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Citation: Kandil, M.; El-Sayed, T.A.;
Kamal, A.M. Comparative Analysis
of Water Hammer Performance in
Different Pipe Parameters with FSI. J.
Exp. Theor. Anal. 2024,2, 58–79.
https://doi.org/10.3390/jeta2030006
Academic Editor: Marco Rossi
Received: 14 May 2024
Revised: 24 July 2024
Accepted: 1 August 2024
Published: 20 August 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Comparative Analysis of Water Hammer Performance in
Different Pipe Parameters with FSI
Mostafa Kandil 1, Tamer A. El-Sayed 1,2,3,* and Ahmed M. Kamal 1
1Department of Mechanical Design, Faculty of Engineering, Mataria, Helwan University, Helmeiat-Elzaton,
P.O. Box 11718 Cairo, Egypt; mustafa_abdulmutalleb@m-eng.helwan.edu.eg (M.K.);
amkamalg@m-eng.helwan.edu.eg (A.M.K.)
2Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen,
Aberdeen AB24 3UE, UK
3School of Engineering, University of Hertfordshire Hosted by Global Academic Foundation, Cairo, Egypt
*Correspondence: tamer_alsayed@m-eng.helwan.edu.eg or tamer.el-sayed@abdn.ac.uk
Abstract:
Water hammer (WH) is a critical phenomenon in fluid-filled piping systems that can
lead to severe pressure surges and structural damage. The characteristics of the pipe material,
geometry, and support conditions play a crucial role in the fluid–structure interaction (FSI) during
WH events. This study investigates the impact of various pipe parameters, including material,
length, thickness, and diameter, on the WH behavior using an FSI-based numerical approach. A
comprehensive computational model was developed based on the algorithm presented in Delft
Hydraulics Benchmark Problem (A) to simulate the WH phenomenon in pipes made of different
materials, such as steel, copper, ductile iron, PPR (polypropylene random copolymer), and GRP
(glass-reinforced plastic). This study examines the influence of pipe parameters on WH performance
in pipelines, utilizing FSI to analyze the phenomenon. The results show that the pipe material has a
significant influence on the pressure wave speed, stress wave propagation, and the overall system
response during WH. Pipes with lower modulus of elasticity, such as PPR and GRP, exhibit lower
pressure wave speeds but higher stress wave speeds compared with steel pipes. Increasing the elastic
modulus, pipe wall thickness, length, and diameter enhances the pipe’s stiffness and impacts the
timing, magnitude of pressure surges, and the likelihood of cavitation. The findings of this study
provide valuable insights into the design and mitigation of WH in piping systems.
Keywords: water hammer; fluid–structure interaction (FSI); pipeline design; pipe parameters
1. Introduction
Fluid–structure interaction (FSI) plays a crucial role in the dynamics of liquid-filled
pipes, particularly during WH events. The interaction between the fluid and the pipe
material can significantly impact the performance of the system, leading to variations in
pressure and stress waves.
The method of characteristics (MOC) is a numerical method that solves the FSI equa-
tions in the time domain by transforming them into ordinary differential equations along
characteristic lines. This method is particularly useful for analyzing the dynamics of
fluid-filled pipes under various operating conditions, including WH events.
WH is a transient phenomenon that arises in fluid systems when a sudden change in
flow rate occurs. This sudden change can be triggered by the rapid closure of a valve, the
start or stop of a pump, or a change in fluid level. The resulting pressure wave propagates
through the fluid system, posing a risk of damage if not properly managed. During WH, the
interaction between the fluid and the tube wall is characterized by FSI. Four key variables
of FSI are affected by WH: water pressure, water velocity, axial stress in the tube wall, and
axial velocity in the tube wall. Figure 1depicts two distinct waves: a fluid wave on the left
J. Exp. Theor. Anal. 2024,2, 58–79. https://doi.org/10.3390/jeta2030006 https://www.mdpi.com/journal/jeta
J. Exp. Theor. Anal. 2024,259
represents the wave generated by WH in the fluid, while a wave in the pipe structure on
the right represents the stress wave generated by WH in the pipe wall.
Figure 1. The propagation of wave speed in liquid and tubes during WH events [1].
For thin-walled tubes, the mass of the tubes can be neglected when analyzing the
effects of WH, as the four-equation model for FSI is sufficient for examining WH on straight
tubes [2].
Several studies have explored the combination of FSI with other phenomena related
to wave dissipation, including the elasticity of wall tubes [
3
–
5
]. Additionally, studies have
examined the phenomenon of cavitation of liquid during WH [
6
] and the effects of unsteady
friction and viscoelasticity in tube wall structures [7].
Tijsseling and Lavooij [
8
] investigated the interaction between pressure surges and
pipe motion, with a focus on the phenomenon of WH and fluid–structure interaction.
The authors aimed to understand how pipes are affected by high pressure during water
hammer, which is not accounted for in classical WH theory. Furthermore, they sought to
distinguish and analyze the three interaction mechanisms—friction, Poisson, and junction
coupling—and to assess their influence on extreme pressures during WH occurrences.
Tijsseling [
9
] presented a rigorous derivation of one-dimensional equations describing
FSI mechanisms in the axial/radial vibration of liquid-filled pipes, accounting for the
thickness of the pipe wall through the averaging of hoop and radial stresses. The author
investigated the influence of wall thickness on wave propagation speeds, natural cycles, and
WH pressure waves, and compared predictions from thin- and thick-walled theories with
experimental data obtained from a relatively thick pipe. The motivation for this research
was the need for a more accurate understanding of WH with FSI in pipes with thick walls,
which is common in practice. Zhang et al. [
10
] conducted an analytical investigation of WH
in a hydraulic pressurized pipe system with a throttled surge chamber for slow closure.
