Evaluating Alternatives for Combined Modeling of Gas Cavities and Unsteady Friction in Closed-Pipe Transients

Abstract

Hydraulic transients in pressurized pipe systems are significantly influenced by the presence of entrapped air, which alters wave propagation through increased compressibility and energy dissipation. Traditional discrete cavity models, such as the Discrete Gas Cavity Model (DGCM), often assume a constant wave celerity, which limits their accuracy under high gas content conditions. This study evaluated different approaches for representing the effects of gas cavities and unsteady friction in closed pipe transients. The work introduces the Adjustable-celerity Gas Cavity Model (AGCM), a formulation that explicitly couples local air volume and pressure to dynamically adjusted celerity values. Two variants are considered, a non-iterative (AGCM.v1) and an iterative approach (AGCM.v2), the latter ensuring consistency between pressure head and celerity at each time step. The models were evaluated through numerical simulations using both experimental datasets and a hypothetical test case with increasing air fractions. Results show that the AGCM was able to represent celerity magnitudes in unsteady flows with large fractions of air. Also, while constant-celerity models perform well under low-air conditions, variable-celerity formulations offer superior consistency in predicting wave amplitudes and celerity dynamics as gas content increases. These findings underscore the importance of dynamic celerity coupling in transient flow modeling and validate the AGCM as a useful approach for transient modeling in conditions with higher air phase fractions.

1. Introduction

Hydraulic transients, commonly referred to as water hammer, are pressure waves generated in pressurized pipe systems due to abrupt changes in flow velocity, typically caused by the sudden closure of valves or pump shutdowns. These events can trigger rapid and intense pressure fluctuations, potentially leading to structural damage such as pipe ruptures or equipment failure. When free air is present in the system, whether due to air entrainment, release of dissolved gases, or the compression of trapped air pockets, the transient response is fundamentally altered. In such conditions, the compressibility of the gas phase introduces additional storage mechanisms that strongly influence wave propagation speed, pressure amplitudes, and energy dissipation. Experimental and numerical investigations have demonstrated that air-pocket compressibility can significantly influence the transient response, surpassing both liquid compressibility and pipe-wall elasticity, resulting in pronounced pressure damping, phase shifts, and delayed pressure peaks [1,2,3,4,5]. These findings suggest that a clear and physically consistent representation of gas–liquid interactions is crucial for accurately describing the transient flows in air-influenced systems.

From a modeling perspective, the need to accurately represent gas–liquid interactions in transient flows motivated the development of several approaches. One class of approaches focuses on lumped air-pocket models, in which trapped air volumes are treated as discrete control volumes governed by thermodynamic relations, commonly assuming ideal gas behavior with isothermal or polytropic compression. These models have been widely applied to rapid filling scenarios and valve-induced transients, particularly to investigate peak pressure surges and pressure damping associated with air-pocket compression [5,6,7,8,9,10,11]. An alternative line of research adopts two-phase formulations, ranging from homogeneous mixture models, where the gas content evolves as a distributed variable, to higher-dimensional CFD-based approaches. While these models offer a detailed description of air-water interactions, their computational cost often limits their applicability in system-scale engineering analyses [10,12,13,14,15,16].

Within this broader landscape, modeling approaches using uniformly spaced gas cavities emerged as a pragmatic and computationally efficient compromise for simulating one-dimensional air-influenced transient flows. Over the past few decades, the Discrete Vapor Cavity Model (DVCM) and the Discrete Gas Cavity Model (DGCM) have been among the most widely used formulations in this class [6,8,17]. The classical DVCM represents liquid-vapor interactions through the formation of vapor cavities whenever the local pressure reaches the vapor pressure, maintaining a minimum pressure plateau until the cavities collapse. Despite its historical importance, this formulation is known to suffer from numerical instabilities and unrealistic pressure spikes in multicavity scenarios [10]. The DGCM was subsequently introduced as a refinement by replacing vapor cavities with small volumes of non-condensable gas located at computational sections. By allowing cavity behavior to be governed by gas compressibility rather than fixed vapor pressure, the DGCM smooths pressure pulses, improving agreement with experimental observations, particularly under mild to moderate cavitation conditions [3,9,18,19].

Recent developments within the discrete-cavity family modeling have followed multiple directions. Efforts have focused on improving numerical robustness and wall-shear representation, including Godunov-type DGCM formulations [19] and DGCM/DVCM models coupled with quasi-2D friction [20]. In parallel, alternative modeling approaches based on higher-fidelity descriptions have been proposed, including two-dimensional and distributed models that better resolve velocity profiles, shear stress, and phase interactions [15,21]. Other studies have extended transient cavitation analysis toward plastic and viscoelastic pipes [11], as well as toward gas-containing pipelines and leak-related applications [22]. These developments indicate that the field has expanded substantially beyond the classical DGCM framework; however, one-dimensional DGCM-based approaches remain particularly attractive for large engineering systems, where computational efficiency and modeling robustness are essential.

Despite the significant decrease in acoustic wave celerities created by gas fractions, most DGCM-based formulations retain the classical assumption of constant wave celerity throughout the pipeline. This assumption becomes increasingly questionable in systems with significant gas content, where local gas volumes evolve dynamically and directly influence wave propagation characteristics [4,10,11]. In this context, different interpretations of the DGCM formulation can be identified in the literature. On one hand, the classical approach assumes constant wave celerity, consistent with the original GCAV framework proposed by Wylie [17]. On the other hand, more recent studies have implicitly or explicitly adopted formulations in which wave celerity varies as a function of local gas conditions, effectively introducing a variable-celerity DGCM framework [22,23].

Although these studies incorporate variable wave celerity, the implications of this modification on the mathematical structure of the DGCM and its interaction with other physical mechanisms are not always well addressed. For instance, the coupling between gas cavity volume, wave celerity, and the governing hydraulic variables introduces an additional level of nonlinearity, as variations in cavity volume affect wave speed, which in turn influences pressure and discharge. This interdependency may impact the consistency and convergence of the numerical solution if not properly treated. A similar issue has been addressed in alternative cavitation formulations, such as that proposed by Vasconcelos and Marwell [24], where an iterative strategy was introduced to ensure consistency between cavity volume and the hydraulic state variables. However, such coupling mechanisms are not explicitly considered in DGCM-based formulations with variable wave celerity.

In this context, the present study introduces two modeling approaches that extend the existing DGCM formulations through the incorporation of variable wave celerity and interactive coupling strategies. The first formulation, referred to as the Adjustable-celerity Gas Cavity Model (AGCM.v1), retains the discrete representation of gas cavities at computational sections while allowing wave celerity to evolve dynamically as a function of local gas volume and pressure. This formulation is consistent with approaches non-interactively adopted in previous studies (e.g., [22,23]), and is herein formalized to provide a clear and unified interpretation. The second formulation, AGCM.v2, extends this framework by introducing an iterative coupling procedure at each time step and computational node. The iterations ensure continuous consistency between piezometric head, cavity volume evolution, and wave propagation characteristics. By explicitly enforcing this coupling, the model surpasses previous non-iterative or loosely coupled approaches, providing a more physically consistent representation of transient behavior under elevated air fractions.

