavity positioned at the highest point of an experimental setup designed to simulate an undulating pipeline, treating it as a boundary condition. The effects of air pockets on fluid transient behaviour are incorporated into the numerical model using the method of characteristics, alongside methodologies outlined by Chaudhry [52] and Wylie et al. [53]. Additionally, specific assumptions proposed by Borrows and Qiu [45] are applied in the formulation of the numerical model, as detailed below:

(1)

The traditional method of characteristics is employed to formulate a system of ordinary differential equations. The equations are solved numerically by employing a finite difference approach aligned with the characteristic directions, utilising a scheme with first-order accuracy. To mitigate numerical dissipation and dispersion errors, interpolation is deliberately omitted during the integration process.

(2)

For the purposes of this analysis, a gas cavity of the specified volume is incorporated at a designated location within the numerical model. Additionally, the gas cavity is considered to occupy a fixed segment of the conduit cross-section remaining stationary throughout the simulation.

(3)

The celerity is assumed to be invariant throughout the analysis. Moreover, the analytical model accounts for energy dissipation by incorporating friction and local losses.

To enhance computational efficiency, the predetermined gas cavity volume is situated at the junction of adjoining pipe segments. In this study, the cavity is represented as an accumulator with uniform pressure distribution throughout its volume. The compressibility of water is deemed negligible when compared to that of air. Furthermore, the influence of inertial forces and frictional resistance are disregarded. The behaviour of the gas is modelled under the assumption of adhering to a reversible thermodynamic process described by a polytropic relationship, which can be expressed as:

$$H_A{V}^{\epsilon }=C_0$$

(1)

In this context, H_A represents the total hydraulic energy (m), V denotes the volume of the entrapped gas cavity (m³), and C is a constant determined from the equilibrium conditions of the gas cavity. The exponent governing the polytropic process ε was assumed to be 1.4, a value commonly used for rapid transient events, such as those caused by a rapid valve closing manoeuvre. This assumption is supported by numerous experimental and numerical studies, which have shown that fast transients involving trapped gas cavities are more accurately modelled utilising ε = 1.4 [54,55,56,57].

Since Equation (1) is applicable at any given moment in time, the relationship is formulated for the joint [j, N(j) + l] at each step t + ∆t.

$$(H_{P_{j,N\left(j\right)+1}}-z+H_b)V_{P_{j,N\left(j\right)+1}}^{\epsilon }=C_0$$

(2)

where $H_{P_{j,N\left(j\right)+1}}$ represents the location of the piezometric line at section (j, N(j) + 1) at t + ∆t (m), z is defined as the vertical displacement from the centreline of the pipe to a reference datum (m), H_b denotes the barometric pressure head (m), and $V_{P_{j,N\left(j\right)+1}}$indicates the volume of the air pocket at the end of the time step (m³).

Figure 1 illustrates a gas cavity situated at (j, N(j) + 1) junction of a pipeline, downstream of which a hydraulic jump is observed. The diagram compares the piezometric line or hydraulic grade line for scenarios both with and without an air pocket. Analysis of the figure reveals that the total head loss in the pipeline section, in the presence of a gas cavity, arises from the combined effects of frictional and local losses within the pipe, the head loss attributable to the gas cavity, and the energy dissipation produced by means of turbulence from the hydraulic jump.

For clarity, the subscript P is utilised to denote variables whose values remain unknown at t + Δt. Conversely, variables excluding this subscript represent known values at the start of the time step, t.

The application of mass conservation principles to the gas cavity yields a continuity equation as follows:

$$V_{P_{j,N\left(j\right)+1}}=V_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}\left[\left(Q_{P_{j+\mathrm{1,1}}}+Q_{j+\mathrm{1,1}}\right)-(Q_{P_{j,N\left(j\right)+1}}+Q_{j,N\left(j\right)+1})\right]$$

(3)

where V_i represents the initial gas cavity volume at t (m³). The duration of the time interval is denoted by Δt (s). The liquid discharge rates at the upstream and downstream boundaries of the gas cavity at (t + Δt) are given by $Q_{P_{j,N\left(j\right)+1}}$ and $Q_{P_{N\left(j\right)+1,1}}$, respectively. Similarly, $Q_{j,N\left(j\right)+1}$ and $Q_{j+\mathrm{1,1}}$ correspond to the liquid discharge rates at the upstream and downstream boundaries of the gas cavity at the start of the time interval (t).

