Finding the most suitable closing law is essential to decrease the shock wave pressure caused by transient flow and minimize the potential damage to equipment. The closure of a valve can occur instantly, rapidly, or gradually, and the appropriate law can be convex, linear, or concave, depending on various factors. These factors include the pipe's characteristics (type, diameter, roughness, and length), the conveyed fluid (nature and temperature), and operating conditions (pressure and flow rate). Other factors that receive less attention, such as the duration of slow closure and the impact of soil load on the pipe, are also considered in this study. The main focus of this article is to investigate how the optimal law evolves based on the time it takes for a valve to gradually close, specifically in the case of a valve located at the end of an underground gravity supply pipe. The findings reveal that when the slow closure time (t) exceeds 0.50 times the return period (t4), the exponent of the optimal law becomes a damped periodic function. Each closure time corresponds to a unique optimal law, and as the valve closure time increases, the range of optimal laws becomes narrower.

• Each slow closing time corresponds to a single optimal convex law.

• As the slow closing time increases, the range of optimal laws decreases and the appearance of maximum loads is delayed.

• The evolution of the optimal pressure at the valve is governed by two models: exponential and linear.

• The characteristic method with mixed scheme is used for the simulation of the transient flow of a long buried pipe.

In hydraulic systems, transient flows occur when there is a disturbance in the initial conditions of steady flow, particularly during valve closure, whether it is slow or rapid. Numerous studies have been conducted in this field. Maurice Gariel's research focused on developing practical rules for calculating the pressure surge induced by water hammer and modeling valve closure to reduce the effects of water hammer, as presented in Remenieras (1961). The study demonstrates the transmission of water hammer along the pipeline, the impact of valve closing time, and the relationship between maximum pressure and the degree of linear valve closure. Hayashi & Eansfoed (1960) studied the effects of sudden opening and closing of fictitious diaphragms located downstream of a very long and highly elastic pipeline. They demonstrated that pressure losses significantly influence the transient phenomenon, resulting in profound changes.

Thanks to the advancements in computing, research has focused on the numerical resolution of the equations governing transient flow, taking into account aspects that can influence this flow. Thirriot (1967) examined approximate methods for calculating water hammer in relatively long pipelines. Streeter & Wylie (1983) and Almeida & Koelle (1992) investigated suitable schemes when friction is the predominant factor. They emphasized the importance of mixed schemes in solving water hammer equations.

The method of characteristics (MOC) has been used for transient analysis (Chaudhry & Hussaini 1985; Chaudhry 2013), where the equations governing the phenomenon are transformed into ordinary differential equations that can be solved through characteristic lines using finite-difference approximations. In their study, Tijsseling & Vardy (2015) examined the application of finite volume and finite difference methods in solving equations related to transient flows. They also highlighted the importance of the MOC for analyzing water hammer phenomena. In fact, the MOC has recently been employed to solve water hammer equations in various types of pipelines, including steel pipelines (studied by BenIffa & Triki (2019)), pipelines with different elastic modulus (studied by Kandil et al. (2019)), and glass fiber–reinforced plastic (GRP) pipelines (examined by Rahul & Arun (2020)).

The effect of conduit nature and network configuration has been duly studied. Trabelsi & Triki (2019) used flexible rubber bypass tubes to mitigate water hammer in pressurized steel pipeline systems. Their method involved inserting these tubes upstream of a control valve, which reduced the surge height (ΔH) and water hammer celerity, even with shorter tubes. In their study, BenIffa & Triki (2019) conducted research to improve water distribution systems that utilize steel pipes. They proposed the addition of two short sections of plastic conduits, namely high-density polyethylene (HDPE) and low-density polyethylene (LDPE). Through numerical simulations, they demonstrated that the composite configuration (HDPE–LDPE) offered an optimal compromise in terms of pressure surge attenuation and limitation of oscillation waves. Compared to the double configuration (LDPE–LDPE), this composite configuration showed better ability to attenuate pressure surges.