The authors aimed to derive analytical formulas for maximum WH pressures and develop
a design/analysis tool for engineers and researchers, particularly in the hydropower
industry. Adamkowski et al. [
11
] conducted an experimental investigation to examine the
effects of pipeline support stiffness on WH phenomena. The authors studied the dynamic
interaction between the liquid and the structure of the pipeline, focusing on the influence
of elastic pipe supports on transient flow parameters and resulting WH pressures. The
study also sought to provide a physical interpretation and explanation of the experimental
results, enhancing the understanding of this complex phenomenon. Pezzinga et al. [
12
]
investigated the reduction of unsteady flow oscillations in pressure pipelines by inserting
in-line sections with low wave speed, specifically using additional pipes made of high-
density polyethylene (HDPE) in pumping installations. The authors experimentally and
theoretically evaluated the effectiveness of these additional pipes in reducing unsteady flow
oscillations and assessed the mechanical parameters of the pipe material using numerical
models. Furthermore, the authors sought to derive analytical solutions for frictionless
pipelines with the additional pipe considered elastic, and to contribute to the design of the
device for reducing unsteady flow oscillations.
J. Exp. Theor. Anal. 2024,260
In their study, Garg and Kumar [
13
] provided analytical solutions for frictionless
pipelines with an additional elastic pipe, contributing to the design of devices for reducing
unsteady flow oscillations. The authors investigated the impact of different materials and
configurations on WH in pipelines, conducting experimental and numerical investigations
on metallic viscoelastic pipelines under transient conditions [
14
]. Keramat et al. [
15
]
presented a mathematical model and a numerical solution that considered support and
elbow motion in FSI with pipe wall viscoelasticity during WH events. By incorporating
both effects simultaneously, the authors derived the governing equations for hydraulics and
structures, offering appropriate numerical solutions and validation from various perspectives.
Two methods for analyzing hydraulic transients using FSI to calculate transient pres-
sure in pipeline systems were discussed [
16
]. The first method, known as the two-mode
method, utilized the MOC-finite element method (FEM) for water and tube wall interac-
tions. The second method, known as the full MOC method, focused on path sides [
17
,
18
].
Lavooij et al. [
19
] applied a two-equation model to solve FSI unknown variables such as
WH pressure, water velocity, axial stress in the tube wall, and axial velocity in the tube wall.
Keramat et al. [
20
] developed a method for detecting leaks in pipes using transient-
based techniques in the frequency domain, taking into account the interactions between
the fluid and the pipe structure, and the viscoelastic properties of the pipe material.
Gao et al. [21]
developed a method for analyzing the frequency domain FSI vibration
characteristics of aircraft hydraulic pipes with complex constraints. The study aimed to
investigate the FSI behavior of these pipes under various operating conditions, including
flow pulsation excitation, and to provide a comprehensive understanding of the vibration
responses and their effects on the pipe’s structural integrity. Li et al. [
22
] developed a
method for analyzing the frequency domain FSI in liquid-filled pipe systems using the
transfer matrix method. The study aimed to investigate the FSI behavior of these systems
under various operating conditions and provide a comprehensive understanding of the
vibration responses and their effects on the pipe’s structural integrity.
Andrade et al. [
23
] investigated the FSI coupling mechanisms in liquid-filled viscoelas-
tic pipes under the influence of fast transients. The study aimed to extend a recently
developed quasi-2D flow model for fluid transients in viscoelastic pipes to handle FSI and
analyze the effects of FSI on the pipe’s behavior during fast transient events.
Bayle et al. [
24
] provided explicit analytic solutions in the Laplace domain for FSI WH
waves within a pipe. The study aimed to transpose the transfer matrix method (TMM)
to the equivalent two-wave propagating problem rather than applying it directly to the
FSI four equations. By using the classical wave matrix diagonalization approach, the
researchers could decouple wave propagation while still coupling boundary conditions in
the diagonal base.
Henclik et al. [
25
] investigated the influence of dynamic FSI on the course of (WH) in a
non-rigid pipeline system. The study focused on a specific model of FSI behavior, where a
straight pipeline with a steady flow was fixed to the floor with several rigid supports and a
quickly closed valve was installed at the end. The valve was attached to the pipeline with a
spring/dashpot system, which allowed for dynamic energy transfer between the fluid and
the structure. The research aimed to analyze the transient pressure changes in the pipeline
for various stiffness and damping parameters of the spring/dashpot valve attachment. The
solutions were found both analytically and numerically using a four-equation model of
WH-FSI and specific boundary conditions at the valve. The study also investigated the
influence of valve attachment parameters on the WH courses and found that the transient
amplitudes could be reduced by optimizing these parameters.
Urbanowicz et al. [
26
]’s development in analytical wall shear stress modeling for water
hammer phenomena was to enhance the understanding and prediction of wall shear stress
effects during water hammer events in fluid systems. By advancing analytical models for
calculating wall shear stress, the study aimed to improve the accuracy and reliability of
simulations, thereby enabling better assessment of the impact of WH on pipeline integrity,
flow stability, and system efficiency. The research contributed to the development of more
J. Exp. Theor. Anal. 2024,261
comprehensive and effective strategies for mitigating the adverse effects of WH in industrial
and municipal piping systems.
Covas et al. investigated the complex interplay between viscoelastic behavior of pipe
walls, unsteady friction, and transient pressures in piping systems. The research aimed
to develop numerical models and simulations to understand the effects of viscoelasticity
and unsteady friction on transient flow behavior, particularly WH in viscoelastic pipes [
27
].
Yazdi et al. [
28
] presented a new modeling to improve WH metamodeling techniques
such as artificial neural network (ANN), support vector regression (SVR), and adaptive
neuro-fuzzy inference system (ANFIS). Results showed that ANN was the most accurate,
followed by ANFIS, while SVR had lower generalization capability. To determine the best
size and location for WH control devices in a pipeline, researchers used a combination of
ANN and differential evolution (DE).
Hyunjun and Kim [
29
] developed a more advanced quasi-2D model for transient
events inducing cavitation in flow. The model aimed to provide a more comprehensive
understanding of the complex interactions between cavitation and WH effects in a reser-
voir/pipeline/valve system. By incorporating the density of the liquid–vapor mixture,
considering radial flux, and utilizing a hybrid solution scheme, the authors aimed to
improve the accuracy and reliability of simulations for transient events with cavitation.