The domain limits are modelled by means of the MOC, with the forward and backward characteristic equations evaluated upon completion of every discrete time interval, expressed as follows:

$$Q_{P_{j,N\left(j\right)+1}}=C_{(+)}-C_{a_j}{H}_{P_{j,N\left(j\right)+1}}$$

(4)

$$Q_{P_{N\left(j\right)+\mathrm{1,1}}}=C_{(-)}+C_{a_{j+1}}{H}_{P_{N\left(j\right)+\mathrm{1,1}}}$$

(5)

where

$$C_{\left(+\right)}=Q_{j,N\left(j\right)+1}+{\displaystyle \frac{g{A}_j}{c_j}}{H}_{j,N\left(j\right)+1}-{{\displaystyle \frac{f_j{∆t}_j}{2D_j{A}_j}}Q}_{j,N\left(j\right)+1}\left|Q_{j,N\left(j\right)+1}\right|$$

(6)

$$C_{\left(-\right)}=Q_{j+\mathrm{1,1}}-{{\displaystyle \frac{g{A}_{j+1}}{c_{j+1}}}H}_{j+\mathrm{1,1}}-{\displaystyle \frac{f_{j+1}{∆t}_{j+1}}{2D_{j+1}{A}_{j+1}}}{Q}_{j+\mathrm{1,1}}\left|Q_{j+\mathrm{1,1}}\right|$$

(7)

where c denotes the velocity of wave transmission in water devoid of any gases (m/s). Following Wylie et al. [53], Equation (8) was employed within the model to calculate c. The variable A signifies the internal transverse area of the pipe (m²), D indicates the diameter of the pipe (m), $f$ is the Darcy–Weisbach resistance coefficient, and $g$ represents gravitational acceleration (approximately 9.81 m/s²).

$$c=\sqrt{{\displaystyle \frac{eE{E}_w}{\rho \left(Ee+D{E}_w\right)}}}$$

(8)

In this context, E_w denotes the compressibility modulus of water (GPa), E represents the elastic modulus of the material constituting the pipe (GPa), e signifies the wall thickness of the pipe (m), and ρ represents the density of the liquid (kg/m³).

Assuming negligible head losses at pipeline junctions, then:

$$H_{P_{j,N\left(j\right)+1}}=H_{P_{j+\mathrm{1,1}}}=H_P$$

(9)

At this stage, the system of equations comprises five unknown variables: $H_{P_{j,N\left(j\right)+1}}$, $H_{P_{j+\mathrm{1,1}}}$, $V_{P_{j,N\left(j\right)+1}}$,$Q_{P_{j+\mathrm{1,1}}}$, and $Q_{P_{j,N\left(j\right)+1}}$. By systematically eliminating the last four unknowns through substitution or other algebraic methods results in

$$\left(H_{P_{j,N\left(j\right)+1}}+H_b-z\right){\left[C_{air}+{\displaystyle \frac{\mathsf{\Delta }t}{2}}(C_{a_j}+C_{a_{j+1}})H_{P_{j,N\left(j\right)+1}}\right]}^{ϵ}=C_0$$

(10)

$$C_{air}=V_i+{\displaystyle \frac{\mathsf{\Delta }t}{2}}{(Q}_{j+\mathrm{1,1}}-Q_{j,N\left(j\right)+1}+C_{\left(-\right)}-C_{(+)})$$

(11)

To solve for $H_{P_{j,N\left(j\right)+1}}$ in Equation (10), it is necessary to employ an iterative technique, such as the Newton-Raphson method. Once$H_{P_{j,N\left(j\right)+1}}$ has been determined, the remaining unknown variables can then be calculated using Equations (2)–(9).

To ensure numerical stability and achieve accurate results, the Courant condition was strictly adhered to throughout all pressure transient simulations conducted using the finite difference scheme

$${\displaystyle \frac{∆x}{∆t}}\ge c$$

(12)

It is crucial to highlight the analysis of surge pressures in flows comprising two distinct components presents a greater complexity and challenge compared to single-component flows. Moreover, the application of the method of characteristics is often limited by the nonlinear effects that cause compression waves to become increasingly sharp. This limitation arises from the necessity to implement shock-governed analytical constraints at internal solution domain interfaces during characteristic-based flow modelling [58]. Consequently, a 1D homogeneous modelling approach is utilised to investigate transient pressure fluctuations in bubbly flows comprising water and air, occurring at the downstream end of trapped air cavities.

In this study of transient two-component bubbly flow within pipelines, a set of partial differential equations may be derived through the application of a control-volume approach, as initially proposed by Yadigaroglu and Leahy [59]. This method involves developing continuity equations for both phases. Notably, this formulation does not adhere to a strictly separated flow model, adopting the assumption of minimal relative motion between the fluid phases. As a result, the continuity equation governing the liquid component may be represented in terms of its mass conservation over time and space in the following manner:

$${\displaystyle \frac{\partial }{\partial t}}\left[\rho (1-\alpha )A\right]+{\displaystyle \frac{\partial }{\partial x}}\left[\rho (1-\alpha )Av\right]=0$$

(13)

where $v$ denotes the velocity of water within the pipe. It is assumed that this velocity is equivalent to that of the gas phase (m/s). Additionally, α represents the air void fraction.

The mass conservation principle applied to the gaseous component may be written in the following form:

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_g\alpha A\right)+{\displaystyle \frac{\partial }{\partial x}}\left({\rho }_g\alpha A{v}_g\right)=0$$

(14)

where ρ_g denotes the density of the gas (kg/m), $v_g$ represents the velocity of the gas phase (m/s), t denotes time (s), and x indicates the longitudinal position along the pipe (m).