A study conducted by Rahul & Arun (2020) analyzed transient pressure generation in metal-viscoelastic pipelines using both experimental tests and numerical simulations. Two types of pipe materials, mild steel (MS) and GRP, were evaluated. The results indicated that the GRP pipeline was more effective in attenuating transient pressure caused by the water hammer. In cases where the exclusive use of the GRP pipeline was limited due to manufacturing constraints, the use of a combination of GRP + MS pipeline (with GRP for the initial part) proved to be a good solution for reducing transient pressure. Kandil et al. (2019, 2021) studied the influence of pipe material properties on hydraulic water hammers and noted that materials with low elasticity modulus had a lower risk of causing water hammer compared to those with a high elasticity modulus. Toumi & Remini (2022) examined pressure, flow, and velocity variations in a pumping system consisting of two and three storage tanks and found that fluctuations were mainly observed in the pipe connected to the tank with a low piezometric head. In addition, the research by Mery et al. (2021) and Mahmoudi-Rad & Najafzadeh (2023) focused on various configurations of air chambers to reduce the water hammer effect. These studies demonstrated that the geometry and configuration of the air chamber, including the diameter of the inlet pipe and the diameter of the air chamber, had a significant impact on this phenomenon.

Furthermore, to reduce surges, Subani & Norsarahaida (2015) studied six different models for valve closure: instantaneous (m = 0), linear (m = 1), concave (m = 0.05), concave (m = 0.5), convex (m = 5), and convex (m = 50). They clearly demonstrated the variation of the pressure wave profile and amplitude for each law and emphasized that the instantaneous and convex closure models produce the minimum and maximum pressures, respectively. The study conducted by Yao et al. (2015) examined the influence of valve closing time on the variation of water hammer pressure. The results of this study revealed a negative correlation between closing time and maximum pressure. On the other hand, Twyman (2018a, 2018b) studied the influence of different types of valves (butterfly, gate, circular, square, ball, needle, and globe) and network configurations on transient pressures. They concluded that extreme pressure values were related to the type of valve while the network configuration was not a significant factor in attenuating transient pressure.

Herasymov et al. (2019) tested linear closure, closure with a breaking point, intermittent closure, and combined closure. They recommended linear closure with a breaking point to reduce surges. However, Wan et al. (2019a, 2019b) emphasized that staggered valve closing times reduce the water hammer phenomenon. According to research conducted by Zhao et al. (2020), it has been proven that the flow rate and valve closing speed have a significant influence on the increase in water hammer pressure. The higher the flow rate and valve closing speed, the greater the surge pressure. Arefi et al. (2021) studied water transport pipelines, which are typically characterized by large diameter and high flow rates and often pose problems during the negative pressure phase. They analyzed the effects of several factors such as abrupt valve closing time, friction coefficient, wave velocity, viscoelasticity, and valve type. The main results indicate that the exclusive use of air valves is not effective; however, the proper installation of equipment such as water flow control devices and hydropneumatic tanks along the pipeline helps prevent the creation of negative pressure.

Jinhao et al. (2021) examined the sensitivity of six key parameters that influence transient flow. The results revealed a positive correlation between the flow rate and Young's modulus with maximum pressure, while pipe diameter, valve closing time, and wall thickness showed a negative correlation. In addition, flow rate, diameter, and valve closing time were identified as key factors impacting water hammer simulation. Yong et al. (2022) demonstrated that extending the valve closing time effectively reduces the maximum water hammer pressure. Moreover, they found that pressure fluctuations are influenced by the valve closing speed, where a faster closing speed initially results in higher water hammer pressure. Zheng et al. (2022) discovered that the optimal nonlinear closure law of the valve exhibits distinct characteristics during fast and slow closure, which provides guidance for real-time valve control and serves as a reference for the design of valves aimed at attenuating surge waves.

It also appears that improving valve closure strategies and coordinating hydraulic system performance are among the most commonly used optimization methods to control the water hammer phenomenon (Bazargan-Lari et al. 2013; Wan & Li 2016; Zhou et al. 2017; Wan & Zhang 2018; Wan et al. 2019a, 2019b). Researchers have focused on optimizing valve closure strategies to mitigate maximum water hammer pressure, as this pressure primarily depends on the valve closure characteristics. Therefore, implementing an optimal valve closure law (OVCL) is a useful approach to control these extreme pressures and mitigate their impact. An OVCL is necessary to design a programmed closure device to control extreme pressure fluctuations, improve operational reliability, and extend the lifespan of the water distribution system.

The aim of this study is to investigate OVCL taking into account additional factors that have received less attention, including the duration of slow closure and the influence of soil load on the pipeline. The study focuses on the specific scenario of a valve situated at the endpoint of an underground gravity supply pipe. By analyzing these factors, the research aims to understand how the optimal closing law develops in such conditions.