To assess the implications of this formulation, the study evaluates four model configurations: (1) the classical DGCM/GCAV; (2) AGCM.v1A, which incorporates variable wave celerity with the unsteady friction model of Vitkovský et al. [25]; (3) AGCM.v1B, which adopts the unsteady friction formulation of Vardy and Hwang [26]; and (4) AGCM.v2, which includes the proposed iterative coupling strategy. The models are first validated against experimental data from Soares et al. [9] and Zhu et al. [23] and subsequently applied to a hypothetical case that enables a sensitivity analysis with increasing gas content. This structured comparison provides insight into the combined effects of discrete gas cavities, wave celerity variation, and unsteady friction interactions under different conditions. The following sections detail the model formulations, present the simulation results, and discuss the key findings, concluding with recommendations for future work.

2. Materials and Methods

2.1. Governing Equations of DGCM

The principles of mass and momentum conservation govern the transient flow in pressurized pipelines. Under one-dimensional assumptions, the governing equations for unsteady flow in a closed conduit are written as:

$${\displaystyle \frac{\partial H}{\partial t}}+{\displaystyle \frac{a_m^2}{gA}}{\displaystyle \frac{\partial Q}{\partial x}}+{\displaystyle \frac{2a^2}{g}}{\displaystyle \frac{\partial {\epsilon }_r}{\partial t}}=0$$

(1)

$${\displaystyle \frac{1}{gA}}{\displaystyle \frac{\partial Q}{\partial t}}+{\displaystyle \frac{\partial H}{\partial x}}=-{\displaystyle \frac{fQ\left|Q\right|}{2Dg{A}^2}}-h_{fu}$$

(2)

where $H$ is the piezometric pressure head, $Q$ is the flow rate, $A$ is the cross-sectional area of the pipe, $D$ is the internal pipe diameter, $a_m$ is the wave pressure celerity, $g$ is the gravitational acceleration, ${\epsilon }_r$ is the retarded strain associated with viscoelastic pipe behavior, $f$ is the friction factor, and $h_{fu}$ is the unsteady friction loss per unit length. These equations form the basis for a wide range of transient flow models.

In the present study, cavitation phenomena are considered through the Discrete Gas Cavity Model (DGCM), as presented by Wylie et al. [6] and implemented in their GCAV.FOR computational code. To solve Equations (1) and (2), the Method of Characteristics (MOC) is applied. The DGCM employs a diamond-shaped computational mesh, where each iteration step advances the solution by $2∆t$.

The DGCM simulates discrete cavitation, in which gas cavities are assumed to form at predefined computational nodes along the pipeline, as schematically illustrated in Figure 1. Importantly, the model assumes that a small amount of free gas is already present at each node from the beginning of the simulation, even under steady-state conditions. Between nodes, the pipe is filled with low-compressibility liquid.

After applying the MOC, the characteristic equations for each node $i$ at the time level $t+2∆t$ are:

$$C^{+}: Q_{Pu (t+2∆t,i)}={\displaystyle \frac{C_P-H_{\left(t+2∆t,i\right)}}{B_P}}$$

(3)

$$C^{-}: Q_{P (t+2∆t,i)}={\displaystyle \frac{H_{\left(t+2∆t,i\right)}-C_M}{B_M}}$$

(4)

In these Equations (3) and (4), $Q_{Pu}$ represents the flow rate upstream of the gas volume within the pipeline section, $Q_P$ is the flow rate downstream of the gas volume, and the coefficients $C_P$, $B_P$, $C_M$ and $B_M$ are calculated according to Equations (5)–(8).

$$C_P={\displaystyle \frac{H_{(t,i-1)}+\frac{a_m}{Ag}{Q}_{P (t,i-1)}+{C_P}_{UF}-{c_1}_{visc}}{(1+{c_2}_{visc})}}$$

(5)

$$C_M={\displaystyle \frac{H_{(t,i+1)}-\frac{a_m}{Ag}{Q}_{Pu \left(t,i+1\right)}+{C_M}_{UF}-{c_1}_{visc}}{(1+{c_2}_{visc})}}$$

(6)

$$B_P={\displaystyle \frac{\frac{a_m}{Ag}+\frac{f∆x}{2gD{A}^2}\left|Q_{P(t,i-1)}\right|+{B_P}_{UF}}{(1+{c_2}_{visc})}}$$

(7)

$$B_M={\displaystyle \frac{\frac{a_m}{Ag}+\frac{f∆x}{2gD{A}^2}\left|Q_{Pu(t,i+1)}\right|+{B_M}_{UF}}{(1+{c_2}_{visc})}}$$

(8)

In the original DGCM, the wave celerity $a_m$ is constant in the solution domain and does not consider the effect of gas content. This simplification is based on the assumption that air is concentrated in discrete pockets at each node and in low fractions, rather than being uniformly distributed along the pipeline. The original DGCM configuration does not consider unsteady friction, and thus $h_{fu}$,${{ C}_P}_{UF}, {C_M}_{UF}$,${{ B}_P}_{UF}$ and ${B_M}_{UF}$ are all zero. Furthermore, viscoelastic effects are not included, and therefore $\partial {\epsilon }_r/\partial t$, ${c_1}_{visc}$, and ${c_2}_{visc}$ are equal to zero.

The piezometric head $H$ at each computational node is implicitly determined based on the evolution of the local air cavity volume, using a gas law formulation under isothermal conditions. The DGCM, following the implementation by Wylie et al. [6], solves a nonlinear relationship between pressure head and air volume derived from the ideal gas law, assuming a constant mass of gas at each node. This formulation also accounts for the geometric and hydraulic characteristics of each node, including local elevation, gas volume, and the hydraulic grade line. Once the updated head $H$ is computed, the upstream and downstream flow rates ($Q_{Pu}$ and $Q_P$) are evaluated using the MOC characteristic equations. The air cavity volume is then corrected based on the net flow into or out of the node over the time step.

2.2. Incorporation of Unsteady Friction Models

In rapid transients, particularly those involving vaporous or gaseous cavitation, the contribution of unsteady friction (UF) can become significant. In this study, two widely adopted UF formulations are incorporated into the DGCM framework:

• The Vitkovský et al. model [25], which includes local acceleration and velocity gradient terms and depends on wave celerity;
• The Vardy and Hwang model [26], which is based on convolution integrals and is independent of celerity, offering a more physically complete representation of wall shear development.

The implementation of each model results in modifications to the momentum equation and the characteristic equations of the MOC, as described below.