The ultimate goal of the research was to contribute to the advancement of hydraulic anal-
ysis and the management of pipeline systems by providing a more effective model for
predicting mass transport and ensuring system safety. Triki et al. [
30
] introduced and
developed an innovative “compound technique” for in-line water hammer control in steel
pressurized-piping systems. The authors aimed to overcome the limitations of the con-
ventional inline strategy by combining the benefits of different polymeric material types
and optimizing the design strategy for improved performance. This approach enhanced
the control of water hammer in pressurized pipe flow by addressing the drawbacks of the
primitive implementation technique. Kandil et al. [
31
] employed an analytical model to
investigate the influence of various tube materials on WH phenomena. The study revealed
that the selection of pipe materials for conveying liquids had a substantial impact on the
WH reaction.
Skalak [
32
] expanded upon the existing theory of WH, providing a more comprehen-
sive solution that included several novel features not present in the elementary theory.
The author re-examined the existing theory and analyzed the propagation of pressure
waves in a cylindrical tube filled with an elastic fluid. The research aimed to enhance the
understanding of WH and improve the accuracy of predictions related to pressure waves
in fluid-filled tubes introduced by Joukovsky in his classical theory [33].
Chaudhry [
34
] analyzed and modeled the behavior of pressure transients, particularly
WH, in piping systems. The author aimed to develop numerical models and simulations
to understand the effects of various factors, including viscoelasticity, unsteady friction,
and transient pressures, on the behavior of pressure transients in piping systems. The
research aimed to provide practical insights into the design and operation of piping sys-
tems, focusing on mitigating the negative effects of pressure transients on the system.
Ferras et al. [
35
] investigated the impact of FSI during hydraulic transients in pipe coils,
developing mathematical models, numerically implementing them, and validating them
with experimental evidence. The goal was to create a comprehensive model that captured
the behavior of pipes during hydraulic transients, incorporating both axial stress waves in
the pipe wall and fluid conservation principles.
Watters [
36
] examined the behavior of PVC pipes under the influence of WH pressure
waves, questioning the validity of applying the classical theory of hydraulic transients to
these pipes. The study was prompted by the growing use of plastic materials, particularly
PVC, in constructing water pipes and the need for a deeper understanding of their behavior
under WH conditions. The research aimed to provide insights into the extent to which
the classical theory of hydraulic transients accurately predicts WH in PVC pipes and to
identify any discrepancies between theoretical predictions and actual behavior.
J. Exp. Theor. Anal. 2024,262
Kubrak et al. [
37
] examined the behavior of WH in steel/plastic pipes connected in
series, analyzing the effects of various steel/plastic configurations on the system. The study
aimed to provide insights into the following aspects: performing WH tests for different
lengths of steel and HDPE sections, considering six different steel/plastic configurations;
examining how unsteady friction and the viscoelastic properties of polymer pipes affect
the system. They used a WH solver and numerical calculations to simulate the behavior of
the system. They compared their results with the classical theory of hydraulic transients to
assess the accuracy and applicability of the theory to steel/plastic pipes connected in series.
Their work showed the complex interplay between steel and plastic pipes, and the effects
of unsteady friction and transient pressures in piping systems, with a focus on mitigating
the negative effects of pressure transients on the system.
Kriaa et al. [
38
] examined the effectiveness of using rubber bypass tubes in reducing
WH effects in uPVC pipes. The study focused on the following aspects: experimentally
demonstrating the reduction of WH effects in uPVC pipes using rubber bypass tubes;
investigating the underlying mechanisms responsible for the reduction of WH effects,
including the damping properties of rubber materials and the role of rubber in attenuating
pressure surges; assessing the feasibility and practical implications of implementing rubber
bypass tubes in uPVC pipes for WH control, considering factors such as cost-effectiveness,
ease of installation, and long-term performance. The research provided insights into the
potential of rubber bypass tubes in mitigating WH effects in uPVC pipes, with a focus on
the practical implications for the design and operation of piping systems.
The present study aims to investigate the comparative performance of waterways in
different pipe parameters including materials, pipe wall thickness, length, and diameter
using FSI. The materials used in this analysis are steel, copper, ductile iron, PPR (polypropy-
lene random copolymer), and GRP (glass-reinforced plastic). This study employs the
method of characteristics (MOC) to analyze the FSI effects in each parameter. The results
provide valuable insights into the performance of each pipe parameter under WH condi-
tions, highlighting the importance of material selection in mitigating FSI effects. This study
is structured as follows: Section 2presents the theory used in the WH model. Section 3veri-
fies the accuracy of the MATLAB model used in this study. Section 4presents the analytical
solution of WH equations. Section 5presents and analyzes the numerical results obtained
for different pipe materials. Section 6summarizes the key findings and recommendations.
2. Theory and Modeling
Lavooij and Tijsseling developed a one-dimensional non-dispersive four-equation
model [
18
], which was employed in this study to incorporate various coupling mechanisms,
including Poisson coupling, junction coupling, and friction coupling.
The authors utilized MOC—a well-established technique for solving hyperbolic par-
tial differential equations—to analyze WH phenomena in pipe systems. This method is
commonly used to obtain exact or approximate solutions for hyperbolic PDE systems, as
detailed in the references [39].
The four-equation model (Equations (1)–(4)) for FSI in WH problems is a fundamental
framework for analyzing the transient behavior of pressurized pipe flows. The model
consists of four hyperbolic equations that describe the fluid motion, fluid continuity, and
structural dynamics. The first equation represents the fluid motion, which is governed by
the Navier–Stokes equation. The second equation ensures mass conservation by describing
the fluid continuity. The third equation accounts for the structural dynamics, including
the effects of pipe stiffness and damping. The fourth equation represents the interaction
between the fluid and the structure, capturing the effects of fluid–structure coupling. These
four equations form the core of the FSI model.