In this paper, the MOC with a mixed scheme is employed to simulate transient flow and investigate the evolution of the optimal valve closing law for a valve situated at the end of a relatively long buried supply pipeline in Guelma, Algeria

Mathematical model

The fundamental hyperbolic equations derived from the continuity and dynamic equations are presented as follows (Streeter & Wylie 1983):
(1)

In the above equations, a is the wave velocity (m/s); P is the pressure (Pa); ρ is the density of the liquid, for water ρ = 103kg/m3; u is the velocity in the x direction (m/s); g is the acceleration due to gravity (m/s2); α is the pipe inclination angle; j is the unit pressure drop; and x and t denote the spatial and time dimensions, respectively. These equations describe the changes in velocity and pressure with respect to time (t) and along the flow direction (x) (Meunier 1980; Ouragh 1971; Toumi & Remini 2022). To determine the values of these parameters at any given time and location along a supply pipeline, the MOC was utilized.

Resolution method

The MOC is a highly effective numerical method used to solve equations that describe transient flow under pressure (Chaudhry & Hussaini 1985; Chaudhry 2013). It converts the two partial differential equations (PDEs) of transient flow into four total differential equations (Pal et al. 2021; Chaudhry 2013; Ouragh 1971; Piskonov 1995; Toumi & Remini 2022).The application of the MOC to the system of equations number one leads to the following equations:
(2)
(3)
where S is the cross section of the pipe (m2), Q is the flow rate (m3/s), and H is the hydraulic head (m).

The differential system is solved by discretizing the pipe into multiple nodes, where these nodes are assumed to be sufficiently close to each other. This assumption allows us to consider the following expression: dF(x, t) = F(i + 1) − F(i) where F represents a function such as H or Q at consecutive discretization points i and (i + 1) (Ouragh 1971).

The total load and flow rate along the pipe are assumed to be known at time t. By integrating the characteristic equation passing through point M with coordinates (i, t) between time instants t and (t + Δt), we move from point i at time t to point (i + 1) at time (t + Δt). For the positive flow direction, Equation (3) is multiplied by dt.

The resulting equation leads us to the following expression:
(4)
Integrating Equation (4) gives us
(5)
When this integral is performed between points i and (i + 1), it can be written as follows:
(6)
For a buried pipe case, the wave celerity is calculated as follows, according to Ait Slimane (2018):
(7)
where
(8)
with ρ being the density of the liquid, for water ρ = 103kg/m3; Em the Young's modulus of metal (Pa); vm the Poisson's coefficient of the metal; em the armor thickness (m); K the modulus of elasticity of water (Pa); r the pipe radius (m); P the hydrostatic pressure (Pa); Pa the backfill soil load (Pa).
The earth fill load is calculated using the formula given by Marston, expressed as follows:
(9)
with γ being the soil density (KN/m3); Bd the trench width (m); H the height of embankment above pipe (m); Cd the load coefficient.
(10)
μ′ is the vertical coefficient friction between the backfill and the trench walls and K is the Rankine's coefficient (Moser 2001).
The variable j is expressed by the following relation:
(11)
where d is the internal diameter of the pipe in m and λ is the coefficient of friction.
Let us define the following variables:
(12)
(13)

In these equations, the parameter R represents the effect of flow resistance, and its elements are defined as follows: a represents the wave velocity in meters per second (m/s), g represents the acceleration due to gravity in meters per second squared (m/s2), and S represents the cross-section area of the pipe in square meters (m2).

By introducing Equations (12) and (13) into Equation (5) and performing integration, we obtain the following equation:
(14)
(15)
where the next term is a mixed scheme to evaluate the flow rate.
(16)
For this scheme, the integration of the resistance part can be written as follows:
(17)

Mixed numerical scheme

To compute the positive characteristic (PC), we can substitute Equation (17) into Equation (14), resulting in
Let us pose: T1 = B(x(i + 1) − x(i)) = BΔx
Let us pose:
(18)
(19)
In the same way, as in the PC, the integration of the negative characteristic (NC) is made passing through N of coordinates (i + 2, t) between two instants t and (t + Δt), as well as by passing from the point (i + 2) to the instant t at point (i + 1) at instant (t + Δt).
(20)
(21)
The mixed scheme is written as follows (Almeida & Koelle 1992):
For this scheme, the last term of the integral can be written as follows:
(22)
Hence Equation (21) becomes
Let us pose: T1 = B.(x(i + 2) − x(i + 1)) = Bx,
Let us pose:
(23)
The integration of the system of Equations (2) and (3) leads to the following four algebraic equations:
(24)
(25)
(26)
(27)
The preceding two equations enable the computation of pressure and flow values for transient flow within the interior nodes. However, determining the values of these boundary parameters necessitates additional expressions that depend on the specific conditions of the case being examined. To calculate the pressure in bar at any node within the hydraulic system, the following expression is employed:
(28)