2.2.1. Unsteady Friction According to Vítkovský et al. [25]

The model proposed by Vitkovský et al. [25], based on the instantaneous acceleration formulation originally introduced by Brunone et al. [27], expresses unsteady friction losses through a corrective term $h_{fu}$ to the momentum equation:

$$h_{fu}={\displaystyle \frac{k_3}{g A}}\left({\displaystyle \frac{\partial Q}{\partial t}}+a_m SGN\left(Q\right)\left|{\displaystyle \frac{\partial Q}{\partial x}}\right|\right)$$

(9)

where $k_3$ is the unsteady friction coefficient, and $SGN( )$ is a function that returns the sign (positive or negative) of the argument. This formulation introduces three correction terms into the MOC equations (Equations (5)–(8)):

$${C_P}_{UF}={\displaystyle \frac{a_m}{g A}}{k}_3\left(Q_{P(t,i)}-SGN\left(Q_{P(t, i-1)}\right){\displaystyle \frac{\left|Q_{P(t, i+1)}-Q_{P(t, i-1)}\right|}{2}}\right)$$

(10)

$${C_M}_{UF}=-{\displaystyle \frac{a_m}{g A}}{k}_3\left(Q_{Pu(t,i)}-SGN\left(Q_{Pu(t, i+1)}\right){\displaystyle \frac{\left|Q_{Pu(t, i+1)}-Q_{Pu(t, i-1)}\right|}{2}}\right)$$

(11)

$${B_P}_{UF}={B_M}_{UF}={\displaystyle \frac{a_m}{g A}}{k}_3$$

(12)

These terms enable the transient model to account for frictional delay effects and internal resistance during transients. As this model depends directly on $a_m$, it is affected by any variation in wave celerity introduced in subsequent model stages.

2.2.2. Unsteady Friction According to Vardy & Hwang [26]

The UF model developed by Vardy and Hwang [26] is based on a convolution approach originally proposed by Zielke [28], and the UF losses are expressed as:

$$h_{fu}={\displaystyle \frac{16\upsilon }{g D^2}}\sum _{k=1}^N{Y}_k$$

(13)

In the DGCM’s diamond mesh, the evolution of each weighting function $Y_k$ is computed recursively:

$$Y_{k (t+2∆t,i)}=Y_{k (t,i)} e^{-2n_k∆\tau }+m_k{\displaystyle \frac{\left({\mathrm{Q}}_{P \left(t+2∆t,i\right)}-{\mathrm{Q}}_{P \left(t,i\right)}\right)}{A}}$$

(14)

where $\upsilon$ is the kinematic viscosity of the fluid, $m_k$ and $n_k$ are exponential sum coefficients and $∆\tau$ is the dimensionless time increment, given by $∆\tau =4\upsilon \mathsf{\Delta }t/D^2$. The number of terms $N$ and the values of $m_k$ and $n_k$ were defined following the approximations proposed by Vitkovský et al. [29], ensuring a good balance between computational efficiency and accuracy.

The resulting MOC correction terms (for Equations (5)–(8)) are:

$${C_P}_{UF}=-{\displaystyle \frac{a_m}{g}}{\displaystyle \frac{16\upsilon ∆t}{D^2}}\sum _{k=1}^N{Y}_k$$

(15)

$${C_M}_{UF}=-{C_P}_{UF}$$

(16)

$${B_P}_{UF}={B_M}_{UF}={\displaystyle \frac{a_m}{g A}}{\displaystyle \frac{16\upsilon ∆t}{D^2}}\sum _{k=1}^N{m}_k$$

(17)

Unlike the formulation by Vitkovský et al. [13], this model is independent of wave celerity, making it especially robust in scenarios where celerity varies over time and space.

2.3. Incorporation of Viscoelasticity Model

In polymeric and flexible pipes, such as PVC and PE, transient flow behavior is significantly influenced by the viscoelastic response of the pipe wall. This effect introduces additional damping and dispersion of pressure waves, which cannot be captured by purely elastic formulations.

In this study, viscoelastic effects are incorporated into the DGCM framework following the formulation proposed by Covas et al. [30], which is based on a generalized Kelvin–Voigt model represented by a series of creep functions. The viscoelastic behavior is described through the retarded strain ${\epsilon }_r$, expressed as a sum of contributions from $N_{kv}$ Kelvin–Voigt elements, each characterized by a creep compliance $J_k$ and a retardation time ${\tau }_k$.

The inclusion of viscoelasticity introduces additional correction terms into the MOC equations. The resulting correction terms in the diamond grid (Equations (5)–(8)) are given by:

$${c_1}_{visc}={\displaystyle \frac{2a^2∆t}{g}}\sum _{k=1}^{N_{kv}}\left[{\displaystyle \frac{J_k}{{\tau }_k}}{e}^{-2\mathsf{\Delta }t/{\tau }_k}{F}_{t,i}+{\displaystyle \frac{J_k{C}_0}{2\mathsf{\Delta }t}}\left(e^{-2\mathsf{\Delta }t/{\tau }_k}-1\right)H_{t,i}-{\displaystyle \frac{e^{-2\mathsf{\Delta }t/{\tau }_k}}{{\tau }_k}}{{\tilde{\epsilon }}_{rk}}_{t,i}\right]$$

(18)

$${c_2}_{visc}={\displaystyle \frac{2a^2∆t}{g}}\sum _{k=1}^{N_{kv}}{\displaystyle \frac{C_0{ J}_k}{\left(2\mathsf{\Delta }t\right)}}\left(1-e^{-2\mathsf{\Delta }t/{\tau }_k}\right)$$

(19)

where $J_k$ denotes the creep compliance of the $k$-th Kelvin–Voigt element, characterizing the deformability of the pipe material, while ${\tau }_k$ represents the corresponding retardation time governing the time-dependent viscoelastic response. The parameter $N_{kv}$ defines the number of elements adopted in the generalized Kelvin–Voigt representation.

The variable ${\tilde{\epsilon }}_{rk}$ corresponds to the partial retarded strain associated with each element and is determined by Equation (20). The function $F_{t,i}$ and the coefficient $C_0$ are defined by Equations (21) and (22), respectively.

$${{\tilde{\epsilon }}_{rk}}_{t+2\mathsf{\Delta }t,i}=J_k{F}_{t+2\mathsf{\Delta }t,i}-J_k{e}^{-2\mathsf{\Delta }t/{\tau }_k}{F}_{i,j}-{\displaystyle \frac{J_k{\tau }_k}{2\mathsf{\Delta }t}}\left(1-e^{-2\mathsf{\Delta }t/{\tau }_k}\right)\left(F_{t+2\mathsf{\Delta }t,i}-F_{t,i}\right)+e^{-2\mathsf{\Delta }t/{\tau }_k}{{\tilde{\epsilon }}_{rk}}_{t,i}$$

(20)

$$F_{t,i}=C_0[H_{t,i}-H_{0, i}]$$

(21)

$$C_0=\left[{\displaystyle \frac{2e}{D}}\left(1+\nu \right)+{\displaystyle \frac{D}{D+e}}(1-{\nu }^2)\right]{\displaystyle \frac{D\gamma }{2e}}$$

(22)

This formulation enables the transient model to capture time-dependent energy dissipation and wave attenuation associated with viscoelastic pipe behavior, providing a more realistic representation of pressure wave propagation in polymeric pipelines.

