Stability of Offshore Risers Conveying Fluid
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,
voorzitter van het College van Promoties,
in het openbaar te verdedigingen
op donderdag 28 februari 2008 om 12:30 uur
door
Guido Leon KUIPER
civiel ingenieur
geboren te Maarheeze
Prof. dr. ir. J. Blaauwendraad Prof. dr. ir. J.A. Battjes
Toegevoegd promotor: Dr. Sc. A.V. Metrikine
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. ir. J. Blaauwendraad, Technische Universiteit Delft, promotor Prof. dr. ir. J.A. Battjes, Technische Universiteit Delft, promotor
Dr. Sc. A.V. Metrikine, Technische Universiteit Delft, toegevoegd promotor Prof.dr. M.P. Païdoussis McGill University, Canada
Prof.dr. H. Nijmeijer Technische Universiteit Eindhoven Prof.dr.ir R.H.M. Huijsmans Technische Universiteit Delft Dr. E. de Langre École Polytechnique, Frankrijk
Dr. Sc. A.V. Metrikine heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
ISBN 978-90-5972-236-1 Uitgeverij Eburon Postbus 2867 2601 CW Delft Tel. 015-2131484 / Fax: 015-2146888 info@eburon.nl / www.eburon.nl
Cover Design: Studio Hermkens
Copyright © 2008 G.L. Kuiper. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission in writing from the proprietor.
“Eppur si muove”
Though a PhD project might be considered as a personal achievement, the work carried out in this project was supported by numerous people. I am grateful to all who have permitted, encouraged and accompanied this work.
This research project was financially supported by the Technology Foundation STW (project DCB.6267), Shell International Exploration & Production and Delft University of Technology. Their contributions are highly appreciated.
Andrei, you have been much more for me than just a co-promoter. You made me enthusiastic to start this project in your established wave mechanics group. I am really proud that I have been taught by such an expert in the field of wave mechanics. Personally, I believe that the mutual trust in each other was the foundation for so many fruitful discussions. At least the result of this PhD project should please you since it perfectly fits with your standard “a good PhD project ends with challenging questions rather than clear answers”.
I like to express my gratitude to my promoters, prof. Johan Blaauwendraad and prof. Jurjen Battjes for their help. I really appreciated that Johan Blaauwendraad showed confidence already at the start of this research project by guaranteeing money for this project, although the research proposal was not yet granted at that time. The critical reviews of my manuscripts by Jurjen Battjes have led, without any doubt, to a higher quality. Since I knew that an “average” piece of work would be commented critically by him, I raised automatically my standards. I am grateful to Jurjen Battjes for his keen eye on my work and our fruitful co-operation.
During my PhD study, Shell offered me the possibility to spend two days a week as offshore structures engineer in the Civil/Marine Group headed by Frank Sliggers. I highly enjoyed exploring this opportunity. I worked on challenging problems primarily related to dynamics of offshore risers. Especially, the in-depth explanations by Mike Efthymiou and the practical insights of Rama Gunturi into offshore problems were of incredible value to me. I am grateful to both of them for their inspiring guidance.
About two times a year I presented my new findings to a users committee, which consisted of both academic and industry representatives. Next to the (co-)promoters the committee consisted of prof. Jan Vugts, prof. Jan Meek, prof. Carl Martin Larsen of Norwegian University of Science and Technology (corresponding member), Frank Lange of Heerema, Mike Efthymiou of Shell, Jaap de Wilde of Marin and Corine Meuleman of STW. The interesting discussions with and the suggestions from this diversified committee improved the quality of the work and
broadened my view. I am grateful to all the members in taking part in this committee.
I thank WL Delft Hydraulics for their great support in building the experimental setup in their tank and helping me during the experiments. In particular, the excellent co-operation with Theo Ammerlaan and Martin Boele is highly appreciated. The surprising observations in the tank forced me to ask them almost daily for drastic changes of the experimental setup in an attempt to unravel these mysteries. Their creative solutions to facilitate this and the enthusiasm of Theo Ammerlaan and Martin Boele really impressed me.
I tender my best thanks to the master students, whose graduation theses I had the pleasure of supervising. I owe a great deal of working with such devoted and enthusiastic colleagues.
I have really benefited from the two comprehensive books about fluid-structure interaction written by professor Païdoussis from McGill University (Canada). Thanks to these books, I saved a lot of time in finding relevant literature. I recommend everybody interested in this field, from student to expert, to consult these books.
Hanna, my gratitude to you is beyond words. Your loving support, continuous encouragement and sincere interest gave me the energy to complete the thesis in time. I also thank my parents for encouraging me to start this project and for supporting me during the study. I like to express my gratitude to Agnes and Gerben for being my ushers at the PhD defence.
Thanks go to all colleagues and roommates, who made me really enjoy working on this PhD project.
Guido Kuiper Delft, January 2008
1. General introduction 1
1.1 Top-tensioned riser in deep water 2
1.2 Free-hanging water intake riser 2
1.3 Mechanisms of instability of offshore risers 4 1.4 Historical overview of work related to cantilever pipes conveying fluid 5
1.5 Aims of the thesis 6
1.6 Outline 8
1.7 Output 9
2. Stability of a pipe conveying fluid – an overview 11
2.1 Introduction 12
2.2 Infinitely long string conveying fluid 14 2.3 Finite-length string conveying fluid with a free end and a fixed end 18 2.4 Infinitely long beam conveying fluid 21 2.5 Finite beam conveying fluid with different supports 23
2.6 Conclusion 26
3. Energy considerations for a pipe conveying fluid 27
3.1 Introduction 28
3.2 The paradox of the simply-supported pipe 28 3.3 Improved energy equation for a pipe conveying fluid 30
3.4 Balance of pseudo-momentum 31
3.5 Energy balance in aperiodic motion 33
3.6 Qualitative analysis of energy increase 34 3.7 Introduction to travelling wave method 38 3.8 Energy of a pulse travelling in one-dimensional pipe conveying fluid 39 3.9 Energy exchange at boundaries of a pipe conveying fluid 42
3.10 Conclusion 46
4. Stability of a free hanging riser conveying fluid
– stability of the straight configuration 47
4.1 Introduction 48
4.2 Assumptions and equation of motion 48
4.3 Description of the internal pressure 51 4.4 Dimensionless form of the equation of motion and the boundary conditions 52
4.5 Characteristic equation 53
4.6 Argand Diagram 55
4.7 D-decomposition method 57
4.8 Effect of fluid pressurization and external damping 62
5. Stability of a free hanging riser conveying fluid
– steady state vibrations with nonlinear damping 65
5.1 Introduction 66
5.2 Equation of motion and boundary conditions 67 5.3 Stability of a pipe conveying fluid in the linear approximation 70 5.4 Hydrodynamic drag on flexible cylinders 72 5.5 Stability of a pipe conveying fluid with nonlinear drag 79
5.6. Conclusion 87
6. Experimental investigation of a cantilever pipe conveying fluid 89
6.1 Introduction 90
6.2 Critical velocities predicted by different theories 91 6.3 Experimental set-up and instrumentation 94 6.4 Observations – qualitative description 98 6.5 Observations – quantitative description 98
6.6 Second set of experiments 101
6.7 Modelling the instability 103
6.8 Possible explanations for the experimentally observed pipe behaviour 109
6.9 Conclusion 114
7. Destabilization of deep-water risers by a heaving platform 115
7.1 Introduction 116
7.2 Assumptions and equation of motion 118 7.3 Stability of the straight configuration 121 7.4 Nonlinear analysis of instability development 127
7.5 Bending stresses in the riser 136
7.6 Conclusion 137
8. Main results, practical relevance and recommendations 139
8.1 Main results 139
8.2 Practical relevance 142
8.3 Recommendations 143
References 145 Appendix A-Spectral energy density of a pulse 151
Appendix B -Depressurization at the inlet 153
Appendix C-Floquet Theory 155
Summary 157 Samenvatting 159
CHAPTER 1
General introduction
The offshore industry is massively involved in exploration and exploitation of oil and gas fields under the sea bed. Up to ten years ago the majority of the oil and gas fields were developed using fixed structures standing on the sea bottom. The tallest fixed structure in the world is located in the Gulf of Mexico based at a depth of 412 m below sea level (Bullwinkle). However, new fields were discovered in deeper water and at greater distance from the shore. Other solutions had to be found since the fixed structures reached their economic limits. The offshore industry responded by designing and constructing floating platforms, like the Tension Leg Platform, the Semi Submersible and the FPSO (Floating Production Storage and Offloading vessel).
