Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment

water

Article Experimental and Numerical Simulation of Water Hammer in Gravitational Pipe Flow with Continuous Air Entrainment

Boran Zhang, Wuyi Wan * ID and Mengshan Shi

Department of Hydraulic Engineering, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China; [email protected] (B.Z.); [email protected] (M.S.) * Correspondence: [email protected]; Tel.: +86-571-8795-1346

Received: 6 June 2018; Accepted: 9 July 2018; Published: 12 July 2018

Abstract: Water hammer is an undesired hydraulic shock phenomenon in water supply pipe systems. It is very important to simulate water hammer for preventing the hazard of over pressure. In order to predict the transient pressure caused by a valve closing in a gravitational pipe with continuous air entrainment, a numerical model based on the Lax-Wendroff format is established, and the matched boundary model is provided. Compared with the traditional methods, this study provides another access by considering the influence of the pipe flow velocity on the wave propagation to simulate transient processes. A corresponding experiment is conducted to optimize the numerical model. Based on the experimental result, an additional friction function is proposed to evaluate the influence of the air content on the attenuation. The result shows that the energy dissipation of the shock waves may be underestimated in air-water mixture flow using the common steady friction. By introducing the additional friction function, the improved model can more accurately simulate the attenuation of the water hammer in the gravitational pipe with continuous air entrainment. As there are plenty of practical water supply systems running with air content, the improved Lax-Wendroff Method (LWM) is valued in accurately predicting water hammer processes especially in those conditions.

Keywords: water hammer; transient ﬂow; numerical simulation; pipe ﬂow; hydraulic experiment

1. Introduction As an undesired hydraulic phenomenon in pipe systems, water hammer is usually caused by a valve closing and opening, a pump shutting and starting, as well as misoperation in pipe systems [1,2]. Various water hammer events commonly occur in water pipe systems, pump systems and hydro-power systems [3]. They can bring many risks, such as overpressures, vibrations, and bursts [4,5]. It is very important to predict water hammer for protecting hydraulic system [6–8]. In the past it has been difﬁcult to simulate water hammer pressure by mathematical algorithm. With the development of the computer, mathematical algorithms become more popular and improved greatly. Numerical simulation has now become the main approach for transient analysis [9–11]. The key of the water hammer calculation is to solve the hyperbolic partial differential equations. Although the water hammer equation is a closed-form expression, no theoretical analytical solution is established at present. In order to obtain the numerical solution, various methods are applied to water hammer simulation, typically including the Finite Difference Method (FDM) [12] and the Finite Element Method (FEM) [13]. According to the existing literatures, FDM is the most favored method. Especially the method of characteristic (MOC), as an FDM, is nowadays the most popular method. It is widely used to simulate the water hammers in various engineering practices, such as valve closing [14–18], and pump failure [19]. Not only that, it has also been applied to special system and boundary

Water 2018, 10, 928; doi:10.3390/w10070928 www.mdpi.com/journal/water Water 2018, 10, 928 2 of 16 conditions. Sibetheros [20] improved the flexibility of numerical simulations by spline polynomials interpolations in MOC. Gong [21] proposed a reconstructed MOC to estimate the pipe parameters. Compared with conducted experiment on a single pipeline with a thin wall thickness, the reconstructed MOC is validated to be of good reliability. Travas [22] proposed a mixed MOC and FDM to consider unsteady friction. Equipped with a feature of relatively low complexity, the mixed method is easier to numerically implement. The Kelvin-Voigt model is introduced to MOC for considering pipe-wall viscoelasticity [23,24]. Nevertheless, based on wave propagation principles, Wood [25] proposed a wave characteristic method (WCM) to simplify the analysis of a water network. According to the comparisons with experimental data and other traditional methods, WCM is confirmed to be of good accuracy except for a significant improvement on convenience. Guo [26] used a sliding mesh method to simulate a water hammer induced by ball-valve closure. They found that Fluent can better simulate the water hammer than the traditional method of characteristics, especially when considering the compressibility of water. In actual water supply systems, there are plenty of pipelines running with air content. In previous research, it has already been ascertained that the numerical solutions for two-phase flow is familiar in format to the model for one-phase liquid flow [27,28]. Bergant et al. [29,30] ascertained that water hammer attenuation may be significantly affected by unsteady friction, cavitation, and trapped air pockets. In their researches, various cases with different conditions are studied and analyzed. It is found that the water hammer attenuation has always been over estimated in traditional simplified methods. Zhou [31] did numerical simulations and conducted an experiment on air leakage effect in a water hammer process, which shows that air leakage has a significant cushioning effect which reduces the intensity of the water hammer. As a conventional numerical method, the Lax-Wendroff Method (LWM) was widely used in computational fluid dynamics (CFD) [32]. However, only a few studies have briefly reported LWM in a water hammer simulation [12,33,34], without considering the convective acceleration terms, and it has hardly been applied in water hammer practices. Considering the real working conditions of water supply systems, it is more important to improve the accuracy of predictions on a water hammer in a pipeline with air content inside. In this paper, water hammer was simulated by LWM for gravitational pipe flow with continuous air entrainment. To analyze the influence of air content on the wave propagation and attenuation, the general numerical model was established by introducing LWM to complete governing equations, and the matched boundary conditions were provided. Then an experiment was conducted to validate and improve the proposed model. Compared with the experiment, the maximum pressure and attenuation were simulated by the proposed model. However, the attenuation is heavily unevaluated in the numerical simulation. Based on the experimental result, an additional friction function was added to the model for approximating the water hammer attenuation. The application shows that the improved model can more accurately simulate the attenuation of water hammer. In water supply systems, especially under low pressure conditions, different air content may occur in the pipelines. It makes the wave speed and energy loss much different from the flows without air. The proposed improved LWM model can well simulate the transient processes of pipe flow with continuous air entrainment. It is valuable in accurately predicting extreme pressure in water hammers with air entrainment, and is helpful in guiding suitable operations in water supply systems for extreme pressure control. Further research of this study will be focused on the physical interactions between entrained air and water mass in air liquid mixture flow. Based on the equipped experimental platform, experiments in an inner visible pipeline will be conducted to trace the air entrainment, delivering, bubble forming and breaking processes by observing the actual inside transient process. Water 2018, 10, 928 3 of 16

2. Numerical Modeling and Meshing

2.1. Governing Equations Many factors can affect water hammer, including the ﬂow velocity, the pressure and the properties of the pipe wall and the ﬂuid. In hydraulic transient, the water hammer is usually considered as an isothermal process, consequently, continuity and momentum equations can describe the dynamics characteristics of the water hammer. For a closed-pipe system, the complete dynamic governing equations of the water hammer can be written as follows [1]

∂h(x, t) ∂v(x, t) ∂v(x, t) v(x, t)|v(x, t)| g + v(x, t) + + f = 0 (1) ∂x ∂x ∂t 2D

∂h(x, t) ∂h(x, t) a2 ∂v(x, t) v(x, t) + − v(x, t) sin α + = 0 (2) ∂x ∂t g ∂x where x and t are independent variables denoting location and time respectively. Correspondingly, h and v are dependent variables denoting hydraulic pressure and ﬂow velocity respectively.

2.2. Discretization Model The dynamic equations system consists of two hyperbolic partial differential equations. It is difficult to establish the general analytical solution [1]. MOC is usually used to establish the numerical solutions, without considering the convective acceleration terms. Considering the influence of convective acceleration terms on water hammer, LWM is used to solve the complete equations system without omitting convective terms. The Lax-Wendroff Method (LWM) is widely applied in CFD [32], but it is hardly used in water hammer simulations. Previously, Chaudhry [12] briefly reported the LWM solution of simplified water hammer equations. Equations (1) and (2) meet LWM in time and space, using a second-order Taylor expansion, then the discretized FDM can be written as follows

∂v(i, t) ∂2v(i, t) ∆t2 v(i, t + ∆t) = v(i, t) + ∆t + (3) ∂t ∂t2 2

∂h(i, t) ∂2h(i, t) ∆t2 h(i, t + ∆t) = h(i, t) + ∆t + (4) ∂t ∂t2 2 In these equations, time and space can be described by discrete nodes and mesh, as seen in Figure1, where i is a series to denote the location of the nodes. When a pipe is divided as n segments, i ∈ {0, 1, 2, . . . , n}. i = 0 and i = n refer to the first and the last node respectively, and the corresponding locations are x = 0 and x = l. ∆t is the time step. Equations (3) and (4) are the discretized second-order Taylor expansion for velocity and hydraulic pressure. The left terms are the variables at time t + ∆t, and the right terms are the variables at time t. The unknown variables at t + ∆t can be solved after the first-order and second-order derivative terms being determined in Equations (3) and (4). According to Equations (1) and (2), the first-order derivative terms can be converted as

∂v(i, t) ∂h(i, t) ∂v(i, t) v(i, t)|v(i, t)| = −g − v(i, t) − f (5) ∂t ∂x ∂x 2D ∂h(i, t) ∂h(i, t) a2 ∂v(i, t) = −v(i, t) + v(i, t) sin α − (6) ∂t ∂x g ∂x Then, the second-order derivative terms can be derived according to Equations (5) and (6). Equations (5)–(8) establish the relationship between time derivatives and spatial derivatives. Water 2018, 10, 928 4 of 16

The ﬁrst-order and second-order space derivatives can be approximated by the central differences as Equations (9)–(12) [32]

2 ( ) ( ) ( ) 2 ( ) h ih ( ) i2 ∂ v i,t = g ∂v i,t ∂h i,t + gv(i t) ∂ h i,t + a2 + v(i t)2 ∂v i,t ∂t2 2 ∂x ∂x 2 , ∂x2 , ∂x ( ) | ( )| | ( )| ( ) ( )| ( )|2 +3v(i, t) ∂v i,t f v i,t + f v(i, t) ∂ v i,t v i,t + f 2 v i,t v i,t (7) ∂x 2D ∂x 2D 4D2 + f v(i,t) ∂h(i,t) + ( ) ∂v(i,t) + v(i,t)|v(i,t)| − ∂v(i,t) 2D g ∂x v i, t ∂x f 2D g ∂x sin α

2 ( ) ( ) 2 ( ) ( ) ( ) ( )| ( )| h i 2 ( ) ∂ h i,t = g ∂h i,t + v(i t) ∂h i,t ∂v i,t + ∂h i,t f v i,t v i,t + v(i t)2 + a2 ∂ h i,t ∂t2 ∂x 2 , ∂x ∂x ∂x 2D , ∂x2 2 ( ) 2 ( ) 2 ( ) 2 2 ( ) 2 | ( )| + 2a v i,t ∂ v i,t + a ∂v i,t + a f ∂v i,t |v(i t)| + a f ∂ v i,t v(i t) (8) g ∂x2 g ∂x 2Dg ∂x , 2Dg ∂x , n ∂h(i,t) ∂v(i,t) v(i,t)|v(i,t)| o + g ∂x + [v(i, t) − 1] ∂x + f 2D sin α ∂v(i, t) v(i + 1, t) − v(i − 1, t) = (9) ∂x 2∆x ∂v(i, t) v(i + 1, t) − v(i − 1, t) = (10) ∂x 2∆x ∂2h(i, t) h(i + 1, t) − 2h(i, t) + h(i − 1, t) = (11) ∂x2 ∆x2 ∂2v(i, t) v(i + 1, t) − 2v(i, t) + v(i − 1, t) = (12) ∂x2 ∆x2

2.3. Meshing and Advancement A time-space differences mesh is needed to simulate the transient process. Figure1 shows a general space and time mesh. In the mesh, x denotes one-dimensional space, representing the distance from the inlet of the pipe, and ∆x is the space step. t is the time variable and ∆t is the time step. There are three kinds of nodes according to time and space: general interior nodes, initial nodes, and side nodes. In Figure1, hollow circles denote the internal nodes, solid circles denote the initial nodes and rectangles denote the side nodes. At the initial state t = 0, the beginning of the first step, all the variables of initial nodes are known or can be determined according to the initial flow conditions. Then the internal nodes at the time t = ∆t, the ending of the first step, can be determined based on the parameters in the beginning of the current step. Then the side nodes will be determined according to the relative boundary conditions. Consequently, all the nodes were solved at the time t = ∆t, which is the ending of the current step and will be considered as the beginning of the next step. Analogously, the water hammer process can be determined by time advancement, if the initial boundary conditions are provided.

Figure 1. Meshes and node types. Water 2018, 10, x FOR PEER REVIEW 5 of 17

   t

tt=Δ3

tt=Δ2

Δt

tt=Δ

Δx t = 0 x 0 i-1 ii+1 n Inlet Outlet Pipeline Water 2018, 10, 928 5 of 16 Figure 1. Meshes and node types.

2.4.2.4. Boundary Boundary Conditions Conditions AsAs shown inin FigureFigure2, the2, the gravitational gravitational pipe pipe system system consists consists of a long of pipe,a long an pipe, upstream an upstream reservoir, reservoir,and a downstream and a downstream valve. Keeping valve. theKeeping upstream the upstream water level water constant, level theconstant, ﬂowis the steady flow atis thesteady initial at thestate. initial The state. water The hammer water will hammer occur will when occur the downstreamwhen the downstream valve close valve suddenly. close Insuddenly. order to In simulate order tothe simulate transient the process transient by the process proposed by the model, proposed the matching model, the boundary matching condition boundary model condition is established model is as establishedfollows. It includesas follows. the It upstream includes reservoirthe upstre andam thereservoir downstream and the control downstream valve. control valve.

Figure 2. Pipe flow system for water hammer analysis. Figure 2. Pipe ﬂow system for water hammer analysis. In this experiment, the detailed parameters of devices and equipment are listed as follows. The pipe connectingIn this experiment, the upstream the detailedand downstream parameters reservoi of devicesrs is a andsteel equipmentpipe 29 m in are length. listed Its as internal follows. diameterThe pipe connectingis 0.107 m, and the upstreamthe thickness and downstreamof the pipe wall reservoirs is 0.005 is m. a steel The pipeelasti 29city m modulus in length. of Its the internal pipe materialdiameter is is 210 0.107 GPa. m, andThe thecontrol thickness valve ofdownstream the pipe wall is isa 0.005butterfly m. Thevalve, elasticity and its modulus characteristics of the pipeare illustratedmaterial is in 210 Section GPa. 3.2. The Along control the valve pipeline, downstream there are is 26 a test butterﬂy nipple valve, joints and distributed its characteristics uniformly areat oneillustrated side of in the Section pipe, 3.2 available. Along thefor pipeline,connecting there pres aresure 26 test sensors. nipple The joints pressure distributed sensors uniformly are of athigh one precision,side of the with pipe, the available time interval for connecting of two captured pressure pressure sensors. Thevalues pressure being sensors0.001 s. areThe of water high precision,hammer experimentalwith the time data interval is obtained of two captured by recording pressure the values pressure being data 0.001 of s. the The sensors water hammerduring the experimental transient processes.data is obtained by recording the pressure data of the sensors during the transient processes.

2.4.1. Upstream Boundary Model Figure3 shows the logic boundary model of the pipe ends. For a large reservoir, the water level can be considered as a constant. Then the boundary model needs to include the constant water level. Thus, the pressure head can be determined by the boundary.

h(0, t + ∆t) = h0 (13)

With ∂h(0, t)/∂t = 0, combining Equation (6), the ﬂow velocity of the upstream side nodes can be solved as

−a2v(1, t) v(0, t + ∆t) = (14) [h(1, t) − h(0, t)]g − a2 Water 2018, 10, x FOR PEER REVIEW 6 of 17

2.4.1. Upstream Boundary Model Figure 3 shows the logic boundary model of the pipe ends. For a large reservoir, the water level can be considered as a constant. Then the boundary model needs to include the constant water level. Thus, the pressure head can be determined by the boundary.

()+Δ = ht0, t h0 (13)

With ∂∂=ht(0, ) t 0 , combining Equation (6), the flow velocity of the upstream side nodes can be solved as

−av2 ()1, t vt()0, +Δ t = −−2 (14) Water 2018, 10, 928 ht()1, h ( 0, tga ) 6 of 16

tt=Δ3

tt=Δ2

tt=Δ

Δt Δx t = 0 x Inlet 132 n-2 n-1 n Outlet

FigureFigure 3. Boundary 3. Boundary modelmodel ofof the the pipe pipe ends. ends.

2.4.2. Downstream2.4.2. Downstream Boundary Boundary Model Model The valve is a flow control device, and its rapidly closing will cause water hammer. The The valveboundary is a flow equation control of the device, valve can and be written its rapidly as closing will cause water hammer. The boundary equation of the valve can be written as ()+Δ = () +Δ − vnt,2, t Cd ghnt t Z (15) q v(n, t + ∆t) = Cd 2g[h(n, t + ∆t) − Z] (15) where Cd is the discharge coefficient, which depends on the valve performance curve and opening ratio at a specific time. The opening ratio is usually known according to the valve-closing process. where Cd is theThus, discharge the discharge coefficient, coefficient may which also be dependsdetermined at on any the time. valve performance curve and opening ratio τ at a specificAs for time. the head The coefficient, opening convert ratio Equation is usually (6) to knownthe different according form of to the valve-closing process. Thus, the discharge coefficient may also be determined at any time. hnt(),1,−− hn ( t )a2 hnt()()(),,,+Δ t = hnt − vnt + vnt() , Δ t As for the head coefficient, convert Equation (6) toΔ the different form of (16) xg h(n, t) − h(n − 1, t) a2 h(n, t + ∆t) = h(n, t) − v(n, t) + v(n, t) ∆t (16) 3. Experimental Measurement ∆x g

3.1. Steady Hydraulic Friction 3. Experimental Measurement The hydraulic head loss is an important factor for water hammer. Traditional water hammer 3.1. Steady Hydraulicsimulations Friction commonly approximate friction resistance based on the steady status. In order to

The hydraulic head loss is an important factor for water hammer. Traditional water hammer simulations commonly approximate friction resistance based on the steady status. In order to consider total head loss,Water including 2018, 10, x FORminor PEER REVIEW loss, various velocity experiments are conducted. Various7 of 17 steady ﬂow conditions ofconsider different total ﬂowhead loss, velocities including are minor accessed loss, various by controllingvelocity experiments the upstream are conducted. reservoir Various water level. Two pressuresteady sensorsflow conditions are set of at different the upstream flow velocities side and are downstreamaccessed by controlling side of thethe pipelineupstream respectively. reservoir water level. The pressure differenceTwo pressure of those sensors two are sensors set at the represents upstream theside totaland downstream head loss underside of thethe correspondingpipeline flow velocity. Figurerespectively.4 shows The the pressure total headdifference loss of changethose two with sensors the represents velocities, the total and head the loss fitting under friction the curve is provided withcorresponding an equivalent flow velocity. friction Figure factor 4 shows 0.269. the It showstotal head that loss thechange steady with the friction velocities, can and be obtainedthe by the fitting friction curve is provided with an equivalent friction factor 0.269. It shows that the steady second powerfriction law. can be obtained by the second power law.

1.0 Experiment measure Fitting in second power law 0.8

0.6 (m) f

h 0.4

0.2

0.0 0.0 0.1 0.2 0.3 0.4 0.5 v (m/s) Figure 4. Measurement of hydraulic head loss in steady flow. Figure 4. Measurement of hydraulic head loss in steady ﬂow. 3.2. Valve Performance Discharge properties of the valve can greatly affect the water hammer process. A relatively smooth valve-closing process can effectively reduce the water hammer strength. To numerically simulate the experimental transient process, it is always necessary preparatory work to measure the valve’s real discharge coefficient curve. To obtain the characteristics of the valve, the experimental data of various steady flow conditions are recorded. Adjusting the open ratio of the valve, according to the corresponding water head difference at two sides and the flow discharge through the valve, several mapping relationship dots are recorded based on Equation (15). After a least square method fitting, the performance curve of the valve in the experiment is shown in Figure 5.

0.4 Experiment Fitting curve 0.3

0.2

0.1

0.0 020406080100 τ%

Figure 5. The experimental discharge coefficient of the valve.

Water 2018, 10, 928 7 of 16

3.2. Valve Performance Discharge properties of the valve can greatly affect the water hammer process. A relatively smooth valve-closing process can effectively reduce the water hammer strength. To numerically simulate the experimental transient process, it is always necessary preparatory work to measure the valve’s real discharge coefficient curve. To obtain the characteristics of the valve, the experimental data of various steady flow conditions are recorded. Adjusting the open ratio of the valve, according to the corresponding water head difference at two sides and the flow discharge through the valve, several mapping relationship dots are recorded based on Equation (15). After a least square method fitting, the performance curve of the valve in the experiment is shown in Figure5.

Figure 5. The experimental discharge coefﬁcient of the valve.

3.3. WaterWater Hammer 2018, 10,Measurements x FOR PEER REVIEW 8 of 17 A related3.3. Water experiment Hammer Measurements was conducted in order to validate and improve the model. As shown in Figure2, the experimentA related experiment consists was of conducted a straight in order pipe, to valves,validate and and improve upstream the model. reservoir. As shown In in the system, the lengthFigure of the 2, pipe the experiment is 29 m and consists the internalof a straight diameter pipe, valves, is 0.107 and upstream m. A water reservoir. hammer In the is system, generated when the length of the pipe is 29 m and the internal diameter is 0.107 m. A water hammer is generated the valvewhen is closed the valve in 2.2 is closed s. Figure in 2.26 s. shows Figure 6 the shows water the water hammer hammer pressure pressure process. process. As As seen seen in inthe the ﬁgure, the wave-shapedfigure, the curve wave-shaped shows curve the stableshows the frequency stable frequency and periods. and periods.

open 100 Valve opening ratio closing

% 50 closed τ 0 012345678 5 closed complete t (s) 4 experimental pressure waves 3

(m) 2 0.74s h 0.74s 1 0.74s 2.23s 0 012345678 t (s) Figure 6. Experimental water hammer progress. Figure 6. Experimental water hammer progress. According to the periods and length of the travel path, the experimental wave speed can be Accordingdetermined to theas follows periods and length of the travel path, the experimental wave speed can be determined as follows = anlte 4 p (17)

The experimental wave speed is only aaboute = 4 151.0nl/t m/s.p However, without taking into account (17) the air, the intrinsic wave speed in the pipe should be about 1340 m/s according to the water hammer wave speed formula. Obviously, the experimental wave speed is far lower than for a pipe without air, because of air entrainment. For pipe flow with air content [1], the relationship between air content and the shock wave speed is

K ρ a = (18) 11++KD Ee mRT() K p − p

where a is the actual wave velocity in a pipe system, K is the volume modulus, ρ is the density, E is the elasticity modulus, and m is the amount of substance of the gas in unit volume. Provided with the actual experimental wave speed, the air content can be evaluated as

2 pK()ρ a−− p KD Ee p m = (19) RT() K p −1

For the case shown in Figure 6, the amount of air per unit volume is about 0.205, and air volume content is about 0.459% according to this measurement. Thus, all the necessary coefficients are established, which provide the basic parameters required to numerically simulate the water hammer process.

Water 2018, 10, 928 8 of 16

The experimental wave speed is only about 151.0 m/s. However, without taking into account the air, the intrinsic wave speed in the pipe should be about 1340 m/s according to the water hammer wave speed formula. Obviously, the experimental wave speed is far lower than for a pipe without air, because of air entrainment. For pipe ﬂow with air content [1], the relationship between air content and the shock wave speed is s K/ρ a = (18) 1 + KD/Ee + mRT(K/p − 1)/p where a is the actual wave velocity in a pipe system, K is the volume modulus, ρ is the density, E is the elasticity modulus, and m is the amount of substance of the gas in unit volume. Provided with the actual experimental wave speed, the air content can be evaluated as

pK/ρa2 − pKD/Ee − p m = (19) RT(K/p − 1)

For the case shown in Figure6, the amount of air per unit volume is about 0.205, and air volume content is about 0.459% according to this measurement. Thus, all the necessary coefﬁcients are established, which provide the basic parameters required to numerically simulate the water hammer process.

4. SimulationWater 2018 and, 10, Analysisx FOR PEER REVIEW 9 of 17

4.1. Simulation4. Simulation and Comparison and Analysis

Above4.1. Simulation we presented and Comparison the LWM model for general internal nodes, and a matching boundary conditions modelAbove waswe presented also established. the LWM model In order for general to debug internal the nodes, proposed and a matching model, theboundary pipe system in Figure2 isconditions simulated model and was compared also established. with In MOC. order Theto debug performance the proposed of model, the valve the pipe is shownsystem in in Figure5. Figure7 showsFigure the2 is simulated numerical and solution compared of with the MOC. water The hammer performance process of the produced valve is shown in the in conditionFigure 5. that the valve is closedFigure 7 in shows 2.2 s.the As numerical shown solution in the of figure, the wate ther hammer proposed process model produced is in in agreement the condition with that the MOC; the valve is closed in 2.2 s. As shown in the figure, the proposed model is in agreement with the indeed, theyMOC; can indeed, almost they can produce almost produce the same the numericalsame numerical simulation. simulation. However, However, a small a small difference difference can be foundcan in thebe found phase in the of thephase wave. of the wave. Obviously, Obviously, the the phase phase difference difference increases increases with with time, time,because because the proposedthe model proposed considers model theconsiders influence the influence of the pipe of the flow pipe velocity flow velocity on the on wave the wave propagation. propagation. With a lower With a lower speed, the flow speed can be considered by the convective term. As shown in Figure 7, speed, the flow speed can be considered by the convective term. As shown in Figure7, both LWM and both LWM and MOC can give out reasonable maximum pressure, which is the key to water MOC canhammer give out prediction. reasonable maximum pressure, which is the key to water hammer prediction.

Open 100 Valve opening ratio Closing

% 50 Closed τ 0 012345 6 t (s) 5 Approximate peak Different peak

(m) 2 h 1

0 experimental MOC present method -1 012345

t (s) Figure 7. The transient-pressure process. Figure 7. The transient-pressure process. 4.2. Improvement to Approximate the Attenuation As seen in Figure 7, the LWM can simulate the actual maximum water hammer pressure. It may meet the common requirement for predicting and preventing water hammer overpressure. However, the numerical results heavily distorted the properties of the attenuation, compared to the experimental result. After the complete closing of the valve, the peak pressure of the water hammer wave decreases rapidly in the experiment, but the water hammer wave decreases slowly in the numerical simulation. Obviously, the attenuation is heavily underestimated in the numerical simulation. In order to modify the deviation, an additional friction function is introduced according to the flow patterns. In Figure 7, the numerical model evaluates the unsteady friction resistance according to the experimental measurement shown in Figure 4, which denotes the steady friction in pipe flow. The model can simulate a water hammer well before the valve has closed completely, and during this period the pipe flow maintained positive velocity. However, after the valve is completely closed, the fluid will have a great impact on the pipeline, and then the steady friction cannot actually approximate the practical energy dissipation. The experiment can actually measure the attenuation

Water 2018, 10, 928 9 of 16

4.2. Improvement to Approximate the Attenuation As seen in Figure7, the LWM can simulate the actual maximum water hammer pressure. It may meet the common requirement for predicting and preventing water hammer overpressure. However, the numerical results heavily distorted the properties of the attenuation, compared to the experimental result. After the complete closing of the valve, the peak pressure of the water hammer wave decreases rapidly in the experiment, but the water hammer wave decreases slowly in the numerical simulation. Obviously, the attenuation is heavily underestimated in the numerical simulation. In order to modify the deviation, an additional friction function is introduced according to the flow patterns. In Figure7, the numerical model evaluates the unsteady friction resistance according to the experimental measurement shown in Figure4, which denotes the steady friction in pipe flow. The model can simulate a water hammer well before the valve has closed completely, and during this period the pipe flow maintained positive velocity. However, after the valve is completely closed, the fluid will have a great impact on the pipeline, and then the steady friction cannot actually approximate the practical energy dissipation. The experiment can actually measure the attenuation process of the water hammer wave, but it cannot measure the impact flow rate and the actual energy dissipation. In order to evaluate the attenuation, an additional friction function is introduced to the numerical model. The friction term is proposed as follows

L v2 h = f + f 0 (20) f D 2g where f 0 is the proposed additional function after the valve’s complete closing, which can be described ( 0 0|t ≤ tc as f = . where tc is the valve closing time. ϕ|t > tc Considering a, m, g and L as the main physical variables, combining the Darcy-Weisbach friction factor, the additional function can be described as

ϕ = ϕ(a, m, g, L, λ) (21) where λ is the Darcy-Weisbach friction factor. According to Rayleigh’s method of dimensional analysis, the exponential function is proposed as ma2 ϕ = C λ (22) gL where C is a dimensionless coefficient that needs to be calibrated by experiment. After a complete valve closing, the flow is only impacted at low velocity. According to the flow patterns, the λ can 64ν √8gn2 separately be described as λ = in laminar model (LM) and λ = 3 in turbulent model (TM) [35]. vD Rp Figure8 shows that the flow velocity may cross laminar and turbulent zones during hydraulic transient. The compound model (CM) can be described as

 2  √8gn >  3 v vc = Rp λ (23)  64ν ≤  vD v vc where vc is the critical ﬂow velocity between laminar and turbulent ﬂow. Water 2018, 10, x FOR PEER REVIEW 10 of 17

process of the water hammer wave, but it cannot measure the impact flow rate and the actual energy dissipation. In order to evaluate the attenuation, an additional friction function is introduced to the numerical model. The friction term is proposed as follows

2 =+' Lv hfff () (20) D 2g

where f ' is the proposed additional function after the valve’s complete closing, which can be ≤ 0 ttc described as f ' = . where t is the valve closing time. ϕ > c  ttc Considering a, m, g and L as the main physical variables, combining the Darcy-Weisbach friction factor, the additional function can be described as

ϕϕ= ()amgL,,,,λ (21)

where λ is the Darcy-Weisbach friction factor. According to Rayleigh's method of dimensional analysis, the exponential function is proposed as

2 ma ϕλ= C (22) gL

where C is a dimensionless coefficient that needs to be calibrated by experiment. After a complete valve closing, the flow is only impacted at low velocity. According to the flow patterns, the λ can 64ν 8gn2 separately be described as λ = in laminar model (LM) and λ = in turbulent model vD 3 R p (TM) [35]. Figure 8 shows that the flow velocity may cross laminar and turbulent zones during hydraulic transient. The compound model (CM) can be described as

8gn2  > vvc  3 R λ =  p (23)  ν 64 ≤  vvc  vD Water 2018, 10, 928 10 of 16 where v is the critical flow velocity between laminar and turbulent flow. c

1.5 6

valve closing valve closed

1.0 4

ht~ = 0.5 tc 2.2s 2 (m) (m/s)

vt~ h v 0.0 0 > ≤ vvc vvc turbulent area laminar area -0.5 -2 0123456 t (s) Figure 8. The flow velocity process during hydraulic transient. Figure 8. The ﬂow velocity process during hydraulic transient.

Water 2018, 10, x FOR PEER REVIEW 11 of 17 In order to calibrate the dimensionless coefﬁcient C, the positive and negative peaks are identiﬁed In order to calibrate the dimensionless coefficient C, the positive and negative peaks are and analyzedidentified in Figure and analyzed9. in Figure 9.

Open 100 Valve opening ratio Closing

% 50 Closed τ 0 012345678 6 t (s)

5 4.77 Experimental Peaks 4 3.50 Attenuation trend 3 2.75 2.41

(m) 2.30

h 2 2.16 2.23 1.95 1.53 1 0.83 0 012345678

t (s) Figure 9. The positive and negative peaks in experimental measurement. Figure 9. The positive and negative peaks in experimental measurement. Based on the identified peaks, the method of least squares is used to obtain the optimal C. The Basedsquared on the residuals identiﬁed between peaks, experimental the method and simulated of least peaks squares are defined is used as to obtain the optimal C. The

2 2 squared residuals between experimental2 ()=− and () simulated () +− peaks () ()are deﬁned as ri hipe hi p hi ve hi v (24) 2 2 2 Then, the sum of squaredr (i) = residualshpe(i is) − hp(i) + [hve(i) − hv(i)] (24)

n 2 Then, the sum of squared residuals is Sri=  () (25) i=1

The optimal C can be determined when the nsum of squared residuals is the least. In order to 2 optimize the dimensionless coefficient C, Sthe= water∑ r hammer(i) processes are simulated by various (25) dimensionless coefficients and three kinds of frictioni=1 models. The relationship between the sum of squared residuals and the dimensionless coefficient C in each model are shown in Figure 10. Table 1 The optimalshows the Cfirstcan 10th be peaks determined of the optimized when simulations the sum in each of squaredmodel. The residualsresult shows isthat the the least.CM In order to optimizemodel the can dimensionless give the least sum coefficient of squaredC residuals,, the water in which hammer the matched processes coefficient are is simulated about 8.48. by various dimensionlessThen the coefficients additional function and three is written kinds as of friction models. The relationship between the sum of squared residuals and the dimensionless coefficient C in2 each model are shown in Figure 10. 8gn 0 tt≤  > c vvc Table1 shows the first 10th peaks of the optimized simulations3 R in each model. The result shows that f ' ==2 , λ p ma λ > (26) 8.48 ttc ν gL 64 ≤   vvc  vD

Figure 11 compares the water hammer simulations. Obviously, the improved model can more actually simulate the attenuation.

Water 2018, 10, 928 11 of 16 the CM model can give the least sum of squared residuals, in which the matched coefﬁcient is about 8.48. Then the additional function is written as    √8gn2 0|t ≤ tc  3 v > vc 0   Rp f = , λ = (26) ma2 > Water 2018, 10, x FOR PEER REVIEW 8.48 gL λ t tc  64ν ≤ 12 of 17  vD v vc

Figure 11 compares the water hammer simulations. Obviously, the improved model can more Water 2018, 10, x FOR PEER REVIEW1.0 TM (Turbulent model) 12 of 17 actually simulate the attenuation. LM (Laminar model) CM (Compound model) 0.8 1.0 TM (Turbulent model) LM (Laminar model) 0.6 CM (Compound model) 0.8

S 0.4 0.6

(11.68, 0.42) S 0.2 0.4

(8.48, 0.12) (10.19, 0.14) 0.0 (11.68, 0.42) 0.25.0 7.5 10.0 12.5 15.0

(8.48, 0.12) C (10.19, 0.14) 0.0 5.0 7.5 10.0 12.5 15.0 Figure 10. The analysis of the Csum of squared residuals.

Table 1. The attenuationFigure 10. ofThe peak analysis pressures ofthe from sum experiment of squared andresiduals. improved simulations. Figure 10. The analysis of the sum of squared residuals. Improved Table 1. The attenuationExperimental of peak pressuresOriginal from experiment and improved simulations. periodsTable 1. The attenuation of peak pressures from experimentLM and improvedTM simulations. CM h h h h h h h h h h pe ve p v p v ImprovedImprovedp v p v ExperimentalExperimental OriginalOriginal periods1st Periods 4.77 0.83 4.96 −0.38 4.93LMLM 0.66 TM4.94TM 1.01 CM 4.93CM 0.66 2nd 3.50 1.53 4.84 −0.27 3.22 1.68 3.08 1.65 3.23 1.68 hpe hpe hve hhpp hhvv hphp hv hv hp hphv hvp hhvp hv 3rd 2.75 1.95 4.74 −0.17 2.61 2.04 2.74 1.86 2.62 2.04 1st 1st4.77 4.770.83 0.83 4.964.96 −−0.38 4.93 4.93 0.660.66 4.94 4.94 1.01 1.01 4.93 0.664.93 0.66 4th 2.41 2.16 4.65 −0.09 2.39 2.18 2.60 1.96 2.39 2.18 2nd 2nd3.50 3.501.53 1.53 4.844.84 −−0.27 3.22 3.22 1.681.68 3.08 3.08 1.65 1.65 3.23 1.683.23 1.68 5th 3rd 2.30 2.75 2.23 1.95 4.74 4.56 − 0.000.17 2.61 2.31 2.042.23 2.74 2.52 1.86 2.02 2.62 2.042.31 2.23 3rd 2.75 1.95 4.74 −0.17 2.61 2.04 2.74 1.86 2.62 2.04 4th 2.41 2.16 4.65 −0.09 2.39 2.18 2.60 1.96 2.39 2.18 4th 5th2.41 2.302.16 2.23 4.564.65 − 0.000.09 2.31 2.39 2.232.18 2.52 2.60 2.02 1.96 2.31 2.232.39 2.18 5th 2.30 2.23 4.56 0.00 2.31 2.23 2.52 2.02 2.31 2.23 Open 100 Valve opening ratio Closing

% 50 Closed τ 0 Open02468 100 Valve opening ratio 6 Closing t (s) % 50 Closed τ 05 02468 64 t (s) 53

(m) 42 h 31

(m) 20

h experimental original LWM Improved LWM -11 02468 0 t (s) experimental original LWM Improved LWM -1 Figure02468 11. Simulation by improved two-phase model. Figure 11. Simulation by improvedt (s) two-phase model.

Figure 11. Simulation by improved two-phase model.

Water 2018, 10, x FOR PEER REVIEW 13 of 17

WaterBy2018 introducing, 10, 928 the additional friction function, the improved model can simulate water12 of 16 hammer more actually, not only on the maximum pressure, but also the attenuation of the water hammer waves. By introducing the additional friction function, the improved model can simulate water hammer For validation, the improved model is applied to various air-water mixtures conditions as more actually, not only on the maximum pressure, but also the attenuation of the water hammer waves. shown in Table 2. Changing the valve closing time and the air mass entrainment, various water For validation, the improved model is applied to various air-water mixtures conditions as hammer processes are produced. These water hammers are experimentally measured and shown in Table2. Changing the valve closing time and the air mass entrainment, various water numerically simulated. Although the actual air content is hard to measure, it can be estimated by hammer processes are produced. These water hammers are experimentally measured and numerically the actual wave speed according to previous research [1]. In the numerical model, the air content is takensimulated. into account Although by the adjusting actual air m content in Equation is hard (26), to measure, the amount it can of be substance estimated of by the the gas actual in waveunit volume,speed according which affects to previous the energy research loss [1 ].degree In the greatl numericaly. This model, is inferred the air contentfrom the is takenmeasured into accountactual m waveby adjusting speed in experimentin Equation using (26), theEquation amount (19). of substance As show ofn in the Figure gas in 12, unit the volume, numerical which simulation affects the resultsenergy using loss degree the improved greatly. This LWM is inferred are in fromagreemen the measuredt with the actual experimental wave speed results in experiment both onusing the maximumEquation (19).pressure As shown and the in attenuation Figure 12, the trends. numerical simulation results using the improved LWM are in agreement with the experimental results both on the maximum pressure and the attenuation trends. Table 2. Various valve closing speed simulations for validation. Table 2. Various valve closing speed simulations for validation. Case Wave Speed (m∙s−1) m Air Content (%) Closing Time (s) #1 Case 183Wave Speed (m·s−1 0.139) m Air Content 0.308 (%) Closing Time (s) 2.8 #2 #1 175 183 0.1520.139 0.308 0.338 2.8 2.6 #3 #2 132 175 0.2690.152 0.338 0.598 2.6 3.0 #4 #3 110 132 0.3880.269 0.598 0.864 3.0 3.2 #4 110 0.388 0.864 3.2

Open Open 100 Valve opening ratio 100 Valve opening ratio Closing Closing % 50 % 50

Closed Closed τ 0 τ 0 012345678 012345678 6 t (s) 6 t (s)

5 5 Experimental (#1) Experimental (#2) Improved LWM Improved LWM 4 4

3 3 (m) (m) h 2 h 2

1 1

0 0 012345678 012345678 t (s) t (s)

Open Open 100 Valve opening ratio 100 Valve opening ratio Closing Closing % % 50 50

Closed Closed τ 0 τ 0 012345678 012345678 6 t (s) 6 t (s)

5 5 Experimental (#3) Experimental (#4) Improved LWM Improved LWM 4 4

3 3 (m) (m) h h 2 2

1 1

0 0 012345678 012345678 t (s) t (s)

FigureFigure 12. 12. ApplicationApplication of of the the improved improved model model to to water water hammer hammer simulations. simulations.

5. Discussion

Water 2018, 10, 928 13 of 16

5. Discussion The LWM is widely used in computational fluid dynamics for fluid field simulation. However, it is uncommonly used in numerical water hammer-simulation for pipe systems. As an alternative approach, LWM can be applied to the water hammer hyperbolic partial differential equations system. In particular, the convective acceleration terms are completely considered in the model, although they only affect the water hammer slightly. Considering the applicability in numerical computation, the LWM can yield the water hammer maximum pressure as the MOC, which shows that the LWM can suit the water hammer equations. However, considering the water hammer experiment in the pipe, LWM may heavily distort the attenuation of the water hammer wave. Based on the experiment, an additional friction function is proposed to simulate the actual wave attenuation. Traditional numerical simulations greatly underestimate air resistance when the valve is closed completely. By introducing the additional resistance function, the proposed model can simulate the water hammer well, both on the maximum pressure and the attenuation process. In particular, it is difficult to reveal the mechanism of the resistance in the shock flow after completely closing valve. Many studies have reported the difference of the water hammer attenuation process in theory and experiment. Traditional models underestimate the resistance, which makes the fluctuation remain longer in the numerical result. Considering the restriction of the experimental scale of the pipe used in this research on the water hammer in gravitational pipe flow system with air entrainment, it may require a different friction coefficient in flow with air content after complete valve closing.

6. Conclusions Based on the Lax-Wendroff difference Method, a numerical water hammer model is established for gravitational flow pipe systems. The proposed model considers the influence of convective acceleration terms, and it also supports flexibly setting the time steps and space steps without the limitation of the characteristics in MOC. With a second-order accuracy, the model is validated by MOC and used in an experiment to predict the maximum transient pressures. The experiment shows that the water hammer processes in water supply systems with air entrainment differ much from those in one phase of pure water, especially on the attenuation processes. The improved model with a modified friction function using a compound model can simulate the water hammer processes well, both on the maximum pressures and the attenuation process. This research reveals that we used to underestimate the energy loss process of a water hammer in pipe systems, especially in conditions with air content. Proposed improved LWM can simulate water hammers much better in such conditions. As plenty of practical water supply systems are running with air content, this research can be helpful on predicting water hammers in actual engineering. In further research, the physical interaction of air entrainment in water hammer processes and the air-delivery process in a water hammer will be researched. To ﬁgure out the air inlet conditions and its physical interactions, a further experiment will be conducted using a tube with a visible interior to observe the air-delivery trail and its shape-changing process in water hammers.

Author Contributions: Methodology, B.Z.; Validation, B.Z.; Investigation, B.Z.; Experiment, B.Z., M.S.; Data-curation, M.S.; Original writing, B.Z.; Resources, W.W.; Visualization, W.W.; Supervision, W.W.; Project administration, W.W.; Funding acquisition W.W. Funding: This research work was funded by the National Natural Science Foundation of China (Grant Nos. 51779216, 51279175) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ16E090001). Conﬂicts of Interest: The authors declare no conﬂicts of interest.

Nomenclature

g = acceleration of gravity (m/s2) h = pressure head (m) x = distance along pipe from inlet (m) t = time, as subscript to denote time (s) v = flow velocity (m/s) Water 2018, 10, 928 14 of 16 f = Darcy friction factor D = main pipe diameter (m) α = the angle between pipe and the horizontal plane. a = speed of pressure wave (m/s) i = serial number of nodes (s) ∆t = time step (s) n = number of sections ∆x = length of segment (m) h0 = head of upstream reservoir (m) Cd = discharge coefficient Z = the height of reference plane h f = head drop due to friction along the pipe (m) τ = valve opening ratio K = volume modulus ρ = density (kg/m3) E = elasticity modulus of steel pipe e = thickness of steel pipe (m) m = air content occupied in mixed fluid R = thermodynamics constant (J/(mol·K)) T = temperature (K) P = absolute pressure (pa) f 0 = additional friction factor in two-phase flow L = length of the pipe (m) ϕ = additional friction factor after valve closed completely tc = time of valve closing operation C = coefficient in additional friction function ν = kinematic viscosity of water (m2/s) Rp = hydraulic radius (m) vc = the critical flow velocity between laminar and turbulent flow (m/s) r = residuals between experimental and simulated peaks (m) S = the sum of squared residuals hpe = experimental positive peak (m) hve = experimental negative peak (m) hp = experimental positive peak (m) hv = experimental negative peak (m)

Acronyms

MOC = Method of Characteristics LWM = Lax-Wendroff Method

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