2.4. Adapting the DGCM to Variable Celerity Conditions

The modeling of transient flows in pressurized pipe systems containing free gas volumes requires proper consideration of the system’s compressibility, which is influenced by both the elasticity of the liquid–pipe system and the presence of entrapped air. In the original DGCM, the pressure wave celerity $a_m$ is treated as constant throughout the simulation. However, when free gas is present, this assumption can introduce errors, particularly in systems with large variations in gas content or during intense transients.

In the present study, the DGCM is extended into the Adjustable-celerity Gas Cavity Model (AGCM). This extension allows the pressure wave celerity to vary spatially and temporally, based on the local air fraction and the pipe wall’s elasticity. The wave celerity for an air–water mixture is computed as [6]:

$$a_m=\sqrt{{\displaystyle \frac{K_m/{\rho }_m}{1+\left(\frac{K_{m}D}{Ee}\right)}}}$$

(23)

where $K_m$ is the bulk modulus of the mixture, ${\rho }_m$ is the mixture density, $D$ is the pipe diameter, $E$ is the pipe wall elasticity modulus, and $e$ is the wall thickness. The mixture modulus, considering an isothermal gas compression/expansion process, is given by:

$$K_m={\displaystyle \frac{K_{liq}}{1+\alpha (K_{liq}/p-1)}}$$

(24)

where $K_{liq}$ is the bulk modulus of the liquid, $p$ is the absolute pressure, and $\alpha$ is the local gas content, defined as the ratio of free gas volume to the total mixture volume. A linear approximation for the mixture density is adopted, as proposed by Wylie et al. [6]:

$${\rho }_m= {\rho }_g\alpha +(1-\alpha ){\rho }_l$$

(25)

where ${\rho }_g$ and ${\rho }_l$ are the densities of the gas and liquid phases, respectively. The gas content $\alpha$ is updated dynamically based on the cavity volume and pressure conditions at each node.

In this work, we propose the AGCM.v1A, which incorporates the unsteady friction model of Vitkovský et al. [25] and extends it by updating the celerity within the friction source terms at each computational node and iteration. This model represents a significant evolution of the traditional DGCM by establishing a distributed representation of compressibility, allowing air effects to influence the wave behavior along the entire reach, not just at discrete nodes.

Due to the dependence of the Method of Characteristics (MOC) on local wave celerity, the time step $∆t$ for each simulation iteration is updated dynamically according to the maximum celerity across the domain:

$$∆t= {\displaystyle \frac{∆x}{\mathrm{m}\mathrm{a}\mathrm{x}\left(a_{m \left(i\right)}\right)}}$$

(26)

This ensures numerical stability and consistent resolution of the wave front, especially under rapidly varying flow conditions.

2.5. Iterative AGCM Approach with Coupled Celerity–Air Volume Calculations

For the presented AGCM.v1, the wave celerity $a_m$ is computed in simplified form as a function of the gas content at each node. However, a strong interdependency arises between pressure, cavity volume, and pressure wave celerity. Specifically, the piezometric head $H$ depends on the local air volume $V_g$, which is itself updated based on flow rates, which in turn are influenced by wave celerity. Since the wave celerity $a_m$ is computed from $V_g$, this results in an iterative process that cannot be converged in a single forward pass.

To address this, the AGCM.v2 introduces an iterative procedure at each time step. For each node, the pressure head, wave celerity, and air cavity volume are updated in a nested loop until convergence is achieved. This procedure ensures that the interdependent variables ($H$, $V_g$, $\alpha$ and $a_m$) remain mutually consistent throughout the simulation. The structure of this iterative coupling was inspired by the transient modeling framework proposed by Vasconcelos and Marwell [24], which similarly enforces convergence among interacting flow variables in mixed-cavity scenarios.

The iterative scheme is outlined as follows:

• Initial values: at time $t$, values of $V_g$ and $a_m$ are obtained from the previous time step;
• First update: Using these values, the MOC equations are solved to compute preliminary values of piezometric head $H$, upstream flow rate $Q_{Pu}$ and downstream flow rate $Q_P$;
• Volume update: The air cavity volume is updated based on the net discharge across the node, computed from the difference between inflowing and outflowing discharges;
• Wave celerity recalculation: The gas content $\alpha$ and wave celerity $a_m$ are recalculated using the updated gas volume $V_g$;
• Pressure and flow recalculation: with the updated celerity $a_m$, the MOC equations are re-solved to obtain new values of $H$, $Q_{Pu}$ and $Q_P$;
• Convergence check: convergence is evaluated by comparing the relative change in piezometric head and celerity between successive iterations. Iteration stops when both relative differences fall below ${10}^{-6}$, or when a maximum of 50 iterations is reached to prevent divergence in stiff conditions. If the tolerance is not met, steps 3–6 are repeated.

This approach maintains the integrity of coupling between compressibility and flow characteristics, allowing for more consistent tracking of pressure waves in systems with significant variations in gas content. Although it increases computational cost, the iterative scheme provides improved fidelity in scenarios where the cavity volume and wave celerity evolve significantly during the transient. The AGCM.v2 retains the unsteady friction formulation of Vitkovský et al. [25], as in AGCM.v1A.

This model configuration is especially important in systems where

• The free gas content is high or highly dynamic,
• The wave celerity varies sharply over space or time,
• Accurate pressure peak prediction is essential (e.g., in surge analysis or protection design).

2.6. Model Comparison

To systematically assess the influence of specific modeling assumptions on transient flow predictions, four numerical configurations were developed by incrementally incorporating additional components into the core structure of the DGCM. These components include unsteady friction models, the treatment of variable wave celerity, and iterative coupling mechanisms between pressure head and air volume.

The four configurations analyzed in this study are:

• DGCM/GCAV: The Discrete Gas Cavity Model, extended with unsteady friction modeled according to Vitkovský et al. [25].
• AGCM.v1A: Adjustable-celerity Gas Cavity Model that integrates the DGCM with variable wave celerity and Vitkovský’s unsteady friction model.
• AGCM.v1B: Adjustable-celerity Gas Cavity Model that integrates the DGCM with variable wave celerity and unsteady friction via the convolution-based model of Vardy and Hwang [26].
• AGCM.v2: Iterative extension of AGCM.v1A, featuring internal coupling between piezometric head, air volume, and wave celerity through a convergence loop at each time step.

Figure 2 presents a flowchart summarizing the computational structure of each configuration. Color-coded arrows indicate the modeling path followed by each configuration and match the visual convention used throughout the Section 3.

2.7. Model Benchmarking and Sensitivity Analysis

The numerical investigation conducted in this study is divided into two main stages: (i) benchmarking against experimental data, and (ii) a parametric sensitivity analysis to assess the role of gas content in transient responses. Three datasets were considered: two experimental configurations reported in the literature [9,23] and one hypothetical system designed to systematically explore air–water interaction effects. The corresponding experimental and numerical setups are summarized in Figure 3.

2.7.1. Benchmarking with Experimental Data

The first stage aims to validate and compare the performance of four model configurations introduced in Section 2.6: DGCM, AGCM.v1A, AGCM.v1B, and AGCM.v2. These models were applied to simulate the experimental test case reported by Soares et al. [9], which involves a pressurized water pipe experiencing vaporous cavitation induced by a rapid downstream valve closure. In that study, the authors concluded that the DGCM provided the best agreement with the experimental measurements under vaporous cavitation conditions [9].