Here, P(i + 1) represents the pressure at the selected node in bar, H(i + 1) denotes the total load at this node in meters of water column, NGL signifies the Natural Ground Level in meters, ρ represents the fluid density in m3/s, and g represents the acceleration due to gravity in m/s2.

Valve closing law

The flow rate at nodal point N, which corresponds to the valve, is closely connected to the average flow velocity passing through the valve. This velocity can take on a concave, linear, or convex shape and is represented by the following equation:
(29)
where 0 ≤ iN, Ct = Nt, 0 ≤ m < ∞, u0 is the initial fluid velocity, and uτ is the fluid velocity at the end of the closing operation.
The exponent m determines the type of valve closing law (Provenzano et al. 2011; Subani & Norsarahaida 2015). When the fluid velocity at the end of the valve closing operation is zero (uτ = 0), Equation (32) can be written as follows:
(30)
In steady flow conditions, the volume flow is expressed as
(31)
This allows us to formulate the expression for the flow rate at node N as follows:
(32)

Exponent m in the expression for Q at node N leads to a multitude of Q functions, making it a crucial task to determine the optimal value of this exponent for efficient management of hydraulic systems.

Program presentation

The calculation code used in this work is the result of original programming performed in a Fortran language environment.

To facilitate understanding of its operation, the flowchart of the computer program is presented below. After deducing the algebraic expressions of the system of PDEs governing transient flow, and based on the characteristics of the pipeline and the transported fluid, as well as the boundary conditions, the program calculates the piezometric head, pressure, velocity, and flow rate at all discretization nodes of the supply pipeline connecting buffer reservoir number 1 to pumping station number 2. The spatial step, Δx, is equal to 0.972 m, and the time step, Δt, is equal to 0.001 s. The stability and convergence of the scheme are verified by the Courant–Friedrichs–Lewy (CFL) condition. This computer code is designed to consider both pumping and gravity-fed scenarios. The originality of this code lies in the introduction of the valve closure function, which allows the deduction of the optimal law to minimize extreme pressures for any closure time. Figure 1 presents a flowchart outlining the calculation steps, while Tables 1 and 2 provide details on the pipe characteristics and the transported liquid, respectively.
Table 1

Pipe characteristics

Kind of materialsFretted concrete
Elasticity modulus in Pascal EFC = 1.962 × 1010
Pipe Poisson's ratio νm = 0.20
Coefficient of vertical friction between the backfill and the walls of the trench μ' = 0.50
Rankine's coefficient K = 0.33
Soil density (KN/m3γ = 19,000
Height of embankment above pipe (m) H = 1.1
Trench width (m) Bd = 2
Exterior diameter in (mm) dext = 1,170
Internal diameter in (mm) dint = 1,000
Pipe thickness e1 in (mm) e1 = 85
Pipe roughness ε in (mm) ε = 0.1
Pipe length (m) L = 11,498
Kind of materialsFretted concrete
Elasticity modulus in Pascal EFC = 1.962 × 1010
Pipe Poisson's ratio νm = 0.20
Coefficient of vertical friction between the backfill and the walls of the trench μ' = 0.50
Rankine's coefficient K = 0.33
Soil density (KN/m3γ = 19,000
Height of embankment above pipe (m) H = 1.1
Trench width (m) Bd = 2
Exterior diameter in (mm) dext = 1,170
Internal diameter in (mm) dint = 1,000
Pipe thickness e1 in (mm) e1 = 85
Pipe roughness ε in (mm) ε = 0.1
Pipe length (m) L = 11,498
Table 2

Liquid characteristics

 Fluid elasticity modulus (Pa) K = 2.07 × 109 Fluid temperature (°C) 15 Kinematic viscosity (m2/s) 1.1445 × 10−6 Density of the liquid (kg/m3) ρ = 103
 Fluid elasticity modulus (Pa) K = 2.07 × 109 Fluid temperature (°C) 15 Kinematic viscosity (m2/s) 1.1445 × 10−6 Density of the liquid (kg/m3) ρ = 103
Figure 1

Flowchart for calculating transient hydraulic parameters: case of valve closure.