The momentum balance for the mixture can be derived using a control volume approach, which excludes the momentum contribution of the gas phase.

$${\displaystyle \frac{\partial }{\partial t}}\left[\rho (1-\alpha )A{v}_{mix}\right]+{\displaystyle \frac{\partial }{\partial x}}\left[\rho (1-\alpha )A{v}_{mix}^2\right]+A{\displaystyle \frac{\partial p}{\partial x}}+\pi D{\tau }_0-\left[g\rho (1-\alpha )Asin\varphi \right]=0$$

(15)

where p represents the pressure at a specified cross-section (Pa). The mixture velocity is denoted by $v_{mix}$ (m/s), which takes up that both water and air travel at equal velocity. τ_o signifies the boundary shear stress (Pa), which can be determined using the Darcy–Weisbach resistance coefficient $f$, and ϕ denotes the angle at which the pipe is inclined, measured from the horizontal plane.

$${\tau }_0={\displaystyle \frac{f}{8}}(1-\alpha )\rho v_{mix}\left|v_{mix}\right|$$

(16)

The density of most water–gas mixtures, denoted as ρ_mix (kg/m³), can be reasonably estimated using the following approximation:

$${\rho }_{mix}=\left(1-\alpha \right)\rho$$

(17)

The equations mentioned above can be reformulated by incorporating the compressibility moduli of water and air, as well as the elastic modulus of the material constituting the pipe. This reformulation assumes that there is no production or absorption of gas within the system. Following this approach, Equations (13)–(15) can be expressed in a form similar to that presented by Wigger et al. [41]:

$$G_1={\displaystyle \frac{\partial p}{\partial t}}+v{\displaystyle \frac{\partial p}{\partial x}}-C_1{C}_2{\displaystyle \frac{\partial v}{\partial x}}=0$$

(18)

$$G_2={\displaystyle \frac{\partial p}{\partial t}}+v{\displaystyle \frac{\partial p}{\partial x}}-C_2\left({\displaystyle \frac{\partial \alpha }{\partial t}}+v{\displaystyle \frac{\partial \alpha }{\partial x}}\right)=0$$

(19)

$$G_3={\displaystyle \frac{\partial v}{\partial t}}+v{\displaystyle \frac{\partial v}{\partial x}}+C_3{\displaystyle \frac{\partial p}{\partial x}}=b$$

(20)

The coefficients$C_1$, $C_2$,$C_3$ and $b$ which account for elastic properties and the force terms, are detailed in the Appendix A provided at the end of this paper.

Equations (18)–(20) are solved for pressure p and velocity v using a four-point centred finite difference scheme. This method is widely used for simulating unsteady flows in both pressurised and open-channel systems due to its robustness [41,42,43]. The continuous spatiotemporal domain is discretised using a rectangular lattice structure illustrated in Figure 2. Each grid point is identified by its spatial index (m) and temporal index (n). Importantly, the spatial spacing Δx and temporal spacing Δt between these points do not necessarily have to be uniform [60].

Implicit four-point difference schemes utilise a finite difference method to discretise time derivatives. A forward difference operator is applied to approximate the time derivative term, with the approximation point situated at the midpoint between grid points m and m + 1 along the x-axis. This approach enhances the accuracy of capturing temporal changes in the solution. For example, when approximating partial derivatives for pressure (p), it can be represented as follows:

$${\displaystyle \frac{\partial p}{\partial t}}={\displaystyle \frac{1}{2∆t}}[p_m^{n+1}+p_{m+1}^{n+1}-p_m^n-p_{m+1}^n]$$

(21)

Similarly, a forward difference operator is employed to approximate spatial derivatives. This operator is applied across two consecutive time steps, incorporating weighting factors θ and (1 − θ) to account for temporal variations.

$${\displaystyle \frac{\partial p}{\partial \mathrm{x}}}={\displaystyle \frac{1}{∆x}}[\theta \left(p_{m+1}^{n+1}-p_m^{n+1}\right)+(1-\theta \left)\left(p_{m+1}^n-p_m^n\right)\right]$$

(22)

Equivalent relationships can be established for the derivatives of velocity. Additionally, coefficients, variables, and functions—excluding derivatives—are discretised in time at the specific temporal level where spatial derivatives are computed. This temporal discretisation employs weighting factors akin to those outlined in Equation (22). In line with recommendations from Wigger et al. [41], a weighting factor of θ = 0.55 is used consistently across all pressure transient simulations.

A finite difference discretisation is applied to the partial derivatives present in Equations (18) and (20). The discretisation process yields a set of coupled nonlinear algebraic equations. These equations are formulated as follows:

$$\begin{gathered} G_1\left(p_m^{n+1},p_{m+1}^{n+1},v_m^{n+1},v_{m+1}^{n+1},\theta \right)={\displaystyle \frac{1}{2∆t}}\left[p_{m+1}^{n+1}+p_m^{n+1}-p_{m+1}^n-p_m^n\right] \\ +{\displaystyle \frac{\overline{v}}{∆x}}\left[\theta \left(p_{m+1}^{n+1}-p_m^{n+1}\right)+\left(1-\theta \right)\left(p_{m+1}^n-p_m^n\right)\right] \\ +{\displaystyle \frac{{\overline{\vartheta }}_1{\overline{\vartheta }}_2}{∆x}}\left[\theta \left(v_{m+1}^{n+1}-v_m^{n+1}\right)+\left(1-\theta \right)\left(v_{m+1}^n-v_m^n\right)\right] \end{gathered}$$

(23)

$$\begin{gathered} G_3\left(p_m^{n+1},p_{m+1}^{n+1},v_m^{n+1},v_{m+1}^{n+1},\theta \right)={\displaystyle \frac{1}{2∆t}}\left[v_{m+1}^{n+1}+v_m^{n+1}-v_{m+1}^n-v_m^n\right] \\ +{\displaystyle \frac{\overline{v}}{∆x}}\left[\theta \left(v_{m+1}^{n+1}-v_m^{n+1}\right)+\left(1-\theta \right)\left(v_{m+1}^n-v_m^n\right)\right] \\ +{\displaystyle \frac{{\overline{\vartheta }}_3}{∆x}}\left[\theta \left(p_{m+1}^{n+1}-p_m^{n+1}\right)+\left(1-\theta \right)\left(p_{m+1}^n-p_m^n\right)\right] \end{gathered}$$

(24)

A crucial aspect of this investigation is the air bubbles (void fraction, α) entrained by the rapid flow regime transitions at the downstream edge of the gas cavity in a pipe with a descending slope. In the numerical model, α—defined as the ratio of gas phase area to total cross-sectional area—is computed using Equation (25). It is essential to note that for this particular study, α values are treated as invariant. This assumption is based on negligible gas absorption or release during transient events.

$$\alpha ={\displaystyle \frac{\beta }{1+\beta }}$$

(25)

β is the air entrainment ratio for hydraulic jumps:

$$\beta ={\displaystyle \frac{Q_a}{Q_w}}$$

(26)

where Q_a is the volumetric airflow rate and Q_w is the volumetric water flow rate.

The air entrainment ratio for hydraulic jumps, denoted as β, was determined using Equation (27). Campbell and Guyton [61] initially proposed this relationship in 1953 to investigate air entrainment requirements arising from turbulent flow phenomena specifically the formation of hydraulic jumps within pressurised tunnels of dams located immediately downstream of gates. Subsequently, Safavi et al. [62] demonstrated that their experimental results for β in a gated tunnel with a circular geometry exhibited high concordance with the correlation developed by Campbell and Guyton [61]. In 1965, Wisner [63] independently arrived at the same relationship based on his own findings and recommended its use for computing air entrainment by hydraulic jumps. Further validation came in 1984 when Ahmed et al. [64] conducted extensive experimental studies in closed conduits with slopes ranging from horizontal to vertical, and found that his proposed equation coincided precisely with the equations put forth by the aforementioned researchers, confirming its applicability for hydraulic jump entrainment. More recently, Pozos et al. [65] and Pozos-Estrada [40] employed Equation (27) to evaluate air entrainment by hydraulic jumps in a real pumping pipeline and in an experimental apparatus with a circular cross-section, respectively. Both studies reported strong agreement between their experimental and numerical results. Based on the consistent validation and widespread applicability of this equation across various studies, the author of the present investigation elected to use this equation to compute the hydraulic jump entrainment coefficient, β.

$$\beta =0.04{(F_1-1)}^{0.85}$$

(27)

where $F_1$ is the Froude number at the beginning of the hydraulic jump.

During simulations of fluid transients, a constant time step Δt is employed to maintain numerical stability. The spatial discretisation Δx is adaptively determined for each pipeline segment by the numerical model, based on the celerity of the water–air mixture a_mix (m/s). For a homogeneous water–air flow where the relative velocity between phases is negligible, the wave speed can be calculated using an expression similar to that proposed by Martin et al. [66]:

$$a_{mix}={\left[{\displaystyle \frac{1}{\rho \left(1-\alpha \right)\left[\left(\frac{D\mu }{Ee}\right)+\left(\frac{\alpha }{E_a}\right)+\left(\frac{\left(1-\alpha \right)}{E_w}\right)\right]}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(28)

In this context, E_a denotes the compressibility modulus of air (GPa), and μ indicates a constraint factor related to how much stress is transferred between fluid pressure and pipe deformation.

3. Experimental Apparatus and Data Acquisition Equipment

A setup was implemented to generate robust and reliable data, with the aim of validating the developed model. The setup comprises an upstream constant-head reservoir, which is a 2.3 m³ hydropneumatic vertical tank. The test section, which simulates an undulating pipeline featuring a high point, is fabricated from transparent PVC piping with a diameter of 200 mm. This section is configured with two inclined pipe segments: an ascending pipe measuring 4.5 metres in length and a descending pipe of 7.5 metres in length. These segments are interconnected by a 60 cm long flexible plastic hose, also with a 200 mm internal diameter. The downstream end of the test section is connected to a gooseneck pipe that recirculates the flow back into an open tank. The choice of a 200 mm diameter was made to minimise the influence of surface tension effects, in accordance with recommendations provided by Pothof and Clemens [67,68] and Pothof [69]. The experimental apparatus is illustrated in Figure 3.

