plan de recherche / research plandidia/fct2014_ariete/22_davidferras_one-year... · février 2013...

Février 2013

EPFL - ENAC - EDCE GC A2 454 - Station 18 - CH-1015 LAUSANNE / SuisseEmail: [email protected]

PLAN DE RECHERCHE /RESEARCH PLAN

Unsteady friction and pipe rheological behaviour in pressurized transient flows

Candidat(e) / Candidate : David Ferras

Directeur de thèse / Thesis director : Prof. Anton Schleiss

Co-directeur de thèse / Thesis co-director : Prof. Dídia Covas

Directeur du Programme/ Program’s Director : ………………………………………………………

Date d'immatriculation / Date of enrolment : 01/10/2012

Nom du mentor / Mentor’s name : Prof. Michael Lehning

Signatures :

Directeur de thèse / Thesis director : ..............................................

Co-directeur de thèse / Thesis co-director : ……………………………………………………..

Candidat(e) / Candidate : ..............................................

Directeur du Programme doctoral /Doctoral program Director : ..............................................

Date : .....................................................................................................

Un exemplaire, maximum 20 pages, de ce plan de recherche doit être adressé, après signatures par le Directeur de thèse et co-directeur (si applicable), le Candidat, au Programme Doctoral Civil et Environnement. Le plan de recherche est envoyé par e-mail en version pdf à [email protected], seule cette page de couverture dûment signée est envoyée au programme par courrier normal.

After signature by the thesis director, the thesis co-director (if applicable) and the candidate, one copy, maximum 20 pages, of this research plan must be sent to Doctoral program Civil and Environmental engineering. The research plan is to be sent by e-mail in pdf version to [email protected]. Only this cover page duly signed is to be sent to the program by regular post.

DOCTORAL PROGRAM CIVIL AND ENVIRONMENTAL ENGINEERINGPROGRAMME DOCTORAL GENIE CIVIL ET ENVIRONNEMENT

Février 2013

DONNEES PERSONNELLES DOCTORANTPersonal data PhD Student

Date de dépôt (plan de recherche) Date limite de soutenance (mois, année) Date of submission (research plan) : Latest possible date for the defense (month, year) :

Nom et prénom / Name and first name : David Ferras ......................................................

Date de naissance / Date of birth : 28/11/1981...............................................................

Origine / Place of origin : Spain .....................................................................................

Adresse privée / Private address : General Prim num.9, 08980 St. Feliu de Llob. (Spain) .......

Diplôme / Diploma : Civil Engineering .............................. Année / Year : 2006........

Etablissement / Institution : Universitat Politècnica de Catalunya (UPC) ..............................

Directeur de thèse / Thesis director : Prof. Anton Schleiss ...............................................

Co-directeur de thèse (éventuellement) / Thesis co-director (if applicable) : Prof. Dídia Covas

Unité d’accueil / Unit : .................................................................................................

Collaborations envisagées / foreseen collaborations : ......................................................

Source de financement / Source of funding : Fundação para a Ciência e a Tecnologia (FCT) .

Dans quel cadre se situe la recherche ? (FNS, CTI, Conseil EPF, Progr. européen, etc.) :

In which framework is done the research ? (FNS, CTI, ETH Board, European program, etc.) :

IST-EPFL JOINT DOCTORAL INITIATIVE

Abstract

Pipeline systems are frequently subjected to pressure surges due to pumps’start-up and trip-off or maneuvers in mechanical devices. Although thecrucial importance of water pipe infrastructures, most of the 1D modellingsoftware used for pipeline design is not accurate enough to represent tran-sient events in real life systems. Transient wave amplitude, shape and timingare not well described by mathematical solvers ending up in over or under-estimated engineering solutions.

Several physical phenomena contribute to the uncertainty of transientsolvers, such as unsteady friction, air entrapment, cavitation and columnseparation, fluid-structure interaction, rheological behaviour of the pipe-wallor leakages and blockages. The importance of these phenomena depends onthe configuration and physical characteristics of the pipe system studied.Notwithstanding this, one of the main sources of uncertainty is the accuratedescription and distinction between rheological behaviour of the pipe walland unsteady friction.

The aim of the present research is to improve the characterization ofthese physical phenomena, both experimentally and numerically, focusingon to unsteady friction and pipe-wall rheological behaviour. The ultimategoal is the development of novel tools for the enhancement of engineeringdesigns. For this purpose, several subtasks can be pointed out:

• Stress-strain analysis: in a first stage, experimental tests will be carriedout in a pipe-coil facility with the goal to understand the axial andcircumferential deformation of the pipe wall during pressure surges.

• Fluid structure interaction modelling: the knowledge acquired fromthe stress-strain analysis will be included in classical water-hammerequations.

• Unsteady friction analysis: state-of-the-art unsteady friction modelswill be incorporated in the mathematical model and compared withexperimental tests.

• Rheological behaviour study: Viscoelasticity and anelasticity will bestudied for different pipe materials (i.e., Copper, HDPE, steel)

Keywords. Fluid-structure interaction, hydraulic transient, hysteresis, rheological be-haviour, stress-strain analysis, unsteady friction.

Contents

1 State of the art 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Pressurized hydraulic transient equations . . . . . . . . . . . . . . . . . . . 31.3 Unsteady friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Physical phenomenon and assumptions in classic water-hammer theory 31.3.2 Unsteady friction in 1D models . . . . . . . . . . . . . . . . . . . . . 41.3.3 Other approaches for unsteady friction . . . . . . . . . . . . . . . . . 8

1.4 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Physical phenomenon and assumptions in classic water-hammer theory 81.4.2 Stress-strain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.3 Structural equations and coupling . . . . . . . . . . . . . . . . . . . 131.4.4 Pipe rheological behaviour . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 State of research conducted by the candidate 202.1 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Definition of the framework of the PhD thesis . . . . . . . . . . . . . . . . . 202.3 Numerical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Experimental facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Copper facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Polyethylene facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.3 Steel facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Stress and strain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.1 Stress-strain models . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.2 Hysteresis effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Detailed research plan 293.1 Objectives and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Next steps on the research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Engineering applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Planning of the project 334.1 Project time-line and milestones . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Expected publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Project consortium and collaborations . . . . . . . . . . . . . . . . . . . . . 354.4 Follow-up of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1

Chapter 1

State of the art

1.1 Introduction

The propagation of pressure waves in fluid-filled tubes depends on the mechanical prop-erties of the fluid and the mechanical properties of the pipe. This phenomena was firstobserved by Helmholtz (1848), who attributed the reduction of the speed wave in confinedfluid to the elasticity of the pipe-wall, thus the concept of elastic wave was introduced inhydraulic transient studies. In other words:

The dynamic behaviour of pressure and flow in a pipe system during a transientevent (e.g., pump trip, emergency valve closure) is dominated by fluid inertia,fluid compressibility and pipe wall elasticity. Pothof (2008).

Taking into consideration only fluid compressibility the pressure wave would be anacoustic wave propagating at the speed of the sound. However, the interaction with theconfining structure arises multiple side-physical phenomena that considerably change thebehaviour of the wave. In classic water-hammer theory, linear elastic behaviour of thepipe-wall and steady or quasi-steady state friction losses are assumed for transient wavedescription (Chaudhry (1987), Wylie et al. (1993), Almeida & Koelle (1992)).

Nowadays, some other factors are identified to have important effects for water-hammerwaveform with regard to attenuation, shape and timing, such as unsteady friction, cavita-tion and column separation, fluid-structure interaction, viscoelastic behaviour of the pipe-wall, leakages and blockages (Bergant et al. (2008a) and Bergant et al. (2008b)). Further-more, in some cases these effects can increase maximum transient pressures (Ramos et al.(2004)). Therefore a clear distinction of these different physical phenomena is challengingnot only from the scientific point of view, but also for being crucial in the development ofreliable designs of pipelines with appropriate surge protection devices, or in leak detectionmethods, operation and maintenance protocols.