Figure 1

Flowchart for calculating transient hydraulic parameters: case of valve closure.

Close modal

Problem statement and boundary conditions

Problem statement

The Guelma region, situated in the northeastern part of Algeria, as depicted in Figure 2, receives its drinking water supply from the Hammam Debagh dam. This supply is facilitated through a mixed hydraulic system that employs both pumping and gravity methods. The pipeline is buried in a 2.35-m-deep trench. The top of the pipe is at a depth of 1.1 m. It extends over a distance of 11,498 m, having a diameter of 1,000 mm and a thickness of 8.5 cm. Its primary function is to convey a flow of 720 L/s, from the buffer reservoir number 1 to the pumping station number 2. Figure 3 illustrates the reservoir, which has a bottom quote of 373 m and a height of 5 m. At an altitude of 338 m, a valve is positioned at the end of the pipe. The disturbance in the initial flow conditions resulted in the occurrence of transient flows, leading to recurrent damage to the pipe, as shown in Figure 4.
Figure 2

Geographical location of the study region.

Figure 2

Geographical location of the study region.

Close modal
Figure 3

Scheme of the water supply pipeline.

Figure 3

Scheme of the water supply pipeline.

Close modal
Figure 4

Repair works of a water supply pipeline in Guelma, Algeria.

Figure 4

Repair works of a water supply pipeline in Guelma, Algeria.

Close modal

Boundary and stability conditions

To simplify the analysis of transient flow, the reservoir is assumed to be large enough that changes in water level within the reservoir and pressure drops at its inlet can be neglected. The valve is located at the downstream end of the pipe and is closed, generating an overpressure wave that propagates toward the reservoir. This wave is reflected as a vacuum wave, and its equation in the (Q, H) plane is represented as
(33)
This allows writing
(34)
Or
(35)
H (2) and Q (2) are quantities calculated at time (t), so
(36)
Due to valve presence at the end of the pipe, the pressure and flow rate calculation are carried out according to the PC of the previous node, whose equations of this node characteristic C+ are written as follows:
(37)
(38)
(39)
The total load at the valve takes the following relationship:
(40)

The convergence and stability of MOC

The convergence of a scheme expresses its tendency to approach the exact solution. To be convergent, a numerical scheme must be both stable and consistent. Consistency is related to truncation error, which is the error incurred during the discretization of derivative terms. This error depends on the order of accuracy of the discretization and decreases as the approximation step (x or t) becomes smaller (Allaire 2016; Sebastien & Victor 2020).
(41)

The stability of a scheme refers to its ability to maintain a consistent behavior even in the presence of numerical disturbances, such as calculation errors in boundary conditions or initial conditions. A stable scheme attenuates disturbances, while an unstable scheme amplifies them over time or in space (Allaire 2016; Sebastien & Victor 2020).

The stability of the MOC calculation scheme is guaranteed by the so-called CFL condition (see Courant et al. (1928)). In the case of linear systems, the stability is studied through the Fourier expansion of the error. The CFL condition is given by stability criterion Sc,
(42)
where a is the wave velocity in (m/s), Δt is the time step, and Δx is the space step.

The MOC gives exact numerical results when Sc = 1.0 (Twyman 2018a, 2018b; Goldberg & Benjamin 1983). Recently, authors such as Pal et al. (2021) and Izquierdo & Iglesias (2002) have used this stability criterion. In this study, the stability criterion was verified and taken equal to one (Sc = 1).

To determine the optimal closing law for the valve, we conducted tests using various closing law functions, including concave functions (m < 1), linear functions (m = 1), and convex functions (m > 1). The objective was to identify the closing law that minimizes the overpressure. Figure 5(a)–5(d) depict the variations of the overpressure in relation to the exponent m for four different closing times: 0.55t4, 0.60t4, 0.65t4, and 0.70t4. Each curve exhibits a distinct minimum value, representing a unique value of the exponent m that defines an optimal closure law, leading to the minimum overpressure.
Figure 5

Pressure versus exponent's value m at the valve for closing times equal to 0.55t4 (a), 0.6t4 (b), 0.65t4 (c), and 0.70t4 (d).