The filling of the test section is facilitated by a water tap connected to a 2.5 cm internal diameter hose, which is connected to a quarter-turn ball valve located at the upstream end of the ascending pipe. Additionally, smaller ball valves with an inner diameter of 1.25 cm are connected at regular intervals (every 10 cm) along the top of the test section. These valves enable air intake and exhaust during both filling and draining processes and serve as designated measurement points (tapping points). Furthermore, these tapping points can be used to inject compressed air via quick-connect fittings.

During the experimental investigations, two distinct methodologies were utilised to determine the water depths beneath the air pockets: (1) A direct measurement approach was employed using an acoustic depth gauge, which was inserted through the tapping points. Upon interacting with the water–air boundary, the device generated discernible acoustic output, and the sensor position was recorded to determine the depth. (2) An indirect measurement technique was also applied, involving a circumferential measurement method. The obtained measurements were subsequently adjusted to incorporate the pipe wall thickness, ensuring accuracy in experimental outcomes.

An ultrasonic flowmeter was utilised to quantify the water discharge within the experimental setup. To quantify frictional and singular pressure losses in the transparent PVC pipe sections, an open-ended water manometer was employed. The manometer was connected to pressure taps situated at the beginning and end of the test section using flexible plastic tubing. This configuration enabled pressure measurements to be taken both with and without the presence of air pockets. The differential pressure (ΔP) was calculated by measuring the vertical displacement (Δh) between the liquid menisci in each limb of the manometer.

A butterfly valve, controlled by a pneumatic actuator positioned at the outlet of the apparatus, was employed to generate transient flow conditions. Pressure transient measurements within the test section were obtained using two piezoelectric pressure transducers (PCB Model 113B26), each with a measurement range of 0 to 690 kPa and an error margin of ±0.5%. These transducers were mounted at distances of 4.0 m and 5.4 m from the constant-head tank. The signals from the transducers were conditioned using a demodulator (Endevco Model 136) and then digitised via a data acquisition board (Data Translation DT9837 24-bit resolution and sampling rates up to 102.4 kS/s). The recorded pressure transient data from both transducers were put in storage on the disk drive of a computer for subsequent study. A pre-triggering routine was employed to capture the initial static pressure within the air pocket or gas cavity before the double-acting pneumatic actuator was activated, ensuring comprehensive recording of the transient pressure data.

4. Experimental Procedure

An investigation into the response of gas cavities and two-component mixtures on pressure surges in pipelines was conducted. Four distinct experimental protocols were implemented using the testing apparatus. The datasets collected from these experiments were subsequently utilised in numerical simulations to model hydraulic transient scenarios both with and without entrapped air. Each experimental run within these protocols was replicated at least three times to minimise errors associated with experimentation. Consistent results showed that the observed pressure transient oscillations exhibited regular periodicity and yielded reproducible pressure waveforms across all tests.

Throughout the experiments, a consistent hydraulic head of 139.7 kPa (20 psi) was sustained at the hydropneumatic vertical tank. Four distinct volumetric flow rates of water—0.020, 0.025, 0.030, and 0.035 m³/s—and three different initial volumes of air pockets were tested. Additionally, to ensure optimal reservoir performance, the tank was kept with minimum air and water volumes of 0.3 m³ and 2.0 m³, respectively. Moreover, during the brief transient period lasting only a few seconds, there was no significant variation observed in the tank pressure.

Throughout the experiments, a consistent hydraulic head of 139.7 kPa (20 psi) was sustained at the upstream reservoir.

The experimental setup featured a test section with a compound slope, simulating a high point in an undulating pipeline profile. The upstream segment exhibited an adverse slope of $S_{up}$ = −0.532, corresponding to an angle of ϕ = −28°. In contrast, the downstream segment had a positive slope of $S_{down}$ = 0.466, with an angle of ϕ = 25°. During the experiments, stable air pockets consistently formed at the high point of the apparatus. These air pockets or gas cavities invariably ended in a rapid transition from super- to sub-critical flow (hydraulic jump) at the descending pipe section. This turbulence generated by the hydraulic jump entrains gas bubbles into the liquid phase, producing a water–air mixture characterised by discrete air bubbles dispersed throughout a continuous liquid medium (see Figure 1).

Extensive research has established that the presence of small gas cavities is capable of substantially increasing hydraulic transients, often to magnitudes that surpass the burst pressure limits of piping systems. Assuming that PVC possesses relatively low tensile strength and limited resistance to fatigue, especially under cyclic loading caused by transient pressures, a precautionary measure was taken. In each experimental run, an adequate amount of air was injected into the test section to deliberately form large and stable air pockets. This method was employed to reduce dynamic pressure fluctuations and thereby prevent catastrophic failure of the PVC piping.

Two distinct configurations for valve placement were examined. In experiments 1 and 2, the butterfly valve was situated at the outlet of the apparatus. In contrast, for experiments 3 and 4, it was positioned at the inlet. The four experimental protocols devised for this study are outlined in detail below.