∂V
∂t+1
ρl
∂p
∂z=0 (1)
J. Exp. Theor. Anal. 2024,263
∂V
∂z+1
K+2R
Eh ∂p
∂t−2v∂U
∂Z=0 (2)
∂U
∂t−1
ρt
∂σ
∂z=0 (3)
∂U
∂z−1
E
∂σ
∂t+Rv
hE
∂p
∂t=0 (4)
a2=ρl
K+2ρlR
hE −1
,c2=E
ρt(5)
The variables used in the analysis of WH phenomena in pipelines are as follows:
axial tube velocity (
U
), velocity of liquid (
V
), time (
t
), classical pressure wave speed in the
water (
a
), axial stress wave speeds (
c
), liquid cross-sectional area (
Al
), tube cross-sectional
area (
At
), modulus of electricity (
E
), thickness of tube wall (
h
), gravity acceleration (
g
),
tube Poisson ratio (
v
), inner radius of the tube (
R
), liquid density (
ρl
), tube density (
ρt
),
liquid bulk modulus (
K
), time (
t
), Cartesian coordinate system axis (
z
), axial stress in tube
wall (
σ
), and liquid transient pressure (
p
). The analysis focused on the WH caused by the
instantaneous closure of the valve in the tank/tube/valve system, as depicted in Figure 2.
Figure 2.
The Benchmark Problem (A) pipeline model comprises a system of valves, pipes, and
a tank.
3. Model Verification
This paper builds upon the same model and algorithms presented in [
40
] and includes
a custom-developed MATLAB code tailored to this specific study. In this section, the
results obtained from the computer program are compared to those presented in [
40
] using
identical variables.
The Delft Hydraulics Benchmark Problem A is a widely recognized test case used to
evaluate numerical methods and FSI models [
8
]. It consists of a tank/tube/valve system
with specific geometric and material properties, as shown in Figure 2. The problem is
defined by parameters such as tube length, radius, wall thickness, material properties,
and fluid velocity. The benchmark problem is characterized by the wave speeds and their
ratio, which are essential for assessing the performance of numerical methods and FSI
models. It serves as a standard test case for validating and comparing different approaches
to simulating WH and FSI phenomena. The Delft Hydraulics Benchmark A has been
extensively used in the research community to advance the understanding and modeling
of WH and FSI in pipeline systems [40].
The results obtained from the program developed for this study were validated
against the Delft Hydraulics Benchmark A, a well-established benchmark case for FSI in
WH analysis. As shown in Figure 3, the pressure profiles generated by the used MATLAB
program closely match the reference data from the Delft Hydraulics Benchmark A. This
J. Exp. Theor. Anal. 2024,264
alignment demonstrates the accuracy and reliability of the used MATLAB numerical model
in capturing the complex FSI phenomena occurring in the pipeline system. The ability
to replicate the benchmark results provides confidence in the applicability of the used
MATLAB program to analyze WH events in a variety of pipe materials and configurations.
This validation step ensures that the insights and conclusions drawn from comparative
analysis of different pipe materials are grounded in a robust and well-tested numerical
framework, further strengthening the overall quality and credibility of this research.
Figure 3.
Verification of proposed MATLAB program with Benchmark Problem (A) for a freely
moving valve (Z/L=1) with reference to absolute pressure.
Table 1presents the complete characteristics of the Benchmark Problem A parameters.
It is crucial to note that the dimensions specified are adopted from [
40
] and that the
validation is based solely on numerical data.
Table 1. Benchmark Problem (A) parameters.
Parameters Benchmark Problem (A)
Tube length (L) 20 m
Tube inner radius (R) 398.5 mm
Tube wall thickness (h) 8 mm
Modulus of elasticity (E) 210 GPa
Water bulk modulus (K) 2.1 GPa
Water initial flow velocity (Vo) 1 m/s
Density of water ( ρl)1000 Kg/m3
Density of tube (ρt)7900 Kg/m3
Tube Poisson ratio 0.3
Pressure behind valve 0 Pa
Intial pressure (reservoir head) (po)1 bar
Pressure wave speeds (η1&η2)±1024.7 m/s
Stress wave speeds (η3&η4)±5280.5 m/s
Valve closure time 0.02 s
To investigate the WH phenomenon in the proposed model, the four system dependent
variables (WH pressure, water velocity (V), axial tube velocity (U), and axial stress in
tube wall (
σ
)) are calculated at four locations along the tube length, and the results are
examined. “Z” is the distance from the main supply header along the tube and “L” is the
total pipe length.
J. Exp. Theor. Anal. 2024,265
4. Analytical Solution of WH Equations
It is assumed that the acoustic phenomenon during the WH being studied can be
represented with the following set of equations.
A∂
∂tφ(z,t)+B∂
∂zφ(z,t)+Cφ(z,t)=0 (6)
Constant matrices
A
and
B
are invertible, and
A−1B
is diagonalizable. Constant matrix
C
, which, when singular, causes frequency dispersion (if
C6=
0). N dependent variables
φi
—constituting the state vector
φ
—are functions of the independent variables z(space)
and t(time), considering N = 4 and C= 0.
In Equation (6), the axial vibration of a liquid-filled tube (Equations (1)–(4)) can be
depicted using a state vector.
φ=
V
p
U
σ
(7)
The coefficients’ matrices
A=
1 0 0 0
0ρla2−10 0
0 0 1 0
0vR(Eh)−10−ρtc2−1
(8)
B=
0ρl−10 0
1 0 −2v0
0 0 0 −ρt−1
0 0 1 0
(9)
The characteristic equation
|B−ηA|=0(10)
In Appendix A, it is shown that characteristic equation
|B−ηA|=
0 has four distinct
real roots, indicating that matrix
A−1B
also possesses four distinct real eigenvalues. These
four characteristic roots or eigenvalues signify ηi.
Equation (6) is referred to in hyperbolic form. In this instance, it can be converted into
a more convenient form by multiplying it with a regular matrix T.
TA ∂
∂tφ(z,t)+TB ∂
∂zφ(z,t)=0 (11)
Based on the principles of linear algebra, it is established that, for any square matrix
with unique real eigenvalues, denoted as
A−1B
, there is a corresponding matrix
S
such that
S−1A−1BS =∆(12)
where ∆is diagonal:
∆=
η10 0 0
0η20 0
0 0 η30
0 0 0 η4
(13)
By considering matrix Tas the following:
T=S−1A−1(14)
J. Exp. Theor. Anal. 2024,266
Substitution of (14) into (12)
TB =∆T A (15)
Substitution of (15) into (11)
TA ∂
∂tφ(z,t)+∆T A ∂
∂zφ(z,t)=0 (16)
Equation (16) is commonly referred to as the standard form of system (11). Introduction
of the vector
v=T A φ(z,t)(17)
Substitution of (17) into (16)
∂v
∂t+∆∂v
∂z=0 (18)
Equation (18) can be simplified to the following:
∂vi
∂t+ηi
∂vi
∂z=0, i=1, 2, 3, 4 (19)
where ηis the eigenvalue.