The present literature review focuses the centre of attention on fluid-structure inter-action, unsteady friction and pipe rheological behaviour. The last two are conceptuallydifferent physical phenomena but produce similar effects to water-hammer wave:

The major challenge of the current and future work is the distinction be-tween frictional and mechanical dampening, as the viscoelastic behaviourof pipe-walls has a dissipative and dispersive effect on the pressure wave,similar to unsteady friction losses. Covas et al. (2005).

A proper quantification of these factors would not be possible without the use of com-puter aid, as numerical techniques are required to solve almost any practical application ofwater-hammer theory. Although the massive development of the information technologyera, 2D or 3D models generate too heavy computations for most engineering problems(Pezzinga (1999)), being reserved basically for scientific purposes. Consequently, greatinterest is invested on the development of 1D water-hammer solvers, and it is withinthis 1D transient solvers context where an accurate computation of the different damping

2

State of the art 3

phenomena is required.

1.2 Pressurized hydraulic transient equations

The derivation of pressurized transient flow equations (water-hammer equations) can beapproached through two different ways: by applying mass and momentum conservation ina control volume (Chaudhry (1987), Wylie et al. (1993)), or by starting from Navier-Stokesequations and applying several simplifying assumptions (Ghidaoui (2004)). The ultimategoal is to obtain the following set of partial differential equations describing motion andcontinuity in fluid-filled pipes:

∂p

∂t+ V

∂p

∂x+ ρa2∂V

∂x= 0 (1.1)

∂V

∂t+ V

∂V

∂x+

1

ρ

∂p

∂x+ g sin θ +

fV |V |2D

= 0 (1.2)

From the derivation of Equations (1.1) and (1.2), the following simplifying assumptionscan be pointed out:

• The flow is considered unidimensional, with uniform pressure and velocity distribu-tions across the cross-section.

• Head losses are assumed to be equal to the head losses corresponding to the steadyflow for a certain instant (quasi-steady assumption).

• One phase and homogeneous fluid during the transient event (no air dissolve and nocavitation are assumed).

• Fluid behaves as quasi-incompressible.

• Temperature remains constant during the transient.

• Pipe is uniform in each reach.

• Pipe material has a linear-elastic rheological behaviour.

• Pipe axis does not move during the transient.

• There is no discharge through the pipe-wall.

1.3 Unsteady friction

1.3.1 Physical phenomenon and assumptions in classic water-hammertheory

In any physical system, friction has a non-linear behaviour. The relation between theparameters defining the unsteadiness of friction has motivated researches of water-hammer

Unsteady friction and pipe rheological behaviourin pressurized transient flows. (Research plan)

State of the art 4

during the last decades. Fast transients have a strong 2D nature of the flow field (Brunoneet al. (1995)). Consequently, a dimension-reduction problem must be added to the alreadymentioned intrinsic non-linear problem of the friction.

Unsteady friction in 1D pipe flow models is one of the factors that generates dissipation,dispersion and shape-change of the pressure wave. Its importance depends on the systemconsidered and the operating conditions. In the majority of the laboratory water-hammertest rigs, unsteady friction dominates over steady friction (Bergant et al. (2008a)), inparticular for fast transients.

After a sudden disruption of the flow, like a valve closure, the velocity profile firstreverses close to the pipe-wall than in the central area. A extreme case is when the meanvelocity is equal to zero, and consequently steady or quasi-steady state friction is null whilethere are very high velocity gradients next to the pipe-wall that create higher shear stressand momentum dissipation. During these instants the steady state friction assumption isfar from reality.

Figure 1.1 depicts observed velocity profiles measured by Zidouh (2009) in an experi-mental apparatus composed of a polypropylene pipe of length L = 2.62 m, D = 61.4 mminternal diameter and e = 6.8 mm wall thickness. The inversion of the flow is evident,strengthening the importance to take into account the 2D nature of the flow field.

An unsteady friction model is beneficial for diagnosis purposes when real data areavailable, for instance for the application of transient-based leak detection methods intransportation pipelines or distribution networks. Unsteady friction plays also an impor-tant role in the prediction of sonic booms, caused by high speed trains in long tunnels.

1.3.2 Unsteady friction in 1D models

This section presents an extensive literature review for the main unsteady friction modelsproposed for hydraulic transient analyses. The different approaches applied by researchersare described and the main models for each method highlighted.

A first contribution of unsteady friction for 1D pipe flow modelling was given by Dailyet al. (1955). The model was based on instantaneous mean flow velocity and instanta-neous local acceleration. After, numerous models have been presented following the sameapproach. A milestone in unsteady friction simulation was presented by Zielke (1968). Ananalytical solution for unsteady friction in laminar flows was presented. The solution wasbased on a weighting function with the aim to represent the 2D behaviour of the velocityprofile. Several models have been developed following the same approach with the aimto better define the weighting function (Trikha (1975)) or to extend its applicability toturbulent flows (Vardy & Brown (1996), Vardy & Brown (1995)). Another importantcontribution in unsteady friction was given by Brunone et al. (1991), a model based onlocal and convective accelerations was developed, an improved version from Daily’s model,simple to implement and with good fitting with experimental data, introducing though, anempirical friction decay coefficient k. Further developments in this approach were carriedout in order to better define the coefficient k or to introduce one more degree of freedom inthe unsteady friction expression with a second k coefficient to better fit the experimentaldata, like Ramos et al. (2004).

Usually, the contribution of the wall shear stress is expressed as the composition of a


State of the art 5

Figure 1.1: Velocity fields over time during a hydraulic transient, Zidouh (2009).

steady part and an unsteady part:

τw = τws + τwu (1.3)

The way the computation of the unsteady term (τwu) is carried out is what defines theapproach and distinguishes the unsteady friction model.

Unsteady friction models based on instantaneous mean flow velocity V

In this approach the unsteady friction term is only dependent on the instantaneous meanflow velocity. Some of the authors that have developed models based on this assumptionare Hino et al. (1977), Brekke (1984) or Cocchi (1988). For instance, the expression forunsteady friction computation with Hino’s model is:

fu = 0.188

(√4L

vπa

) −12.85

(1.4)


State of the art 6

Unsteady friction models based on instantaneous mean flow velocity V and

instantaneous local acceleration ∂V∂t

This approach was the first used to include unsteady friction in hydraulic transient anal-ysis with Daily’s model Daily et al. (1955). After him, many other researchers proposedimprovements and developments of this kind of models, such as Carstens & Roller (1959),Safwat & Van der Polder (1973), Kurokawa et al. (1986), Shuy & Apelt (1987), Golia(1990) or Kompare et al. (1995). Two of these models are discussed in the followingparagraphs.

Daily’s model formulation:

fufs

= 1 + c22D

fsV 2

dV

dt(1.5)

where, c2 = 0.01 for accelerated flow and c2 = 0.62 for decelerated flow.

Shuy’s model formulation:

fu = fs + kD

V 2

dV

dt(1.6)

where, k = −0.33 for accelerated flow and k = −52 for decelerated flow.