Experiment 1: Pressurised flow conditions in the test section (no air accumulation).

In this experiment, the transparent pipe section was divided into two distinct segments with opposing slopes. The upstream segment was set at an upward slope of $S_{up}$ = −0.532, while the downstream segment was configured with a downward slope of $S_{down}$ = 0.466. The valve, integrated with a pneumatic actuator, was mounted at the downstream end of the apparatus. The procedure began by introducing water into the measurement section through a 2.5 cm diameter hose. Throughout the filling process, the absolute pressure in the upstream hydropneumatic tank was carefully maintained at a constant value of 139.7 kPa (20 psi). To ensure the complete removal of air, small 1.25 cm diameter quarter-turn valves, positioned at the top of the test section, were left open during the filling phase. Once the measurement section was entirely purged of air, the pressure in the hydropneumatic tank was confirmed to be stable, and the fluid discharge rates of water flowing across the pipe were recorded, the actuator was engaged to close the valve over a duration of 0.15 s, adhering to a linear valve closure.

Experiment 2: Stationary air pocket at the high point of the apparatus.

This experiment examined the impact of entrapped air on hydraulic transients produced by the rapid closure of a valve (0.15 s, linear closure). The methodology was as follows: With the test section operating under full flow conditions, air was introduced through a ball valve placed at the inlet of the test section. This resulted in the formation of a stationary gas cavity at the high point of the slope transition. A gooseneck pipe, installed beyond the valve, induced a hydraulic jump within the supercritical flow of the downward-sloping pipe. This created a bubbly flow regime downstream, where gas bubbles merged into larger bubbles that returned in counterflow without interruption. As a result, the air volume within the system remained constant throughout the test. Once the gas cavity stabilised at the high point, the water depths along the water–air boundary beneath the gas cavity were measured. The pressure in the hydropneumatic tank was then verified, and the water flow rate was recorded. Finally, the butterfly valve was closed to generate a hydraulic transient, which was captured and recorded using a computer system for subsequent processing and analysis.

The methodology employed for experiments 3 and 4 was similar to that used in the preceding two experiments, with the notable exception that the butterfly valve was positioned at the upstream end of the test section.

In experiments 2 and 4, the water depths at the point of hydraulic jump initiation were recorded to estimate both the entrained air content β and void fraction α. Furthermore, measurements were taken of air pocket lengths in the test section. Following each experimental run, the test section was thoroughly purged of air, after which the procedure was repeated with adjustments made to both water discharge rates and injected air volumes.

Table 1. Variables measured during experiments 2 and 4.

Volume of Air	Liquid Depth	β	α	Total Pressure in the Air Pocket	Length of the Air Cavity Profiles	Length of the Air Cavity Profiles
(m3)	y(m)	(-)	(%)	(kPa)	(m)	(m)
					Profile Upstream	Profile Downstream
Qw = 0.02 (m3/s)
0.0328	0.044	0.101	9.2	138.9	0.559	0.497
0.0381	0.041	0.117	10.4	139.7	0.559	0.675
0.0462	0.038	0.135	11.9	140.2	0.559	0.923
Qw = 0.025 (m3/s)
0.0323	0.046	0.116	10.4	138.8	0.452	0.581
0.0373	0.041	0.143	12.5	139.2	0.452	0.785
0.0471	0.038	0.164	14.1	139.8	0.452	1.101
Qw = 0.03 (m3/s)
0.0357	0.051	0.119	10.7	139.1	0.287	0.887
0.0473	0.042	0.165	14.2	139.6	0.287	1.191
0.0625	0.038	0.196	16.4	140.2	0.287	1.688
Qw = 0.035 (m3/s)
0.0279	0.060	0.104	9.4	139.9	0.203	0.721
0.0373	0.052	0.134	11.8	140.1	0.203	0.974
0.0478	0.050	0.145	12.7	140.5	0.203	1.331

provides a summary of the variables measured during experiments 2 and 4. These include air pocket volumes, air void fractions α, liquid depths at the beginning of the jumps, and the entrained air content β. Additionally, measurements were taken for volumetric flow rates, total pressure within the gas cavity under steady-state conditions, and the length of flow profiles beneath the air pockets.

5. Results

5.1. Results with the Valve at the Outlet of the Test Section

The discussion presents a comparative analysis of the experimental and numerical results derived from pressure transient analysis. Figure 4 illustrates the contrast of outcomes obtained from experimental measurements and simulations using the MOC following the closure of a butterfly valve. This particular simulation was conducted with a water-filled test section, intentionally excluding air entrapment at the high point, thereby validating the sole application of the MOC. A slight overestimation of pressure amplitudes is noted in the results achieved with the model in comparison with the experimental outcomes. This discrepancy may be attributed to measurement uncertainties inherent in the experimental setup. Nonetheless, despite this minor deviation, a robust correlation between the numerical and experimental findings is evident, demonstrating a high level of agreement overall.