Hence, the total derivative is given as
dvi(z,t)
dt=0 (20)
along dz
dt=ηi(21)
The solution to ordinary differential Equations (20) and (21) is:
vi(z,t)=vi(z−ηi∆t,t−∆t)(22)
When a numerical time step ∆tis utilized, or, more generally, according to Figure 4,
vi(P) = vi(Ai)or v(P)=
v1(A1)
v2(A2)
v3(A3)
v4(A4)
(23)
The value of the unknown variable
η
does not change along the line
Ai
P, as shown in
Figure 4:
Characteristic Equation (10), corresponding to matrices (8) and (9), is
η4−q2η2+a2c2=0 (24)
where q2=1+2v2ρl
ρt
R
ha2+c2
η1=1
2q2−q4−4a2c21/21/2
=ˆ
a(25)
η2=−η1=−ˆ
a(26)
η3=1
2q2+q4−4a2c21/21/2
=ˆ
c(27)
η4=−η3=−ˆ
c(28)
J. Exp. Theor. Anal. 2024,267
In the
z−t
plane,
P
and
Ai
are points, as depicted in Figure 4. The integrated compati-
bility Equations (12)–(15) reveal that for point
P
not on a boundary, the unknowns
V
,
p
,
U
,
and
σ
are entirely determined by their values in the “upstream” points
Ai
. Therefore, if the
boundaries are disregarded, the solution of system (11) at any point
P
can be derived from
Equations (12)–(15), provided that initial values for V,P,U, and σin Pare known.
Figure 4.
Point P and lines characterizing “feeding” in the distance–time plane. Reprinted with
permission from [40] © Elsevier.
Determining the variables’ values at known point (P) on the computational grid
depicted in Figure 4uses the data from the previous points.
The MATLAB program is used to solve the prementioned mathematical equations,
and the outcomes are shown in the following Section 5.
Initial and Boundary Conditions
To simulate WH with FSI, it is essential to define the initial conditions of pressure and
velocity of the water in the tube, and the initial position and deformation of the structure
(e.g., tube or valve) interacting with the water. In the FSI simulation, the interaction between
the water and the tube is accounted for in the boundary conditions, which include the
transmission of forces and displacements at the interface between the fluid and structure.
J. Exp. Theor. Anal. 2024,268
In this study, it is assumed that the pipe system filled with liquid is initially in a state
of equilibrium (steady state) before the occurrence of a transient event. The initial values
of key variables, including pressure, velocity, and tube displacement, are calculated using
one-dimensional equations.
In turbulent tube flow, the pressure loss (
∆p0
) across a fully open valve is given by the
orifice equation [35].
∆p0=ℵ0
1
2ρl(V0−U0)|V0−U0|(29)
where
ℵ0
represents an empirical loss coefficient. It is assumed that the same relation holds
for a closing valve [35].
∆p=ℵ1
2ρl(V−U)|V−U|(30)
where
ℵ
depends on the valve position and therefore on time. Dividing Equation (29)
by Equation (30) yields the dimensionless valve closure coefficient
τ=√ℵ0/ℵ
and the
nonlinear boundary condition
po(V−U)|V−U|=τ2(t)(V0−U0)|V0−U0|p(31)
Quadratic Equation (31) is solved concurrently with three linear equations (two Rie-
mann invariants and one boundary condition) [36].
The initial conditions are used for the water and the tube in the analysis as follows:
V=V0=1 m/s (32)
p=po=1 bar (33)
U=U0=0 m/s (34)
σ=Alpo/At(35)
5. Results and Discussion
This study investigated the impact of various pipe parameters, including material,
length, thickness, and diameter, on WH behavior using an FSI-based numerical approach.
All the pipe materials examined were considered within their elastic zones. The results show
that pipe material significantly influences pressure wave speed, stress wave propagation,
and the overall system response during WH events. This study underscores the importance
of considering pipe parameters in the design and mitigation of WH in piping systems, as
changes in these parameters can significantly affect the timing and magnitude of pressure
surges and the likelihood of cavitation.
5.1. Effect of Changing the Pipeline Material
The magnitude and propagation of WH pressure waves are strongly influenced by the
properties of the pipe material and the FSI between the pipe wall and the flowing liquid.
The pipe material affects the wave speed, which determines the timing and magnitude of
the pressure surges and the potential for cavitation and pipe failure.
Previous studies have investigated the effect of pipe material on WH using various
analytical and numerical approaches. However, most of these studies have focused on
a limited number of pipe materials, such as steel and plastic, and have not provided a
comprehensive comparison of the WH behavior across a wide range of pipe materials
commonly used in engineering applications.
This research aims to investigate the impact of various pipe materials, including steel,
copper, ductile iron, PPR (polypropylene random copolymer), and GRP (glass-reinforced
plastic), on the WH phenomenon with FSI.
Table 2presents the various pipe material properties that are relevant to the problem
scenario involving a tank/tube/valve system, as illustrated in Figure 2.
J. Exp. Theor. Anal. 2024,269
Table 2. Mechanical properties of the pipes used in the numerical analysis [41].
Tube Material Tube Modules of
Elasticity (GPa)
Density of Tube (ρt)
(kg/m3)
Poisson’s
Ratio
Steel tube (ST) 210 7900 0.30
Copper (CU) 110 8900 0.36
Polypropylene random copolymer (PPR) 0.850 909 0.36
Ductile iron (DI) 170 7050 0.27
Glass fiber-reinforced plastic (GRP E “glass”)
4.27 2100 0.30
All other dimensions and properties, including tube radius, thickness, length, and
water density, are kept constant for all materials, as specified in Table 1.