During the production process, the hydrocarbons from the oil and gas fields flow (rise) through pipes to the floating unit. The pipe connecting the wellhead at the sea bottom to the floating platform is called the riser. For fixed structures the risers are supported by the substructure. For floating units the riser is suspended from the main deck of the floating platform and has no supports along its length.
1.1 Top-tensioned riser in deep water
In the last years, drilling activities took place in waters as deep as 2000 meter. It is expected that this is not the limit, since fields at water depths of over 3000 meter have already been discovered. For exploitation activities a straight top-tensioned riser (Fig. 1.1) or a catenary-shaped riser may be used, depending on the floating unit from which the riser is suspended. If the platform is not moving significantly in the vertical direction (heave), it is preferable to use a top-tensioned riser due to its direct access to the well. The lowest point of a top-tensioned riser is connected to the wellhead at the sea bottom. Since the diameter of a steel top-tensioned riser is about 0.25 m, the diameter over length ratio in these extreme water depths is in the order of 1/10,000. This low aspect ratio causes the spectrum of the natural frequencies to be quite dense in the sense that the natural frequencies are located close to each other. Hence, instability phenomena and forced vibrations can easily occur at a natural frequency, which might result in significant vibrations of these long risers.
1.2 Free-hanging water intake riser
Besides going deeper, a second trend in the offshore industry is to develop large gas fields much further away from the shore. As a result, the cost of the infrastructure to connect newly found reserves to onshore facilities is increasing. To circumvent this, a new concept has been developed, in which the gas is liquefied offshore, on a barge (Fig. 1.2), thereby achieving a substantial cost reduction for remote fields.
Fig 1.2 - Artist impression of a barge on which the natural gas is liquefied.
In order to liquefy the gas on such a barge, a great volume of cooling water is required. Depending on the location this water, to be cold enough, has to be pumped up from depths ranging from 150 to 500 meters below the sea level. The pumping takes place through a set of steel intake risers, which are suspended from the barge. In contrast to the top-tensioned riser, this free hanging riser is not connected to the sea bottom (Fig. 1.3). The current concept of a floating LNG plant requires large quantities of cooling water with a maximum rate of water intake in the order of 75,000 m3/hour. Consequently, a large diameter riser (about 1.0 meter diameter) in
combination with a high flow velocity is needed to limit the number of water intake risers to less than ten. This type of risers with a free end and a high internal fluid velocity has so far not been applied in the offshore industry.
Fig. 1.3 - Sketch of a water intake riser suspended from a barge. Barge
Water intake riser
1.3 Mechanisms of instability of offshore risers
It is common practice in the offshore industry to analyse forced vibrations of the riser. The riser is subjected to current and waves that act on the riser both directly and through the floating platform, imposing oscillations on the riser top.
Besides conventional issues associated with forced vibrations, the offshore industry considers one important instability mechanism: vortex-induced-vibrations (VIV). VIV is a complex phenomenon, which can occur for pipes exposed to steady current flows. The riser vibrates mainly perpendicular to the flow direction. These vibrations are caused by unsteady lift and drag forces generated by shedding of vortices. In practice, theoretical prediction of vortex-induced oscillations proceeds hand in hand with experimental verifications.
However, the free-hanging water intake riser and the top-tensioned riser in deep water are also sensitive to less known instability mechanisms. The free-hanging
water intake riser might lose stability due to internal fluid flow. It is well known
that a cantilever pipe, which conveys fluid from the fixed end to the free one, becomes unstable when the fluid velocity exceeds a critical value. A good example is the chaotic motion of the loose end of a garden hose. However, for the reversed flow direction, i.e. the flow enters the pipe at the free end and leaves the pipe at the fixed end, there is no consensus among researchers as to the mechanism of instability and the water velocity at which the instability occurs. In this thesis this configuration is referred to as a cantilever pipe aspirating fluid. For the offshore industry the latter situation is of clear relevance, since this corresponds to the water intake risers that will be used in the floating LNG concept.
A possible mechanism of instability of top-tensioned risers in deep water is associated with fluctuation of the axial tension in the riser caused by vertical motion (heave) of the floating platform in waves. Although this fluctuation is significantly reduced by heave compensators through which the riser is connected to the platform, it can be dangerous. The danger is that the fluctuating tension might destabilize the equilibrium of the straight riser and cause it to vibrate at a dangerously high level. Obviously, there is also an internal fluid flow through these top-tensioned risers, which, in principle, can destabilize the system. However, owing to the low fluid speed and the fixed ends, the internal fluid flow cannot destabilize these top-tensioned risers in practice.
1.4 Historical overview of work related to cantilever pipes conveying fluid
Dynamics of pipes conveying fluid has been studied by many researchers. Self-excited oscillations of a cantilevered pipe conveying fluid were observed in the laboratory by Brillouin as early as in 1885 but remained unpublished1. The first
serious study of the dynamics of pipes conveying fluid is due to Bourrières (1939), who derived the correct equations of motion and reached accurate conclusions regarding stability of the cantilevered system. This study, published in the year of the outbreak of the Second World War, was effectively ‘lost’, and researchers in the years thereafter re-derived everything in ignorance of its existence.
Like with this thesis, a practical application initiated the study of pipes conveying fluid. In the 1950s the vibrations observed in the Trans-Arabian Pipeline were the cause to study pipes conveying fluid. Ashley and Haviland (1950) were the first who attempted to explain these vibrations. Soon hereafter, Feodos’ev (1951), Housner (1952) and Niordson (1953) studied the dynamics of pipes supported at both ends.
