TRANSSIENT ANALYSIS OF FLUID STRUCTURE INTERACTION IN STRAIGHT
PIPE
BADREDDIN GIUMA S.K ELGHARIANI
A project report submitted in partial fulfillment of the
Requirements for the award of the degree of
Master of Engineering (Mechanical)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
NOVEMBER 2007
ii
To my father, mother, brothers and sisters
iii
ACKNOWLEDGEMENT
I would like to express my deepest gratitude to Dr. Kahar Osman my supervisor
for his continued support during my study and the encouragement, guidance and
dedication he provided for this project. Without him this project would have not been
possible
I wish to express my gratitude also to all who teach me during my study in the
Universiti Teknologi Malaysia
My deep gratitude goes to my whole family, especially to my father and mother
for their encourage and support through the years
iv
ABSTRACT
Water hammer phenomenon is a common problem for flows in pipes. Water
hammer usually occurs when transfer of fluid is quickly started, stopped or is forced to
make a rapid change in direction. The aim of this study is to use method of characteristics
to study water hammer phenomenon. In this study, computational method is used to
investigate the transient water hammer problem in a straight pipe. Method of
characteristics is applied to constant density flow in a simple reservoir-pipeline-valve
system. The water hammer effect is produced via suddenly closing the valve located at
the upstream and downstream ends, respectively. Quasi steady shear stress is assumed for
the flow. This study also considers steady and unsteady friction. Fluid structure
interaction will also be analyzed. The results obtained show slightly higher pressure than
that of published experimental data. This could be due to the Quasi steady shear stress
assumption. Final results show that when fluid structure interaction is considered, more
accurate answers were determined.
v
Abstrak
Fenomena ‘Water Hammer’ adalah satu masalah yang biasa dalam aliran dalam
paip. ‘Water Hammer’ biasanya berlaku apabila perpindahan bendalir berlaku dengan
cepat, berhenti atau dipaksa untuk melakukan perubahan arah dengan tiba-tiba. Tujuan
kajian ini adalah untuk menggunakan ‘Method of characteristic’untuk mengkaji
fenomena ‘Water Hammer’. Di dalam kajian ini, kaedah berkomputer digunakan untuk
mengkaji masalah aliran peralihan “water hammer’ di dalam paip. ‘Method of
characteristic’ di aplikasikan kepada ketumpatan malar di dalam takungan-paip-injap
mudah. Kesan ‘Water hammer’ di hasilkan melalui penutupan injap yang berada di atas
dan di bawah takungan secara tiba-tiba. Tegasan ricih di anggap tidak berubah dengan
masa dalam kajian ini. Geseran tidak bergantung pada masa dan bergantung pada masa
juga digunakan dalam kajian ini. Struktur interaksi bendalir juga akan di analisis. Hasil
kajian menunjukan tekanan sedikit tinggi jika dibandingkan dengan kajian melalui
eksperimen yang sudah di publikasikan. Ini mungkin kerana anggapan tegasan ricih tidak
bergantung kepada masa. Keputusan akhir menunjukan apabila struktur interasi bendalir
diambil kira dalam kajian akan menghasilkan keputusan yang lebih jitu.
vi
CONTENTS
CHAPTER SUBJECT PAGE
ABSTRACT iv
LIST OF FIGURES viii
CHAPTER 1 INTRODUCTION 1
1.1 Introduction 1
1.2 Objective 3
1.3 Scope 3
CHAPTER 2 LITERATURE REVIEW 5
2.1 history of water hammer analysis 5
2.2 unsteady friction 7
2.3 fluid structure interaction 8
2.4 basic equations 11
2.4.1 Classical water hammer theory 11
2.4.2 Brunone unsteady friction model 12
2.4.3 Fluid structure interaction 13
2.4.4 Initial and boundary conditions 15
2.4.4.1 Initial conditions 15
vii
2.4.4.2 The boundary conditions 15
CHAPTER 3 Methodology 17
3.1 Numerical solution of the classical water 17
hammer with quasi steady shear stress
3.2 Numerical solution of the classical water 25
with unsteady steady shear stress
3.3 Numerical solution of fluid structure 27
interaction with quasi steady shear stress
3.4 Numerical solution of fluid structure 32
with unsteady shear stress
CHAPTER 4 Result and Discussions 39
4.1 Comparison of numerical and experimental result 39
4.2 effect of time closure and initial velocity 41
CHAPTER 5 Conclusions 43
REFERENCES 44
APPENDIX A 46
APPENDIX B 72
viii
LIST OF FIGURES
FIGURE TOPIC PAGE
3.1 interpolations of H and V values on the Δx – Δt 21
3.2 flow chart of the methodology 38
A.1 the flow chart of the programs that is used to 46
solve water hammer and FSI with steady and
unsteady friction
A.2 Computer program for solving the classical 47
water hammer by using MOC
A.3 Computer program for solving the water hammer 50
with Brunone Unsteady Friction Model by using MOC
A.4 Computer program for solving the fluid structure 55
interaction by using MOC
A.5 Computer program for solving the fluid structure 62
interaction with Brunone Unsteady Friction
Model by using MOC
B.1 Variation of piezometric head with time at: (a) the 72
downstream end; and (b) the mid-point for
V0=0.1 m/s. Experiment (black line) classical water
hammer (green line), water hammer with unsteady
friction (red line)
ix
B.2 Variation of piezometric head with time at: (a) the 73
downstream end; and (b) the mid-point for V0=0.1 m/s.
Experiment (black line) FSI with steady state friction
(blue line) FSI with unsteady friction (yellow line)
B.3 Variation of piezometric head with time at: (a) the 74
downstream end; and (b) the mid-point for
V0=0.2 m/s. Experiment (black line) classical water
hammer (green line), water hammer with unsteady
friction (red line)
B.4 Variation of piezometric head with time at: (a) the 75
downstream end; and (b) the mid-point for
V0=0.2 m/s. Experiment (black line) FSI with steady
state friction (blue line) FSI with unsteady friction
(yellow line)
B.5 Variation of piezometric head with time at: (a) the 76
downstream end; and (b) the mid-point for
V0=0.3 m/s. Experiment (black line) classical water
hammer (green line), water hammer with unsteady
friction (red line)
B.6 Variation of piezometric head with time at: (a) the 77
downstream end; and (b) the mid-point for
V0=0.3 m/s. Experiment (black line) FSI with steady
state friction (blue line) FSI with unsteady friction
(yellow line)
B.7 Variation of piezometric head with time at: (a) the 78
downstream end; and (b) the mid-point for
L=143.7 m. Experiment (black line) classical water
hammer (green line), water hammer with unsteady
friction (red line)
x
B.8 Variation of piezometric head with time at: (a) the 79
downstream end; and (b) the mid-point for
L=143.7 m. Experiment (black line) FSI with steady
state friction (blue line) FSI with unsteady friction
(yellow line)
B.9 Variation of piezometric head with time at: (a) the 80
downstream end; and (b) the mid-point for
L=77.8 m. Experiment (black line) classical water
hammer (green line), water hammer with unsteady
friction (red line)
B.10 Variation of piezometric head with time at: (a) the 81
downstream end; and (b) the mid-point for
L=77.8 m. Experiment (black line) FSI with steady
state friction (blue line) FSI with unsteady friction
(yellow line)
B.11 Variation of piezometric head with time at (a, c) the 82
downstream end and (b, d) the mid-point for
V0 =0.1 m /s. classical water hammer (green line),
water hammer with unsteady friction (red line) FSI
with steady friction (blue) and FSI with unsteady
friction (yellow).