Figures 5–8presents the transient WH pressure distributions at four distinct locations
along the pipeline (1/4 L, 1/2 L, 3/4 L, and total L), showcasing the variation of WH
pressure, water flow velocity, axial tube velocity, and axial stress in the tube wall over time
for five different pipe materials (steel (ST), copper, polypropylene (PPR), ductile iron, and
glass-reinforced plastic (GRP)). The figures visually depict the variables at four locations
for each material. The comparison between materials is based on the maximum value and
the number of fundamental wave frequencies occurring within a 1 s time frame, as listed in
Table 3.
Figure 5.
Results of transient pressures at four specific locations: (
a
) steel (ST), (
b
) copper (Cu),
(
c
) polypropylene random copolymer (PPR), (
d
) ductile iron (DI), and (
e
) glass-reinforced plastic
(GRP) pipelines.
J. Exp. Theor. Anal. 2024,270
Figure 6.
Results of fluid velocity at four specific locations: (
a
) steel (ST), (
b
) copper (Cu),
(
c
) polypropylene random copolymer (PPR), (
d
) ductile iron (DI), and (
e
) glass-reinforced plastic
(GRP) pipelines.
Figure 7.
Results of axial tube velocity at four specific locations: (
a
) steel (ST), (
b
) copper (Cu),
(
c
) polypropylene random copolymer (PPR), (
d
) ductile iron (DI), and (
e
) glass-reinforced plastic
(GRP) pipelines.
J. Exp. Theor. Anal. 2024,271
Figure 8.
Results of axial stress wave in the tube wall at four specific locations: (
a
) steel (ST), (
b
) copper
(Cu), (
c
) polypropylene random copolymer (PPR), (
d
) ductile iron (DI), and (
e
) glass-reinforced plastic
(GRP) pipelines.
Figure 9shows the fast Fourier transform (FFT) results for the numerical simulations
of water hammer events in different types of pipeline materials at a location of
Z/L
= 0.5,
where
Z
represents the distance along the pipeline length L. The five subfigures (a) through
(e) present the FFT results for steel (ST), copper (Cu), polypropylene random copolymer
(PPR), ductile iron (DI), and glass-reinforced plastic (GRP) pipelines, respectively. The
FFT analysis allows for the identification of the dominant frequencies present in the WH
pressure oscillations for each pipeline material.
Figure 9.
FFT results for numerical results at Z/L = 0.5: (
a
) steel (ST), (
b
) copper (Cu), (
c
) polypropy-
lene random copolymer (PPR), (
d
) ductile iron (DI), and (
e
) glass-reinforced plastic (GRP) pipelines.
J. Exp. Theor. Anal. 2024,272
Table 3. Summarized WH pressure results for the five tube materials.
Tube Material Maximum WH
Pressure (MPa)
Maximum Water
Flow Velocity
(m/s)
Maximum Axial
Tube Velocity
(m/s)
Maximum Axial
Stress in Tube
Wall (MPa)
Fundamental
Wave Frequency
Steel (ST) 1.686 0.477 0.477 41.57 13
Copper (Cu) 1.335 0.55 0.55 32.88 10
Polypropylene
Random Copolymer
(PPR)
0.149 0.65 0.65 3.27 1
Ductile iron (DI) 1.475 0.48 0.48 36.64 12
Glass fiber-reinforced
plastic (GRP E “glass”)
0.579 0.56 0.56 14.378 2
As shown in Figure 5, the results indicate that steel (ST), ductile iron, and copper
exhibit higher maximum pressure values and frequencies compared with PPR and GRP.
This suggests that they are less affected by WH due to the higher WH pressure over a
certain number of cycles compared with PPR and GRP tube materials under the same
operating conditions. Conversely, PPR displays a lower maximum pressure and fewer
cycles, potentially indicating a different response to WH and suggesting a greater effect
on it.
As shown in Figure 6, the results indicate that steel (ST), ductile iron, GRP, and copper
exhibit lower maximum water flow velocity values compared with PPR. This suggests that
they are more susceptible to WH due to the lower WH water flow velocity compared with
the PPR tube material under the same operating conditions. Conversely, PPR displays a
higher maximum water flow velocity and relatively low cycles.
As shown in Figure 7, the results indicate that steel (ST), ductile iron, GRP, and copper
exhibit lower maximum axial tube velocity values compared with PPR. This suggests that
they are more susceptible to WH due to the lower WH axial tube velocity compared with
the PPR tube material under the same operating conditions. Conversely, PPR displays a
higher maximum water axial tube velocity and relatively low cycles. Notably, the maximum
values of water flow velocity are equivalent to the maximum axial tube velocity.
As shown in Figure 8, the results indicate that steel, ductile iron, and copper exhibit
higher maximum axial stress in the tube wall values and frequencies compared with PPR
and GRP. This suggests that they are less affected by WH due to the higher WH pressure and
number of cycles compared with PPR and GRP tube materials under the same operating
conditions. Conversely, PPR displays a lower maximum axial stress in the tube wall and
has relatively low cycles, indicating a greater susceptibility to WH.
The FSI model for WH in five pipe materials (steel, copper, DI, PPR, and GRP) was
analyzed to provide a more comprehensive understanding of the phenomenon. The results
show that the pressure wave speed and natural frequencies of the pipes decrease as the
stiffness and density of the pipe material decrease.
The wave speeds in the liquid and pipe wall during a WH event vary significantly
depending on the pipe material used, as listed in Table 4.
The key observations from these data are that the wave speed in the pipe wall is
significantly higher than the wave speed in the liquid for all the pipe materials. This
difference in wave speeds can lead to complex fluid–structure interactions during WH
events. The wave speed in the liquid is relatively consistent across the different pipe
materials, ranging from around 1020 m/s for steel to 71 m/s for GRP. This is due to the
similar density and bulk modulus of the liquid (water) in all cases. The wave speed in
the pipe wall varies much more widely, from around 5280 m/s for steel to 510 m/s for
GRP. This is due to the significant differences in the elastic modulus and density of the
pipe materials.
J. Exp. Theor. Anal. 2024,273
Table 4. WH wave speeds for the five tube materials.