The next study related to cantilevered pipes conveying fluid appeared in 1961 and was performed by Benjamin. He treated in two papers the dynamics of articulated cantilevers conveying fluid and concluded with a discussion of the corresponding continuous system. He was the first who established the Lagrangian equations for this open system, where fluid flows through the two boundaries. From then on, a lot of researchers studied dynamics of a pipe conveying fluid. Up to 2003, approximately 550 significant publications appeared2. Professor Païdoussis of
McGill University, Montreal, Canada, deserves special attention in this historical overview since he published a large number of authoritative papers in this field. In addition, he wrote two comprehensive books (Païdoussis, 1998 and 2003) about fluid-structure interactions of slender structures and axial flow.
Most studies of dynamic stability of cantilever pipes conveying fluid are devoted to pipes discharging the fluid from the free end. An overview of these studies can be found in Païdoussis (1998). For fluid velocities below a critical value such pipes are stable, whereas beyond this critical velocity they show a self-excited oscillatory motion (flutter). Theoretical models predict the same critical velocity as observed in experiments.
A closely related problem of a submerged cantilever pipe aspirating water through its free end received less attention in the past. Again, the study was initiated by a
1 Mentioned by Païdoussis (1998) in an overview of researchers working on pipes conveying fluid. 2 Païdoussis mentioned this number in his article in Journal of Sound and Vibration (2008) – “The
canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across Applied Mechanics”.
practical problem; cantilever pipes were used in ocean mining to transport a mixture of water and nodules from the sea bottom to a ship (Chung et al., 1981). The early theory predicted unstable behaviour for undamped cantilever pipes at infinitely small fluid velocity (Païdoussis and Luu, 1985; Sällström and Åkesson, 1990; Kangaspuoskari et al., 1993). In contrast, in experiments with cantilevered pipes pumping up water (Hongwu and Junji, 1996; Païdoussis, 1998) no instability was observed. In 1999, Païdoussis formulated a hypothesis why these cantilevered pipes are expected to be stable. The main reason, according to this explanation, was the negative pressurisation of the fluid at the inlet of the pipe. This was also related to Feynman’s dilemma on the direction of rotation of an S-shaped lawn sprinkler sucking up water instead of spewing it out (see also Forrester, 1986; Hsu, 1987). The Nobel Prize winner could argue convincingly both directions of rotation3. For
some years, the explanation in Païdoussis’ paper was considered as satisfactory and no papers appeared on stability of cantilever pipes aspirating fluid in that period. In 2005 Kuiper and Metrikine pointed out that this “proof” was at best incomplete. They showed that there is still a contradiction between reported experiments, where no instability was observed, and theory, which predicted instability beyond a critical velocity. This paper initiated a re-thinking of the contradiction and resulted in several papers from different research groups in the last two years. These recent papers are discussed in detail in this thesis.
This thesis should not be considered as the final answer to this contradiction and, hence, it is to be expected that more papers will appear on this interesting topic. This statement has been confirmed by Païdoussis who in an overview article (Païdoussis, 2005) classified this problem as one of the unresolved issues in the area of fluid-structure interactions.
1.5 Aims of the thesis
The study presented in this thesis is devoted to the stability of a submerged riser. We consider both the free-hanging water intake riser aspirating fluid and a top-tensioned riser subject to excitation by a heaving platform. A riser might lose stability because of the fluid flow through the riser (pipe flow instability) and/or because of the wave-induced variation of the riser tension in time (parametric instability). The main focus of this thesis is on risers conveying fluid and to a lesser extent on parametric excitation of top-tensioned risers. Vortex-induced-vibrations of risers are not considered in this thesis.
The main objective of the thesis is to understand the phenomenon of instability of a
cantilever pipe aspirating fluid. A number of experiments were carried out in the
past (Hongwu and Junji, 1996; Païdoussis, 1998) in order to experimentally observe and study the instability. However, in all these experiments the pipe remained stable. According to all known theories, a pipe aspirating fluid should become unstable in one way or another, at a certain fluid speed. This contradiction between experiments and theory triggered us to explore whether in all previous experiments the internal fluid velocity has apparently not exceeded the “critical fluid velocity” or the pipe aspirating water is unconditionally stable and hence all existing theories are incorrect.
In particular, the thesis has the following objectives for the water intake riser: • to investigate whether instability of a pipe aspirating fluid can be observed
experimentally;
• to predict quantitatively the onset and the type of instability (if instability exists);
• to predict the pipe amplitude of the steady-state vibrations in the unstable regime (if instability exists);
• to understand the main parameters which govern the pipe flow instability and, hence, to conclude whether a free hanging riser of 150 m or more suffers from flow-induced instability.
To bridge the gap between theory and experiments we worked on both aspects. Hence, in this thesis one can find both improvements of existing theory and new experiments in which much higher internal fluid velocities were attained than before. For the straight top-tensioned riser in deep water the primary aim is to find the range of practically relevant amplitudes and frequencies of the vertical vibration of the platform which may destabilize the riser and cause significant transverse vibrations. There are several original components in this work in determining the instability zones. The model accounts for viscous hydrodynamic damping, depth-dependent axial tension and a high modal density of deep-water risers. Conventionally-used analytical methods are not applicable in this case and, hence, other semi-analytical methods were developed.
The second aim is to identify possible mechanisms of the destabilization and to distinguish these mechanisms which may lead to dangerously high dynamic amplification of the stresses in the riser.
In this thesis the research on top-tensioned risers is restricted to a theoretical part only.
1.6 Outline
This thesis consists of two parts. The larger part (Chapters 2-6) deals with instability of slender pipes induced by internal, axial flow.
In Chapter 2 the fundamentals of instability of pipes conveying fluid is explained. Simple models are used to familiarize the reader with the physical background of the phenomenon. For experts in the field the subsequent chapters might be of more interest. Similar simplistic models are treated in Chapter 3, now considering such pipes from an energy point of view. In order to understand the energy transfer of a pipe conveying fluid, the energy exchange at a support is analyzed. The analysis reveals the large impact of the type of support and the fluid direction on the critical fluid velocity (internal fluid velocity beyond which the pipe behaves unstable). Readers with a more practical interest in water intake risers are advised to start at Chapter 4. In this chapter a more sophisticated model is developed for a submerged, free hanging offshore riser that is aspirating fluid. As a first step, the derived equation is linearized around the straight equilibrium. It is proven that the hydrodynamic drag caused by the surrounding water prohibits the pipe from unstable behaviour at low fluid velocities, in contrast to the commonly assumed cause of the negative fluid pressurization at the inlet of the pipe. In Chapter 5 a nearly identical model for the free hanging offshore riser is considered. The only difference is related to the hydrodynamic drag, which is described by a nonlinear time-domain description and is based on experimental data available in literature. All existing theories, whether described in these chapters or published in literature, predict instability beyond a critical fluid velocity. However, reported experiments did not show any instability. To investigate this apparent discrepancy between theory and experiments, a new test set-up was built in which theoretically predicted critical velocities were achievable. The experimental set-up and the test results are described in Chapter 6.