The experimental setup consists of a horizontal copper pipeline, 15.22 m in length, with an internal diameter of 20 mm and a wall thickness of 1 mm. A steady-state flow of 0.156 L/s is imposed at the upstream reservoir, and the transient is initiated by a fast-acting closure at the downstream valve. The unsteady friction model adopted in each simulation followed the specifications in Section 2.6. When the Vítkovský et al. [25] formulation was used, the coefficient $k_3 = 0.016$ was applied, as calibrated by Soares et al. [9].

Model performance was assessed by comparing the simulated pressure histories at the downstream end with the experimental measurements. For this purpose, two statistical indicators were used:

• Root Mean Square Error (RMSE):

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(H_{sim, i}-H_{exp,i}\right)}^2}$$

(27)

• Nash–Sutcliffe Efficiency (NSE):

$$NSE=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left(H_{sim,i}-H_{exp,i}\right)}^2}{{\sum }_{i=1}^N{\left(H_{exp,i}-{\overline{H}}_{exp}\right)}^2}}$$

(28)

where $H_{sim,i}$ and $H_{exp,i}$ represent the simulated and measured piezometric heads at time step i, respectively, and ${\overline{H}}_{exp}$ is the average of the measured values. The NSE ranges from $-\mathrm{\infty }$ to 1, with values closer to 1 indicating better model performance, while lower RMSE values indicate improved agreement between simulated and measured data. These indicators were used to quantify the ability of each model to reproduce pressure peaks, damping trends, and general waveform fidelity.

In addition, the models were benchmarked against the experimental dataset reported by Zhu et al. [23], which investigates transient air-water mixing flows under varying gas contents. The laboratory setup also consists of a reservoir-pipe-valve configuration. The test pipeline is made of Plexiglas, with a total length of 36.0 m, an inner diameter of 90 mm, and a wall thickness of 10 mm. The elastic modulus of the pipe material was reported as approximately 2.684 GPa, with a Poisson’s ratio of about 0.358. During the experiments, a steady upstream pressure head of approximately 5.3 m was maintained. In the models where unsteady friction was simulated using the formulation of Vítkovský et al. [25], the coefficient $k_3=0.016$ was applied.

Given that the experiments were conducted in a viscoelastic pipe, the inclusion of pipe-wall viscoelasticity is required to ensure a physically consistent comparison between numerical and experimental results. Accordingly, the viscoelastic model described in Section 2.3 was implemented using parameters calibrated by Zhu et al. [23]. The pipe-wall behavior was represented using a generalized Kelvin–Voigt model with three elements, characterized by retardation times ${\tau }_1=0.05\hspace{0.17em}\mathrm{s}$, ${\tau }_2=0.5\hspace{0.17em}\mathrm{s}$, and ${\tau }_3=1.5\hspace{0.17em}\mathrm{s}$, and corresponding creep compliances $J_1=0.00839\times {10}^{-9}\hspace{0.17em}{\mathrm{Pa}}^{-1}$, $J_2=0.3504\times {10}^{-9}\hspace{0.17em}{\mathrm{Pa}}^{-1}$, and $J_3=0.3552\times {10}^{-9}\hspace{0.17em}{\mathrm{Pa}}^{-1}$.

Model performance was assessed by comparing wave celerities obtained from numerical simulations with the corresponding experimental measurements reported by the authors. For the dataset of Zhu et al. [23], the analysis focused on the downstream end location, where experimental pressure signals were used to infer wave propagation characteristics. The experimentally measured wave celerities were compared with: (i) the average celerities reported by Zhu et al. [23] based on their numerical model, and (ii) the celerities obtained from the present simulations using AGCM.v1A and AGCM.v2.

For AGCM-based formulations, the average wave celerity was computed as the arithmetic mean of the instantaneous celerity values calculated at the analysis point using Equation (23) over the simulation period. In this context, the average celerity represents a time-averaged quantity derived directly from the local pressure-density relationship resolved by the model.

However, in any DGCM-based formulation, where discrete air cavities constitute the core representation of the air-water interaction, the instantaneous local wave celerity computed from Equation (23) represents an internal model variable and is not directly comparable to wave speeds inferred from experimental pressure measurements.

For this reason, an apparent average wave celerity was computed for each model using the time interval between successive positive pressure peaks at the downstream end of the pipe. The average time interval ($t_m$) was used to compute the apparent wave celerity according to the formula $c_{apparent}=4L/t_m$, where $L$ is the pipe length. This procedure aims to reproduce how wave celerity is effectively inferred from experimental pressure measurements and enables a consistent comparison between experimental observations and numerical predictions across different modeling approaches.

2.7.2. Sensitivity Analysis with Varying Gas Content

In the second stage, a sensitivity analysis was carried out to investigate how the initial air volume fraction (${\alpha }_0$) affects transient wave behavior. The goal was to evaluate model responses under progressively increasing the gas content, which impacts local compressibility and wave propagation speed.

To isolate the effects of air entrainment, a hypothetical pipeline was designed using the same configuration as the experimental setup but scaled in length and diameter, as also indicated in Figure 3. The larger domain ensures that the pressure waves undergo sufficient propagation time before reaching the reservoir, thereby enabling more precise analysis of transient storage effects and model consistency. The hypothetical pipe has a total length of 1000 m, an internal diameter of 0.10 m, and a wall thickness of 2.5 mm. In the models where unsteady friction was simulated using the formulation of Vítkovský et al. [25], the coefficient $k_3=0.016$ was applied.

3. Results

3.1. Model Validation with Experimental Datasets

To assess the performance of the proposed AGCM configurations, a comparative validation was conducted using the experimental dataset from Soares et al. [9], which describes a transient event generated by the rapid closure of a valve in a pressurized pipeline system.

Figure 4 presents the numerical results obtained using four different modeling approaches: the classical DGCM/GCAV with unsteady friction, AGCM.v1A, AGCM.v1B, and AGCM.v2. The AGCM.v1A and AGCM.v1B formulations correspond to DGCM implementations with variable wave celerity, differing only in the adopted unsteady friction model, namely the Vitkovský et al. [25] formulation and the Vardy and Hwang [26] model, respectively. The AGCM.v2 formulation further extends this framework by incorporating the proposed interactive iterative coupling strategy. All models that require calibration of the unsteady friction coefficient employed $k_3=0.016$, as recommended by Soares et al. [9].

The accuracy of each model was evaluated using the Root Mean Square Error (RMSE) and the Nash–Sutcliffe Efficiency (NSE), as summarized in

Table 1. Model performance comparison against experimental pressure signals [9].

Model	RMSE (m)	NSE
DGCM	7.90	0.898
AGCM.v1A	9.30	0.858
AGCM.v1B	14.79	0.642
AGCM.v2	14.05	0.676

. These metrics were calculated by comparing numerical pressure head predictions with experimental measurements at the downstream valve.