Although both formulations do not differ in the approach, there is a clear controversyin the sign of the coefficients. For a positive accelerated flow the unsteady term will be pos-itive in Daily’s formulation but negative in Shuy’s formulation. And the other way around,for positive decelerated flows the unsteady term will be negative in Daily’s formulationand positive in Shuy’s. Both agree, though, that the magnitude of the unsteady part willbe more important when decelerating than accelerating flows. The reason to explain suchdifferences could be that Daily experiments where carried out for low Reynolds numbers(Re = 150 to 770, laminar flow), whereas Shuy’s model was based on turbulent flow. Afriction coefficient of an accelerated flow tends to be larger than that of a quasi-steadyflow in the laminar region, while it has a reverse tendency in the turbulent region. For adecelerated flow, a turbulent region is maintained almost all over the deceleration periods,and the friction coefficient becomes larger than that of a quasi-steady flow (Kurokawaet al. (1986)).

Unsteady friction models based on instantaneous mean flow velocity V , in-

stantaneous local acceleration ∂V∂t and instantaneous convective acceleration

∂V∂x

With the idea that during fast transients both, local and convective accelerations are cor-related to friction forces, Brunone et al. (1991) proposed a single expression that requiresan empirical coefficient k, referred here as Brunone’s model:

fu =kD

V |V |

(∂V

∂t− a∂V

∂x

)(1.7)

Vitkovsky et al. (2000) improved Brunone’s formula taking into account the sign of theconvective term (−a∂V∂x ) for particular flow and wave directions in acceleration and decel-


State of the art 7

eration phases. Vıtkovsky’s model:

fu =kD

V |V |

(∂V

∂t+ a · sign(V )

∣∣∣∣∂V∂x∣∣∣∣) (1.8)

Finally, Ramos et al. (2004) proposed a second coefficient in order to differentiate localand convective accelerations, with an extra degree of freedom. Ramos’ formulation enablesa better fitting between numerical and experimental data.

Ramos’ model:

fu =1

gA

(kv1

∂Q

∂t+ kv2 · a · sign(Q)

∣∣∣∣∂Q∂x∣∣∣∣) (1.9)

Unsteady friction models based on instantaneous mean flow velocity V andweights of past velocity changes W (τ)

From this approach derive the so called weighting-function based models (WFB). WFBmodels take into account the 2D nature of the velocity profile that causes the frequency-dependent attenuation and dispersion of the hydraulic transient. The first model of thiskind was developed by Zielke (1968), who developed an analytical solution for unsteadyfriction in laminar flows, where the unsteady head loss term is a convolution of past fluidaccelerations with a weighting function(full convolution method):

hfu(t)exact =16ν

gD2

(∂V

∂t∗W

)(t) (1.10)

where ∗ indicates convolution and W the weighting function. The convolution in Zielke’smodel is approximated using the rectangular rule and the acceleration term is approxi-mated using a central difference. However, this scheme is very expensive from the compu-tational point of view. Trikha (1975) simplified this computation reducing the weightingfunction to the sum of three exponential terms and eliminating the need for convolutionwith an approximate recursive relationship:

hfu(t)app. =16ν

gD2

N∑k=1

Yk(t) (1.11)

where, Yk = function that represents the exponential terms

Yk(t+ ∆t) = mk[V (t+ ∆t)− V (t)] + e−nk∆τYk(t) (1.12)

N = the number of exponential terms (N = 3 in the case of Trikha formulation); τ =is the dimensionless time step, τ = 4ν∆

D2 ; nk = coefficient of the exponential sum, nk =(26.4, 200, 8000); mk = coefficient of the exponential sum, mk = (1, 8.1, 40).

After Trikha, other authors have tried to improve the accuracy by changing the re-cursive relation or adding more iterations. Kagawa et al. (1983) proposed a recursionup to N = 10 for a different expression of Yk. Schohl (1993) proposed a different con-volution algorithm by assuming that the acceleration term in the convolution integral isconstant between time steps. Vitkovsky et al. (2004) presented an accurate approximationof Zielke’s analytical solution. Vardy & Brown (1996), Vardy & Brown (1995), Vardy &Brown (2003) and Zarzycki (1997), Zarzycki (2000) developed WFB models for smoothpipe turbulent flows, and for rough pipe turbulent flows, Vardy & Brown (2004) proposeda suitable model as well.


State of the art 8

1.3.3 Other approaches for unsteady friction

Unsteady friction in axisymmetric models

Fast hydraulic transients have a strong 2D nature of the flow field. As such, unsteadyfriction computation in 1D models is subjected to a dimension-reduction problem. Modelsbased on the calculation of instantaneous velocity profiles avoid this problem by com-puting the whole velocity profile. These models, called axisymmetric models, assumeaxisymmetry of the flow field, not being purely 2D as pressure-head is considered constantin each cross-section (being for this reason referred as quasi-2D models). During the lastdecades, many authors worked on the development of such models: Vardy & Brown (2003),Eichinger & Lein (1992), Silva-Araya & Chaudhry (1997) or Pezzinga (1999). This kind ofmodels require heavy computations compared to 1D models and may not be justified formany practical engineering applications. Nevertheless, their good results led researchersto base 1D modelling validation on quasi-2D models in a similar manner as it is done withexperimental data. An assessment on the reliability of axisymmetric models was carriedout by Ghidaoui et al. (2002) and some conditions were stated with the aim to ensure theresults of such calibration/validation practice. In Pezzinga (2000), a 1D unsteady frictionmodel was compared with a quasi-2D model and with experimental measurements as well.Although the attempt to exclude the dependence on experimental conditions, Pezzingaemphasized the need for further experimental validation either for 1D and 2D models.

1.4 Fluid-structure interaction

1.4.1 Physical phenomenon and assumptions in classic water-hammertheory

Fluid-structure interaction (FSI), as the name states, is any interaction between the liquidand its containing structure (Wiggert & Tijsseling (2001)). Helmholtz (1821-1894) was thefirst suggesting that the speed of pressure disturbance wave propagation in fluid-filled tubesdepends on the deformability of both the fluid and the tube wall (Anderson & Johnson(1990)). FSI problem can be subdivided in three main coupling mechanisms (Tijsseling(1996)): Poisson coupling, friction coupling and junction coupling. The first mechanismis based on the axial deformation of the pipe caused by the radial load produced by theinner pressure. Friction coupling, arises from the shear stress between the pipe-wall andthe fluid. Junction coupling results from unbalanced pressure forces and by changes in thefluid momentum, that occur in pipe curves, Tee junctions or cross section changes.

Pipe systems experience severe dynamic forces during water-hammer events. Whenthese forces make the system move, significant FSI may occur, so that liquid and pipesystems cannot be treated separately in a theoretical analysis: interaction mechanismsmust be taken into account (Tijsseling (2007)). The need for an accurate understandingof FSI phenomena has been increasing during the past thirty years accordingly to a greaterconcern on safety and reliability of pipe systems.

For the sake of simplicity, classic water-hammer equations are derived assuming astraight conduit with expansion joints throughout its length and a linear-elastic behaviourof the pipe wall (Chaudhry (1987)). Hence, anelastic/viscoelastic rheological behaviour,deformation in the axial direction, and any other kind of displacement dependent on the


State of the art 9

system geometry (elbows, connections, reductions...) are not taken into account. In thepresent research work, phenomena excluded from classic theory and associated to the pipemovement will be considered as fluid-structure interaction.

The following scheme represents the different sources of interaction:

Figure 1.2: Sources of fluid transients and pipe motion (Wiggert (1986)).

Over the past ten years FSI has experienced renewed attention because of safety andreliability concerns in power generation stations, environmental issues in pipeline and thewill to tight pipe designs (Wiggert & Tijsseling (2001)). However, the complexity of FSIdoes not allow the development of a general theory. FSI is very dependent on the pipeconfiguration and the problem must be approached case by case (Locher et al. (2000)).

