TRANSSIENT ANALYSIS OF FLUID STRUCTURE INTERACTION ...

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

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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

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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.

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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.

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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


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)





friction (red line)



V0=0.2 m/s. Experiment (black line) FSI with steady

state friction (blue line) FSI with unsteady friction

(yellow line)





friction (red line)



V0=0.3 m/s. Experiment (black line) FSI with steady


(yellow line)



L=143.7 m. Experiment (black line) classical water


friction (red line)

x



L=143.7 m. Experiment (black line) FSI with steady


(yellow line)



L=77.8 m. Experiment (black line) classical water


friction (red line)



L=77.8 m. Experiment (black line) FSI with steady


(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

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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)

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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

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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

“ Eindhoven University of Technology,

[10] YongLiang Zhang and K. Vairavamoorthy (2005). Analysis of transient flow in

pipelines with fluid structure interaction using method of lines, Int. J. Numer.

Meth. Engng 2005; 63:1446–1460

[11] Arris Tijsseling, Martin Lambert, Angus Simpson, Mark Stephens, john Vitkovsky,

and Anton Bergant. “wave front dispersion due to fluid-structure interaction in

long liquid-filled pipelines’

[12] B Sreejith, K Jayaraj, N Ganesan, and Padmanabhan (2004). Finite element analysis

of fluid-structure interaction in pipeline systems, nuclear engineering and design

[13] David C Wiggert and Arris S Tijsseling. (2001) fluid transients and fluid-structure

interaction in flexible liquid-filled piping’. American Society of Mechanical

Engineers

[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

systems” nuclear engineering and design

[16] C.J. Greenshields and H. G. Weller (2005). A unified formulation for continuum

mechanics applied to fluid structure interaction in flexible tubes. Int. J. Numer.

Meth. Enging 2005;64:1575-1593. Published online 29 July 2005 in Wiley

interscience

[17] K. Namkoong, H. G. Choi, and J. Y. Yoo (2005). Computation of dynamic fluid

structure interaction in two dimensional laminar flows using combined

formulation. Journal of fluids and structures 20(2005)51-59

[18] S.Mttal, and T. E. Tezduyar (1995). Parallel finite element simulation of 3D

incompressible flows: fluid structure interactions. International Journal for

numerical methods in fluids, vol. 21, 933-953 (1995)

[19] M. P. Paidoussis (2005). Some unresolved issues in fluid structure interactions.

Journal of fluids and structures 20 (2005) 871-890

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


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




50




51




52




53


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


the mid-point for L=143.7 m Experiment (black line)[10] FSI with steady state friction


55


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


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


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


(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


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


(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


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


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)

TRANSSIENT ANALYSIS OF FLUID STRUCTURE INTERACTION ...

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