The influence of varying volumes of gas cavities and the downstream water–air mixture void fraction on pressure transients during test runs, conducted with a constant liquid flow rate of 0.02 m³/s, is depicted in Figure 5. The analysis indicates that the increase in gas cavity volumes and void fraction considerably reduces the magnitude of pressure surges while simultaneously extending the wave period. Experimental validation using piezoelectric transducers produced consistent results, supporting the observed trends. Moreover, numerical simulations of hydraulic transients—both with and without entrapped air pockets—across various water flow discharges exhibited similar pressure wave patterns. As a result, only a representative selection of the simulation outcomes is presented in this study.

Figure 6 provides a comparative analysis of the simulated fluid transient behaviour within a test section that contains a gas cavity volume of V = 0.0328 m³, with a downstream water–air mixture of 9.2%. The outcomes obtained with the model are contrasted with experimental pressure transient measurements. The analysis reveals a significant reduction in the magnitude of pressure variations owing to the presence of the gas cavity. Additionally, the downstream water–air mixture contributes to a further substantial decrease in the pressure transient envelope compared to conditions where no air is present.

The analysis of the data shown in Figure 7 and Figure 8 indicates a robust correlation between the water–air mixtures and the behaviour of pressure transients. In particular, an increase in both the air in the mixtures and the volume of gas pockets is associated with a notable decrease in the magnitude of pressure oscillations throughout the test section. Figure 8 presents an examination of the numerical simulations, highlighting a significant reduction in the reflection of transient pressure waves when an air-water mixture is present. In this instance, an air volume of 0.0462 m³, along with its corresponding air void fraction of 10.4%, demonstrates a beneficial damping effect.

The analysis of the preceding graphical data suggests that the presence of gas cavities, along with a downstream void fraction, markedly reduces pressure transients within the test section. Additionally, the observed damping of pressure fluctuations can be attributed to the considerable air volumes and associated void fractions, leading to a more pronounced attenuation of transients compared to scenarios without any air content.

The observed attenuation of both experimental and simulated pressure transients can be associated with the effect of air pocket volume on the characteristics of hydraulic jumps. An increase in the volume of the air pocket results in a decrease in the water depth at the point of hydraulic jump initiation (as shown in Table 1). This reduction in depth enhances turbulence within the jump, promoting the entrainment of gas bubbles from the gas cavity. The presence of this air-water mixture significantly decreases wave celerity, thereby diminishing the propagating pressure wave. This phenomenon has been substantiated by prior research findings [38,70,71].

5.2. Results with the Valve at the Inlet of the Test Section

Consistent with the approach taken for hydraulic transient simulations involving the valve positioned at the downstream end, only a representative selection of simulations for the valve located at the upstream end is presented here. These simulations focus on a water flow rate of 0.02 m³/s and explore variations in both the volume of trapped gas pockets and the air void fraction.

Figure 9 presents a comparative analysis of the measured and numerically simulated pressure transients within the test section when the valve closes suddenly at the upstream end. The numerical model, which utilises the MOC, was run without considering trapped gas pockets at the test section. Both the experimental and simulated data exhibited an immediate drop in pressure from the steady-state condition upon valve closure. Although the computed pressure amplitudes were marginally greater in comparison with the experimental outcomes, the discrepancy can be attributed to the inherent uncertainties associated with experimental measurements. Nonetheless, a satisfactory correlation between the numerical and experimental results is evident.

Figure 10 shows that numerical and experimental results obtained with the valve positioned at the upstream edge of the experimental apparatus demonstrate the initial pressure peak is considerably greater in the presence of air. Specifically, with a gas cavity volume of 0.328 m³ and an air void fraction of 9.2%, the initial pressure peak reached an initial pressure peak of 189.5 kPa. This represents a substantial increase compared to the 76.3 kPa observed under conditions without trapped air. Furthermore, it was noted that increasing the volume of air pockets and the air void fraction contributes to a reduction in the magnitude of pressure oscillations. In this particular case, the presence of air pockets provides a degree of protection against severe negative pressure events. It can be concluded that air pockets function effectively as air chambers, eliminating the risk of cavitation and preventing potential pipe collapse.

Figure 11, Figure 12 and Figure 13 demonstrate that an increase in both the gas cavity volume and the void fraction results in a reduction in the amplitude of pressure transients. Nevertheless, the initial peak pressure value was observed to be greater in scenarios involving air pockets compared to those without air. This indicates that the presence of gas cavities and the consequent void fraction partially attenuate the propagation of the pressure wave, thereby influencing the dynamic response of the system.

An examination of the computational results presented in Figure 11 reveals that a gas cavity of 0.328 m³, corresponding to an air void fraction of 9.2%, generates the highest initial peak pressure of 189.5 kPa. This value represents a substantial increase from the 76.3 kPa recorded without an entrapped gas cavity (i.e., full pipe flow). The amplification of this pressure peak is primarily attributed to the dynamic compression and expansion of the gas cavity, combined with reflections of the transient pressure wave at the butterfly valve, the gas cavity interface, and the downstream edge of the test section. These interactions collectively contribute to the observed pressure surge.