1.4.2 Stress-strain analysis

Part of this Ph.D. research is focused on the study of the influence of the pipe deformationduring hydraulic transients for different sets of pipe systems. For this purpose and beforeany hydraulic analysis, the stress-strain behaviour of the pipe system must be well knownin regard to the nature of the load to be applied, the inner pressure and the pipe constraintsas presented bellow.

Some definitions must be considered in stress-strain analyses:

Stress: Stress is the term used to define intensity and direction of the internal forces ata particular point and acting on a given plane (Juvinall (1967)). Therefore, the stress ata certain point over a plane is

σ = limdA→0

dF

dA(1.13)

where dF is the resultant of the internal forces acting on a small area dA. Frequently,stresses are decomposed into two components: normal stress σ, which is perpendicular


State of the art 10

to the plane, and shear stress τ , which is parallel to the plane. Stress is a second ordertensor σij , and a complete description of the stress magnitude and direction for all possibleplanes of a certain point constitutes the state of the stress at that point.

Strain: Stress and strain are directly related. Stress is not directly measurable andwhat is measured in experimental tests is either the force or the strain. Strain is thedirection and intensity of the deformation at any given point with respect to a specificplane passing through that point. Strain is therefore a quantity analogous to stress. Stateof strain is a complete definition of the magnitude and direction of the deformation at agiven point with respect to all the planes passing through the point (Juvinall (1967)). Forconvenience, strains are always resolved into normal components ε and shear componentsγ. For instance, normal strain in the x direction is defined as:

εx = limx→0

dx

X(1.14)

and for shear strain in the yx plane:

γyx = limy→0

dx

Y(1.15)

Poisson’s ratio: The relation between transverse strain to axial strain during uniaxialloading is known as the Poisson’s ratio (see Figure 1.3), which is defined as:

ψ = −εtrans/εaxial (1.16)

As it has been mentioned in the previous section, the Poisson’s ratio is a very importantfactor in fluid-structure interaction when the axial deformation of the pipe is assessed for aradial load like inner pressure. Table 1.1 presents the values of Poisson’s ratio for differentstandard materials.

Figure 1.3: Graph representing deformation according to Poisson’s ratio

Young’s modulus: The linear relationship between uniaxial stress and strain due to anelastic behaviour is represented by a constant of proportionality which is a characteristicproperty of an elastic material. This constant of proportionality is called the Young’s


State of the art 11

modulus or the modulus of elasticity E. The rheological behaviour linear-elastic can bemathematically described by the Hooke’s law which is a linear law relating both variables:

σx = Eεx (1.17)

Table 1.1 presents Poisson’s ratios and Young modulus of elasticity for the materialsused in the experimental facilities:

Material Young modulus E (GPa) Poisson’s ratio ψ

Copper 110-120 0.33-0.36Polyethylene 0.7-1.4 0.4Steel 190-210 0.27-0.30

Table 1.1: Poisson’s ration and Young modulus of elasticity values extracted from Gree &Timoshenko (1997).

Creep function: Creep compliance function or simply creep function, (J(t)), is themathematical expression that characterizes the time-dependant strain behaviour for aconstant stress. This function can be estimated by a creep test or by dynamical testingover a certain range of loading frequencies (Covas (2003)). In a creep test, a constantstress σ0 is applied to the material while the resulting time-dependent strain is measured.The strain function J(t) resulting from the unit step stress (i.e., σ0 = 1) is called creepcompliance (Banks et al. (2011)).

Bulk modulus: The modulus of volume expansion K is the ratio between hydrostaticstress and volumetric strain. For normal structural materials, this parameter is definedwith the following expression:

K =E

3(1.2ψ)(1.18)

Membrane theory of shells of revolution: general solution

In this research, fluid-structure interaction during hydraulic transients will be assessed fordifferent sets of pipe systems: a copper coil of 20 mm of pipe diameter, a polyethylene coilof 44 mm of pipe diameter and a steel facility of 200 mm of pipe diameter. Not just thematerial, but also the geometrical singularities of each facility must be considered in orderto assess the strain-stress laws of each pipe system. For instance, during the manufactoryprocess of the coiled copper pipe, the tube is bended deforming its cross section to an ovalshape instead of circular, leading to a much different stress-strain law.

The approach to derive the stress-strain laws for the different pipe systems has beenbased on the membrane theory of shells of revolution. A general solution proposed byZingoni (1997) for axisymmetric loading is presented. This general law is the startingpoint, and henceforth the specific solutions for each one of the case-studies can be derivedby adopting the correct assumptions.


State of the art 12

A surface of revolution is generated by rotating a plane curve through 360◦ over an axisof revolution. Usually, it is convenient to dispose the axis of revolution in a vertical positionand to define parallel planes of latitude and meridional planes, respectively disposed inhorizontal and vertical positions.

Figure 1.4 presents a scheme of a shell element bounded by two adjacent circles oflatitude and two adjacent meridians, where φ is the meridional angle (with the axis ofrevolution) and θ the parallel angle (in the horizontal plane); and therefore, σφ is themeridional stress and σθ the parallel stress. The loads Fr and Fφ correspond respectively tothe normal direction and meridional direction. The radius r1 denotes the principal radiusof curvature in the meridional plane, and r2 is the second principal radius of curvaturegiven by the distance from the shell to the intersection with the axis of revolution. Finally,the radius R denotes the radius of revolution through a given point.

Figure 1.4: Element of the axisymmetrically loaded shell of revolution

As it can be seen in the Figure 1.4, by symmetry and by definition of shell of revolution,both meridional boundaries of the shell element have the same extension. In consequence,and considering static equilibrium of the element, the stresses in the parallel plane haveto be constant for any θ. On the other hand, the meridian stresses will vary in function ofφ, as the extension of the boundaries of the shell in such direction vary as well in regardto the shell geometry.

The basic assumption in membrane theory is that the shell cannot transmit bendingmoments, twisting moments and transverse shearing forces. For axisymmetric conditions,in plane shearing forces cannot exist either. Consequently, the solution for membrane shellsof revolution axisymmetrically loaded consists of determining σθ and σφ. The expressionsof such solution, after applying equilibrium of forces in the direction of the tangent to ameridian and in the direction of the normal to the shell midsurface, are presented bellowand the correspondent demonstration can be found in Zingoni (1997):

σφ =1

e · r2 · sin2φ

[∫r1 · r2 (Fr · cosφ− Fφ · sinφ) sinφ · dφ+ k

](1.19)


State of the art 13

σθ =r2

e· Fr −

r2

e · r1Fφ (1.20)

where k is an integration constant to be determined for a specific boundary condition.

1.4.3 Structural equations and coupling

The classic water-hammer equations implicitly include the elastic interaction between thefluid and the pipe concerning the radial expansion of the pipe wall. However, they do notinclude waves from a different nature that can also propagate along the pipe material andinteract and perturb the propagation of the water-hammer wave. These waves are due tothe axial, bending, shear and torsional movements in the pipe wall (Heinsbroek (1997)).The equations of such movements describe the structural behaviour of the pipe systemand these should be coupled to the classic water-hammer equations.

• Pipe axial motion equation:

ρtAt∂2uz∂t2

− EAt∂2uz∂z2

=

=ψDAt

2e

∂p

∂z+ρfAffVr|Vr|

2D+ ρtAtgsinζ

(1.21)

where ρt = pipe mass density; ρf = fluid mass density; At = cross-sectional pipe wallarea; Af = cross-sectional fluid area; uz = axial displacement; ψ = Poisson’s ratio;f = Darcy-Weisbach friction factor; ζ = pipe elevation angle; and Vr = relative fluidvelocity in the axial direction (with respect to the pipe-wall); p = pressure of thefluid; z = elevation.