B.12 Variation of piezometric head with velocity at (a) the 83
downstream end and (b) the mid-point for
Tc = 0.009 sec. Classical water hammer (green line),
water hammer with unsteady friction (red line) FSI
with steady friction (blue) and FSI with unsteady
friction (yellow).
B.13 Effect of time of close in the maximum pressure 84
with different initial velocity (classical water hammer
xi
B.14 Effect of time of close in the maximum pressure with 85
different initial velocity (classical water hammer with
unsteady friction)
B.15 Effect of time of close in the maximum pressure with 86
different initial velocity (FSI with unsteady friction)
B.16a Effect of time of close in the maximum pressure with 87
different initial velocity (classical water hammer,
water hammer with unsteady friction and FSI)
B.16b Effect of time of close in the maximum pressure with 88
different initial velocity (classical water hammer,
water hammer with unsteady friction and FSI)
1
CHAPTER 1
INTRODUCTION
1.1 Introduction
Pipes installed in water supply systems, irrigation networks, hydropower stations,
nuclear power stations and industrial plants are required to convey liquid reliably, safely
and economically. Modern hydraulic systems operate over a broad range of operating
regimes. Any change of flow velocity in the system induces a change in pressure. The
sudden shut-down of a pump or closure of a valve causes a pressure wave develops
which is transmitted in the pipe at a certain velocity that is determined by fluid properties
and the pipe wall material. This phenomenon, called water hammer, can cause pipe and
fittings rupture. The intermediate stage flow, when the flow conditions are changed from
one steady state condition to another steady state, is called transient state flow or transient
flow; water hammer is a transient condition caused by sudden changes in flow velocity or
pressure.
The classical theory of water hammer [1, 2] describes the propagation of
pressure waves in fully liquid filled pipe system. The theory correctly predicts extreme
pressures and wave periods, but it usually fails in accurately calculating damping and
dispersion [3] of wave fronts. In particular, field measurements usually show much more
damping and dispersion than the corresponding standard water-hammer calculations. The
2
reason is that a number of effects are not taken into account in the standard theory for
example:
Generally friction losses in the simulation of transient pipe flow are estimated by
using formulae derived for steady state flow conditions, this is known as the quasi-steady
approximation. This assumption is satisfactory for slow transients where the wall shear
stress has a quasi-steady behaviour. Experimental validation of steady friction models for
rapid transients [4, 5, 6, 7] previously has shown significant discrepancies in attenuation
and phase shift of pressure traces when the computational results are compared to the
results of measurements. The discrepancies are introduced by a difference in velocity
profile, turbulence and the transition from laminar to turbulent flow. The magnitude of
the discrepancies is governed by flow conditions (fast or slow transients, laminar or
turbulent flow) and liquid properties (viscosity) [7].
Also the waves have an acoustic pressure that acts against the surface of the pipe.
Consequently, the fluid flow and the solid surface are coupled through the forces exerted
on the wall by the fluid flow. The fluid forces cause the structure to deform, and as the
structure deforms it then produces changes in the flow. As a result, feedback between the
structure and flow occurs: action-reaction. This phenomenon what is call fluid structure
interaction that can be attributed to three coupling mechanisms [8] Friction coupling is
due to shear stresses resisting relative axial motion between the fluid and the pipe wall.
These stresses act at the interface between the fluid and the pipe wall. Poisson coupling is
due to normal stresses acting at this same interface. For example, an increase in fluid
pressure causes an increase in pipe hoop stress and hence a change in axial wall stress.[8]
The third coupling mechanism is junction coupling, which results from the reactions set
up by unbalanced pressure forces and by changes in liquid momentum at discrete
locations in the piping such as bends, tees, valves, and orifices. These include unsteady
friction and fluid structure interaction which are taken into account in this study.
In addition the discrepancies between the computed and measured water hammer
waves may originate from some other assumptions in standard water hammer, i.e. the
3
flow in the pipe is considered to be one-dimensional (cross-sectional averaged velocity
and pressure distributions), the pressure is greater than the liquid vapour pressure, the
pipe wall and liquid behave linearly elastically, and the amount of free gas in the liquid is
negligible. Also from discretization error in the numerical model, approximate
description of boundary conditions and uncertainties in measurement and input data. In
this study unsteady friction and fluid structure interaction are taken into account.
Because of the interaction between the fluid flow and the solid surface the
equations of motions describing the dynamics are coupled. This makes the problem more
challenging, and even worse when the flow is turbulent. In addition, this means that the
Navier-Stokes equation and the structure equation for the solid surface must be solved
simultaneously with their corresponding boundary conditions [9]. In this project Method
of characteristics is used to solving classical water hammer with unsteady friction and
fluid structure interaction which solved one-dimensional, four-coupled first- order, non-
linear hyperbolic partial differential equation (PDE) model, which governs axial motion
and includes Poisson, junction and friction coupling.
1.2 Objective
The objective of this project is to investigate the unsteady friction and fluid-
structure interaction that may affect water hammer wave attenuation, shape and timing
for single phase fluid in a simple reservoir-pipeline-valve system by using the method of
characteristics which compared with experimental result [3, 10]
1.3 Scope
We consider cylindrical pipes of circular cross-section with thin linearly-elastic walls and
filled with incompressible liquid, the flow velocities are small, the absolute pressures are
4
above vapour pressure and the pipe is thin walled and linear, homogeneous and isotropic
elastic.
The method of characteristics (MOC) is used to solve classical water hammer
with quasi-steady shear stress, and with unsteady shear stress. To solving FSI, we used
single procedure which treats the whole fluid–structure domain as a single entity and
describes its behaviour by a single set of equations. these are solved using a single
numerical method (MOC-MOC) The main focus will be in compare between water
hammer with and without unsteady friction and FSI at different initial velocity and time
closure and compare both with experimental results [3, 10]
5
CHAPTER 2
Literature review
In this section we apply some previous works done by other researchers which
has used as reference but most of them are briefly mentioned us, however we have
divided this section into fourth parts History of water hammer analysis, some previous
unsteady shear stress and fluid structure interaction research and the fourth part is Basic
equations.
2.1 History of water hammer analysis.
During the second half of the 19th century and the first quarter of the 20th
century, the majority of the publications on water hammer came from Europe. The
conception of the theory of surges can, amongst others, be traced to Ménabréa (1858,
1862), Michaud (1878), Von Kries (1883), Frizell (1898), Joukowsky (1900) and Allievi
(1902, 1913) [1, 2, 3, 4, 5]. Joukowsky performed classic experiments in Moscow in
1897/1898 and proposed the law for instantaneous water hammer in a simple pipe
system. This law states that the (piezometric) head rise ∆H resulting from a fast (Tc <
2L/a) closure of a valve, is given by: [1]
(1.1)
6
In which, a = pressure wave speed, V0 = initial flow velocity, g = gravitational
acceleration, L = pipe length and Tc = valve closure time. The period of pipe, 2L/a, is
defined as the return time for a water hammer wave to travel from a valve at one end of
the pipeline to a reservoir at the other end, and back to the valve.
The theoretical analyses performed independently by Joukowsky and Allievi
formed the basis for classical water-hammer theory. Joukowsky’s work was translated by
Simin in 1904. Allievi's work was not known generally outside Europe until Halmos
made an English translation in 1925. Gibson (1908) presented one of the first important
water hammer contributions in English. He considered the pressures in penstocks
resulting from the gradual closure of turbine gates [2].
In the 1930s, friction was included in the analysis of water hammer problem and
first symposium of water hammer was held in Chicageo in 1933. Topics covered included
high-head penstocks, compound pipes, surge tanks, centrifugal pump installations with
air chambers, and surge relief valves [1].
In 1937, the second water hammer symposium was held in New York with
presentations by both American and European engineers. The leaders in the field were in
attendance as paper were presented on air chambers, surge valves, water hammer in
centrifugal pump lines, and effects of friction on turbine governing [1].
During these period graphical techniques of analysis thrived under the work of
Allievi, Angus, Bergeron, Schnyder, Wood, Knapp, Paynter, and Rich. In later years
moves were made to more accurately incorporate frictional effects into the equations.
Also more sophisticated boundary conditions were employed and more general forms of
basic equations were used in analysis [1, 2].
7
2.2 Unsteady friction
Several researchers have proposed the unsteady friction models for transient pipe
flow, the early model developed by Daily, Hankey, Olive, and Jordaan [7] in which the
unsteady friction is dependent on instantaneous mean flow velocity and instantaneous
local acceleration.
In (1973) Safwat and Polder developed unsteady loss models by adding correction
terms that were proportional to fluid acceleration for laminar pipe flow [6]. Brunone,
Golia, and Greco, [7] deduced an improved version in which the convective acceleration is
added to Golia’s version of the basic Daily model. The Brunone model is relatively simple
and gives a good match between the computed and measured results using an empirically
predicted (by trial and error) Brunone friction coefficient k.
Zielke [7] found a frequency dependent model for laminar transient flows. He
applied the Laplace transform to the momentum equation for parallel axisymmetric flow
of an incompressible fluid and derived the equation which relates the wall shear stress to
the instantaneous mean velocity and to the weighted past velocity changes. The advantage
of this approach is that there is no need for empirical coefficients that are calibrated for
certain flow conditions.
The Zielke model has been modified by several researchers to improve
computational efficiency and to develop weights for transient turbulent flow. In (1993)
Vardy, Hwang, and Brown extended Zeilke’s model to moderately turbulent, unsteady
flow [7] and. In (1995, 1996) Vardy and Brown to turbulent flow at high Reynolds
numbers. The approach was to split the flow into an outer viscosity-dominated region and
an inner turbulent region characterized by uniform velocity.
Other researchers have proposed different models [6], in 1989 Suo and Wylie
proposed a frequency dependent friction factor model for application with the impedance
8
method, in 1989 Jelev argued that it was reasonable to assume that energy dissipation is
proportional to internal forces in the liquid and at the pipe wall, but with a quarter period
phase shift. Motivated by experimental results, and in 1991 Brunone assumed unsteady
shear was proportional to the mean local and convective accelerations of the fluid.
From previous research can classify the unsteady friction into six groups: [7]
1- The friction term is dependent on instantaneous mean flow velocity V (Hino et al.,
Brekke , Cocchi )
2- The friction term is dependent on instantaneous mean flow velocity V and
instantaneous local acceleration / (Daily et al., Carstens andRoller , Safwat and van
der Polder , Kurokawa and Morikawa , Shuy and Apelt , Golia , Kompare ,
3- The friction term is dependent on instantaneous mean flow velocity V,
instantaneous local acceleration / and instantaneous convective acceleration /
(Brunone et al., Bughazem and Anderson )
4- The friction term is dependent on instantaneous mean flow velocity V and
diffusion (Vennatrø , Svingen )
5- The friction term is dependent on instantaneous mean flow velocity V and weights
for past velocity changes W(τ) (Zielke , Trikha , Achard and Lespinard, Arlt , Kagawa et
al., Brown , Yigang and Jing-Chao, Suzuki et al., Schohl , Vardy, Vardy et al., Vardy and
Brown, Shuy, Zarzycki.
6- The friction term is basedon cross-sectional distribution of instantaneous flow
velocity (Wood and Funk , Ohmi et al. Bratland, Vardy and Hwang, Eichinger and Lein
Vennatrø, Silva-Araya and Chaudhry, Pezzinga).
2.3 Fluid structure interaction
9
Three coupling mechanisms determine fluid structure interaction in pipelines.
Friction coupling is due to shear stresses resisting relative axial motion between the fluid
and the pipe wall. These stresses act at the interface between the fluid and the pipe wall.
Poisson coupling is due to normal stresses acting at this same interface. For example, an
increase in fluid pressure causes an increase in pipe hoop stress and hence a change in
axial wall stress. Junction coupling takes place at pipe boundaries that can move, either in
response to changes in fluid pressure or because of external excitation. The literature
review concerning these follows.
In liquid-filled pipes, Poisson coupling results from the transformation of the
circumferential strain, caused by internal pressure, to axial strain and is proportional to
Poisson's ratio. Skalak [11] was among the first to extend Joukowski's method to include
Poisson coupling. Williams [8] conducted a similar study. He found that structural
damping caused by longitudinal and flexural motion of the pipe was greater than the
viscous damping in the liquid. These researchers did not include the radial inertia of the
liquid or the pipe wall. Lin and Morgan [8] included the pipe inertia term and the
transverse shear in their equations of motion. Their study was restricted to waves which
have axial symmetry and purely sinusoidal variation along axis. Walker and Phillips [8,
12] extended the study by Lin and Morgan to include both the radial inertia of the pipe
wall in the fluid and the axial equations of motion. Their interest in short duration,
transient events produced a one-dimensional, axisymmetric system of six equations.
Vardy and Fan [8, 13, and 14] conducted experiments on a straight pipe, generating a
pressure wave by dropping the pipe onto a massive base. Their results showed good
agreement with the analytical model by Wilkinson and Curtis. Wiggert and Otwell [8]
neglected the radial acceleration in their studies using the six equation model of Walker
and Phillips. This simplification reduced the mathematical model to four equations. Budny
[8, 13] also reduced the six-equation model, but he included viscous damping and a fluid
shear stress term to account for the structural and liquid energy dissipation. Experimental
tests verified that the model satisfactorily predicts the wave speeds, fluid pressure, and
10
structural velocity of a straight pipeline for several fluid periods after a transient has
excited the fluid.
The junction coupling mechanism is generated by the pressure resultants at
elbows, reducers, tees, and orifices act as localized forces on the pipe. For pipes with only
a few bends, a continuous representation of the piping was devised by Blade, Lewis, and
Goodykoontz [8]. Experimental tests were conducted to analyze the response of an L-
shaped pipe to harmonic loading. The experimental setup included a restricting orifice
plate at the downstream end of the pipe. They concluded that an uncoupled analysis does
not produce accurate estimates of natural frequencies, and that the elbow, which provides
coupling between the pipe motion and liquid motion, causes no appreciable reflection,
attenuation, or phase shift in the fluid waves . Wiggert, Hatfield, Lesmez, and Wiggert,
Hatfield and Stuckenbruck[8, 12, 13, and 15] used a one dimensional wave formulation in
both the liquid reaches and the piping structure resulting in five wave components and
fourteen variables. The method of characteristics was used to solve for the fourteen
variables and to find the expressions for the wave speeds. Joung and Shin [8] developed a
model that takes into account the shear and flexural waves of an elastic axisymmetric tube.
The method of characteristics was used in the solution for four families of propagating
waves. Their results compared closely to Walker and Phillips results for relatively small
pipe deformations.
Wood [8] studied a pipe structure loaded with a harmonic excitation. He found that
the natural frequencies of liquid were shifted, especially when the frequency of the
harmonic load is near one of the natural frequencies of the supporting structure. Ellis [8]
reduced a piping structure to equivalent springs and masses by selectively lumping mass
and stiffness at fittings and releasing specific force components at bends, valves and tees.
His formulation of axial response was a modification of the method of characteristics and
included pipe stresses and velocities. The finite element method is used to model the
structure, treating each pipe element as a beam. Schwirian and Karabin [8] generalized
this approach by using a finite element representation of the liquid and the piping. Their
11
studies imposed coupling at fittings only. The effect of the supports and piping stiffness
was shown to be significant.
2.4 Basic equations
2.4.1 Classical water hammer theory
The classical theory of liquid transient flow in pipelines (water hammer) usually
draws on the following basic assumptions [1, 2]
1-The liquid flow is one-dimensional with cross-sectional averaged velocity and pressure
distributions
2- Unsteady friction losses are approximated as quasi-steady state losses.
3- The pipe is full and remains full during the transient.
4- There is no column separation during the transient event, i.e. the pressure is greater than
the liquid vapour pressure.
5- Free gas content of the liquid is small such that the wave speed can be regarded as a
constant.
6- The pipe wall and the liquid behave linearly elastically.
7- Structure-induced pressure changes are small compared to the water hammer pressure
wave in the liquid.
Water hammer equations include the continuity equation and the equation of motion. [1,
2]
(2.2)
(2.3)
12
Where, P=pressure, V=averaged fluid velocity, =the pipe inclination angle,
=acceleration due to gravity, =density of fluid, =velocity of pressure wave is defined
by
(2.4)
Where k is a parameter depending on the constraint conditions [1, 2]
: is the shear stress between the pipe and fluid and expressed as the quasi steady part
(2.5)
Where f =friction factor
2.4.2 Brunone Unsteady Friction Model
The shear stress explicitly used in equations (2.2) and (2.3) is expressed as the sum
of the quasi-steady part and the unsteady part (called the full friction
coupling model hereinafter). The computation of the quasi-steady shear stress is
straightforward, whereas the unsteady shear stress is related to the instantaneous local
(temporal) acceleration and instantaneous convective (spatial) acceleration [3], i.e.
(2.6)
In which sign (V) = {+1 for V ≥ 0 and -1 for V < 0} the Brunone friction coefficient can
be predicted either empirically by the trial and error method or analytically using Vary,s
shear decay coefficient C
13
(2.7)
The Vardy,s shear decay coefficient C from[7] is:
- Laminar flow C=0.00476
- Turbulent flow
In which Re=Reynolds number (Re=VD/v)
2.4.3 Fluid-structure interaction
Normally when it is desired to obtain the fluid velocity in a pipe, equations are
applied with the assumption of no wall deformation. If the walls deform, the deformation
will affect fluid thus creating a fluid structure interaction. The following extended water
hammer equations are obtained [9, 10]
(2.8)
(2.9)
Where
v : is the Poisson,s ratio
is the axial pipe velocity
14
E: is Young,s modulus
Kf: is bulk modulus
In the pipe domain, the equations of motion for the pipe in the axial direction is given by
Wiggert et al [9, 10]
(2.10)
(2.11)
Where
: is the density of the pipe
: is the axial stress wave (precursor wave) speed i.e.
: is the shear stress between the pipe wall and fluid, and expressed as
the sum of the quasi-steady part and unsteady part
the computation of the quasi-steady shear stress is
(2.12)
And the unsteady shear stress is
(2.13)
The last terms on the left-hand side of Equations (2.8) and (2.9) represent shear stress
coupling whereas the last terms in the left-hand side of Equations (2.7) and (2.10) denote
Poisson coupling.
15
2.4.4 Initial and boundary conditions
All these equations are solved subject to boundary conditions at the upstream and
downstream ends of the pipeline and initial conditions
2.4.4.1 Initial conditions
2.4.4.1 The boundary conditions
For a pipeline connected to a reservoir and a valve at the upstream and
downstream ends, respectively.
While the pipeline with a fixed valve and a reservoir at the upstream and
downstream ends, respectively,
16
17
CHAPTER 3
Methodology
The basic equations presented in the previous chapter can be numerically solved in
many ways. For fluid-structure interaction problems the fluid equations are often solved
by the method of characteristics (MOC) and the structural equations by the finite method
(FEM) but it is advantageous to use one method for all equations. In this study we have
used method of characteristics (MOC) to solve both the fluid domain and pipe domain.
This chapter is divided into four sections to show how all these equations can be
solved by method of characteristics which the partial differential equations transform into
ordinary differential equations along characteristics line. First the classical water hammer
is described and then water hammer with unsteady friction, and fluid structure interaction.
The fourth one is fluid structure interaction with unsteady friction.
3.1 Numerical solution of the classical water hammer with quasi-steady shear
stress
From equation of motion and equation of continuity (2.2, 2.3), there are two
independent variables, x and t, and two dependent variables, P and V. Other variables are
characteristics of the conduit system and are time invariant but may the function of x.
Laboratories test have shown that wave velocity, af , is significantly reduced by reduction
18
of pressure even when it remains above the vapor pressure. The friction factor, f, varies
with the Reynold number, but f is considered constant because the effects of such a
variation on the transient state conditions are negligible. With a nonlinear resistance terms
for friction and other effects, no general solution to these equations is known, but they are
solved by the method of characteristics for a convenient finite-difference solution with the
digital computer due to its large storage capacity and its ability to operate at very high
computing rates. In the method of characteristics, the partial differential equations are first
converted into ordinary differential equations, which are then solved by an explicit finite-
difference technique. Since each boundary condition and each conduit section are
analyzed separately during a time step,
The simplified equations of motion and continuity are identified as L1 and L2 from
equations 2.2 and 2.3 as
(3.14)
(3.15)
Where
(3.16)
These equations are combined linearly using an unknown multiplier λ
Regrouping terms, in the equation in a particular manner
19
(3.17)
Both variables V and P are functions of x and t, and the independent variable x is a
functions of t, from calculus
(3.18)
(3.19)
Note that if
Is to be replaced by
Then
And if
20
Is to replaced by
Then
Rewriting the restriction equations for dx/dt,
And
Equating the two expressions for dx/dt and solving we get
Our characteristic equations are in this case
And
The result is that we have replaced two partial differential equations with two
ordinary differential equations provided we follow certain rules which relate the
independent variables x and t in each case. If, in addition, we replace P with then
we can visualize better the propagation of the pressure waves because H is the height of
the EL-HGL above the datum. This substitution gives
only if (3.20)
only if (3.21)
21
The numerical solution procedure first assumes that we can approximate the
characteristic curve as straight lines over each time interval. This assumption appears
to be promising because a>>V, however, it should be carefully noted that the slope of
each characteristic is generally slightly different than that of any other. The problem this
creates in the finite difference approximation to the differential equations can be seen on
figure 3.1. The procedure is to find the value of V and H at new point (P) the curved
characteristics intersecting at P are approximated by straight lines. The slope of straight
lines is determined by the known value of velocity at the earlier time. It is important to
note that the characteristics passing through P do not pass through the grid points Le and
Ri, but pass through the t= constant line at points L and R somewhere in between.
Figure 3.1 interpolations of H and V values on the Δx – Δt
In this case the finite difference approximation to equations (2.20) and (3.21) becomes
(3.22)
(3.23)
The values VL, HL, VL, and HL are not known. However the values of VLe, HLe,
VRi, HRi, Vc and Hc are known. The unknown values of H and V at points L and R can be
22
estimated by interpolation in this case we will use linear interpolation and the sketch
below illustrates the relationships
Considering the C+ characteristic
Where
Solving for VL and HL gives
And
Substituting the value of gives
23
(3.24)
And
(3.25)
A similar analysis for the C- characteristic gives
(3.26)
(3.27)
Because is of order which is very small compared to one it
is good approximation to neglect the second terms in the denominators of equations (3.24)
and (3.26). The result is
(3.28)
(3.29)
The simultaneous solution of equations (3.22) and (3.23) for VP and HP gives
(3.30)
(3.31)
24
The value of is still determined by the number of sections into which we have
chosen to divide the pipe. Because our interpolation procedure implies that the points R
and L are between points Ri and Le, we must choose so small as to guarantee this
always occurs. The equations suggest that [2]
(3.32)
Where is the maximum expected absolute value of the sum of the
wave speed and flow velocity. If locations of points R and L fall outside the grid points Le
and Ri, numerical stability problems and accuracy problems begin to develop.
The computer program which is used to solve the water hammer equations by using
method of characteristics is presented in Figure (A.2)
In summary the program reads in the basic information, generates steady state H
and V values at the grid intersection points (nodes) along the pipe and then begins the
unsteady flow calculations. The interior grid intersection points are first calculated using
equations (3.30) and (3.31). The upstream and downstream boundary conditions are used
to get values of HP and VP at each end of pipe to simulate transient problems with other
boundary conditions, it is necessary only to change those parts of the program listed under
upstream and downstream boundary conditions. The whole process begins again using the
just- computed values of HP and VP as the known values. The process continues to loop
until the time has reached Tmax. Before execution is terminated values of H and V are
printed in the matrices H_OUT and V_OUT respectively for each node. The output of this
program is the variation of piezometric head with time at the downstream end and mid
point for different time closure and initial velocity.
25
3.2 Numerical solution of the classical water hammer with unsteady-steady shear
stress
In this case the shear stress is expressed as the sum of the quasi-steady part
equation (2.5) and unsteady part equation (2.6) and the friction coefficient for unsteady
shear stress has to determined as demonstrated by Brunone [7], so the equation of motion
and equation of continuity are
(3.33)
(3.34)
Where ks is the sign of i.e. for whereas for
Again the multiplier λ is used to combine the partial differential equations.
Multiplying λ by equation (3.34) and adding the result to equation (3.33) gives
To carry forth the same procedure as was used previously, we must break and
down into component parts. The result is
As before,
26
If
And
If
Rewriting the restriction equations for
And
Equating the two expressions for and solving, we get
So our characteristic equations are in this case
(3.35)
(3.36)
27
The final set of equations which compare with equations (3.20) and (3.21) after
replacing P with , are
(3.37)
(3.38)
The computer program for solution of the water hammer with unsteady friction is
similar in most respects to the program used with the classical water hammer the main
differences occur in the requirements to include the Brunone friction coefficient . The
computer program is shown in Figure (A.3)
3.3 Numerical solution of Fluid-structure interaction with quasi-steady shear
stress
The four-coupled first-order, non-linear hyperbolic partial differential equation
(2.8), (2.9), (2.10), and (2.11) which the friction coupling is mainly based on the
assumption where shear has a quasi steady behaviour can be transform into ordinary
differential equations to determine V, P, , and as following
(3.39)
(3.40)
(3.41)
28
(3.42)
Where
Both variables V, P, , and σ are functions of x and t, and the independent variable x is a
functions of t, from calculus
(3.43)
(3.44)
(3.45)
(3.46)
Equations (3.39) through (3.46) can be expressed in matrix form as
29
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
dtdx
dtdx
dtdx
dtdx
ahEvD
V
vaaV
PP
P
ff
10.00.00.00.00.00.0
0.00.010.00.00.00.0
0.00.00.00.010.00.0
0.00.00.00.00.00.01
0.0110.00.00.00.02
10.00.010.00.00.00.0
0.00.00.00.0110.0
0.00.020.00.01
2
f
2f
2f
ρ
ρ
ρ
ρρ
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂
x
t
xutuxVtVxPtP
σ
σ
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
dtddtdudtdVdtdP
C
σ
ω
0.0
0.0
(3.47)
Or
[ ]Q
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂
x
t
xutuxVtVxPtP
σ
σ
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
dtddtdudtdVdtdP
C
σ
ω
0.0
0.0
(3.48)
30
Where
This system will have a unique solution if the determinant of the matrix [Q] is
nonzero. On the other hand, the system will have an infinite number of solutions, if the
determinant of the matrix [Q] is zero.
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
dtdx
dtdx
dtdx
dtdx
ahEvD
V
vaaV
PP
P
ff
10.00.00.00.00.00.0
0.00.010.00.00.00.0
0.00.00.00.010.00.0
0.00.00.00.00.00.01
0.0110.00.00.00.02
10.00.010.00.00.00.0
0.00.00.00.0110.0
0.00.020.00.01
2
f
2f
2f
ρ
ρ
ρ
ρρ
=0.0 (3.49)
A characteristic is defined as a curve along which the determinant of the matrix
[Q] is zero. Thus the direction of the characteristics can be found from equation (3.49).
The slopes of characteristics are defined by
(3.50)
31
The four roots of this equation (3.50) are denoted as and
. The curve along which the slope equal to , - , or - is
termed as the characteristic. Thus, there will be four families of characteristic curves in the
domain (x, t-plane) of the problem
When equation (3.48) holds, Cramer’s rule implies that a solution of equation
(3.47) cannot be obtained unless the determinants of the matrices obtained by substituting
the right hand column of equation (3.47) into the first, second, third, or fourth column of
the matrix [Q] are equal to zero
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
dtdx
dtd
dtdx
dtdu
dtdx
dtdV
dtdx
dtdP
a
C
V
vaaV
PP
P
ff
10.00.00.00.00.0
0.00.010.00.00.0
0.00.00.00.010.0
0.00.00.00.00.00.0
0.0110.00.00.00.00.0
10.00.010.00.00.0
0.00.00.00.0110.00.020.00.00.0
2
f
2f
2f
σ
ρ
ρ
ρω
ρρ
=0.0 (3.51)
The equation (3.51) yield after replacing P with
(3.52)
32
(3.53)
Where and
It can see that the hyperbolic partial differential equations (3.39) to (3.42) are now
replaced by the ordinary differential equations (3.52) and (3.53). The method of
characteristics involves the determination of the characteristic curves as x=x(t) by
integrating equation (3.50) and the solution the solution of equations (3.52) and (3.53) by
integration along the characteristic curves. The computer program for solving these
equations is illustrated in the Figure (A.4)
3.4 Numerical solution of Fluid-structure interaction with unsteady shear stress
In this case we consider both the unsteady friction and fluid structure interaction
which the continuity equations and motions equations for the fluid and structural are
(3.54)
(3.55)
33
(3.56)
(3.57)
Both variables V, P, , and σ are functions of x and t, and the independent
variable x is a functions of t, from calculus
(3.58)
(3.59)
(3.60)
(3.61)
Equations (3.54) through (3.61) can be expressed in matrix form as
34
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
−Δ
−
dtdx
dtdx
dtdx
dtdx
a
vaaV
PP
P
ff
10.00.00.00.00.00.0
0.00.010.00.00.00.0
0.00.00.00.010.00.0
0.00.00.00.00.00.01
0.0110.00.00.00.0
10.010.00.0
0.00.010.0
0.00.020.00.01
2
f
2f
2f
ρθ
ρψφδψφδ
ζδβρ
ρρ
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂
x
t
xutuxVtVxPtP
σ
σ
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
dtddtdudtdVdtdP
C
σ
ω
0.0
0.0
(3.62)
Where
35
The direction of the characteristic curve can be obtained by setting the determinant
of the coefficient matrix in equation (3.62) equal to zero
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
−Δ
−
dtdx
dtdx
dtdx
dtdx
a
vaaV
PP
P
ff
10.00.00.00.00.00.0
0.00.010.00.00.00.0
0.00.00.00.010.00.0
0.00.00.00.00.00.01
0.0110.00.00.00.0
10.010.00.0
0.00.010.0
0.00.020.00.01
2
f
2f
2f
ρθ
ρψφδψφδ
ζδβρ
ρρ
=0.0 (3.63)
Expanding the determinant we obtain
(3.64)
The four roots of this equation (3.64) are denoted as and
. The curve along which the slope equal to , ,
36
, or is termed as the characteristic. Thus, there will be four
families of characteristic curves in the domain (x, t-plane) of the problem.
As in the case of previous section, if the determinant of the coefficient matrix is zero in
equation (3.62), the right-hand side must be compatible with this in order to have a
solution to equation (3.62). This implies that when the right-hand side is resulting matrix
must be zero. For example, when the fourth column of the matrix on the left side is
replaced by right-hand-side column of equation (3.62) and the determinant is set to zero,
we obtain
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
−Δ
−
dtdx
dtd
dtdx
dtdu
dtdx
dtdV
dtdx
dtdP
a
C
vaaV
PP
P
ff
10.00.00.00.00.0
0.00.010.00.00.0
0.00.00.00.010.0
0.00.00.00.00.00.0
0.0110.00.00.00.00.0
10.010.0
0.00.010.00.020.00.00.0
2
f
2f
2f
σ
ρ
ρψφδψφδ
ζδβρ
ω
ρρ
=0.0 (3.65)
Which yields, upon expansions
37
V
(3.66)
V
(3.67)
When and . Are substituted in equations (3.66) and
(3.67), and replace P with , we obtain four ordinary differential equations to
determine H, V, , and along the and characteristics. Thus the system of
hyperbolic partial differential equations and solved by solving four ordinary differential
equations. The computer program for solving these equations is shown in the Figure (A.5)
The summary of the this chapter has be illustrated in the Figure (3.2) which we
have started with classical water hammer to find the numerical expression by using
method of characteristic and used same procedure to find the numerical expressions for
38
the water hammer with unsteady shear stress , fluid structure interaction with quasi-steady
shear stress and with unsteady shear stress. Brunone Unsteady Friction Model had been
used. Figure (A.1) show the flow chart of the programs that is used to solve these
problems which the main different between these programs are the numerical expressions.
Figure (3.2) flow chart of the methodology
39
CHAPTER 4
RESULT AND DISCUSSIONS
In the first part of this chapter will be presenting the numerical results of water
hammer with and without unsteady friction and fluid structure interaction with and
without unsteady friction that are compared with experimental result [3, 10]. The effect of
time closure and initial velocity are presented in the second part.
\4 -1 comparison of numerical and experimental result.
In the first example we considers a straight sloping copper pipeline connected with
a tank and a ball valve located at the upstream and downstream ends, respectively,
Pressures in the pipeline evolves under the action of valve closure with closure time Tc
=0.009s. The physical and geometric parameters, used in the experimental tests [3, 10],
are as follows:
40
First we examine the variation of pressure with time for transient laminar flow at
initial steady flow velocity with Reynolds number . The
computational results obtained by the classical water hammer and water hammer with
unsteady friction figures B.1(a) and B.1(b) agree well with the experimental results for the
first and the second pressure head rise. The discrepancies between them and the
experimental results are magnified for later time. Also can be seen the water hammer with
unsteady friction is better agreement than the classical water hammer in phase shift and
there is slight difference between them in amplitude. Figures B.2 (a) and B.2(b) show the
comparison of experimental results [3, 10] with theoretical results obtained using the fluid
structure interaction with full-friction coupling model and the partial-friction coupling
model. It can be seen that there is better agreement between the experiment and the full-
friction coupling model (there is almost no phase shift and slight difference in amplitude).
Discrepancies between the experiment and the theoretical results increase with time.
Second comparison of numerical and experimental results for transient turbulent
flow at initial steady flow velocity (law Reynolds number
turbulent flows). Figures B.3 and B.5 show the comparisons of the experimental results [3,
10] with the theoretical results obtained using the classical water hammer and water
hammer with unsteady friction for low Reynolds number turbulent flow
respectively. The computational results from the FSI with full-
friction coupling model and the partial-friction coupling model for low Reynolds number
turbulent flow are compared with results of measurements and
are depicted in figures B.4 and B.6 respectively. Similar trends in the variation of pressure
with time hold, but discrepancies between the experiment and the partial-friction coupling
41
model are magnified compared to the results for laminar flow. And also between FSI with
full- friction coupling model and classical water hammer.
The second example we consider is a pipeline equipped with a ball valve and a
tank located at the upstream and downstream ends, respectively (the opposite of the
previous example). Its geometrical and physical parameters, used by Pezzinga and
Scandura [10], are:
, , , ,
, , , ,
, , , ,
, .
The variation of pressure with time for transient turbulent flow at initial steady
flow velocity is examined. Figures B.7, B.8, B.9 and B.10 show the
comparison between our numerical results and the experimental results of Pezzinga and
Scandura [10] for L=143.7 and 77.8 m, respectively. Water hammer event starting with an
initial small magnitude of pressure is initiated by rapid valve closure. Similar behaviour,
which is illustrated in Figures B.1-B.6, is observed.
4-2 effect of time closure and initial velocity
The physical and geometric parameters given in the first example are used with
different time closure of valve to see the effect of time closure in the pressure traces.
Figure B.11 shows the comparison of pressures obtained using the classical water
42
hammer, water hammer with unsteady friction, and FSI with steady friction and unsteady.
It can be seen that transient shear stress damps pressure fluctuation and decelerates
pressure wave propagation more at a smaller valve closure time. Hence the influence of
transient shear stress can be significant and varies considerably, depending on the valve
closure time.
To obtain the effect of initial velocity we used the physical and geometric
parameters given in the first example with different initial velocity [0.1, 0.2, and 0.3] as
shown in the Figure B.12. It can be seen the effect of unsteady friction and fluid structure
interaction increase as the initial velocity increase
Figures B.13, B.14, and B.15 show the maximum pressure as a function of initial
velocity for different closing time. For classical water hammer, water hammer with
unsteady friction, and fluid structure interaction with unsteady friction. They can be seen a
linear behavior between maximum pressure and the initial velocity. In these Figures are
evident that the faster the close time higher is the pressure. Also, there isn’t different
between closing valve at 0, 0.005 and 0,009 and the difference between closing the valve
at 0.35 and 0.4 second is minimal thus, .009and 0.34 second may be taken as the critical
values. As expected the fluid will tend to increase it pressure at higher velocities. Figures
B.16, and B.17 compare between the maximum pressure of classical water hammer, water
hammer with unsteady friction and fluid structure interaction with unsteady friction at
different initial velocity and time closure from these figures we can see the higher
maximum pressure occur with classical water hammer than water hammer with unsteady
friction and fluid structure interaction.
43
CHAPTER 6
CONCLUSIONS
The method of characteristics is employed to solve the water hammer with quasi-
steady friction model and Brunone model and FSI with quasi-steady friction model and
Brunone model which compared with published data from a fast valve closure for laminar
flow (0.1 m/s) and low Reynolds number turbulent flows ( 0.2, and 0.3 m/s). The
unsteady shear stress part, which are related to the instantaneous relative local acceleration
and the instantaneous relative convective acceleration, and the steady shear stress part, act
collectively to damp pressure fluctuation and reduce phase shift of pressure traces. FSI
with full friction coupling is showed better result than that of FSI with quasi steady
friction and its effect improved as initial velocity increase. The influence of transient shear
stress, which consists of the steady and unsteady shear stress parts, can be significant and
varies considerably, depending on the valve closure time. The agreement between
experiment and theory shows that the MOC offers an alternative way to investigate the
behaviour of transient flow in pipe system with fluid–structure interaction.
44
LIST OF REFERENCES
[1] M. Hanif Chaudhry (1979), applied hydraulic transients. New York_Van Mostrand
Reinhond, 1997
[2] Gary Z.Watters (1984). Analysis and control of unsteady flow in pipelines. 2nd
Boston,Mass_buttbrworths
[3] Anton Bergant and Arris Tijsseling.(2001). Parameters Affecting Water Hammer Wave
Attenuation, Shape and Timing
[4] Anton Bergunt, Arris Tijsseling, John Vitkovsky, Didia Covas, Angus Simpson, and
Martin Lambert.(2003). Further Investigation of Parameters Affecting Water
Hammer Wave Attenuation, Shape and Timing part 1: Mathematical Tools”
[5] Anton Bergunt, Arris Tijsseling, John Vitkovsky, Didia Covas, Angus Simpson, and
Martin Lambert. (2003). Further Investigation of Parameters Affecting Water
Hammer Wave Attenuation, Shape and Timing part 2: case studies”
[6] Bruno Brunone, Bryan W.Karney, Michele Mecarelli, and Marco Ferrante (2000).
Velocity Profiles and Unsteady Pipe Friction in Transient Flow. Journal of water
resources planning and management
[7] Anton Bergant, Angus Ross Simpson, and John Vttkovsky (2000, 2001).
Developments in Unsteady Pipe Flow Friction Modeling. Journal of Hydraulic
Research, vol.39,2001, No, 3
45
[8] Arris Tijsseling (1996) fluid-structure interaction in liquid-filled pipe systems: a
review. Journal of fluids and structures (1996) 10,109-146
[9] A.S. Tijsseling. “Water hammer with fluid-structure interaction in thick-walled pipes
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[10] YongLiang Zhang and K. Vairavamoorthy (2005). Analysis of transient flow in
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[11] Arris Tijsseling, Martin Lambert, Angus Simpson, Mark Stephens, john Vitkovsky,
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[13] David C Wiggert and Arris S Tijsseling. (2001) fluid transients and fluid-structure
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[14] L. Zhang, A. S. Tijsseling , and A. E. Vardy(1999). “FSI analysis of liquid-filled
pipes” journal of sound and vibration
[15] A. G. T. J. Heinsbroek(1997) ”fluid-structure interaction in non-rigid pipeline
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46
Figure (A.1) the flow chart of the programs that is used to solve water hammer and FSI
with steady and unsteady friction
47
Figure-B.1 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.1 m/s. Experiment (black line)[3, 10] classical water hammer
(green line), water hammer with unsteady friction (red line)
48
Figure-B.2 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.1 m/s. Experiment (black line)[3, 10] FSI with steady state friction
(blue line),FSI with unsteady friction (yellow line)
49
Figure-B.3 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.2 m/s. Experiment (black line)[3, 10] classical water hammer
(green line), water hammer with unsteady friction (red line)
50
Figure-B.4 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.2 m/s. Experiment (black line)[3, 10] FSI with steady state friction
(blue line),FSI with unsteady friction (yellow line)
51
Figure-B.5 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.3 m/s. Experiment (black line)[3, 10] classical water hammer
(green line), water hammer with unsteady friction (red line)
52
Figure-B.6 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for V0=0.3 m/s. Experiment (black line)[3, 10] FSI with steady state friction
(blue line),FSI with unsteady friction (yellow line)
53
Figure-B.7 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for L=143.7 m. Experiment (black line)[10] classical water hammer (green
line), water hammer with unsteady friction (red line)
54
Figure-B.8 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for L=143.7 m Experiment (black line)[10] FSI with steady state friction
(blue line),FSI with unsteady friction (yellow line)
55
Figure-B.9 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for L=77.8 m. Experiment (black line) [10] classical water hammer (green
line), water hammer with unsteady friction (red line)
56
Figure-B.10 Variation of piezometric head with time at: (a) the downstream end; and (b)
the mid-point for L=77.8 m Experiment (black line) [10] FSI with steady state friction
(blue line), FSI with unsteady friction (yellow line)
57
Figure-B.11 Variation of piezometric head with time at (a, c) the downstream end and (b,
d) the mid-point for V0 =0.1 m /s. classical water hammer (green line), water hammer with
unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady
friction (yellow).
58
Figure-B.12 Variation of piezometric head with velocity at (a) the downstream end and (b)
the mid-point for Tc = 0.009 sec. Classical water hammer (green line), water hammer with
unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady friction
(yellow).
59
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120
40
60
80
100
120
140
160
180
initial velocity m/sec
max
imum
pre
ssur
e he
ad m
Tc=0.0secTc=0.005secTc=0.009secTc=0.1secTc=0.15secTc=0.2secTc=0.25secTc=0.35secTc=0.4
Figure B.13 Effect of time of close in the maximum pressure with different initial velocity
(classical water hammer)
60
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120
40
60
80
100
120
140
160
180
initial velocity m/sec
max
imum
pre
ssur
e he
ad m
Tc=0.0secTc=0.005secTc=0.009secTc=0.1secTc=0.15secTc=0.2secTc=0.25secTc=0.35Tc=0.4sec
Figure B.14 Effect of time of close in the maximum pressure with different initial velocity
(classical water hammer with unsteady friction)
61
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120
40
60
80
100
120
140
160
180
initial velocity m/sec
max
imum
pre
ssur
e he
ad m
Tc=0.0secTc=0.005secTc=.009secTc=0.1secTc=0.15secTc=0.2secTc=0.25secTc=0.35secTc=0.4sec
Figure B.15 Effect of time of close in the maximum pressure with different initial velocity
(FSI with unsteady friction)
62
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120
40
60
80
100
120
140
160
180
initial velocity m/sec
max
imum
pre
ssur
e he
ad m
(W H)Tc=0.009sec(W H)Tc=0.1sec(W H)Tc=0.15sec(UNF)Tc=0.009sec(UNF)Tc=0.1sec(UNF)Tc=0.15sec(FSI)Tc=0.009sec(FSI)Tc=0.1sec(FSI)Tc=0.15sec
Figure B.16a: Effect of time of close in the maximum pressure with different
initial velocity (classical water hammer, water hammer with unsteady friction and
FSI)
63
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 130
35
40
45
50
55
60
65
70
initial velocity m/sec
max
imum
pre
ssur
e he
ad m
(W H)Tc=0.2sec(W H)Tc=0.25sec(W H)Tc=0.35sec(UNF)Tc=0.2sec(UNF)Tc=0.25(UNF)Tc=0.35sec(FSI)Tc=0.2sec(FSI)Tc=0.25sec(FSI)Tc=0.35sec
Figure B.16b: Effect of time of close in the maximum pressure with different
initial velocity (classical water hammer, water hammer with unsteady friction and
FSI)