Figure 11, Figure 12 and Figure 13 demonstrate a notable increase in the initial pressure peak. However, it is worth mentioning that the existence of gas pockets and their corresponding air void fractions play a significant role in alleviating pressure drops. In scenarios devoid of air, a minimum pressure of −74.7 kPa was recorded, while when air was present, gas pockets worked as air chambers, elevating the lowest pressure to less negative values when compared to conditions without air.

The analysis of transient pressures highlights the necessity of safeguarding the pipeline against both overpressure and negative pressure events. To address these risks effectively, the strategic installation of air/vacuum valves is strongly advised. These valves should be positioned at the inlet of the piping network, preceding the flow control valve, as well as at other key locations such as high points along the pipeline. A typical example of their application is at the discharge point of a pump. The air/vacuum valves facilitate the release or admission of air as required, thereby preventing issues arising from air pockets within the pipeline.

Experimental and numerical results have revealed that the positioning of the valve plays a pivotal role in influencing transient pressure dynamics within pipeline systems. The findings highlight that pressure waves undergo significant damping as they propagate through the turbulent zone downstream of the gas pocket in the bubbly flow regime. This turbulence, particularly at the hydraulic jump, markedly reduces the wave velocity and attenuates the amplitude of pressure peaks. Importantly, an increase in air volume intensifies turbulence and agitation at the hydraulic jump, resulting in the detachment of additional bubbles from the primary air pocket. This heightened turbulence further decelerates the wave speed and diminishes the magnitude of pressure peaks. As a result, when the valve is situated downstream, pressure waves are effectively attenuated. In contrast, positioning the valve upstream leads to an increase in pressure peak magnitudes.

6. Conclusions

In pipeline systems with undulating topography, air pockets frequently accumulate at elevated sections. Hydraulic jumps typically form at the downstream end of these gas cavities, generating turbulent energy at the jump interface. The turbulence entrains small air bubbles into the flowing liquid, creating a two-phase bubbly flow regime characterised by a dispersed air phase within a continuous liquid phase. This research introduces a numerical model that accounts for the integrated influence of gas cavities and the resulting void fractions on pressure surge behaviour within pipelines. The validity of the numerical model is confirmed through comparisons with experimental data collected from a specially designed hydraulic transient test rig.

The impact of air pocket volume and subsequent void fraction on pressure surges was investigated through both numerical simulations and experimental techniques. Transient conditions were initiated when a valve was closed suddenly. Experiments were conducted with the valve situated both at the outlet and inlet of the test section, with the air pocket volume systematically varied for each valve position.

The outcomes obtained from laboratory testing and computational modelling with the valve positioned downstream demonstrate a clear attenuation of pressure surges owing to the existence and characteristics of gas cavities. Specifically, gas cavities consistently decrease the magnitude of hydraulic transients. The magnitude of this damping effect is directly correlated with the volume of the gas cavity and the resulting water–air mixture. As shown in Figure 6, Figure 7 and Figure 8, increasing both air volume and air void fraction leads to a more pronounced reduction in pressure transients. This attenuation is attributed to the rapid transition from super- to sub-critical flow at the descending pipe section, which is promoted by the gas cavity. The enhanced turbulence leads to increased entrainment of gas bubbles, forming a water–air mixture that significantly reduces the wave celerity, thereby dissipating the energy of the propagating pressure wave. These findings confirm that strategically introducing and managing air pockets upstream of a control valve can be an effective strategy for mitigating transient pressure surges in pipeline systems.

The results obtained with the valve positioned upstream reveal a more complex interaction between air pockets and transient pressures. While the presence of air pockets generally reduces the overall amplitude of pressure oscillations and mitigates negative pressure events, the initial pressure peak can be significantly amplified compared to scenarios without air. As illustrated in Figure 11, Figure 12 and Figure 13, increasing the volume of the gas cavity and the percentage of gas in the water–air mixture contributes to a reduction in the magnitude of pressure oscillations. However, the first pressure peak is consistently higher in the presence of gas, with a gas cavity volume of 0.328 m³ and corresponding air void fraction of 9.2% inducing the highest initial peak pressure of 189.5 kPa, a substantial increase from the 76.3 kPa observed in full pipe flow (i.e., without air entrapment). This amplification is attributed to the dynamic compression and expansion of the gas cavity, coupled with reflections of the pressure wave at the valve, air pocket interface, and downstream boundary (Gooseneck pipe). Crucially, however, the presence of pockets provides a degree of protection against severe negative pressure events. The minimum pressure recorded in the absence of air (−74.7 kPa) is substantially lower than the minimum pressures observed when air pockets are present, thereby indicating that these air pockets effectively function as air chambers, mitigating the potential for cavitation and pipeline collapse.

The findings emphasise the importance of protecting pipelines against both overpressure and sub-atmospheric pressure conditions. The strategic positioning of air/vacuum valves at critical locations is strongly recommended. These locations should include positions upstream of flow control valves and at high points along the pipeline. Such valves facilitate the controlled release or ingress of air as required, thus mitigating problems arising from air entrapment and ensuring the operational stability of the pipeline system. This preventative measure is particularly crucial in systems where transient pressure events are anticipated.