Tube Material Pressure Wave Speeds
(η1&η2) (m/s)
Stress Wave Speeds
(η3&η4) (m/s)
Steel (ST) ±1024 ±5280
Copper (Cu) ±848 ±3660
Polypropylene random
copolymer (PPR) ±92 ±1070
Ductile iron (DI) ±969 ±5020
Glass fiber-reinforced plastic
(GRP E “glass”) ±71 ±510
The lower wave speeds in the PPR and GRP pipes indicate that these materials may be
more susceptible to WH effects, as the pressure waves can propagate more slowly through
the system. This could lead to higher pressure surges and increased risk of pipe damage or
failure. In summary, the choice of pipe material has a profound impact on the wave speeds
and, consequently, the WH behavior in a piping system. Understanding these material-
dependent wave speed characteristics is crucial for the proper design and mitigation of
WH effects.
These analyses and results provide a more comprehensive understanding of the FSI
model for WH in five pipe materials. The results show that the pressure wave speed
and natural frequencies of the pipes decrease as the stiffness and density of the pipe
material decrease.
5.2. Effect of Changing the Pipeline Length
This study investigates the impact of pipe length on the WH behavior in steel pipes
using an FSI-based numerical approach. This study considers steel pipes with lengths of
10 m, 20 m, and 30 m to analyze the effect of pipe length on the WH phenomenon. All other
dimensions and properties, including tube radius, thickness, and water density, are kept
constant for all materials, as specified in Table 1.
Figure 10 presents the variation of WH pressure, water flow velocity, axial tube velocity,
and axial stress in the tube wall over time for the steel pipeline for pipe lengths of 10 m,
20 m, and 30 m.
Figure 10.
WH pressure, fluid flow velocity, axial tube velocity, and axial stress in the tube wall over
time for the steel pipeline for pipe lengths of 10 m, 20 m, and 30 m.
J. Exp. Theor. Anal. 2024,274
The results show that the pipe length has a significant influence on the WH behavior
in steel pipes. The pressure wave speed is found to be independent of the pipe length, as it
is primarily determined by the properties of the fluid and the pipe material.
The magnitude of the pressure surges increases with pipe length. For the 10 m pipe,
the peak pressure surge is around 1.50 MPa, while for the 30 m pipe, the peak pressure
surge reaches 1.60 MPa. The longer pipe length allows the pressure wave to travel a greater
distance, resulting in higher pressure amplification due to the reflection and superposition
of the waves. The time taken for the pressure wave to travel from the valve to the upstream
end and back to the valve increases linearly with pipe length. For the 10 m pipe, the time
for the pressure wave to complete one round trip is around 0.5 ms, while, for the 30 m pipe,
it is approximately 0.4 ms. The higher pressure surges in the longer pipes increase the risk
of cavitation, as the pressure can drop below the vapor pressure of the fluid.
5.3. Effect of Change the Pipeline Radius
This study investigates the effect of pipe radius on the WH phenomenon with FSI in
steel pipes, considering a range of pipe sizes commonly used in engineering applications.
The pipe radii analyzed in this research are 300 mm, 400 mm, and 500 mm. All other
dimensions and properties, including tube thickness, length, and water density, are kept
constant for all materials, as specified in Table 1.
Figure 11 presents the variation of WH pressure, water flow velocity, axial tube velocity,
and axial stress in the tube wall over time for the steel pipeline for pipe radii of 300 mm,
400 mm, and 500 mm.
Figure 11.
WH pressure, water flow velocity, axial tube velocity, and axial stress in the tube wall over
time for the steel pipeline for pipe radii of 300 mm, 400 mm, and 500 mm.
The results show that the pipe radius has a significant influence on the WH behavior
in steel pipes. The pressure wave speed is found to be inversely proportional to the square
root of the pipe radius (refer to Equation (5)), with the 300 mm pipe having a pressure
wave speed of approximately 1095 m/s, while the 500 mm pipe has a pressure wave speed
of around 966 m/s. This is due to the relationship between the pressure wave speed
and the effective bulk modulus of the fluid pipe system, which is affected by the pipe
radius. The magnitude of the pressure surge increases with the pipe radius, with the
300 mm pipe experiencing a peak pressure surge of around 1.7 MPa, while the 500 mm
J. Exp. Theor. Anal. 2024,275
pipe reaches a peak pressure surge of 1.48 MPa. The larger pipe radius results in higher
fluid inertia, leading to more severe pressure surges during the WH event. The time taken
for the pressure wave to travel from the valve to the upstream end and back to the valve
is independent of the pipe radius, as it is primarily determined by the pipe length and
the pressure wave speed. The higher pressure surges in the larger pipes increase the risk
of cavitation, as the pressure can drop below the vapor pressure of the fluid, potentially
leading to the formation of vapor bubbles.
5.4. Effect of Change on Pipeline Thickness
The pipe wall thickness is another important parameter that can significantly affect
the WH behavior in steel pipes.
Figure 12 presents the variation of WH pressure, water flow velocity, axial tube velocity,
and axial stress in the tube wall over time for the steel pipeline for pipe thicknesses of 4 mm,
8 mm, and 12 mm. All other dimensions and properties, including tube radius, length, and
water density, are kept constant for all materials, as specified in Table 1.
Figure 12.
WH pressure, water flow velocity, axial tube velocity, and axial stress in the tube wall over
time for the steel pipeline for pipe thicknesses of 4 mm, 8 mm, and 12 mm.
The pressure wave speed is proportional to the square root of the pipe wall thickness
(refer to Equation (5)), with a 4 mm thick pipe having a pressure wave speed of approxi-
mately 838 m/s, while a 12 mm thick pipe has a pressure wave speed of around 1123 m/s.
This is due to the relationship between the pressure wave speed and the effective bulk mod-
ulus of the fluid pipe system, which is affected by the pipe wall thickness. The magnitude
of the pressure surges increases with increasing pipe wall thickness, with a 4 mm thick
pipe experiencing a peak pressure surge of up to 1.25 MPa, and a 12 mm thick pipe having
a peak pressure surge of around 1.75 MPa. Increasing the steel pipe wall thickness from
4 mm to 12 mm effectively reduces the magnitude of the longitudinal stress waves during
a WH event. The thicker pipe walls have higher longitudinal stress wave speeds and lower
peak longitudinal stresses due to their increased resistance to deformation. The reduced FSI
and lower longitudinal stresses in thicker steel pipes can enhance the overall reliability and
safety of the piping system under WH conditions. The wave frequency based on the wave
periodic time for WH with FSI in steel pipes decreases as the pipe wall thickness increases
from 4 mm to 12 mm. The thicker pipe walls result in lower pressure wave speeds, leading
to longer wave periodic times and lower wave frequencies. The FSI can also influence the
wave characteristics, but the general trend of decreasing frequency with increasing pipe
J. Exp. Theor. Anal. 2024,276
thickness remains. Thicker pipe walls provide more resistance to deformation, resulting in
lower pressure surges during the WH event. The time taken for the pressure wave to travel
from the valve to the upstream end and back to the valve is independent of the pipe wall
thickness, as it is primarily determined by the pipe length and the pressure wave speed.
The lower pressure surges in thicker pipes reduce the risk of cavitation.
6. Conclusions
This research aims to present the distinct outcomes for the four unknowns of FSI,
which are depicted through curves illustrating the extent of the changes occurring in them,
a novel aspect that has not been explored in previous research. This study also compares
the responses of different parameters to WH and identifies which parameter exhibits the
most significant effect. For this purpose, a numerical model has been developed. The
model results provide insights into the axial stress, axial tube velocity, and water velocity,
which were not readily accessible before.
The conclusions from this analysis are that the choice of pipe material significantly
affects the performance of the pipeline under WH conditions. The lower pressure wave
speeds and higher stress wave speeds in the PPR and GRP pipes suggest that these materials
may be more suitable for applications where high-frequency vibrations are a concern. On
the other hand, the higher natural frequencies of the steel pipe suggest that it may be more
suitable for applications where high-frequency vibrations are not a concern.
The findings of this study demonstrate the significant impact of pipe parameters on
the WH behavior in pipes. Larger pipe lengths, radii, and thicknesses are more susceptible
to higher pressure surges and increased risk of cavitation, which should be considered in
the design and mitigation of WH effects in fluid transport systems.
The key findings are as follows:
1.
Based on the analysis, PPR and GRP show better resistance to WH than steel. Copper
and ductile iron have lower resistance to WH due to the lower WH pressure and
frequencies under the same operating conditions.
2.
PPR is strongly recommended for use in environments where WH may occur due to
its minimal susceptibility to the phenomenon.
3.
In situations where WH is possible, it is advisable to choose plastic pipes because
of their superior impact absorption compared with rigid materials. This helps to
decrease the risk of any damage that may result from WH.
4.
The pressure wave speed remains independent of pipe length, but the time for a
pressure wave to complete a round trip increases with length, increasing the risk
of cavitation.
5.
Larger pipe radii result in higher pressure surges and fluid inertia, leading to more
severe WH effects.
6.
Thicker pipes have higher pressure wave speeds, peak pressure surges, and lower
longitudinal stress, providing increased resistance to deformation and enhancing
system reliability under WH conditions.
Author Contributions:
Conceptualization, M.K., A.M.K. and T.A.E.-S.; methodology, M.K.; software,
M.K. and T.A.E.-S.; validation, M.K., A.M.K. and T.A.E.-S.; formal analysis, M.K.; investigation, M.K.;
writing—original draft preparation, M.K.; writing—review and editing, M.K., A.M.K. and T.A.E.-S.;
supervision, A.M.K. and T.A.E.-S. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received no external funding.
Data Availability Statement:
The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Acknowledgments:
The authors would like to express their sincere gratitude to Arris Tijsseling for
his invaluable help in preparing the research.
Conflicts of Interest: The authors declare no conflict of interest.
J. Exp. Theor. Anal. 2024,277
Nomenclature
Symbol Description Units
AlCross-sectional area of liquid m2
AtCross-sectional area of tube m2
aFluid wave speed m/s
cAxial stress wave speed m/s
EModulus of elasticity Pa
gAcceleration of gravity m/s2
GShear modulus Pa
HHead m
hTube wall thickness m
hfFriction losses m
IArea moment of inertia m4
JPolar moment of inertia m4
KWater bulk modulus Pa
LTube length m
MMoment N.m
pWH pressure Pa
poPressure upstream the tank m
QWater discharge m3/s
RInner radius of tube m
tTime s
UAxial tube velocity m/s
VWater velocity m/s
VrRelative velocity m/s
ρlDensity of water kg/m3
ρtDensity of tube wall kg/m3
σAxial stress in tube wall Pa
Appendix A
Solving the Matrix Equation (TB =∆TA)
TB =∆T A (A1)
To solve this matrix equation, we begin by defining the given matrices.
(T)
is the
unknown matrix, while
(A)and (B)
are known square matrices, and
(∆)
is a known
eigenvalues diagonal matrix; see Equations (8), (9) and (13).
T=tij ,B=bij ,A=aij ,∆=diag(η1,η2, . . . , ηn)(A2)
The subsequent step involves expanding both sides of the equation, yielding a system
of sixteen equations for the elements
tij
of the matrix
(T)
. These sixteen equations are
then categorized into four groups, each corresponding to a row of the 4
×
4 matrix. This
process results in a simplified matrix equation of the following form:
(B−ηiA)ti=0 (A3)
where ti=[ti1ti2ti3ti4]T.
To avoid trivial solutions where
(T)
would be a zero matrix, we need to ensure that
the matrix
(B−ηiA)
is singular by appropriately selecting eigenvalues
(ηi)
. This selection
is crucial to guarantee non-trivial solutions for (T).
The eigenvalues
(ηi)
should be selected in accordance with Equations (25)–(28), en-
suring that Equation (A2) is satisfied. Consequently, the system of Equation (A3) becomes
dependent, allowing one of the four equations to be omitted in each iteration. This process
is repeated for each row of equations. The resulting system of linear equations is then
solved for the elements
tij in (T)
. This approach ensures that the original matrix equation
J. Exp. Theor. Anal. 2024,278
(TB =∆TA)
is satisfied. For further details of the solution, refer to Tijsseling’s thesis (see
ref. [6]).
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