The second part of this thesis (Chapter 7) deals with the stability of a top-tensioned riser, subject to a time-varying axial tension in the riser. A possible and undesirable consequence is the excitation of a transverse riser vibration caused by this fluctuation. Although this topic differs from that discussed in the preceding chapters, similar solution techniques are used to find the instability zones.
Chapter 8 summarizes the main conclusions and provides some thoughts for challenging research directions related to the stability of a free hanging pipe aspirating fluid.
1.7 Output
Most of the results of the research presented in this thesis have been or will be published in the papers listed below:
Kuiper, G.L. and Metrikine, A.V. (2005) Dynamic stability of a submerged, free hanging riser conveying fluid. Journal of Sound and Vibration 280, 1051-1065. Metrikine, A.V., Battjes, J.A. and Kuiper, G.L. (2006) On the energy transfer at
boundaries of translating continua. Journal of Sound and Vibration 297, 1107-1113.
Kuiper, G.L., Metrikine, A.V. and Battjes, J.A. (2007) A new time-domain drag description and its influence on the dynamic behaviour of a cantilever pipe conveying fluid. Journal of Fluids and Structures 23, 429-445.
Kuiper, G.L., Brugmans, J. and Metrikine, A.V. (2008) Destabilization of deep-water risers by a heaving platform. Journal of Sound and Vibration 310, 541-557. Kuiper, G.L. and Metrikine, A.V. (2008) Experimental investigation of dynamic
stability of a cantilever pipe aspirating fluid. Journal of Fluids and Structures 24 (in press).
Metrikine, A.V, and Kuiper, G.L. (submitted to Journal of Fluids and Structures) Energetics of simply-supported pipe conveying fluid, revisited.
CHAPTER 2
Stability of a pipe conveying fluid
– an overview
The objective of this chapter is to familiarize the reader with the phenomenon of instability of pipes conveying fluid. To this end, simplistic models of the pipe conveying fluid are considered to enable a clear mathematical definition of instability. The physical background of the phenomenon is discussed by comparing infinitely long systems with finite systems. It appears that the type of boundary supports and the flow direction have a large impact on the critical fluid velocity, which identifies the transition from stable to unstable behaviour. To explain this, the energy exchange at the boundaries is studied in more detail in the next chapter. This introductory chapter is only meant to explain the fundamentals of stability of pipes conveying fluid and does, therefore, not present new contributions in this field. Experts in the field may skip this chapter and proceed directly to remaining chapters.
2.1 Introduction
In this introduction the term “stability” in linear systems is explained. We consider subsequently a one-mass-spring-system, a finite system and an infinitely long system. The stability for a linear system is not affected by external forces, and hence, they are disregarded. A linear system is unstable if its displacement grows in time exponentially or, more generally, can be represented as a superposition of exponentially growing functions.
First, the stability of a one-mass-spring-system is considered. Its motion is described by the following ordinary differential equation:
2
2 + + =0
d w dw
m c kw
dt dt , (2.1)
where m is the mass of the system, c is the damping coefficient, k is the spring stiffness, w is the distance from the equilibrium position and t is the time. The general solution of this second order differential equation is:
2 1 ( ) λ = =
∑
jt j j w t C e , (2.2)where Cj are constants and λj are the complex eigenvalues of the system.
Substitution of the general solution (2.2) in the equation of motion (2.1) yields the characteristic equation:
2 0
mλ +cλ+ =k . (2.3)
The two eigenvalues are determined by the characteristic equation:
2 1,2 4 2 c c mk m λ =− ± − . (2.4)
For positive values of the damping coefficient the real part of the eigenvalues is negative, and hence, according to Eq. (2.2) the displacement of the system decays in time. As expected, a “standard” system (positive values for mass, damping coefficient and spring stiffness) behaves stable. On the contrary, if in the hypothetical case the damping coefficient is negative the eigenvalues are positive and hence the displacement grows exponentially in time, i.e. the system is unstable for all arbitrary initial conditions.
Normally, the dashpot takes energy out of the system. However, a negative value of the damping coefficient implies that energy is continuously put into the system, obviously resulting in unstable behaviour.
Stability of a finite dimension linear non-parametric continuum is fully determined by its natural frequencies. The procedure of finding these frequencies is well-known
(similar as for the one-mass-spring-system) and can be found in any text-book on the subject (for example Païdoussis, 1998). Analysis of stability of infinite linear continua is somewhat more cumbersome and, therefore, is shortly outlined below. If a one-dimensional system is infinitely long, then any physical realizable initial conditions (initial displacement ϕ
( )
x and initial velocity ψ( )
x ) can be represented as:( )
x( )
k e ikxdk ϕ ∞ϕ − −∞ =∫
, (2.5)( )
x( )
k e ikxdk ψ ∞ψ − −∞ =∫
, (2.6)where ϕ
( )
k and ψ( )
k are the initial displacement and initial velocity in the wavenumber domain, respectively, k is the real wavenumber and i= −1.The initial conditions initiate a wave process in the system. Each pulse comprising this process can be presented as the following superposition of waves:
( )
,( )
i( ( )k t kx) w x t w k e ω dk ∞ − −∞ =∫
, (2.7)where w x t
( )
, is a displacement of the system, w k( )
is the complex amplitude of the wave with the wavenumber k and ω( )
k is a radial frequency corresponding to this wavenumber k. The complex amplitude w k( )
is determined by the initial conditions, Eqs. (2.5) and (2.6). The wavenumber k and the radial frequency ω are related to each other through the dispersion equation. Since we consider a linear system one may consider the evolution of each wave-component,( )
i( ( )k t kx)w k e ω − , of the pulse separately and then superpose the results.
Each real wavenumber corresponds to a number of radial frequencies, which are generally complex. From Eq. (2.7) it is clear that if the imaginary part of one of the frequencies is negative, the wave-component at this specific wavenumber grows exponentially in time. Since the total displacement is an integral over the wave components, if at least one of these grows, the total displacement grows off bounds as well, and hence, the system is considered unstable. In short, for a linear infinite system one may analyse all individual waves separately. If at least one wave grows in time while propagating (the imaginary part of the frequency of this wave is negative) and the initial conditions give rise to this specific wave, one should conclude that the system is unstable.
In the remaining sections different models are discussed for an infinitely long and finite-length pipe conveying fluid.
2.2 Infinitely long string conveying fluid
In an infinitely long system boundary effects are not considered. To explain the mechanism of instability of an infinitely long pipe conveying fluid, a tensioned pipe is considered, whose flexural rigidity can be disregarded. This model of the pipe is referred to as a “string” to express that in the absence of the fluid the governing equation reduces to that of a taut string. The equation of motion of an infinitely long tensioned pipe conveying fluid (see Fig. 2.1) can be written as (Chen and Rosenberg, 1971): 2 2 2 2 2 2 2 p 2 f f 2 2 f 2 0 w w w w w T m m u u x t x x t t ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ − + + ⎜ + + ⎟= ∂ ∂ ⎝ ∂ ∂ ∂ ∂ ⎠ , (2.8)
where T is the constant tension in the pipe, mpis the mass of the pipe per unit length,
f
m is the mass of fluid per unit length, flowing with a steady flow velocity uf , and
w is the transverse deflection of the pipe; x and t are the axial coordinate and time,
respectively.