• Pipe lateral motion equation (Timoshenko & Young):

EI∂4uy∂z4

+ (ρtAt + ρfAf )∂2uy∂t2−

− ρtI∂4uy∂z2∂t2

EI

k2GAt

∂2

∂z2

((ρtAt + ρfAf )

∂2uy∂t2

)+

+ρtI

k2GAt

∂2

∂t2

((ρtAt + ρfAf )

∂2uy∂t2

)= 0 (1.22)

where uy = horizontal lateral displacement; E = Young’s modulus of the pipe ma-terial; G = Shear modulus of the pipe material; I = moment of inertia; k2 = shearcoefficient.

• Pipe torsional motion equation:

ρtJ∂2Θz

∂t2−GJ ∂

2Θz

∂z2= 0 (1.23)

where Θz = torsional rotation; and J = polar moment of inertia.


State of the art 14

Coupling

As referred, there are three kinds of coupling: Poisson coupling, friction coupling andjunction coupling. The first one couples the axial motion (Equation 1.21) with the con-tinuity equation from the classic hydraulic transient equations. Friction coupling, on theother side, couples also axial motion (Equation 1.21) but with momentum conservationequation. Junction coupling couples all the equations (Heinsbroek (1997)).

In the present research, FSI will be used to increase the accuracy of 1D water-hammermodels applied in coiled pipes. Due to the geometry of such systems coils, the pipesystem can be considered a composition of toroids and be modelled as a shell of revolutionapplying membrane theory. On the other side, membrane theory assumes that no bendingmoments, twisting moments and transverse shearing forces exist in the shell. Hence, inorder to compute FSI in coiled pipes just the axial motion equation will be required.

1.4.4 Pipe rheological behaviour

FSI is considered as an extension of the classic water-hammer theory, adapting its equa-tions to take into account the pipe movements. The radial expansion of the pipe wallis included in the derivation of the basic equations and pipe behaviour is assumed to belinear-elastic. Whilst this is verified for concrete and metal pipes, plastic pipes present astrongly non-elastic behaviour (Covas (2003)).

Experimental evidence has shown that the elastic assumption for plastic pipes canlead to an underestimation of the wave damping effect (Fox & Merckx (1973), MeiBner& Franke (1977), Williams (1977), Sharp & Theng (1987), Mitosek & Roszkowski (1998),Pezzinga & Scandura (1995), Covas et al. (2004c), Covas et al. (2004b), Soares et al.(2008)). The viscoelastic behaviour of pipe walls has a dissipative and dispersive effect inthe pressure wave, similar to unsteady friction losses. Although the viscoelastic behaviourof polymers is well known, this behaviour tends to be forgotten in hydraulic transientanalysis in plastic pipes (Covas et al. (2005)). Furthermore, the damping of the water-hammer wave is higher due to viscoelasticity than to unsteady friction (Ramos et al.(2004)). Duan et al. (2009) assessed a quasi 2D water-hammer model taking into accountunsteady friction and viscoelasticity and compared with experimental data from Covaset al. (2005). This author concluded that the viscoelastic effect is more sever for lowfrequencies whereas unsteady friction is more intense for high frequencies of the waveoscillation.

The phenomenon of viscoelasticity and assumptions

Viscoelasticity is the property of the materials that exhibit both viscous (dashpot-like) andelastic (spring-like) characteristics when undergoing deformation. Viscoelastic materialsare those for which the relationship between stress and strain depends on time, and theypossess the following three important properties: stress relaxation, creep, and hysteresis(Banks et al. (2011)).

Stress relaxation: When a viscoelastic material sample is submitted to a constantstrain, its stress decreases over time (see Figure 1.5). This property is called stress relax-ation, and the time dependent change in stress is the relaxation modulus which defines


State of the art 15

the stress function G(t). For a purely elastic material, the stress will remain constant asthe strain-stress response is instantaneous.

Figure 1.5: Stress and strain histories in the stress relaxation test (Banks et al. (2011)).

Creep: This property of the viscoelastic materials refers to the variation of strain whena constant stress is applied (Figure 1.6). The time dependent function describing thestrain is called creep compliance function, J(t).

Figure 1.6: Stress and strain histories in the creep test (Banks et al. (2011)).

Hysteresis: A hysteretic system depends not only on its current state but on its pastconditions. In viscoelastic materials, the hysteresis arises from the difference between theloading and unloading process. Due to its time dependency, a viscoelastic material has“memory” and it behaves differently whether it is ascending or descending the stress-straincurve (see Figure 1.7-b).

Figure 1.7: Stress and strain curves during cyclic loading-unloading. (a): Hookean elasticsolid; (b): viscoelastic material (Banks et al. (2011)).

Linear viscoelastic models

Linear viscoelastic materials are those that present a linear response between stress andstrain history. This assumption is usually applicable only for small deformations. Twocomplementary approaches have been developed for linear viscoelastic models: mechanicalanalogs and the Boltzmann superposition models.


State of the art 16

Mechanical analogs models: This kind of models are based on the assumption thatviscoelastic behaviour can be described by a linear combination of spring and dashpots,respectively representing the elastic and the viscous behaviour. The elastic part is definedby Hooke’s law:

σ = E · ε (1.24)

and the viscous part is described by the Newtonian fluid:

σ = ηdε

dt(1.25)

where σ = the stress; ε = the strain; E = the modulus of elasticity; and η = the viscositycoefficient.

Figure 1.8: (a): Schematic representation of the Hookean spring; (b): Schematic repre-sentation of Newtonian dashpot.

There are two basic mechanical analog models, Maxwell model and Kelvin-Voigt model,and the standard linear solid model (SLS or three-element model). In the Figure 1.9 ascheme of each model is presented.

Figure 1.9: (a): Mechanical analogue model schemes and their basic equations: (a)Maxwell model, (b) Kelvin-Voigt model and (c) Standard linear solid model.

These basic mechanical models can be combined, either rearranging Maxwell or Kelvin-Voigt elements in parallel or in series, obtaining generalized models that allow a betterrepresentation of the viscoelastic behaviour depending on the material studied. The me-chanical model of a generalized viscoelastic solid (Figure 1.10) is typically used to describethe creep function, which expression can be represented in the following way:

J(t) = J0 +

N∑k=1

Jk(1− e− tτk ) (1.26)

where N is the number of Kelvin-Voigt elements used, Jk the creep-compliance of thespring of the Kelvin-Voigt k-element defined by Jk = 1/Ek, and Ek the modulus of elas-ticity of the spring of k-element, τk the retardation time of the dashpot of k-element,τk = ηk/Ek and ηk the viscosity of the dashpots of k-element.


State of the art 17

Figure 1.10: Generalized Kelvin-Voigt Model for viscoelastic solids (Covas et al. (2005)).

These different linear combinations generate constitutive relations that are ordinarydifferential equations of the type:

a0σ + a1dσ

dt+ a2

d2σ

dt2+ ...+ an

dnσ

dtn= b0ε+ b1

dε

dt+ b2

d2ε

dt2+ ...+ bn

dnε

dtn(1.27)

where parameters an and bn are related to the elastic modulus and viscosity of the material,respectively, and which are usually determined from physical experiments.

Boltzmann superposition models: Instead of the previous linear combinations thatlead to ordinary differential equations, integral equations can be used as well. Basicallythe Boltzman superposition principle states: if the stress due to a strain ε1(t) is σ(ε1)and that due to a different strain ε2(t) is σ(ε2), then the stress due to both strains isσ(ε1 + ε2) = σ(ε1) + σ(ε2). Combining these terms:

σ(aε1 + bε2) = aσ(ε1) + bσ(ε2) (1.28)

written in a more general form:

σ(t) = krε(t) +

t∫0

K(t− s)dε(s)ds

ds (1.29)

where kr represents an instantaneous relaxation modulus, and K is the “gradual” relax-ation modulus function. Notice that any formulation of the form presented in mechanicalanalog models can be also expressed in the form of Boltzman superposition models. Theother way around is only possible for some specific conditions of the stress relaxationmodulus.