The original DGCM/GCAV showed the best performance in this case, with the lowest RMSE (7.90 m) and highest NSE (0.898), closely matching the experimental pressure peaks and wave attenuation. This result is consistent with the characteristics of the experimental setup, which corresponds to a vapor cavitation event occurring under very low air concentration conditions (${\alpha }_0={10}^{-7}$). Although the DGCM/GCAV is not specifically designed to model vapor cavitation, it can represent such scenarios when the initial air fraction is sufficiently small, causing the gas cavities to behave similarly to vapor cavities. Under these conditions, the assumption of constant wave celerity remains valid, which favors the performance of the traditional DGCM/GCAV formulation.

The AGCM.v1A, which introduces wave celerity variability, produced very good results, albeit with slightly less accuracy (RMSE = 9.30 m; NSE = 0.858), indicating that while it still captures the overall pressure pattern reasonably well, the added complexity of variable celerity may reduce its precision when modeling pure vapor cavitation. The AGCM.v2, which also incorporates an iterative coupling between celerity and cavity volume, further deviated from experimental data, likely due to overcorrection in the pressure-celerity feedback loop under low $\alpha$ conditions.

Notably, the AGCM.v1B model, which adopts UF computation using the convolution-based unsteady friction model and variable celerity (Equations (13) and (14)), yielded the lowest performance metrics, with an RMSE of 14.79 m and an NSE of 0.642. It should be noted that, differently from the Vitkovský et al. formulation [25], the unsteady friction model of Vardy and Hwang [26] does not require ad hoc calibration for the specific test case. The model results showed a tendency to underpredict pressure damping, resulting in peak pressures that were slightly higher than those observed experimentally. Still, the overall agreement remained acceptable, with transient wave patterns captured adequately

The slightly reduced performance observed for the models incorporating variable celerity does not indicate a lack of robustness, as all configurations still achieved high statistical agreement with the experimental data. Rather, it reflects the fact that the additional model complexity introduced to represent air-related compressibility effects becomes less influential when the air fraction is negligible.

These results underscore the importance of aligning model structure with the physical characteristics of the flow regime. In low-air or vapor-dominated cases, simpler cavitation representations, such as DGCM/GCAV with steady celerity, yield superior agreement. However, as explored in the next section, these conditions may not reflect the full range of air-water interactions found in real-world systems, motivating the development and testing of more adaptable models.

To further contextualize the benchmarking results, the models were also evaluated against the experimental dataset reported by Zhu et al. [23], focusing on the prediction of wave celerity under air-water mixing conditions. The comparison includes experimentally measured wave celerities, the average celerities reported by Zhu et al. [23] for their DGCM formulation, and the apparent and average celerities obtained in the present study using AGCM.v1A and AGCM.v2. The resulting celerity values for all tested cases are summarized in

Table 2. Comparison of experimental, apparent, and average wave celerities obtained from the Zhu et al. [23] dataset for different air fractions and numerical models.

Case No	${\mathit{\alpha }}_0$	Water Velocity (m/s)	Experimental Wave Celerity [23] (m/s)	Apparent Celerity of AGCM.v1A (m/s)	Apparent Celerity of AGCM.v2 (m/s)	Average Celerity of AGCM.v1A (m/s)	Average Celerity of AGCM.v2 (m/s)	Zhu’s [23] Average Celerity for Their DGCM (m/s)
1	0.0125	1.73	90.25	76.67	76.47	101.27	101.31	107.88
2	0.0138	1.56	86.07	72.97	72.77	96.59	96.62	100.45
3	0.0165	1.30	79.23	66.67	66.47	88.65	88.67	88.26
4	0.0193	1.11	73.16	61.69	61.53	82.21	82.22	80.33
5	0.0237	0.90	65.35	55.68	55.57	74.46	74.46	70.83

. The discrepancy between celerity values was computed by subtracting the numerical predictions from experimental values, and dividing by the latter, with all results presented in

Table 3. Relative discrepancy between experimental celerity values and the corresponding numerical predictions for apparent and average celerity values using the conditions presented in Zhu et al. [23].

Case No	${\mathit{\alpha }}_0$	AGCM.v1A Apparent Celerity Discrepancy	AGCM.v2 Apparent Celerity Discrepancy	AGCM.v1A Average Celerity Discrepancy	AGCM.v2 Average Celerity Discrepancy	Zhu’s [23] Average Celerity Discrepancy
1	0.0125	15.0%	15.3%	−12.2%	−12.3%	−19.5%
2	0.0138	15.2%	15.5%	−12.2%	−12.3%	−16.7%
3	0.0165	15.9%	16.1%	−11.9%	−11.9%	−11.4%
4	0.0193	15.7%	15.9%	−12.4%	−12.4%	−9.8%
5	0.0237	14.8%	15.0%	−13.9%	−13.9%	−8.4%

.

The results indicate that the apparent wave celerities predicted by both AGCM.v1A and AGCM.v2 are consistently lower than the experimentally measured values across all cases. The discrepancies remain relatively uniform, ranging from approximately 14.8% to 15.9% for AGCM.v1A and from 15.0% to 16.1% for AGCM.v2. This systematic underestimation suggests that both formulations exhibit similar limitations in capturing the effective wave propagation speed under gas–liquid conditions.

When considering average wave celerities, however, the opposite trend is observed. Both AGCM.v1A and AGCM.v2 overestimate the experimental values, with discrepancies ranging between approximately 11.9% and 13.9%. The closer agreement between the two formulations reflects the fact that the wave celerities were computed following the same local formulation (Equation (23)) across the different models, rather than indicating differences in physical wave propagation behavior. A similar behavior is observed in the results reported by Zhu et al. [20], whose DGCM-based formulation also exhibits discrepancies in the prediction of average wave celerity, with deviations ranging from approximately 8.4% to 19.5%. This consistency suggests that such differences are not model-specific, but rather inherent to DGCM-based approaches under these flow conditions.

This behavior is further illustrated by the pressure head evolution shown in Figure 5 for Case 1 (${\alpha }_0=0.0125$), where noticeable discrepancies between numerical predictions and experimental data can be observed across all model configurations. In particular, while the models capture the general damping trend, differences in peak amplitude, phase, and wave attenuation remain evident.

A similar pattern is observed in the results reported by Zhu et al. [23], whose DGCM-based formulation also does not fully reproduce the experimental pressure response, exhibiting discrepancies in both peak attenuation and phase alignment. This behavior is consistent with the results obtained using AGCM.v1B in the present study, which follows a similar modeling framework. Such agreement between independent implementations reinforces that the observed deviations are not model-specific, but are instead inherent to DGCM-based approaches under these flow conditions.

These results highlight the challenges associated with accurately representing wave celerity and transient response in gas-containing flows using discrete-cavity formulations.