Figure 5

Pressure versus exponent's value m at the valve for closing times equal to 0.55t4 (a), 0.6t4 (b), 0.65t4 (c), and 0.70t4 (d).

Close modal
An extended study was conducted to investigate a wide range of valve closing times between t = 26.0304 s (t = 0.51t4) and t = 473.28 s (t = 10t4) using a time step of 0.01t4 (resulting in 950 points). Figure 6 illustrates the changes in the optimal values of the exponent m for each time instant t. For each instant, the value of m was determined to minimize the overpressure.
Figure 6

Optimal values of exponent (m) versus valve closing time.

Figure 6

Optimal values of exponent (m) versus valve closing time.

Close modal

For valve closing times ranging from 0.51t4 to 10t4 (corresponding to t = 26.0304 s to t = 473.28 s) and using a time step of 0.01t4, the optimum values of the exponent m range from 1.126 to 1.446. These optimal values of m correspond to convex closing laws (m > 1), indicating that a single optimum closing law exists for each time interval, resulting in the minimum overpressure. It can be observed that the optimal values of the exponent m vary periodically as the closure time changes. The amplitude of variation is larger for shorter closure times and decreases as the closure time increases. A larger amplitude indicates a wider range of optimal closure laws, while a smaller amplitude narrows down the range of optimal laws. In addition, within each period, multiple optimal closing laws are repeated, but with a progressively narrower distribution.

Figure 6 demonstrates that for the studied closure times, the exponent m of the optimal closure law falls within the interval defined by the two straight lines. This variation can be expressed as
The values of the exponent m vary within the range of
Figure 7 illustrates the changes in overpressure and depression under optimal conditions of slow valve closing laws. It is evident that as the closing time increases, both overpressure and depression decrease. However, there are occasional regressive amplitudes observed. These short-lived protuberances, averaging around 6 s, result in increased optimum pressures (minimum overpressure and minimum depression). This increase is accompanied by significant and rapid variations in the exponent m, leading to a higher number of optimal valve closing laws (averaging 14 laws).
Figure 7

Optimal over pressure and depression variation versus valve closing time.

Figure 7

Optimal over pressure and depression variation versus valve closing time.

Close modal
In Figure 8, two curves representing overpressure and depression are displayed under optimal conditions, along with corresponding curves plotted for a quadratic valve closing law. As the closing time progresses, specifically after 274 s (4t4), the pressure deviation between the quadratic law and the optimal laws diminishes, resulting in a diminishing effect of the closing time.
Figure 8

Maximum and optimal pressure variation versus valve closing time.

Figure 8

Maximum and optimal pressure variation versus valve closing time.

Close modal

Transient flow is a complex phenomenon characterized by significant pressure fluctuations that can pose serious risks to hydraulic systems. Numerous parameters influence this phenomenon, and it is crucial to minimize the potential risks associated with overpressure, such as pipe rupture, valve deformations or failures, leaks, and equipment malfunctions. Achieving this requires determining the optimal valve closing law, considering various often overlooked or less studied parameters, such as the interaction between the valve closing law and the closing delay, as well as the effect of ground load on the pipe. To investigate this, we conducted a study on a real case involving a long buried gravity pipe that supplies water to the Guelma region in Algeria.

Based on the results, it is evident that convex closure laws (m > 1) are superior, resulting in lower overpressure compared to other laws. The optimal closing law depends on the specific characteristics of the hydraulic system. For each closing time, there exists a unique optimal closure law that minimizes overpressure. Furthermore, the optimal values of the exponent m vary periodically over time. When the closure time is short, the variation in m is significant, but it decreases as the closure time increases. Larger amplitudes lead to a broader dispersion of the m coefficient and a wider range of optimal closure laws, whereas smaller amplitudes narrow down the margin of optimal laws. Over time, it appears that the variation in the optimal law tends to stabilize toward a single value.

Simulation of valve overpressure over time demonstrates a smooth regression curve with occasional pressure peak protrusions. Similarly, the simulation of valve depression over time exhibits an increasing curve with peaks of pressure. These curves align well within the envelope of the overpressure and underpressure curves of the quadratic valve closure law. As the slow-closing time exceeds 4t4, the pressure deviation between the maximum pressure of the quadratic law and the minimum pressure of the optimal law decreases, and the impact of the slow-closing time becomes less pronounced.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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