The internal fluid flow is approximated as a plug flow, i.e. as if it were an infinitely flexible rod travelling through the pipe, all points of the fluid having a velocity uf
relative to the pipe. This is the simplest possible form of the slender body approximation for the problem at hand. A fluid particle of mass mf experiences three acceleration components as measured by a stationary observer:
• centripetal acceleration, 2 2 2 f w u x ∂ ∂ , • Coriolis acceleration, 2uf 2w x t ∂ ∂ ∂ , • local acceleration, 2w2 t ∂ ∂ .
Another, identical representation of the equation of motion can be achieved by making use of two observers; one stationary observer and one observer, who moves with the flowing fluid:
2 2 2 2 2 2 0 f p f x u t w w d w T m m x t dt = ∂ ∂ − + + = ∂ ∂ . (2.9)
It can easily be shown that Equations (2.8) and (2.9) are identical by making use of the material derivative of the lateral displacement:
f f f x u t x u t dw w w dx w w u dt = t x dt = t x ∂ ∂ ∂ ∂ = + = + ∂ ∂ ∂ ∂ , 2 2 2 2 2 2 2 2 2 f f f f f x u t x u t d w d w wu w u w u w dt = dt t x = t t x x ∂ ∂ ∂ ∂ ∂ ⎛ ⎞ = ⎜ + ⎟ = + + ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ .
Fig. 2.1 - Sketch of a tensioned pipe conveying fluid.
A solution of Eq. (2.8) is sought in the form of a travelling harmonic wave: i.e.,
( )
, i( jt k xj)j
w x t =A e ω − , (2.10)
where Aj is the complex amplitude of the wave j, ωj and kj are the radial frequency and the wavenumber of this wave, respectively. Substituting expression (2.10) into the second-order partial differential equation (2.8) yields the following algebraic equation:
(
)
2 2 2 2 2 0 j f f j f f j j p f j Tk m u k m uω k m m ω − + − + + = . (2.11)This is the dispersion equation for a string conveying fluid, which relates the frequencies and the wavenumbers of waves that may be excited in the system. When solved, equation (2.11) yields the following two radial frequencies:
2 2 2 1 ω = f f + f f + t − t f f t m u m u m T m m u k m , (2.12) 2 2 2 2 ω = f f − f f + t − t f f t m u m u m T m m u k m . (2.13)
where mt =mp+mf. From the linear relation between the radial frequency and the wavenumber, it can be concluded that the string conveying fluid is non-dispersive, i.e. the shape of a pulse remains the same as it propagates along the medium. The phase velocity, (the propagation velocity of points of constant phase of a harmonic wave) follows directly from Eqs. (2.12) and (2.13):
2 2 2 1 f f f f t t f f d t m u m u m T m m u c k m ω + + − = = , (2.14) 2 2 2 2 f f f f t t f f u t m u m u m T m m u c k m ω − + − = = , (2.15) uf T T x w(x,t)
Fig. 2.2 - Definition sketch of the downstream and upstream direction.
where cd and cu are the downstream and upstream phase velocities, respectively. The downstream direction is defined as the direction of the fluid flow (see Fig. 2.2). For small values of the fluid velocity, uf < T mf , an initial distortion of the string results in two pulses that propagate in opposite directions, since the two phase velocities are of opposite sign.
As can be seen from Eqs. (2.14) and (2.15), the pulse in the downstream direction, i.e. in the direction of the fluid flow, travels faster than the one in the upstream direction. In systems with a non-moving medium the modal vibrations can be thought of as an interaction between two counter propagating harmonic waves with the same frequency. The modal vibration can only occur since the phase velocities of the harmonic waves in opposite directions have the same magnitude. In a moving medium the upstream and downstream travelling waves have different phase velocities, and consequently, such a finite system does not possess classical normal modes. The modal displacement patterns contain both stationary and travelling-wave components.
It is interesting to see what happens if the fluid velocity increases. A sufficiently large fluid velocity results in a positive numerator of Eq. (2.15). Physically, this means that an initial distortion results in two pulses both travelling in the downstream direction, though with different phase speeds. Hence, all the points situated at the upstream side of an initially excited area remain undisturbed. The transition from two pulses travelling in opposite direction to two pulses travelling in the same direction occurs at the following fluid velocity:
, . f crit low f u = T m . (2.16) uf T T Upstream Downstream Downstream wave Upstream wave
Fig 2.3 - Impulse response (δ( ) ( )x δ t ) of a pipe conveying fluid. (a) absolute instability (b) convective instability. Each trace corresponds to successive instants (taken from De Langre
and Ouvrard, 1999).
In this chapter this velocity is referred to as the lower critical velocity. By increasing the fluid velocity further, the expression under the square root in Eq. (2.12) becomes negative, indicating complex radial frequencies. This occurs for the first time at
(
)
(
)
, . f crit high t f t f u = m T m m −m . (2.17)This velocity is referred to as the higher critical velocity. In the flow regime exceeding the higher critical velocity, there is always a frequency with a negative imaginary part, irrespective of the wavenumber of the wave. From Eq. (2.10) it can be concluded that in this case an initial distortion grows exponentionally in time, indicating an unstable system. In this regime two unstable cases may then be distinguished by considering the long-time behaviour of an impulse response: absolute and convective instability (De Langre and Ouvrard, 1999).
The instability is said to be absolute in the case where the response is contaminating the entire medium after an infinitely long time (see Fig. 2.3.a). Conversely, convective instability refers to the case where the wavepacket is being convected away from the source in the downstream direction (see Fig. 2.3.b). For the string conveying fluid both instabilities can occur, depending on both the fluid velocity and the mass ratio of fluid and pipe.
2.3 Finite-length string conveying fluid with a free end and a fixed end
In order to study a more realistic model, a finite length string conveying fluid is analysed by imposing two supports on the system. The simplest model to represent a cantilever pipe aspirating fluid is the finite length string with an upstream free support and a downstream fixed support separated by a distance L. Like in the preceding section the internal flow is approximated as a plug flow. The equation of motion is the same as used for the infinitely long string, Eq. (2.8). For a finite system, this model is valid only for fluid velocities smaller than the lower critical velocity, Eq. (2.16). For higher fluid velocities no waves can travel in the direction opposite to the fluid flow. Hence, this model is unable to describe reflection of a wave incident on a downstream end at these super-critical velocities (in this case the equation of motion should be extended with the term representing the bending stiffness).