3.2. Assessing the Role of Air Fraction in Pressure Wave Behavior

To investigate how entrapped air influences pressure wave behavior, a sensitivity analysis was conducted using the hypothetical dataset presented in Section 2.7. Simulations were run for four models (DGCM/GCAV, AGCM.v1A, AGCM.v1B, and AGCM.v2) across a range of initial gas contents $\alpha$ from $1 \times {10}^{-6}$ to $1 \times {10}^{-3}$. Results are shown in Figure 6, representing the pressure head at the downstream end of the pipeline. It is important to note that, in this case, no viscoelastic effects are considered, in contrast to the previous experimental dataset. For models employing the unsteady friction formulation of Vítkovský et al. [25], the coefficient $k_3=0.016$ was adopted, consistent with Section 3.1, where this value had been previously calibrated by Soares et al. [9] using experimental data. Although the current analysis uses a hypothetical domain, the same value was retained to preserve consistency across simulations.

At lower air concentrations (e.g., ${\alpha }_0=1 \times {10}^{-6}$), the influence of free air on wave celerity is minimal, and assuming a constant celerity remains a reasonable approximation. In this regime, the compressibility introduced by discrete gas cavities, as modeled by the DGCM/GCAV, is sufficient to represent the transient behavior, and the differences among models remain small. As the gas content increases, more pronounced differences between models begin to emerge. As ${\alpha }_0$ values increased, the DGCM/GCAV approach, which is the only one without celerity variation, significantly overestimated the first pressure peak and the frequency of pressure oscillation. This result underscores the importance of including wave celerity variability when modeling highly compressible flows.

For subsequent pressure oscillation cycles, the following observations are noted:

• AGCM.v1 formulations (AGCM.v1A and AGCM.v1B) produce nearly identical results across all cases, even at higher ${\alpha }_0$ values, suggesting a shared sensitivity to mixture compressibility and wave celerity changes, despite differing implementations of the unsteady friction calculation.
• AGCM.v2, which recalculates wave celerity and head within each time step as explained in Section 2.5, also shows close agreement with the AGCM.v1 at low gas fractions. However, as the gas content increases, systematic differences become more noticeable; particularly in terms of phase alignment, it diverged slightly. These differences are likely due to small differences in the converged values of celerity and $\alpha$ at each node over time.
• DGCM/GCAV shows increasing deviation with higher gas content, both in terms of amplitude and phase, due to its assumption of constant celerity.

Additionally, phase shifts between the models become more pronounced as ${\alpha }_0$ increases, affecting both wave arrival times and peak alignment. These shifts reveal not only amplitude-related discrepancies but also changes in wave propagation celerity, strongly influenced by how each model handles air-induced compressibility.

To complement the evaluation of wave attenuation mechanisms, Figure 7 presents an analysis of the pressure-damping behavior in AGCM.v2 for different $\alpha$ values. Two indicators were used to characterize attenuation: the ratio between successive pressure peaks $H_i/H_{i-1}$ and the relative damping $(H_{i-1}-H_i)/H_{i-1}$, both plotted as functions of dimensionless time $t/(2L/a)$, where $a$ is the apparent average celerity for each case (see

Table 4. Average time between successive positive pressure peaks and corresponding apparent celerity values, for different ${\alpha }_0$ and modeling configurations.

Parameters	${\mathit{\alpha }}_0$	DGCM/ GCAV	AGCM.v1A	AGCM.v1B	AGCM.v2
Time Between Positive Peaks (s)	1 × 10−6	4.037	4.044	4.044	4.044
	1 × 10−4	4.043	4.168	4.170	4.162
	5 × 10−4	4.075	4.634	4.652	4.612
	1 × 10−3	4.115	5.164	5.196	5.128
Apparent Average Celerity (m/s)	1 × 10−6	990.9	989.1	989.1	989.1
	1 × 10−4	989.3	959.7	959.1	961.0
	5 × 10−4	981.6	863.1	859.6	867.3
	1 × 10−3	972.1	774.6	769.8	780.0

). As $\alpha$ increases, the curves indicate more rapid stabilization of wave amplitudes, reflecting enhanced energy dissipation due to higher system compressibility. This effect is particularly pronounced for ${\alpha }_0= 1\times {10}^{-3}$, where damping accelerates in the early wave cycles and progressively smooths out as the system approaches equilibrium.

Figure 8 compares these same indicators against the experimental data reported by Soares et al. [9]. While some variability is present in the experimental measurements, especially in early wave peaks, the numerical trends captured by AGCM.v2 with ${\alpha }_0= 1\times {10}^{-7}$ follow the general attenuation pattern closely. The agreement between the experimental and numerical damping trajectories reinforces the physical consistency of the proposed model, even under low air concentrations.

3.3. Influence of Air Fraction on Celerity Variation

To explore how models with variable wave celerity respond to changes in air concentration, the temporal evolution of the relative celerity was analyzed for three different initial values of $\alpha$: $1 \times {10}^{-6}$,$1 \times {10}^{-4}$, and $1 \times {10}^{-3}$. As shown in Figure 9, the results focus exclusively on the three models that incorporate celerity variation: AGCM.v1A, AGCM.v1B, and AGCM.v2.

The celerity is shown in relative terms, normalized by its initial value for each case. For very low gas content ($\alpha = 1 \times {10}^{-6}$), the relative celerity remains nearly constant throughout the simulation, indicating that the compressibility of the gas phase plays a negligible role and the wave celerity is primarily governed by the pipe-wall elasticity and the liquid phase compressibility. As ${\alpha }_0$ increases to ${\alpha }_0= 1 \times {10}^{-4}$, subtle oscillations begin to emerge, revealing that local gas volume changes begin to influence wave propagation. At ${\alpha }_0= 1 \times {10}^{-3}$, the effect becomes dominant: celerity oscillations are pronounced and tightly follow the pressure wave patterns, reflecting the strong coupling between gas compressibility and transient flow dynamics.

Figure 9d directly compares the celerity behavior of the three models for $\alpha = 1 \times {10}^{-3}$. Overall, AGCM.v1A, AGCM.v1B, and AGCM.v2 exhibit very similar oscillation amplitudes and temporal trends, indicating that all three formulations predict a comparable evolution of wave celerity under this gas-content condition. Minor differences can still be observed in the local alignment and shape of the peaks and troughs, particularly during the early stages of the transient, but these discrepancies remain limited throughout the simulation. This result suggests that, although the iterative coupling adopted in AGCM.v2 affects the transient solution, its influence on the overall celerity evolution is relatively small when compared with the two non-interactive AGCM.v1 formulations.

To further quantify the influence of gas content on wave propagation, the apparent average celerity was computed for each model using the time interval between successive positive pressure peaks at the downstream end of the pipe. The average time interval ($t_m$) was used to compute the apparent wave celerity according to the formula $c_{apparent}=4L/t_m$, where $L$ is the pipe length. The results, summarized in Table 4, show a consistent reduction in wave celerity as the air fraction $\alpha$ increases, confirming the expected compressibility effects. However, while all four models reflect this trend, the magnitude of the reduction differs substantially. The three models that incorporate variable celerity (AGCM.v1A, AGCM.v1B, and AGCM.v2) demonstrate a much more pronounced decrease in average wave celerity compared to the classical DGCM/GCAV, which retains constant celerity. Among the variable-celerity models, the apparent celerities are similar in magnitude, with AGCM.v2 tending to yield slightly smaller values across all $\alpha$ levels. These results reflect the direct influence of dynamic gas volume on wave propagation speed and highlight the role of celerity adjustment in modeling transient flows under high gas content conditions.