Viscoelastic behaviour in hydraulic transient solvers

Several authors proposed mathematical models to describe the viscoelasticity of pipe-wallduring fluid transients (Gally et al. (1979), Rieutford & Blanchard (1979), Rieutord (1982),Franke (1983), Suo & Wylie (1990), Covas et al. (2002), Covas (2003), Covas et al. (2004a),Covas et al. (2005), Keramat et al. (2012), Duan et al. (2009)). Two different approachescan be followed to describe the viscoelastic behaviour of plastic pipes in hydraulic transientsolvers: a frequency-dependant wave speed and an additional viscoelastic term added tothe mass balance fluid equation (Covas et al. (2005)).


State of the art 18

Frequency-dependent wave speed: As the viscoelastic behaviour of the pipe wall istime dependent, it can be described in a frequency domain in terms of angular frequency.Hence the modulus of elasticity of the pipe material used in the wave speed computation isreplaced by the inverse of the creep function, J (Covas et al. (2004b)). MeiBner & Franke(1977) and Franke (1983) studied the dampening of steady-oscillatory flows in PVC andsteel pipes, deriving wave speed and dampening formulae. Rieutord (1982) proposed a“one Kelvin-Voigt element” model to describe creep and include it in the wave speedformula. Suo & Wylie (1990) modelled pipe wall viscoelasticity in both, oscillatory andnon-periodic flows. Covas et al. (2004b) and Covas et al. (2005) compared the results ofthis approach with experimental data.

Additional viscoelastic term: This approach incorporates an additional term to themass balance equation of the fluid. The rheological viscoelastic behaviour of the pipe wallin the classic water-hammer equations is divided in two parts, an instantaneous elasticresponse (accounted for in the elastic wave speed) and a retarded-viscoelastic responsedue to the creep property of the viscoelastic material added to the mass balance equation.This formulation has been proposed by Gally et al. (1979) and Rieutord (1982). Rachidet al. (1992) implemented several types of non-elastic rheological behaviour.

Metal at elevated temperatures or when subjected to fast loading-rates presents a hys-teretic behaviour. Anelasticity is the term used to describe the hysteretic elastic behaviourof metals. The analysis of structural and atomistic features responsible for anelasticity hasshown that metallic atoms are capable of moving relative to one another in much the sameway that long polymer chains (Courtney (1990)). This phenomenon, which has not beenstudied yet in the field of hydraulic transients, can cause damping of the water-hammerwave in a similar manner as viscoelasticity in plastic pipes, though with lower intensity.

1.5 Motivation

As discussed in previous sections, water-hammer wave can be very sensitive to pipe rhe-ology and unsteady friction damping effects. Both depend on the nature of the hydraulictransient itself and produce similar effects in the wave propagation in regard to shapeand timing. Hence, the good characterization of one of the two phenomena will drive abetter identification of the other. The present Ph.D. research work intents to approachthis problem from both points increasing the accuracy in unsteady friction computationand improving the characterization of the pipe-wall rheological behaviour.

Several state-of-the-art unsteady friction models will be implemented and simulationswill be compared and, if required, calibrated and validated with experimental data, notwith the target to fit with measurements but to fit with reality (i.e., it is preferable a non-accurate model but with the right assumptions, than an accurate model but with wrongassumptions). The ultimate goal will be to identify the best models for different setsof transient conditions: laminar/turbulent flow, smooth/rough pipe-wall, plastic/metallicmaterials, coiled/straight pipes. In case for any of the conditions a good fit to reality isnot achieved, efforts will be invested to seek for improvements either assessing the modelconcepts, assumptions or their parametrization.

Once unsteady friction problem will be optimized for different case studies, a good


State of the art 19

base will have been set for the assessment of FSI analysis and pipe rheological behaviour.Fluid-structure interaction will be analysed from different approaches. Wave speed andclassic water-hammer equations will be assessed and updated to take into account piperheology. Not just for plastic pipes, but also in the metallic facilities anelastic/viscoelasticbehaviour will be studied. It is presumed that for “instantaneous” loading processes (i.e.forces varying in a very short time scale over a structure, such us pressure surges causedduring hydraulic transients in pressurized flows), metals can behave in an anelastic manner.

Fluid-structure interaction will also be used to assess the effect of coiled pipes geometry.Two of the three experimental facilities that will be used to test hydraulic transients arecoiled pipes (polyethylene and copper facilities). The unbalanced forces during a transientover the coil curvature plane will produce an axial strain that will have to be accountedfor. Additionally, the cross-section of circular tubes when are curved become slightly oval.The effect of ovality will be to increase the apparent elasticity of the pipe wall becausechanges in cross-sectional area will arise from not only hoop but also bending stresses thatalter the cross-sectional shape wall, in consequence the wave speed will be accordinglyaltered (Anderson & Johnson (1990)). The slightly deformation of the cross-section inthe polyethylene and copper pipe facilities will have to be carefully assessed. This is alsoa kind of FSI that so far, and from the knowledge of the author, has not been deeplyinvestigated.


Chapter 2

State of research conducted by thecandidate

2.1 Education

During the first year of the Ph.D. work (academic year 2012/2013), the candidate attendedcourses strongly related to the research topic with the aim to improve his research skillseither in numerical and experimental analysis.

Courses attended and passed:

• C++, A Comprehensive Hands-On Introduction (DIT-EPFL).

• Object-Oriented Design and programming in LabVIEW (DIT-EPFL).

• Design of experiments (EPFL Doctoral School).

• Advanced Course on Computational Hydraulics (IST Doctoral School).

2.2 Definition of the framework of the PhD thesis

The candidate carried out an extensive literature review which has allowed him to identifythe current research gaps from where to base the foundations of the research hypotheses.Through the literature review, the candidate increased his knowledge on:

• Classic water-hammer theory, derivation of the basic equations and main assump-tions associated to the pipe, fluid and flow.

• Different approaches for unsteady friction models.

• Fluid-structure interaction.

• Pipe wall rheological behaviour, viscoelasticity and anelasticity.

• Stress-strain formulations and membrane theory of shells of revolution.

The candidate has already carried out experimental tests in the copper coil facility.Strains of the pipe-wall either in axial and circumferential direction have been measured,as well as pressure data in different sections of the pipe and for different flow conditions.The goal at this stage is to stablish a relation between inner pressure and pipe expansion(radially and axially) and, subsequently, modify water-hammer equations (FSI coupling).

Find below attached a pie chart representing the work distribution of the Ph.D. re-search and its progress so far:

20

State of research conducted by the candidate 21

Figure 2.1: Work load and progress of the Ph.D. research by chapters (striped area: workcarried out so far).

2.3 Numerical modelling

As a first exercise and with the aim to learn and understand traditional theory on water-hammer, the Ph.D. candidate carried out an implementation of the classical water-hammerequations using the method of characteristics (MOC). Once the code was ready it enabledhim to build a basic model of the experimental facilities and compare numerical outputwith measured data, having a first feeling on how traditional theory differs from reality.

Following this learning approach, the Ph.D. student also implemented two unsteadyfriction models (Brunone’s and Trikha’s formulations) and included the viscoelastic rheo-logical behaviour of the pipe qall in the transient solver.

2.4 Experimental facilities

In the present section, technical data of the experimental facilities is presented and somepictures are attached.