At both ends one boundary condition is formulated. Like in the case of a string without internal fluid, it is assumed that at the free end the transverse force is equal to zero4. The fixed support requires zero deflection. This yields the following
system of equations for the cantilever string aspirating fluid:
2 2 2 2 2 2 2 p 2 f f 2 2 f 2 0 w w w w w T m m u u x t x x t t ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ − + + ⎜ + + ⎟= ∂ ∂ ⎝ ∂ ∂ ∂ ∂ ⎠ , (2.18)
( )
0 , 0 x w x t x = ∂ = ∂ and w x t( )
, x L= =0. (2.19)We seek for a solution to this problem having the following form:
( )
,( )
i tw x t =W x eω . (2.20)
If the imaginary part of at least one of the eigenfrequencies ω is negative, the displacements grow exponentially in time, i.e. the system becomes unstable. Inserting solution (2.20) into the differential equation (2.18) yields:
(
)
2 2 2 2 2 ω ω 0 − + − = f f f f t d W dW m u T m u i m W dx dx . (2.21)The general solution to this ordinary differential equation is:
( )
2 1 j ik x j j W x C e = =∑
, (2.22)in which Cj are constants. Substitution of this solution into Eq. (2.21) gives:
4 At a free end the influence of fluid intake on the balance of transverse forces is investigated in detail
(
2)
2 2 2 0f f j f f j t
T m u− k − m u ωk −mω = . (2.23)
From this algebraic equation the wavenumbers kjcan be found as functions of the frequency ω: 2 2 2 1 2 f f f f t t f f f f m u m u m T m m u k T m u ω⎛⎜ + + − ⎞⎟ = − ⎜ ⎟ ⎝ ⎠ , (2.24) 2 2 2 2 2 f f f f t t f f f f m u m u m T m m u k T m u ω⎛⎜ − + − ⎞⎟ = − ⎜ ⎟ ⎝ ⎠ . (2.25)
To find the natural frequencies Eq. (2.22) is inserted in the boundary conditions, Eq. (2.19), to give the following system of algebraic equations:
1 1 2 2 0
C k +C k = and 1 2
1 ik L 2 ik L 0
C e +C e = . (2.26)
A non-trivial solution of these equations exists if:
2 1
1 ik L 2 ik L 0
k e −k e = . (2.27)
Substituting expressions (2.24) and (2.25) into Eq. (2.27) and carrying out some mathematical manipulations the natural frequencies can be expressed as:
( ) ( ) ln 2π ω =− − − n i a b n a b L , (2.28) where f f f2 f2 t2 t f f2 f f m u m u m T m m u a T m u + + − = − , 2 2 2 2 f f f f t t f f f f m u m u m T m m u b T m u − + − = − .
In Eq. (2.28) n is an integer. A common way to analyse the natural frequencies is to make use of an Argand Diagram (Païdoussis, 1998). In this Diagram the real and imaginary parts of the natural frequency ω are plotted parametrically, as they depend on one of the system parameters. The flow velocity uf is used as such a
parameter and is gradually increased, starting from zero. At zero fluid velocity, the system possesses only real natural frequencies. These natural frequencies equal the resonance frequencies of a fixed-free string including the enclosed fluid. From this starting point, the fluid velocity is gradually increased and the accompanying complex values of the natural frequencies are computed and plotted in the Argand Diagram.
For the finite string with an upstream free end and a downstream fixed end the Argand Diagram is shown in Fig. 2.4. Only the first four natural frequencies are plotted. The paths of the other natural frequencies have a similar shape. As explained before, the system behaves unstable if the imaginary part of at least one
of the eigenfrequencies is negative. Except for zero fluid velocity, the imaginary parts of all natural frequencies are negative, implying that the pipe becomes unstable at any infinitesimal fluid velocity! As the fluid velocity approaches the lower critical velocity, uf crit low, . = T mf , all paths approach the origin of the Argand
Diagram. In the beginning of this section it was noted that the string model is only valid for fluid velocities below this critical velocity. For higher fluid velocities no waves can travel in the direction opposite to the fluid flow. Hence, the boundary condition at the downstream end cannot be satisfied all the time. For these super-critical velocities another description should be used, e.g. the Euler-Bernoulli beam (see sections 2.4 and 2.5).
If the fluid velocity is reversed, i.e. a cantilever pipe discharging fluid, the paths in the Argand Diagram are mirrored with respect to the horizontal axis, as shown in Fig 2.5. For such a cantilever pipe, i.e. with an upstream fixed end and a
downstream free end, all paths have a positive imaginary part, indicating a stable
system for fluid velocities below the lower critical fluid velocity.
Fig. 2.4 – Argand Diagram for a string conveying fluid with an upstream free end and a
downstream fixed end (first four natural frequencies)
(
6 3 3)
1.00 10 N, 1.00 10 kg/m, 2.00 10 kg/m, =100 m = ⋅ f = ⋅ t= ⋅ T m m L . 0 0.5 1 1.5 2 2.5 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05 0.075 0.1 1 2 3 4 Re(ω) in rad/s Im( ω) in r ad/s uf = 0 uf = 0 uf = 0 uf → √(T/mf) uf = 0Fig. 2.5 – Argand Diagram for a string conveying fluid with a downstream free end and an
upstream fixed end (first four natural frequencies)
(
=1.00 10 N, ⋅ 6 =1.00 10 kg/m, ⋅ 3 =2.00 10 kg/m, =100 m⋅ 3)
f t
T m m L .
This above analysis of the string conveying fluid leads to the conclusion that the physical mechanism causing instability of the free-fixed string is different from the instability mechanism of an infinite string. In the infinitely long string unstable waves are generated in the medium for large fluid velocities. Since the infinitely long string is stable at low fluid velocities, the instability of the free-fixed string at low velocities should be due to another mechanism, namely due to energy exchange at the boundaries. Wave reflections at the boundaries result in an energy gain or loss of the total system. Depending on the type of boundary support and fluid flow direction, instability may or may not occur at low fluid velocities. Energy transfer at the boundaries is explained in detail in Chapter 3.
2.4 Infinitely long beam conveying fluid
As explained in the previous section, the string model cannot be used for fluid velocities exceeding the lower critical velocity. In addition, for physical systems where the bending energy dominates, the critical velocity is incorrectly determined if the string model is used. In order to overcome this limitation and to establish a more realistic model, the bending stiffness of the pipe, EI, is incorporated.
0 0.5 1 1.5 2 2.5 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05 0.075 0.1 1 2 3 4 Re(ω) in rad/s Im (ω) in r ad /s uf = 0 uf = 0 uf = 0 uf → √(T/mf) uf = 0
Considering a pipe, modeled as a tensioned Euler-Bernoulli beam, the linear equation of motion takes the form:
(
)
(
)
4 2 2 2 2 4 f f 2 2 f f p f 2 0 w w w w EI T m u m u m m x x x t t ∂ − − ∂ + ∂ + + ∂ = ∂ ∂ ∂ ∂ ∂ . (2.29)In literature, stability analysis of infinitely long beams conveying fluid has been discussed. Roth (1964) and Stein and Tobriner (1970) were the first who analysed this topic. In many studies this “classical” equation is taken as a starting point for analysing stability of slender pipes conveying fluid. Païdoussis (2008) has explained the popularity of this equation: “The governing equation of motion is simple enough
to solve, yet can demonstrate generic features of much more complex dynamical systems.”
In various articles (e.g. Païdoussis, 1970), Eq. (2.29) has been derived by considering an element dx of the pipe and the enclosed fluid. The forces acting on elements of the fluid and the pipe are balanced in the transverse and longitudinal directions. Assuming no fluid acceleration in the longitudinal direction, the static forces in the longitudinal direction are substituted into the equation of motion in the transverse direction, resulting in Eq. (2.29).