Importantly, this analysis also demonstrates that the magnitude and dynamics of wave celerity are primarily governed by the gas content, with only a secondary influence from the specific modeling strategy adopted. While differences between formulations can be observed, particularly in the local evolution of peaks and troughs, these variations remain relatively small when compared to the overall trends. Consequently, for the conditions analyzed, the choice of modeling approach (e.g., non-interactive versus iterative coupling) has a limited impact on the global evolution of wave celerity and the timing of transient responses.

3.4. Computational Performance and Mesh Dependency

In addition to the physical behavior of the model, a combined assessment of mesh dependency and computational performance was conducted for the AGCM.v2. model. The analysis was performed using the experimental dataset of Soares et al. [9]. Four spatial resolutions were considered: a finer mesh ($0.5\Delta x$), the reference mesh ($\Delta x$), and two coarser meshes ($2\Delta x$ and $4\Delta x$). The numerical pressure head at the valve was interpolated to the experimental time base, and model performance was evaluated using the Nash–Sutcliffe efficiency coefficient (NSE) and the root mean square error (RMSE). These statistics were computed over the time interval from 0 to 2 s of simulation, using the experimental sampling interval ($∆t\approx 3.34\times {10}^{-4}$ s) as the reference time step for interpolation. As summarized in

Table 5. Mesh dependency analysis for the Soares et al. dataset [9]: spatial resolution, Nash–Sutcliffe efficiency (NSE), and root mean square error (RMSE) computed at the valve.

Mesh	$∆\mathit{x}$(m)	RMSE (m)	NSE
0.5∆x (Finer)	0.24	13.56	0.699
1∆x (Reference)	0.47	14.05	0.676
2∆x (Coarser)	0.94	13.22	0.713
4∆x (Very coarse)	1.88	14.53	0.654

, both performance metrics remained very similar across all tested meshes, with NSE values ranging from 0.654 to 0.713 and RMSE values between approximately 13.22 m and 14.53 m. These results indicate a weak sensitivity of the numerical solution to the spatial discretization within the investigated range. The corresponding pressure head time histories are shown in Figure 10, where a close agreement among all mesh resolutions can be observed, limited by the dimension of $\mathsf{\Delta }x$.

In parallel, a computational performance comparison was conducted to evaluate the computational cost of AGCM.v2 relative to AGCM.v1A. This analysis was carried out using the hypothetical dataset, in which four initial air fractions were tested (${\alpha }_0 = 1\times {10}^{-6}$ to $1\times {10}^{-3}$). Under these conditions, AGCM.v2 required significantly higher computational effort, with total simulation times approximately 20 to 35 times greater than those of AGCM.v1A. This increase is primarily associated with the internal iterative procedure used to enforce coupling between pressure head and wave celerity at each time step. The iterative scheme required, on average, between approximately 44 and 47 iterations per time step, with maximum values reaching up to 49 iterations, depending on the air fraction. A convergence tolerance of ${10}^{-6}$ was used, applied to the relative difference between successive values of pressure head and wave celerity. Despite the substantial increase in computational cost, the iterative procedure remained stable and robust across all tested conditions. However, given that AGCM.v2 does not provide a clear improvement in predictive accuracy relative to AGCM.v1, its application should be carefully considered, particularly in large-scale simulations where computational efficiency is a critical factor.

4. Discussion

The results demonstrate that the performance of each model is strongly dependent on the gas content in the system. Under low air concentrations, such as in the experimental case from Soares et al. [9], the classical DGCM/GCAV, which assumes constant wave celerity, yielded the best agreement with measured data. This outcome reflects the fact that, for lower ${\alpha }_o$ conditions, the added complexity of dynamic celerity and iterative coupling offers limited benefits and may slightly reduce numerical accuracy.

However, as the air fraction increases, this trend reverses. The constant-celerity DGCM/GCAV increasingly overestimates pressure peaks and underrepresents damping effects. In contrast, the three variable-celerity models (AGCM.v1A, AGCM.v1B, and AGCM.v2) provided more physically consistent results. These models more accurately captured the pressure amplitude under high gas content when compared with the DGCM/GCAV algorithm presented in Wylie et al. [6]. These improvements highlight the importance of accounting for gas-induced compressibility effects in transient flow modeling.

Among the variable-celerity models, AGCM.v2 exhibits differences primarily in the temporal evolution of the solution, including smoother pressure transitions and slight variations in phase alignment. These effects result from the iterative coupling between piezometric head, cavity volume, and wave celerity. However, the overall agreement between AGCM.v2 and the non-interactive formulations (AGCM.v1A and AGCM.v1B) remains very close across all tested conditions, indicating that the iterative coupling does not lead to a significant improvement in predictive accuracy for the cases analyzed.

Damping behavior, assessed through peak ratio analysis, confirmed the strong influence of gas content on wave energy dissipation. All variable-celerity models reproduce the general experimental damping trends, with only minor differences between formulations. Although AGCM.v2 provides a more consistent coupling between the governing variables, this does not translate into a clear advantage in terms of accuracy when compared to AGCM.v1. Considering that AGCM.v2 introduces a substantially higher computational cost, with simulation times increasing by more than one order of magnitude relative to AGCM.v1A, its application should be carefully evaluated, particularly in large-scale simulations where computational efficiency is critical.

5. Conclusions

This study introduces the Adjustable-celerity Gas Cavity Model (AGCM) as an extension of the Discrete Gas Cavity Model, incorporating the effects of gas–liquid interactions on wave celerity values. By allowing wave celerity to evolve dynamically with local gas volume, the AGCM yields a more consistent representation of compressibility effects in gas–liquid systems. The AGCM was validated using experimental datasets with low gas content [9] and the occurrence of vaporous cavitation, as well as high gas content [23]. The results obtained indicated that the AGCM is applicable for such flow conditions.

Among the tested configurations, the iterative version (AGCM.v2) provided results that are consistent in terms of the coupling between pressure head, local air fraction, and celerity. Despite a greater computational effort, such consistency is important in conditions with large air phase fractions. Conversely, this research indicated that for low ${\alpha }_o$ fractions (i.e., ${\alpha }_o\le {10}^{-6}$), there are no clear benefits in using approaches that adjust the wave celerity value in the solution domain. In terms of reproducing flow conditions, it is anticipated that the AGCM could be employed for simulations of transient flow in which wave speed variability in both time and space is relevant. Applications include surge protection, pipeline design, and forensic analysis, among others.

Overall, the AGCM framework provides a structured approach for incorporating variable wave celerity into DGCM-based models, clarifying different formulations found in the literature. While the AGCM.v2 formulation offers improved numerical consistency, the AGCM.v1 formulations represent a more efficient alternative for practical engineering applications.