2.4.1 Copper facility

Relevant parameters of the copper pipe facility:

• Material: Copper

• Nominal diameter: ND = 20 mm

• Pipe length: L = 103.26 m

• Minimum radius of cross-sectional ellipse (external): a = 0.0106 m



• Maximum radius of cross-sectional ellipse (external): b = 0.0114 m

• Torus mean radius of revolution: R = 0.5 m

• Pipe-wall thickness: e = 0.001 m

• Young’s modulus (copper): E = 110 GPa

• Poisson’s ratio (copper): ψ = 0.33

Figure 2.2: Pictures of the copper coil facility.

2.4.2 Polyethylene facility

Relevant parameters of the polyethylene pipe facility:

• Material: Hight Density Polyethylene (HDPE)


• Pipe length: L = 200 m

• Cross-section radius: r = 0.043 m

• Torus mean radius of revolution: R = 0.5 m


• Young’s modulus (HDPE): E = 0.8 GPa

• Poisson’s ratio (HDPE): ψ = 0.45



Figure 2.3: Pictures of the polyethylene coil facility.

2.4.3 Steel facility

Relevant parameters of the steel pipe facility:

• Material: Steel


• Pipe length: L = 115 m

• Cross-section radius: r = 0.206 m


• Young’s modulus (steel): E = 0.8 GPa

• Poisson’s ratio (steel): ψ = 0.45

2.5 Stress and strain analysis

2.5.1 Stress-strain models

Strain gauges were installed in the polyethylene and copper pipesto measure deformationsduring hydraulic transients. The goal is to assess the elastic or viscoelastic behaviour ofthe pipe wall through measurements in the circumferential direction. However, a coiledpipe submitted to a pressure surge will experience also strain in the axial direction. Notonly for the Poisson effect coming from the circumferential strain component, but also dueto a balance of forces over the coil curvature plane. Therefore, also strain gauges in theaxial direction were installed. The exact location of the strain gauges is represented in



Figure 2.4: Pictures of the steel pipe facility.

the Figure 2.5 and, also, in the Figure 2.2 there is a picture of the facility where it can beobserved the installation of the strain gauges.

A coil structure can be seen as a composition of toroids. A toroid can be described as ashell of revolution. There is a general solution from membrane theory to solve stress-strainsin function of axisymmetric load (Equations 1.19 and 1.20). From these equations axialand circumferential stress expressions for toroids where derived either assuming circularor elliptic cross section of the pipe.

It is important to mention that, at this point of the research, hydraulic transients itselfare not being assessed. They are generated in the experimental facilities with the onlyend to have a big range of pressure and strain measurements. The goal at this phase is tounderstand the relationship between pressure and strain.

Model-1: circular torus

In a toroid with circular cross-section, the meridional radius will be constant (r1 = a) andr2 will be:

r2 =R

sinφ=A+ a sinφ

sinφ(2.1)

Substituting r2 in Equation 1.20 it, yields (see Zingoni (1997) for a full derivation),

σφ =pa

e

(A+ a

2 sinφ

A+ a sinφ

)(2.2)



Figure 2.5: Location of the strain gauges used for axial and circumferential strain mea-surements in the copper coil facility

and in Equation 1.19

σθ =pa

2e(2.3)

Figure 2.6 depicts measured strains versus computed strains (using Equations 2.2 and 2.3)at different locations of the cross-section for a water-hammer wave raised from a fast valveclosure during a steady discharge of 500 l/h (0.14 l/s) in the copper coil facility. Thegraphs show, in general, a reasonable agreement between numerical results and measureddata in terms of phase and shape of the pressure wave, although computed values areslightly lower. The only exception is graph c) (circumferential strain in the top side)where computed values are much higher and inverted compared to measurements. Thisshows that Model-1 is not capable of accurately describing the pipe coil circumferentialand axial deformation.

Model-2: elliptic torus

For a toroid with elliptic cross-section, the radius r1 and r2 have more complex expressions:

r1 =a2b2(

a2 sin2 φ+ b2 cos2 φ) 3

2

(2.4)

r2 =A

sinφ+

a2(a2 sin2 φ+ b2 cos2 φ

) 12

(2.5)

where a and b are the main axes of the cross-sectional ellipse and A the mean radius ofthe elliptic torus. As it was done previously, substituting r1 and r2 at Equation 1.20, ityields (see Zingoni (1997) for a full derivation),

σφ =pa2

e(a2 sin2 φ+ b2 cos2 φ

) 12

·

A (a2 sin2 φ+ b2 cos2 φ) 1

2 + a2

2 sinφ

A(a2 sin2 φ+ b2 cos2 φ

) 12 + a2 sinφ

(2.6)



Figure 2.6: Measured and computed strains using Model-1 during a hydraulic transient inthe copper coil: (a) circumferential strain in the inner side of the torus; (b) circumferentialstrain in the outer side; (c) circumferential strain in the top side; (d) axial strain in theouter side.

and, applying Equation 1.19

σθ =pa2

eb2·

A(b2 − a2

a2

)sinφ+

b2 − 12

(a2 sin2 φ+ b2 cos2 φ

)(a2 sin2 φ+ b2 cos2 φ

) 12

(2.7)

Notice that in the case of a circular cross-section (b = a) both expressions can be obtainedthe Equations 2.2 and 2.3.

Results considering elliptic torus are shown in Figure 2.7:As it can be observed, comparing Figures 2.6 and 2.7, computed results significantly

differ from experimental data for both circular or elliptic cross-sections. Comparing re-sults with measured data, circular cross-section assumption seems to give a better fitting.However, surprisingly, the results in terms of circumferential strain in the top position ofthe cross-section (Figure 2.6-c) are inverted in comparison to measurements. On the otherhand, huge discrepancies can be observed in Figure 2.7 when comparing computed withmeasured strains. For instance axial strains (Figure 2.7-b) seem not to have any relation,as both traces present inverted oscillations and with different order of magnitude.

Therefore, assuming the correct development of both expressions, the question remainsin regard to how these expressions are applied. Looking at Equation 2.7, it can be seen theright hand side is composed of the pressure term followed by a product of the summation



Figure 2.7: Measured and computed strains during a hydraulic transient in the coppercoil: (a) circumferential strain in the inner side of the torus; (b) circumferential strain inthe outer side; (c) circumferential strain in the top side; (d) axial strain in the outer side.

of two terms, where the first one ( b2−a2a2

) is very related to the eccentricity of the ellipse.For an A relatively large (A >> a), a low eccentricity of the elliptic cross section isenough to get large strains. Hence, strains are very sensitive to the elliptic eccentricityhighly varying its stress-strain behaviour. Equations 2.6 and 2.7 can be true for a fixedgeometry, but what might be the reason of such discrepancy is the dynamical effect ofthe cross section. Due to bending forces, elliptic eccentricity changes in function of theload, and strains might be very sensitive to this varying eccentricity, therefore in thestrain equation parameters are changing in regard to the output (non-linear problem).The membrane theory used can be valid, but its implementation is incomplete for thepresent case study. Currently, the Ph.D. candidate is working on the determination ofthese pipe-wall displacements with the aim to define the right expression that will allowto stablish the correct relation between pressures and strains.

2.5.2 Hysteresis effect

Strain gauges were installed not only in the copper facility, but also in the polyethylenesystem. Pressure-strain graphs were generated from both facilities obtaining a cloud ofpoints which trend indicates the modulus of elasticity. A selection of those points wascarried out in order to observe only one loading-unloading cycle, and with the aim tocompare both curves. The pressure and strain values were normalized, adjusting values to



a common unitary scale, and then put together in the same graph (Figure 2.8). In bothfacilities, circumferential strain was measured in the top side of the cross-section.

Figure 2.8: Hydraulic transient test in (a) copper pipe and in (b) the HDPE pipe. (c)depicts the normalized stress-strain curves for both tests, and the hysteresis effect.

Figure 2.8-c shows that, copper pipe presents a hysteretic behaviour like the polyethy-lene pipe as the loading curve follows a marked different path than the unloading curve,there is a delay on the strain response. It is important to point out that the stress-straincurves from both materials belong to different time scales. The water-hammer wave in thecopper pipe is much faster than in the polyethylene pipe (around 5 times faster), conse-quently the loading-unloading cycle observed in the copper pipe is, in the same proportion,for a much shorter period of time than in the polyethylene.

Another aspect to point out is the concave shape of the polyethylene graph in theloading curve, which differs on what was explained in the state-of-the-art section (seeFigure 1.7). It is presumed that the reason could be the geometrical constraints of thecoil structure (FSI), however further tests must be carried out in order to determine theexact reason of such phenomena.


Chapter 3

Detailed research plan

3.1 Objectives and methodology

The present research work aims to contribute to the better understanding and improve-ment of gaps in water-hammer theory identified and stated in the literature review and toprovide guidelines for an enhanced use of transient solvers. There is a need to distinguishfrictional and mechanical damping effects and to clarify both physical phenomena. Animprovement on unsteady friction will retrieve a feedback for rheological behaviour of thepipe wall and vice-versa. A more accurate distinction of both phenomena will enhancethe capability of engineers and consultants for a better diagnosis, design and operationalguidelines based on improved transient tools.

To this end, several milestone achievements must be overcome:

• Stress-strain analysis: Stress-strain analysis is the first step to understand thedisplacements of the pipe, either due to lateral, torsional, axial or radial movementsof the pipe-wall and the whole structure. This analysis will be first approachedstatically, measuring strains for specific pressures and stress-strain relationships willbe theoretically developed based on collected data.

• Fluid-structure interaction: The knowledge acquired in the previous step willbe used to carry out coupling between water-hammer classical equations and thestress-strain-displacements laws. In this part of the research a good characterizationof the FSI phenomena should help to isolate the damping effect raised from unsteadyfriction.

• Unsteady friction: With an improved FSI model, the distinction of unsteadyfriction phenomena should be possible from some specific cases on the experimentalfacilities, i.e. for metallic pipes with elastic behaviour of the pipe-wall. Additionally,shear stress will be measured in the steel pipe wall simultaneously with pressureand strain during hydraulic transients. These data will be used to compare thecurrent state-of-the-art unsteady friction models, to select the best for the specificflow conditions and to implement improvements if necessary.

• Rheology of metals: Hysteresis property of metals for very fast loads is a phe-nomena that has not been taken into account yet in hydraulic transient analyses.Currently, this possible source of extra damping effect could be embedded as un-steady friction losses. In this part of the research, and with the improved transientsolver, effort will be invested to distinguish rheological behaviour effect.

Figure 3.1 shows a schematic representation of the methodology to be followed duringthe Ph.D. research.

29

Detailed research plan 30

Figure 3.1: Scheme of the methodology to be followed.

3.2 Thesis outline

Find below attached the provisional table of contents of the Ph.D. dissertation document:



Figure 3.2: Provisional table of contents of the Ph.D. thesis.

3.3 Next steps on the research

Following there is presented a brief summary of the steps to be followed from now on:

• Determination of the best fit stress-strain laws in coils by carrying experimental testsin the copper coil facility and validation of theoretical stress-strain models.

• Inclusion of the strain-stress laws in water hammer equations (FSI coupling), nu-merical and experimental analysis.

• Analysis of the rheological behaviour during hydraulic transients for different mate-rials: copper, polyethylene and steel.



• Unsteady friction analysis by means of shear stress experimental data and velocityprofiles.

• Assessment of mechanical and frictional dissipation in hydraulic transients

3.4 Engineering applications

The present research will contribute to the enhancement of knowledge in fluid mechanicsto improve engineering designs in regard pressurized water pipes. More accurate transientsolvers are required for an improvement of pipe systems safety to prevent pipe collapse orundesired intrusion of contaminants in the pipe. Engineers, water utilities, environmentand society in general will be beneficed by the contribution of the present research.

Results of this project will have a direct impact on:

• engineers and consultants that will have enhanced computational tools and guidelinesfor carrying out hydraulic transient analysis in pressurised water pipes;

• water utilities, as fewer accidents and disruptions caused by extreme pressures willoccur allowing a reduction of operational and maintenance costs associated with pipefailures;

• the consumers, as there will be less failures of service and disruptions;

• the society, as the use of more reliable tools will enhance protection of public health(less frequent accidents and contaminant intrusion);

• the environment, as there will be less water losses, energy consumption and residuesgeneration.

Furthermore, this research work will also contribute to the enhancement of knowledgeon the behaviour of hydraulic transients within coiled pipes. Coil systems have manyapplications in industrial engineering practices. Because its optimal shape they are mostlyused for any heat exchange process like cooling systems in power plants, industrial andcommercial refrigerators, solar water heaters, radiators for automocion industry, etc.


Chapter 4

Planning of the project

4.1 Project time-line and milestones

During the Ph.D. work several milestones must be achieved:

• M1: Stress-strain analysis of a coiled copper pipe for inner pressure loads.

• M2: Fluid-structure interaction in coiled pipes during hydraulic transients.

• M3: Analysis of the rheological behaviour of the pipe-wall during hydraulic tran-sients.

• M4: Mechanical and frictional dissipation in hydraulic transients.

33

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ork

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Planning of the project 35

4.2 Expected publications

During the thesis, several scientific publications in peer-reviewed journals in the field ofhydraulics, fluid mechanics or applied engineering are expected:

• Publication-1: Stress-strain analysis of a coiled copper pipe for inner pressure loads.

• Publication-2: Fluid-structure interaction in coiled pipes during hydraulic transients.

• Publication-3: Analysis of the rheological behaviour of the pipe-wall during hydraulictransients.

• Publication-4: Mechanical and frictional dissipation in hydraulic transients.

Moreover, the candidate intends to have an active participation at national and interna-tional conferences in regard to the research topic.

4.3 Project consortium and collaborations

The PhD research work is embedded in a larger project entitled: “Friction and mechanicalenergy dissipation in pressurized transient flows: conceptual and experimental analysis”.The project comprehends two master theses and two Ph.D. dissertations and its ultimategoal is the publication of design guidelines (methodologies) for using and calibrating devel-oped and commercial hydraulic transient software. Research members of this project are:Antonio Patrıcio de Sousa Betamio de Almeida; Dıdia Isabel Cameira Covas; Nelson JorgeGaudencio Carrio; Alexandre Almeida Mendes Borga; Fabio Verıssimo Gonalves; HelenaMargarida Machado Silva Ramos (Ferreira); Rui Miguel Lage Ferreira. This project isfounded by “Fundacao para a Ciencia e a Tecnologia” (FCT).

The present Ph.D. research, in the framework of the joint doctoral initiative IST-EPFL, is funded by “Fundacao para a Ciencia e a Tecnologia” (FCT) from Portugal andLaboratory of Hydraulic Constructions (LCH) from Lausanne (Switzerland).

4.4 Follow-up of the project

This research work is being supervised by two co-supervisors, Prof. Dıdia Covas and Prof.Anton Schleiss. Accordingly, progress meetings are set regularly either during the timebeing at EPFL or IST. Annual internal conferences are planned at LCH department andat the half of the thesis is expected a milestone internal conference at IST where thecandidate has the chance to expose the work carried out and discuss further steps in frontof a jury and colleagues. Furthermore, annual reports will be sent to the“Fundacao paraa Ciencia e a Tecnologia” (FCT) in order to describe the work completed during everyyear.


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