If the pipe displacement is sought for in the form of w x t
( )
, =Aei(ωt kx− ), the lineardispersion relation is readily obtained as (de Langre and Ouvrard, 1999)
(
)
(
)
4+ − 2 2+2 ω − + ω2=0
f f f f p f
EIk T m u k m u k m m . (2.30)
From this dispersion relation the radial frequency can be written as a function of the wavenumber
(
)(
) (
)
2 2 2 2 1,2 ω = ± + + − + + + f f f f p f f f p f p f m u k k m u m m T m u m m EIk m m . (2.31)In contrast to the string model, the Euler-Bernoulli beam is a dispersive system (harmonic waves with different frequency propagate with a different phase velocity), since there is no longer a linear relation between the radial frequency ω and the wavenumber k, as shown in Eq. (2.31). A second difference between the infinite string model and the infinite Euler-Bernoulli beam is the response of the system to a harmonically vibrating point load. The beam response can be represented as a superposition of four waves in contrast to two waves that comprise the string response. As long as the fluid velocity is below the lower critical velocity, defined in Eq. (2.16), the beam response to the load is a superposition of two propagating waves travelling in opposite directions and two spatially decaying waves. In the string model this lower critical velocity marks the onset of the regime in which the waves can travel only in the downstream direction. Exceeding this lower critical
velocity the beam response consists of either two propagating waves and two spatially decaying waves or four propagating waves. This transition depends on the forcing frequency of the harmonically vibrating point load. For zero forcing frequency this transition occurs at the lower critical velocity as defined in Eq. (2.16), whereas for higher forcing frequencies this transition starts at higher fluid velocity. By increasing the fluid velocity further, the infinitely long beam may become unstable (depending on whether the initial conditions excite this specific wave). In contrast to the infinitely long string, where all radial frequencies become complex at the same fluid velocity, the critical velocity for the beam depends on the frequency via the wavenumber k:
(
)
(
)
2 , . t f crit beam f t f m k EI T u m m m + = − . (2.32)This expression indicates that only a limited band of wavenumbers becomes unstable. The smaller the wavenumber, the smaller the critical velocity. In an infinite beam, in which all wavenumbers can exist, the critical velocity approaches the higher critical velocity of the string, Eq. (2.17).
2.5 Finite beam conveying fluid with different supports
The simple example with the string demonstrates that the type of support and the flow direction have a large impact on the stability of the system. The same holds for the Euler-Bernoulli beam conveying fluid. A lot of research is documented in the literature about the influence of the supports on stability. Païdoussis and Issid (1974) have shown that pinned-pinned and clamped-clamped beams lose their stability at high fluid velocities in their first mode by divergence (buckling) via a pitchfork bifurcation. For fluid velocities in the post-divergence regime the first- and second-mode loci coalesce, corresponding to the onset of coupled-second-mode flutter (oscillatory instability). It is known that in case of symmetrical boundary conditions (clamped-clamped, pinned-pinned, etc.), a (tensioned) beam conveying fluid behaves as a gyroscopic conservative system, implying that the total energy of the system varies periodically in time (Païdoussis and Issid, 1974). In the past, stability of a clamped-pinned pipe conveying fluid was often considered numerically, sometimes leading to controversial results (Païdoussis, 2004). Kuiper and Metrikine (2004) proved analytically the stability of a clamped-pinned pipe conveying fluid at a low velocity and showed that the pipe loses stability at high fluid velocities through divergence. The main focus of this thesis is on the stability of a cantilever pipe (free-clamped) conveying fluid. Gregory and Païdoussis (1966a) determined the conditions of
stability in case of an upstream clamped end and a downstream free end. Depending on the mass ratio between the fluid and the pipe, the system loses stability in a particular “mode” through flutter. In an accompanying paper, Gregory and Païdoussis (1966b) showed a satisfactory agreement between experimental observations and theoretical predictions. Incorporating the effect of gravity, Païdoussis (1970) extended the theory of cantilever pipes discharging fluid by considering a hanging cantilever and a standing cantilever. In both situations the fluid exits the pipe at the free end. It was shown that for sufficiently high flow velocities both hanging and standing cantilevers become subject to oscillatory instability. In addition, it was experimentally observed that standing cantilevers, which would buckle under their own weight in the absence of flow, in some cases were stable within a certain range of flow velocities.
The boundary conditions at the free end (x=L) assume no change of fluid momentum as the fluid exits the pipe. As used by e.g. Païdoussis (1998), the boundary conditions for a cantilever Euler-Bernoulli beam with constant pretension read:
( )
, 0 0 x w x t = = ,( )
0 , 0 x w x t x = ∂ = ∂ ,( )
2 2 , 0 x L w x t x = ∂ = ∂ and( )
( )
3 3 , , 0 x L w x t w x t EI T x x = ∂ ∂ − = ∂ ∂ . (2.33)An example of the Argand Diagram for a pre-tensioned cantilever pipe with an upstream clamped end and a downstream free end, modeled according to Eqs. (2.29) and (2.33), is presented in Fig. 2.6. In this Diagram, the real and imaginary parts of the first four natural frequencies are plotted parametrically, as they depend on the fluid velocity. The system becomes unstable through flutter. The critical velocity beyond which the cantilever pipe becomes unstable is higher than for a clamped-clamped beam with the same parameters. At first sight it might look contradictory that a system gains stability by removing a support! This is explained in Chapter 3 where the energy exchange at different boundaries is considered.
Païdoussis and Luu (1985) were the first to consider a cantilever pipe aspirating fluid, i.e. a pipe conveying fluid from a free end to a clamped end. Essentially, the same equation of motion, Eq. (2.29), and the same boundary conditions, Eqs. (2.33), were assumed, only the flow was reversed by using -uf instead of uf. The resulting
Fig. 2.6 – Argand Diagram for a pre-stressed beam conveying fluid with an upstream
clamped end and a downstream free end (EI=1.00 10 Nm ,⋅ 9 2
6 3 3
T =1.00 10 N, ⋅ L=100 m, mf =1.00 10 kg/m, ⋅ mt=2.00 10 kg/m⋅ ).
Fig. 2.7 – Argand Diagram for a pre-stressed beam conveying fluid with an upstream free
end and a downstream clamped end (EI=1.00 10 Nm ,⋅ 9 2
6 3 3 T =1.00 10 N, ⋅ L=100 m, mf =1.00 10 kg/m, ⋅ mt=2.00 10 kg/m⋅ ). -2 0 2 4 6 8 10 -0.5 0 0.5 1 1.5 2 1 2 3 4 Re(ω) in rad/s Im (ω) in r ad/s uf = 0 uf = 0 uf = 0 uf,crit 0 2 4 6 8 10 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 1 2 3 4 Re(ω) in rad/s Im( ω) in r ad /s uf = 0 uf = 0 uf = 0 uf = 0
Remarkably enough, all “modes” become unstable at infinitesimally small flow velocity. This conclusion was also drawn by Païdoussis and Luu (1985) even though they incorporated a spatially varying tension. However, experimental verification could not be given.
