# kuiper_20080228.

Stability of Offshore Risers Conveying Fluid 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 Dit proefschrift is goedgekeurd door de promotoren: 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 [email protected] / 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” Galileo Galilei (1564-1642) Acknowledgements 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 interaction written by professor Thanks to these books, I saved recommend everybody interested these books. two comprehensive books about fluid-structure Païdoussis from McGill University (Canada). a lot of time in finding relevant literature. I in this field, from student to expert, to consult 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 Contents 1. General introduction 1.1 Top-tensioned riser in deep water 1.2 Free-hanging water intake riser 1.3 Mechanisms of instability of offshore risers 1.4 Historical overview of work related to cantilever pipes conveying fluid 1.5 Aims of the thesis 1.6 Outline 1.7 Output 1 2 2 4 5 6 8 9 2. Stability of a pipe conveying fluid – an overview 2.1 Introduction 2.2 Infinitely long string conveying fluid 2.3 Finite-length string conveying fluid with a free end and a fixed end 2.4 Infinitely long beam conveying fluid 2.5 Finite beam conveying fluid with different supports 2.6 Conclusion 11 12 14 18 21 23 26 3. Energy considerations for a pipe conveying fluid 3.1 Introduction 3.2 The paradox of the simply-supported pipe 3.3 Improved energy equation for a pipe conveying fluid 3.4 Balance of pseudo-momentum 3.5 Energy balance in aperiodic motion 3.6 Qualitative analysis of energy increase 3.7 Introduction to travelling wave method 3.8 Energy of a pulse travelling in one-dimensional pipe conveying fluid 3.9 Energy exchange at boundaries of a pipe conveying fluid 3.10 Conclusion 27 28 28 30 31 33 34 38 39 42 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 4.9 Conclusion 62 5. Stability of a free hanging riser conveying fluid – steady state vibrations with nonlinear damping 5.1 Introduction 5.2 Equation of motion and boundary conditions 5.3 Stability of a pipe conveying fluid in the linear approximation 5.4 Hydrodynamic drag on flexible cylinders 5.5 Stability of a pipe conveying fluid with nonlinear drag 5.6. Conclusion 65 66 67 70 72 79 87 6. Experimental investigation of a cantilever pipe conveying fluid 6.1 Introduction 6.2 Critical velocities predicted by different theories 6.3 Experimental set-up and instrumentation 6.4 Observations – qualitative description 6.5 Observations – quantitative description 6.6 Second set of experiments 6.7 Modelling the instability 6.8 Possible explanations for the experimentally observed pipe behaviour 6.9 Conclusion 89 90 91 94 98 98 101 103 109 114 7. Destabilization of deep-water risers by a heaving platform 7.1 Introduction 7.2 Assumptions and equation of motion 7.3 Stability of the straight configuration 7.4 Nonlinear analysis of instability development 7.5 Bending stresses in the riser 7.6 Conclusion 115 116 118 121 127 136 137 8. Main results, practical relevance and recommendations 8.1 Main results 8.2 Practical relevance 8.3 Recommendations 139 139 142 143 References 145 Appendix A - Spectral energy density of a pulse Appendix B - Depressurization at the inlet Appendix C - Floquet Theory 151 153 155 Summary Samenvatting 157 159 Curriculum Vitae 161 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. 2 Chapter 1 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 1.1 - A top-tensioned riser in deep water. General introduction 3 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. Barge Water intake riser Fig. 1.3 - Sketch of a water intake riser suspended from a barge. 4 Chapter 1 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 toptensioned risers in practice. General introduction 5 1.4 Historical overview of work related to cantilever pipes conveying fluid Dynamics of pipes conveying fluid has been studied by many researchers. Selfexcited 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 appeared 2 . 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. 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”. 2 6 Chapter 1 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 toptensioned 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. 3 A more detailed description of this nice anecdote can be found in Païdoussis (1998). General introduction 7 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, depthdependent 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. 8 Chapter 1 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. General introduction 9 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, 11071113. 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 deepwater 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. 12 Chapter 2 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: m d 2w dw +c + kw = 0 , 2 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 w(t ) = ∑ C j e λ jt , (2.2) j =1 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: mλ 2 + cλ + k = 0 . (2.3) The two eigenvalues are determined by the characteristic equation: λ 1,2 = −c ± c 2 − 4mk . 2m (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 Stability of a pipe conveying fluid – an overview 13 (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: ∞ ϕ ( x ) = ∫ ϕ ( k ) e − ikx dk , (2.5) ψ ( x ) = ∫ ψ ( k ) e − ikx dk , (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: w ( x, t ) = ∞ ∫ w ( k ) e i (ω ( k ) t − kx ) 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, w ( k ) ei(ω ( k )t − kx ) , 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. 14 Chapter 2 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): −T ∂2w ∂2w + + mf m p ∂x 2 ∂t 2 ⎛ 2 ∂2w ∂2w ∂2w ⎞ + 2 + u u ⎜ f ⎟=0, f ∂x 2 ∂x∂t ∂t 2 ⎠ ⎝ (2.8) where T is the constant tension in the pipe, m p is the mass of the pipe per unit length, m f is the mass of fluid per unit length, flowing with a steady flow velocity u f , 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 u f 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 m f experiences three acceleration components as measured by a stationary observer: • • • ∂2w , ∂x 2 ∂2w Coriolis acceleration, 2u f , ∂x∂t ∂2w local acceleration, 2 . ∂t centripetal acceleration, u f 2 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: −T ∂2w ∂2w d 2w + mp 2 + m f = 0. 2 ∂x ∂t dt 2 x = u t (2.9) f 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: dw ∂w ∂w dx = + dt x = u f t ∂t ∂x dt = x =u f t ∂w ∂w + uf , ∂t ∂x 2 d w d ⎛ ∂w ∂w ⎞ ∂2w ∂2w 2 ∂ w = + u = + u + u 2 . f f f ⎜ ⎟ ∂t∂x ∂x 2 dt 2 x = u t dt ⎝ ∂t ∂x ⎠ x = u f t ∂t 2 f 2 Stability of a pipe conveying fluid – an overview 15 w(x,t) x T T uf 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., w ( x, t ) = A j e ( i ω jt −k j x ) , (2.10) where Aj is the complex amplitude of the wave j, ω j and k j 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: −Tk j 2 + m f u f 2 k j 2 − 2m f u f ω j k j + ( m p + m f ) ω j 2 = 0 . (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: ω1 = ω2 = m f u f + m f 2 u f 2 + mt T − mt m f u f 2 mt m f u f − m f 2 u f 2 + mt T − mt m f u f 2 mt k, (2.12) k. (2.13) where mt = m p + m f . 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): cd = cu = ω1 k ω2 k = = m f u f + m f 2u f 2 + mt T − mt m f u f 2 mt m f u f − m f 2 u f 2 + mt T − mt m f u f 2 mt , (2.14) , (2.15) 16 Chapter 2 Upstream wave T T uf Upstream Downstream Downstream wave 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, u f < T m f , 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 travellingwave 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: u f ,crit .low = T m f . (2.16) Stability of a pipe conveying fluid – an overview 17 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 u f ,crit . high = mt T (m (m − m )) . f t f (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. 18 Chapter 2 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 zero 4 . The fixed support requires zero deflection. This yields the following system of equations for the cantilever string aspirating fluid: −T ∂2w ∂2w m + + mf p ∂x 2 ∂t 2 ⎛ 2 ∂2w ∂2w ∂2w ⎞ u u + 2 + ⎜ f ⎟=0, f ∂x 2 ∂x∂t ∂t 2 ⎠ ⎝ ∂w ( x, t ) = 0 and ∂x x = 0 w ( x, t ) x = L = 0 . (2.18) (2.19) We seek for a solution to this problem having the following form: w ( x, t ) = W ( x ) eiωt . (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: (m u f 2 f −T ) d 2W dW + 2m f u f iω − mtω 2W = 0 . 2 dx dx (2.21) The general solution to this ordinary differential equation is: 2 W ( x ) = ∑C je ik j x , (2.22) j =1 in which C j 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 in Chapter 5. Stability of a pipe conveying fluid – an overview (T − m u ) k 2 f f 2 j 19 − 2m f u f ωk j − mtω 2 = 0 . (2.23) From this algebraic equation the wavenumbers k j can be found as functions of the frequency ω: ⎛ m u + m 2u 2 + m T − m m u 2 f f f f t t f f k1 = ω ⎜ T − mf uf 2 ⎜ ⎝ ⎞ ⎟, ⎟ ⎠ (2.24) ⎛ m u − m 2u 2 + m T − m m u 2 f f f f t t f f k2 = ω ⎜ T − mf uf 2 ⎜ ⎝ ⎞ ⎟. ⎟ ⎠ (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: C1k1 + C2 k2 = 0 and C1eik L + C2 eik L = 0 . 1 (2.26) 2 A non-trivial solution of these equations exists if: k1eik2 L − k2 eik1L = 0 . (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: ωn = where a = −i ln ( a b ) − 2π n m f u f + m f 2 u f 2 + mt T − mt m f u f 2 T − mf uf 2 (a − b) L , b= , (2.28) m f u f − m f 2 u f 2 + mt T − mt m f u f 2 T − mf uf 2 . 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 20 Chapter 2 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, u f ,crit .low = T m f , 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 supercritical 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. 0.1 0.075 0.05 uf → √(T/mf) Im(ω) in rad/s 0.025 0 uf = 0 uf = 0 uf = 0 1 2 3 uf = 0 4 -0.025 -0.05 -0.075 -0.1 0 0.5 1 1.5 2 2.5 Re(ω) in rad/s Fig. 2.4 – Argand Diagram for a string conveying fluid with an upstream free end and a ( downstream fixed end (first four natural frequencies) ) T = 1.00 ⋅ 106 N, m f = 1.00 ⋅ 103 kg/m, mt = 2.00 ⋅ 103 kg/m, L=100 m . Stability of a pipe conveying fluid – an overview 21 0.1 0.075 0.05 Im(ω) in rad/s 0.025 1 2 3 uf = 0 uf = 0 4 0 uf = 0 -0.025 uf = 0 uf → √(T/mf) -0.05 -0.075 -0.1 0 0.5 1 1.5 2 2.5 Re(ω) in rad/s Fig. 2.5 – Argand Diagram for a string conveying fluid with a downstream free end and an upstream fixed end (first four natural frequencies) (T = 1.00 ⋅ 106 N, m f = 1.00 ⋅ 103 kg/m, mt = 2.00 ⋅ 103 kg/m, L=100 m ) . 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. 22 Chapter 2 Considering a pipe, modeled as a tensioned Euler-Bernoulli beam, the linear equation of motion takes the form: EI 2 ∂4w ∂2w ∂2w 2 ∂ w T m u m u m m − − + 2 + + =0. ( ) ( ) f f f f p f ∂x 4 ∂x 2 ∂x∂t ∂t 2 (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 linear dispersion relation is readily obtained as (de Langre and Ouvrard, 1999) EIk 4 + (T − m f u f 2 ) k 2 + 2m f u f ωk − ( m p + m f ) ω 2 = 0 . (2.30) From this dispersion relation the radial frequency can be written as a function of the wavenumber ω1,2 = m f u f k ± k m f 2u f 2 + ( m p + m f ) (T − m u ) + ( m mp + m f 2 f f p + m f ) EIk 2 . (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 Stability of a pipe conveying fluid – an overview 23 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: u f ,crit .beam = mt ( k 2 EI + T ) m f ( mt − m f ) . (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 secondmode loci coalesce, corresponding to the onset of coupled-mode flutter (oscillatory instability). It is known that in case of symmetrical boundary conditions (clampedclamped, 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 clampedpinned 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 24 Chapter 2 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: w ( x, t ) x = 0 = 0 , ∂ 2 w ( x, t ) = 0 and ∂x 2 x=L EI ∂w ( x, t ) = 0, ∂x x = 0 ∂ 3 w ( x, t ) ∂w ( x, t ) −T =0. ∂x 3 ∂x x = L (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 clampedclamped 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 Argand Diagram is displayed in Fig. 2.7. Stability of a pipe conveying fluid – an overview 25 2 Im(ω) in rad/s 1.5 1 0.5 2 1 0 uf = 0 3 4 uf = 0 uf = 0 uf,crit -0.5 -2 0 2 4 6 8 10 Re(ω) in rad/s 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 ⋅ 109 Nm 2 , T = 1.00 ⋅ 10 N, L = 100 m, m f = 1.00 ⋅ 10 kg/m, mt = 2.00 ⋅ 10 kg/m ). 6 3 3 0.2 0.1 uf = 0 0 uf = 0 uf = 0 uf = 0 2 3 4 Im(ω) in rad/s 1 -0.1 -0.2 -0.3 -0.4 -0.5 0 2 4 6 8 10 Re(ω) in rad/s 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 ⋅ 109 Nm 2 , T = 1.00 ⋅ 10 N, L = 100 m, m f = 1.00 ⋅ 10 kg/m, mt = 2.00 ⋅ 10 kg/m ). 6 3 3 26 Chapter 2 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). 28 Chapter 3 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: Energy considerations for a pipe conveying fluid EI 29 ∂4w ∂2w ∂2w ∂2w + m f u f 2 2 + 2m f u f + (mp + m f ) 2 = 0 , 4 ∂x ∂x ∂x∂t ∂t w ( 0, t ) = w ( L, t ) = 0 and ∂2w ∂x 2 = x =0 ∂2w ∂x 2 = 0, (3.1) (3.2) x=L 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 u f ( ) = π EI m f 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 simplysupported 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: L ∆E = − m f u f ∫ T 0 ⎛ ⎛ ∂w ⎞ 2 ∂w ∂w ⎞ ⎜⎜ ⎜ ⎟ dt . ⎟ + uf ∂t ∂x ⎟⎠ ⎝ ⎝ ∂t ⎠ 0 (3.3) where x 0 = x ( L ) − x ( 0 ) . 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. L At a pinned (or clamped) support the transverse velocity of the pipe is zero, ∂w ∂t = 0 , 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 30 Chapter 3 formulated paradox has not been resolved, for the following question remained unanswered: How is it possible that the energy equation, Eq. (3.3), and the equation of motion, Eq. (3.1), which are based on exactly the same energy functional, do not give the same stability predictions? In view of the preceding discussion, the obvious answer to the above question is that Eq. (3.3) is inapplicable to aperiodic processes, to which both divergence and coupled-mode flutter belong. It has to be noted that Eq. (3.3) was never intended to decide pipe stability, but merely the possible existence of flutter. The purpose of the next section is to obtain a generalized, unified energy criterion, that is valid for both periodic and aperiodic processes. The origin and the physics of the additional term in this equation that vanishes in case of periodic motion are discussed. 3.3 Improved energy equation for a pipe conveying fluid In this section the energy equation is derived for a pipe conveying fluid based on the linear equation of motion, Eq. (3.1). A standard procedure (Elmore and Heald, 1969; Rowland and Pask, 1999) is to multiply the equation of motion (3.1) by the transverse velocity, ∂w ∂t , and rearrange to the following divergence form that constitutes the differential form of the energy equation: ∂e ( x, t ) ∂F ( x, t ) ∂R ( x, t ) , + = ∂t ∂x ∂t (3.4) where e ( x, t ) = F = EI 2 2 2 1 ⎛ ∂w ⎞ 1 ⎛ ∂ w ⎞ 1 mp ⎜ + EI ⎜ 2 ⎟ + mf ⎟ 2 ⎝ ∂t ⎠ 2 ⎝ ∂x ⎠ 2 ∂w ∂ 3 w ∂2w ∂2w − + mf uf EI ∂t ∂x 3 ∂x 2 ∂x∂t 2 ∂w ⎞ ⎛ ∂w + uf ⎜ ⎟ , ∂x ⎠ ⎝ ∂t (3.5) ⎛ ⎛ ∂w ⎞ 2 ∂w ∂w ⎞ ⎜⎜ ⎜ ⎟, ⎟ + uf ∂t ∂x ⎟⎠ ⎝ ⎝ ∂t ⎠ (3.6) ∂w ⎞ ∂w ⎛ ∂w R = mf uf ⎜ . + uf ⎟ t ∂ ∂x ⎠ ∂x ⎝ (3.7) In Eqs. (3.4)-(3.7), e ( x, t ) and F ( x, t ) are the energy of the pipe-fluid system per unit length and the energy flux through a cross-section x of the system (both excluding the constant terms associated with the fluid convection), and R ( x, t ) is a function with the dimension of force. The physical meaning of the latter term is discussed later in this chapter. Integrating Eq. (3.4) on the element from x = x1 to x = x2 the “physical” interpretation of this equation becomes clear; the time rate of change of energy within the element can only change due to the flux through the Energy considerations for a pipe conveying fluid 31 two boundaries at x = x1 and x = x2 and due to the rate of work of an external force at the segment. The energy density e ( x, t ) is composed of the kinetic energy of the pipe (first term in Eq. (3.5)), the potential bending energy of the pipe (second term) and the kinetic energy of the flowing fluid (third term), all per unit length. The energy flux F ( x, t ) is composed of the flux associated with the shear force (first term in Eq. (3.6)), the flux associated with the bending moment (second term) and the flux associated with the fluid flow. It has to be mentioned that Eq. (3.1) multiplied by the transverse velocity ∂w ∂t can be rearranged not only to the form given by Eq. (3.4). For example, the following equation can be obtained as well: ∂e pseudo ( x, t ) ∂t + ∂F ( x, t ) ∂x =0, (3.8) 2 ⎛ ⎛ ∂w ⎞ 2 ⎞ 2 ⎛ ∂w ⎞ u − ⎜⎜ ⎜ f ⎜ ⎟ ⎟ ⎟⎟ , ⎝ ∂x ⎠ ⎠ ⎝ ⎝ ∂t ⎠ (3.9) where 2 e pseudo = 2 2 1 ⎛ ∂w ⎞ 1 ⎛ ∂ w ⎞ 1 mp ⎜ EI + ⎜ 2 ⎟ + mf ⎟ 2 ⎝ ∂t ⎠ 2 ⎝ ∂x ⎠ 2 and F ( x, t ) is the energy flux given by Eq. (3.6). The function e pseudo ( x, t ) is the socalled pseudo-energy (Goldstein, 1980), which is obviously different from the ‘true’ energy e ( x, t ) . The pseudo-energy is not positive-definite and cannot be considered as a measure of the mechanical energy of a system. Nor can Eq. (3.8) be used for calculating the energy of continua based on the energy flux, see Metrikine et al. (2006)5. Thus, despite of the possibility to obtain from Eq. (3.1) a number of pseudo-energy equations (such as Eq. (3.8), for example), Eq. (3.4) is the only ‘true’ energy equation for the system at hand as it relates the ‘true’ energy to the ‘true’ flux. The term ∂R ( x, t ) ∂t on the right-hand side of Eq. (3.4) describes, therefore, the rate of work per unit length of a certain external (to the pipe-fluid system) force that acts at every cross-section of the system. The origin and the physical meaning of this force can be identified from the balance of longitudinal pseudo-momentum that is derived in the next section. 3.4 Balance of pseudo-momentum In the linear equation of motion of the beam-flow system, Eq. (3.1), axial displacement does not appear explicitly but its existence is nevertheless implied (Païdoussis, 2005). The forces associated with this mathematically absent but 5 See also Sections 3.7 and 3.8. 32 Chapter 3 physically existent motion (every transverse motion generates an axial counterpart) can sometimes be found by using the balance of longitudinal pseudo-momentum (Vesnitskii et al. 1983). The rate of change of pseudo-momentum is not always applicable as the measure for the force exerted by transverse waves (Rowland and Pask, 1999). However, see Pippard (1992), it is applicable to mechanical systems with ideal constraints 6 provided that the force is analyzed using the approach proposed by Rayleigh (1902). In any case, the balance of pseudo-momentum is a useful tool to uncover energy sources hidden in the models. To derive the pseudo-momentum equation, