Fluid-structure Interaction Adopting Full Water Hammer Model

A full water hammer model is developed and coded in the Mathcad software environment. The developed model constitutes a general platform for studying the water hammer transient incorporating the most legitimate parameters that affect the phenomenon. In this study, the ﬂ uid e structure interactions during water hammer transients were investigated through the affecting parameters such as pipeline material hysteresis behavior what is so-called viscoelasticity (VE) behavior, and unsteady friction (UF) with different coupling mechanisms such as friction coupling, Poisson coupling, and junction coupling which may be due to a free-to-move valve for system with a reservoir-pipe-valve. Based on the numerical results corresponding to the variation of parameters such as ﬂ uid ﬂ ow rate, pipe length, pipe size, pipe material, pipe Poisson ratio, and the effect of different coupling mechanisms for the aforementioned system. The present results highlight the following: for shorter pipes, the frequency of the ﬂ uid wave is increased, and the wave amplitude attenuates faster than for longer pipes. The ﬂ uid wave characteristics depend on the pipe material. The maximum head with polyvinyl chloride (PVC) material is higher than that with High Density Polyethylene (HDPE) material, which has more hysteresis dissipation, as HDPE material has more viscoelastic behavior than PVC material, and so HDPE material has more damping effects than PVC material. Also, HDPE has a lower wave celerity than the PVC material, which leads to low frequency. For a higher Poisson ratio, the coupling between the ﬂ uid transient and pipe transient is strongly entrained; consequently, the wave amplitudes are ampli ﬁ ed.


Introduction
T he water hammer problem impacts a variety of industrial sectors, including water supply networks, industrial conduits, cooling circuits in thermal and nuclear power plants, etc.Any rapid changes in the flow velocity can trigger the flow transients (Kan et al., 2021).A severe water hammer may cause service pipe failures, piping components rapture, joint failure, and other hydraulic system damage.That is why pressure and stress waves must be continuously predicted and controlled, as they cause loads that are higher than those observed during steady-state.Water hammer behavior has been recognized since the earliest research, which began with the Joukowsky equation about a century ago and later developed into more complex fluid transient models.The remarkable impact of the mechanical behavior of the pipe wall on the transmission of pressure pulses is one of the first observations in the area.In the literature, a variety of approaches have been used to examine the effects of elastically deforming pipes on pressure transients.
Pipe systems are never completely rigid in practice.In classical fluid transient analysis, the pressure wave propagation speed incorporates the pipe elasticity in a very approximated form due to pipe supports.The classical water hammer model uses two hyperbolic partial differential equations (PDE) of first order (Ghidaoui et al., 2005;Koelle and Almeida, 1991;Wylie and Streeter, 1993).In rigidly anchored pipe systems, axial pipe motion and pipe inertia are not considered.But in a less restrained system, axial pipe motion, and pipe inertia become more important (Tijsseling, 1996).Therefore, it is important to take into account both the fluid pressure wave and the pipe wall stress wave and their interactions.The water hammer equations should be appropriately changed if only longitudinal pipe motion is considered, and so the model incorporates two more PDEs for the structure.To solve those PDEs in the time domain, a method of characteristics has been widely used (Henclik, 2015;Tijsseling, 2003;Wiggert et al., 1987).When designing piping systems, the crucial issue is how significant fluidestructure interactions are.Three time-scales can be compared with one another, according to the research by (Lavooij and Tusseling, 1991) that was done to address this topic.When the effective valve closure timescale is shorter than the water hammer wave period, the investigation of the water hammer in the hydraulic system is necessary.When the structure timescale is longer than the effective valve closure time and lower than the fluid timescale, the fluidestructure interaction becomes significant.
The fluidestructure interaction four equation model was developed by (Skalak, 1955) as an extension of the Joukowsky methodology.According to his model, arbitrary-shape waves move without spreading at the phase of either the pipe's or the fluid's velocity (Stuckenbruck et al., 1985).Investigated the impact of pipe motion on acoustic wave propagation and discovered that; for diameter to thickness ratios greater than 100, the actual wave speed in the liquid is 4e7 % lower than that indicated by the conventional water hammer analysis (Budny et al., 1991).Investigated the impact of structural damping on internal pressure during a transient pipe flow and discovered that; by removing the high-frequency components, structural damping lowers the pressure maxima during a transient event.The model (Lavooij and Tusseling, 1991) contains all three fundamental coupling processes (Poisson, junction, and friction coupling).
There are three fluidepipe interaction mechanisms: Poisson coupling, junction coupling, and friction coupling (Keramat and Tijsseling, 2012).The axial-longitudinal stresses in the pipe are related to the pressure in fluid through Poisson coupling due to the contraction or expansion of the pipe wall radially (Tijsseling and Lavooij, 1990).The stress waves induce disturbances in the fluid at a wave speed faster than and ahead of the conventional water hammer waves.While the Poisson coupling acts along the entire pipe, the junction coupling acts at specific points in the pipe system such as bends, tees, and unrestrained valves (Tijsseling, 1996;Henclik, 2015;Ahmadi and Keramat, 2010;Andrade et al., 2022;Wiggert and Tijsseling, 2001).Friction coupling is the mutual friction between fluid and pipe.Though unsteady flow is considered, the quasi-steady friction model is often used due to its simplicity.Although the unsteady friction models employed in traditional water hammer analysis (Adamkowski and Lewandowski, 2006;Bergant and Ross Simpson, 2001;Urbanowicz and Zarzycki, 2012) would undoubtedly perform better, their implementation required significant modifications for the FSI scenario (Tijsseling, 1996).Concluded that the fluidestructure interactions main effects depend on the particular system under investigation, and analysis predictions that include fluidestructure interaction compared with classical water hammer prediction may lead to lower or higher stresses and extreme pressures, nature frequency changes of the systems, and more dispersion and damping in the stress and pressure waves (Guo et al., 2020).Studied the influence of junction coupling such as surge chamber, elbows, and valves on fluid structure interaction.This study showed that the most important coupling mechanism are Poisson coupling and junction coupling.There is a stronger pressure pulsation produced at the valve, resulting in a more complex fluidestructure interaction in the system.Moreover, frictional coupling leads to the lower pressure wave amplitude.The underestimation of the dispersion, dissipation, and alteration of fluid transient head waves in 1D models is related to unsteady friction.This dispersion happens in fast fluid transient events or highfrequency oscillation of the unsteady-friction losses in materials that have linear elastic behavior such as metal, concrete, and asbestos cement (Bergant et al., 2008;Ferr a et al., 2016).Showed experimentally the importance of unsteady friction, especially in systems with fast transient.In laminar flow (Trikha, 1975;Zielke, 1968), well described the unsteady friction.In transient flow analysis the unsteady friction is incorporated with different approaches (Ferras et al., 2017).These approaches can be classified into four different categories: (i) instantaneous mean flow velocity (Hino et al., 1976); (ii) instantaneous mean flow velocity and local acceleration (Carstens and Roller, 1959;Daily et al., 1956;Shuy, 1996); (iii) instantaneous mean flow velocity, convective acceleration and local acceleration (Brunone et al., 1991;Vitkovsky et al., 2000); (iv) instantaneous mean flow velocity and past velocity changes weights (Trikha, 1975;Zielke, 1968;Suzuki et al., 1991;Vardy andBrown, 2003, 2004;Vardy and Hwang, 1993).(Abdeldayem et al., 2021) investigated the performance of different unsteady friction models in terms of accuracy, reliability, and efficiency to determine the most-suitable engineering practice.As a result of this comparison, Vitkovsky unsteady friction model was found to be the best fit.
It was noticed that this model fell short in simulating pipes with viscoelastic behavior where, viscoelastic effects dominate.
Polyethylene pipes have a different rheological behavior in comparison to metal and concrete pipes (Brinson and Brinson, 2008).When subjected to a certain instantaneous stress, s 0 , polymers do not respond according to Hooke's law, plastics have an immediate-elastic response and a retarded-viscous response.In this way, strain can be decomposed into an instantaneous-elastic strain, e e and a retarded strain, e r (Brinson and Brinson, 2008).
The VE behavior of polymers such as polyvinyl chloride (PVC), polyethylene (PE), acrylonitrile butadiene styrene (ABS) affects conventional water hammer results (Keramat et al., 2010).Because of the advantages of these polymer pipes, such as fast and easy installation, high chemical resistance, and low price, these polymers are being used in pipe systems in many areas of the industry.As a result, it is now more important than before to take into consideration their effects in the water hammer model.VE is due to the hysteresis loss of the material, which leads to delay in the material 's mechanical response.In a viscoelastic material, for example, creep behavior occurs when sudden application of constant stress or sudden relaxation from constant strain (Covas et al., 2004b).In a piping system when the pipe is loaded with high and low pressures due to water hammer, the viscoelastic behavior becomes significant.There is a delay between the radial expansion and contraction of the pipe wall relative to the transient pressure wave.This behavior can dampen the water hammer pressure wave.There is no delay between the water hammer pressure wave and the pipe radial expansion or contraction in the elastic pipes (Keramat et al., 2010).In order to account for the dynamic effects resulting from the pipe wall's VE (Covas et al., 2005) included a factor to the continuity equation that described the creep behavior in the pipe wall's radial deformation (Kandil et al., 2020).Studied the pressure fluctuations by changing some effective variables such as pipe's elastic modulus and Poisson ratio.The results demonstrate how, under the same operating conditions, materials with a lower elastic modulus are less likely to experience a water hammer than those with a higher elastic modulus.The water hammer phenomenon in viscoelastic pipes was investigated by (Aliabadi et al., 2020), who took fluidestructure interaction into consideration in the frequency domain.The generalized Kelvin-Voigt model is used to simulate the viscoelastic behavior during water hammer phenomena.In the case of VE, natural frequencies are reduced in contrast to the elastic pipe.Although the absolute difference between VE and elastic at natural frequencies for short pipes is generally minor, it is more noticeable for long pipes, especially at higher frequencies (Zhang et al., 2022).Conduit experiment to study the direct water hammer generated by rapid closure of the downstream valve in pipes that had viscoelastic behavior with six flow velocities in nearly 70 tests.Results showed that the maximum amplitude of the water hammer wave generated in the viscoelastic pipe at all flow velocities was twenty percent greater than the traditional value of the Joukowsky formula.
Designers can more easily identify the water hammer phenomenon with the help of sensitivity analysis.Changing a few chosen parameters affects the outcomes of both positive and negative pressure.These parameters divide into two categories: (a) the system's initial conditions (such as the pipe flow rate) and (b) the model's parameters (such as the pipe's diameter, length, wall thickness, material, wave celerity, and roughness) (Emadi and Solemani, 2011).Studied the effects of changing some parameters, such as water temperature, flowing through the pipeline pipe material, pipe internal diameter, and pipe internal thickness (Mansuri, 2014).Conducts a sensitivity analysis by changing some variables, such as pipe length, pipe diameter, and wave velocity in the pipe (Urbanowicz and Firkowski, 2018).Investigated the effect of changing some parameters related to the pipe and the flowing fluid, such as water density, water bulk modulus, pipe thickness, pipe Young's modulus, and the kinematic viscosity of the water (Nile et al., 2020).Some parameters are studied to investigate their effects on water hammer pressure fluctuations and the maximum pressure head in both cases: a single pipe and three pipes in series.Investigate the effect of changing some parameters on the water hammer results in both a single pipe system and three pipes in series system.The previous sensitivity analyses did not adapt to the full water hammer model.
Throughout the present paper, the Method of Characteristics (MOC) is used to solve the governing equations for the water hammer phenomenon in viscoelastic pipes while considering fluidestructure interaction and unsteady friction.A Mathcad code is developed to solve the equations resulted from the MOC for a single pipeline.According to the literature, the model used for unsteady friction in the present work is different from the one used by (Keramat and Tijsseling, 2012).In addition, using FSI, VE, and UF models, they do not carry out sensitivity analysis with those models.Also (Keramat and Tijsseling, 2012), did not study the effect of coupling on water hammer results with different parameters such as FSI, VE, and UF.The boundary condition equations used in the developed code are stated in the mathematical model.Then, the code is validated with experimental data extracted from previous work.By the end of the paper, a sensitivity analysis is conducted to examine the impact of several important parameters, related to the water hammer phenomenon in light of FSI, UF, and VE, on a system with a single pipe without junction coupling.These parameters include fluid flow rate, pipe length, pipe size, pipe material, and pipe Poisson ratio.After that, employing the free-tomove valve as junction coupling, the impact of various coupling methods is investigated for both the classical model and the FSI model.

Mathematical model
The full water hammer model incorporates the effects of FSI, UF, and pipe material constitutive equation of elasticity such as ideal elastic or VE.The present model consists of four first order coupled partial differential equations, two of which are for fluid flow dynamics and the other two equations are for pipe motion.The following four equations govern the water hammer phenomenon, considering the behavior of the pipe material under applied stress, the unsteady friction, and the fluidstructure-interaction: The momentum equation for fluid (Keramat et al., 2012) The continuity equation of fluid (Keramat et al., 2012) For pipe with material that has elastic behavior described by equations ( 3) and ( 4) By neglecting s r and use s f ¼ R e r f gH (Tijsseling, 1996) and the relation between the axial stress and axial strain equation ( 2) for elastic pipe can be rewritten as: For pipe with material that has viscoelastic behavior described by equations ( 7) and ( 8) (Keramat et al., 2012) in which '*' is the convolution operator and '*d' is the Stieltjes convolution operator.The creep compliance function J(t) corresponding to the generalized KelvineVoigt model is given by where J 0 ¼ Jð0Þd1 =E 0 represents the immediate response of the material; N KV represents the number of Kelvin Voigt elements; J k defined by J k d1=E k is the creep compliance of the spring of the kth KelvineVoigt element; E k is the modulus of elasticity of the kth spring and t k is the retardation time of the kth dashpot.The time scale is t k dm k =E k in which m k is the viscosity of the kth dashpot.Also neglecting s r and use s f ¼ R e r f gH, equation (2) becomes as adopted (Keramat et al., 2012) where the term I H represents the retarded hoop strain defined by (Keramat et al., 2012) The momentum equation of pipe is given by (Ferr a et al., 2016) The fourth equation is a relationship between the stress and displacement.For the pipe with elastic behavior, it is obtained by taking the time derivative of equation ( 4) (Ferr a et al., 2016) For the pipe with viscoelastic behavior the time derivative of equation ( 8) (Keramat et al., 2012) And the term I sz represents the retarded axial strain defined by (Keramat et al., 2012) The term t 0 in equations ( 1) and ( 11) is shear stress which has two parts, first part is steady shear which is given by Darcy Weisbach equation and the second part is the dynamic shear t wu as given by equation ( 17) (Vardy and Brown, 2004) The values of n i and m i can be found from (Vardy and Brown, 2004).
The previous four partial differential equations (Eqs.(1), (2), ( 12) and ( 13)) are converted into four ordinary differential equations odes along the characteristic lines of both fluid and pipe.The four odes are then numerically integrated that to yield four numerical equations.Then these numerical equations are solved along with the appropriate initial and boundary conditions.

Initial and boundary conditions
The initial conditions for the pipeline system are determined from the steady state analysis, considering steady shear formulas such as HazeneWilliams or Darcy-Weisbach friction losses.
The boundary conditions for a system having a reservoir, pipes and valves considered in this work are: Reservoir: The location of the reservoir is at the upstream end of pipe and has a head of h r that is assumed to be constant during the water hammer event and the upstream end of the pipe is fixed, so that the structure speed is zero.Both the equations of the negative characteristic line of the fluid wave with the negative characteristic line of the structure wave, and the constant tank head are solved together.
Valve: The location of the valve is at downstream end of the pipe where the closure time is assumed to be instantaneous in this work.If the valve is fixed, the velocity of the fluid at the valve is zero, as is the structure velocity.For an unfixed valve, the flow velocity equals the structure velocity.With these conditions, the equation of the positive characteristic line of fluid wave and the equation of the positive characteristic line of the structure wave can be solved.

Code validation
A computer program is developed to solve the water hammer equations along with pre-defined initial and boundary conditions using Mathcad scripts.The code was validated using three cases as in the literature.These cases are: 1. FSI validation with the Delft Hydraulic Benchmark Problem.2. Viscoelastic model validation with the case carried by (Weinerowska-Bords, 2006).3. Unsteady friction model validation with the case carried by (Vitkovsky et al., 2004).

Delft hydraulic benchmark problem
The problem is a single pipe that has a tank at one end and a valve at the other end, which is free to move and has an instantaneous closing time that triggers the water hammer phenomenon as specified in Table 1.As shown in Fig. 1

Viscoelastic model validation
The problem, carried out by (Weinerowska-Bords, 2006), consists of a single pipe with a tank at the upstream end and a valve at the downstream end that is fixed and with instantaneous closing time to trigger the water hammer phenomenon with specifications as given in Table 2.In the calculation, the FSI and the unsteady shear are not taken into  account, only the viscoelastic model is considered.Fig. 2 shows that the present results are in a good agreement with those due to (Weinerowska-Bords, 2006).The traditional method (elastic model) is insufficiently precise in the case of polymer pipe, whereas the viscoelastic water hammer model enables the attainment of an acceptable solution.Due to the viscoelastic behavior of the pipeline material, there is an increase in wave dispersion, which causes a change in wave frequency and, as a result, oscillation period.There is also additional surge dumping, which is evidently stronger than in the case of elastic material, like steel, and the phenomenon occurs faster.

Unsteady friction model validation
The problem carried out by (Vitkovsky et al., 2004), consists of a single pipe with a tank at the upstream end and a valve at the downstream end which is fixed and has an instantaneous closing time to trigger the water hammer phenomenon with specifications as given in Table 3.In the calculations only the unsteady friction model is considered.Fig. 3 shows that the present numerical calculation are in good agreement with those carried outby the (Vitkovsky et al., 2004).These simulations show that the steady friction approximation used in the classical model overestimates the damping and dispersion predicted by the physically more realistic unsteady friction model.The stable friction model is unable to explain the strong frequency-dependent attenuation, as demonstrated by the inability of establishing the typical shape of the evolution of the pressure oscillations.

Results and discussions
In this section, the effect of some parameters on the water transient wave and maximum head at the valve is studied, using a single pipe with a tank at upstream and a valve at downstream with an instantaneous closing time to trigger the water hammer wave (Table 4).

Single pipe without junction coupling
The fluidestructure interaction, viscoelastic behavior, unsteady shear without junction coupling, and other factors are taken into consideration while studying the impacts of various parameters on the water transient wave.These parameters include flow velocity, pipe length, variation in pipe size, pipe material, and Poisson ratio.The system specifications are given in Table 4, some parameters are changed as stated in each subsection.

Effect of flow velocity
The effect of flow velocity is studied using four different velocities V 0 ¼ f0:1; 0:5; 1; 1:5g (m/s) on the same single pipe.Fig. 4 shows the effects of variation of flow velocity on water hammer head wave form for four different initial velocities.As shown in Fig. 4, increasing the flow velocity increases the maximum pressure head H max .Also, as noticed, increasing the flow velocity leads to only an increase in wave amplitude but does not affect the wave frequency.Based on these findings, the maximum head at the valve and the DH max is gathered and  displayed in Fig. 5. DH max is equal to the difference between the maximum transient head at the valve and the steady-state head at the valve.Fig. 5 also, shows that increasing flow velocities increases the maximum H max and the maximum head difference DH max .

Effect of pipeline length
The effect of pipeline length is studiedusing five different pipeline lengths L ¼ f50; 100; 200; 250; 300g m with the same velocity.The effect of pipeline length on the head wave form is illustrated in Fig. 6.Based on the numerical results, increasing the pipeline length decreases both the maximum pressure and the wave frequency Based on the results, the maximum head at the valve H max and the DH max are illustrated in Fig. 7.The pipe with a 50 m length has the highest head, the highest wave frequency, and the highest wave damping.The pipe with a 300 m length has the lowest head, wave frequency, and the lowest wave damping.

Effect of pipe size
The effect of the pipe size is studied using six different pipe sizes with the same initial velocity (0.3 m/s), the same length (300 m) and the same Poisson ratio (0.49) as listed in Table 5. Fig. 8 shows that changing pipe size changes the fluid celerity and the structure speed.As the wave speed of the fluid and the structure increase the wave frequency and amplitude increase.The pipe with a diameter of 26.6 mm and a thickness of 3.38 mm has the highest wave amplitude and maximum head.Also, it has the highest wave frequency.The pipe with a diameter of 77.5 mm and a thickness of 5.4 mm has the lowest wave amplitude and lowest wave frequency (Table 5).

Effect of pipe material
The effect of pipe material on the pressure wave is studied with two types of materials (High Density Polyethylene (HDPE) and PVC) and the specifications of these materials are listed in Table 6.This study is carried out with two velocities of 0.5 and 1.0 m/s.As shown in Fig. 9 From these values, it can be noticed that the maximum head in the case of PVC material is higher than that of HDPE material.That is due to two reasons, the wave speed which depends on the material density and the inelastic deformation behavior of each

Effect of Poisson ratio
The effect of the pipe Poisson ratio on the pressure wave is studied using ten Poisson ratios n ¼ f0::; 0:5g.Pipes with a high Poisson ratio have high stress wave speed and thus high fluid wave frequency and higher fluid wave amplitude and consequently higher maximum head.As shown in

Effect of junction coupling
The effect of junction coupling is studied first with the classical model (two equations model), one without junction coupling and the other with free movable valve as junction coupling.Also, the four equations model is used to simulate the effect of the structure, one without the junction coupling and the other with a free movable valve as the junction coupling.The results are illustrated in Fig. 12 which show that the classical model without junction coupling has the lowest maximum head, then the classical model with junction coupling, and then the four equations model with only Poisson coupling.The four equations model with both the Poisson coupling and the free-to-move valve as junction coupling has the highest maximum head because the free-to-move junction tends to act like a turbine in its  move in the positive x direction and acts as a pump in its negative direction, thus increasing the head.

Summary and conclusions
The fluid structure interaction, pipe elasticity behavior, and unsteady friction are all included in the mathematical model for water transient analysis that is given.Also, the three fluid structure interaction's mechanisms, which are Poisson, friction, and junction couplings are considered.Junction coupling could be due to free to move valve or free to move junction or both of them together.The MOC is used to solve the system of the four equations that describe the fluid and structure interactions.The appropriate boundary conditions have been used to simulate the source of fluid structure transients.The developed code in Mathcad environment is used to simulate different dynamical parameters that affect the transient system under consideration.This code is validated with three cases to test the code in different aspects.The test cases are chosen from previous work for standard reservoir-single-pipevalve system.The obtained results from the developed code are in good match with the standard previous work.The viscoelastic behavior of the pipes and unsteady friction are taken into consideration in a parametric study of fluidestructure interaction.The variation of parameters such as fluid flow rate, pipe length, pipe size, pipe material, and pipe Poisson ratio one without considering the junction coupling and one with considering the junction coupling due to the free-to-move valve.The obtained results are summarized as follows: Increasing the fluid flow velocity leads to an increase in wave amplitude but has no effect on the wave frequency.For shorter pipes, the frequency of the fluid wave is increased, and the wave amplitude is attenuated faster than the longer pipes.The shortest pipe has the highest head, the highest wave frequency, and the highest wave damping.The longest pipe has the lowest head, lowest frequency, and lowest wave damping.
The fluid wave characteristics depend on the pipe material.The maximum head with PVC material is higher than that with HDPE material, due to higher hysteresis dissipation of HDPE.
Smaller pipe size increases the wave celerity and consequently the wave frequency increased and so does the maximum head.
The pipes with high Poisson ratio have high structure wave speed and thus high fluid wave frequency and amplitude, consequently higher maximum head.It is found that as the Poisson ratio increases, the maximum head decreases and reaches minimum value after which the maximum head increases.
The coupling mechanism between the fluid and structure can be due to an inherent parameter in the governing equations such as Poisson ratio and shear model or due to the boundary condition at the boundary sections such as free-to-move valve.The coupling mechanism affects the fluid wave head, frequency, and the maximum head.In the classical solution of water hammer, with free-to-move valve, the wave frequency decreases and the maximum head increases.The present model that incorporates the FSI with, free to move valve, show that both the wave frequency and the maximum head increase.
For low fluid flow rates, the effect of free to move valve becomes significant in the fluid wave form, such that, after the instantaneous closure of valve there is a consequent sudden increase in pressure.The free-to-move junction tends to act like an expander in its move at the end of pipe and thus decreases the head, in the contrary, it acts as a pump in its negative direction and thus increases the head.
Finally, it is recommended to include the total effect column separation (CS) while studying the water hammer phenomenon.Also, considering all the degree of freedom of pipe motion and studying the effect of series junctions employing the present full water hammer model.
, the results of the present model are in a good agreement with delft problem result.Poisson coupling, junction coupling, and stable friction coupling are all taken into account in the calculation of the results shown in Fig. 1.At t ¼ 0 s, and z ¼ 20 m, an axial stress wave in the pipe and a pressure wave in the fluid are both started by the instantaneous valve closure.The initial pressure rise is less than predicted by the Joukowsky formula since the valve is moving in a positive axial direction.The effect of the valve's motion is clearly visible in the (coupled) solutions.Poisson coupling has an important impact.The classically expected maximum pressure heads are shown to be exceeded by up to 50 % as a result of Poisson and junction coupling.

Fig. 1 .
Fig. 1.Comparison between the numerical simulation from present model and the Delft benchmark problem (Tijsseling, 2003) at the end point (valve) for both Poisson and Junction coupling.

Fig. 2 .
Fig. 2. Comparison between the numerical simulation from present model and the (Weinerowska-Bords, 2006) for the end point (valve) for viscoelastic behavior of the pipe only.
, the PVC material with initial velocity of 1 m/s has H max ¼ 70.58 m, DH max ¼ 62:11 m and the PVC material with an initial velocity of 0.5 m/s, has H max ¼ 46.76 m, DH max ¼ 33:66 m.Whereas the HDPE material with an initial velocity of 1 m/s has H max ¼ 54.69 m, DH max ¼ 46:22 m and the HDPE material with an initial velocity of 0.5 m/s, has H max ¼ 35.84 m, DH max ¼ 22:74 m.

Fig. 3 .
Fig. 3. Comparison between the numerical simulation from present model and the(Vitkovsky et al., 2004) for the end point (valve) for unsteady friction only.
material.Pipe with PVC material has the highest wave amplitude and wave frequency.

Fig. 4 .
Fig. 4. Influence of initial flow velocity on water hammer head wave form.

Fig. 5 .
Fig. 5. Maximum head and maximum head difference at valve for different flow velocity.

Fig. 6 .
Fig. 6.Influence of pipe length on water hammer head wave form.

Fig. 7 .
Fig. 7. Maximum head and maximum head difference at valve for different pipe lengths (fixed valve).

Figs. 10
Figs. 10 and 11, increasing the Poisson ratio decreases the maximum pressure head at the valve but at a certain value of the Poisson ratio, this head is

Fig. 11 .
Fig. 11.Effect of Poisson ratio on the maximum head at the valve (fixed valve).

Table 2 .
Experiment specifications carried by (Weinerowska-Bords, 2006).fluid density; r f Steady state velocity; V 0 Reservoir head; H r Pipe material; r p Length; L Internal diameter; D Wall thickness; e

Table 4 .
Specifications of the single pipe system.

Table 5 .
Data used to study the effect of variation of pipe sizes for the single pipe without junction coupling and the corresponding H max and D H max at valve.