The theoretical prediction of dynamic instability at low fluid velocities should be taken as quite alarming. Although effects like hydrodynamic damping and fluid depressurization were not taken into account, this simple model was considered as fairly adequate for the free hanging riser aspirating fluid as applied in the offshore industry. The threat of unstable behaviour of cantilevers at low fluid velocities is the main motivation of this thesis.
2.6 Conclusion
Infinite pipes conveying fluid lose stability at relatively high fluid velocities. Obviously, in this case the unstable waves are generated in the medium (there are no boundaries). The same mechanism, i.e. instability in the medium, can occur for finite systems. However, for finite systems there is a second possible source of instability; wave reflections at the boundaries can result in an energy gain or loss of the total system. Depending on the type of boundary support and fluid flow direction, instability might occur at low fluid velocities. The cantilever pipe aspirating fluid is an example of this category and is further investigated in Chapters 4 to 6.
CHAPTER 3
Energy considerations for a pipe conveying
fluid
Energetics of translating continua has attracted considerable attention of researchers because of the profound effect the movement of the continua can have. A pipe conveying fluid belongs to the class of one-dimensional translating continua. The linearized equation of motion of a simply-supported pipe that conveys fluid at a constant speed predicts first divergence and then coupled-mode flutter as the speed is increased. These instabilities cannot be predicted by the commonly accepted energy equation that governs variation of the pipe motion, despite the fact that this equation is based on exactly the same energy functional. This inconsistency originates from the fact that the commonly accepted energy equation is derived assuming that the pipe moves periodically, which is not applicable to the instabilities mentioned above.
In the first part of this chapter, the energy equation is derived that is in one-to-one correspondence with the linearized equation of motion and is capable of predicting the above mentioned, aperiodic instabilities. This energy equation shows that the energy of the pipe-flow system can change not only due to energy flux through the end cross-sections of the pipe but also due to work of an external longitudinal force that acts at every cross-section of the pipe-flow system.
In the second part of this chapter the energy exchange at a boundary of a pipe is investigated. For this purpose, the “travelling wave method” as developed by Lee and Mote (1997a,b) is used. The expression for the energy reflection coefficient at a boundary, as introduced by Lee and Mote, is corrected in this chapter to make it applicable to dispersive translating continua. With this method we are able to explain the large impact of the type of support and the fluid direction on the critical fluid velocity, as observed in Chapter 2.
The main results of this chapter related to the travelling wave method have been published by Metrikine et al. (2006).
3.1 Introduction
By definition, the variation of energy of a continuum over a time interval is given by the difference in the energy at the end and at the beginning of the chosen interval. The energy at any instant can be calculated by integrating the energy density of the continuum over the occupied volume. If there is no energy source or sink within the volume of the continuum, the energy can vary only due to the energy flux through the boundary of the continuum. In this case, the energy variation can be calculated by integrating the energy flux over the chosen time interval.
When considering a one-dimensional axially translating continuum such as a pipe conveying fluid, two approaches can be applied for modeling axial translation of the continuum:
1. The motion of the continuum is modeled by prescribing the forces that cause translation. In this case the translational velocity of the continuum is
a priori unknown and should be found by solving coupled (as a rule,
nonlinear) equations of motion in both transversal and axial directions. 2. The translational velocity is prescribed a priori and only transverse motion
of the continuum is considered. This motion can be relatively well described by a linear equation of motion. This so-called kinematic prescription of the velocity introduces a possibility for the energy gain or loss at every cross-section of the continuum, as an external force may be necessary to maintain this velocity.
Therefore, in the latter case, one cannot predict the variation of energy of the continuum considering only energy flux through the boundaries. Somewhat strikingly, the commonly accepted energy equation does not take into account the possibility of energy input in the bulk of the continuum.
We start this chapter with deriving an energy equation based on first principles for a well-studied system, the simply-supported pipe conveying fluid. After that, the energy exchange at boundaries of a pipe conveying fluid is studied. To this end, the travelling wave method (Lee and Mote, 1997a,b) is employed in a corrected form that is applicable for translating dispersive continua.
3.2 The paradox of the simply-supported pipe
In this section, one the most basic models of a simply-supported pipe conveying fluid is adopted. The pipe is modelled as an Euler-Bernoulli beam, while a uniform plug-flow model is employed for the fluid flow description. The system of equations that govern small transverse vibration of this model is given as:
(
)
4 2 2 2 2 4 f f 2 2 f f p f 2 0 w w w w EI m u m u m m x x x t t ∂ + ∂ + ∂ + + ∂ = ∂ ∂ ∂ ∂ ∂ , (3.1)( )
0,( )
, 0 w t =w L t = and 22 22 0 0 x x L w w x = x = ∂ =∂ = ∂ ∂ , (3.2)where the same notations are used as in Chapter 2.
This “classical” system of equations (Eqs. (3.1) and (3.2)) has been studied by many researchers (e.g. Païdoussis and Issid, 1974). They proved that the pipe loses stability in its first mode via a pitchfork bifurcation at a flow velocity given by
( )1 =π
(
)
f f
u EI m L. At higher fluid velocities the loci of the first and second modes in the complex frequency plane coalesce on the imaginary or real axis (depending on the mass ratio of fluid and pipe) and thereafter leave the axis, indicating the onset of coupled-mode flutter.
Complementary to the analysis of the equation of motion, stability of the simply-supported pipe conveying fluid has been studied based on an energy balance, which is based on exactly the same energy functional. Benjamin (1961) was the first who for periodic motion derived an expression for the work done by the fluid on the pipe over a period of oscillation T:
2 0 0 d ⎛⎛∂ ⎞ ∂ ∂ ⎞ ∆ = − ⎜⎜⎜∂ ⎟ + ∂ ∂ ⎟⎟ ⎝ ⎠ ⎝ ⎠
∫
L T f f f w w w E m u u t t t x . (3.3) where 0L=( ) ( )
− 0x x L x . This expression has also been derived using an extended form of Hamilton’s principle (McIver, 1973) and has been used in several papers for further analyses (e.g. Païdoussis and Issid, 1974; Païdoussis, 1998; Païdoussis, 2005). It is important to underline that Eq. (3.3) has been obtained under the assumption that the pipe performs a periodic motion. Thus, this equation is inapplicable to aperiodic motions such as divergence and coupled-mode flutter. Given that divergence is the primary mechanism of instability of a simply-supported pipe, it is remarkable that Eq. (3.3) has never been generalized to become applicable to stability analysis of aperiodic motions, as far as the author is aware.
At a pinned (or clamped) support the transverse velocity of the pipe is zero,
0
w t
∂ ∂ = , and hence, in case of a simply-supported pipe, the right-hand side of Eq.
(3.3) is trivially zero. This, according to Païdoussis (2005) “… constitutes a paradox: for how is it possible for the pipe to flutter if the system is conservative and no energy is supplied to sustain the oscillation?” This paradox was addressed in a number of papers, such as of Done and Simpson (1977) and Holmes (1977, 1978), where it was shown that the post-divergence flutter ceases to exist if a proper nonlinear model of the pipe and its fixations is considered. However, the above
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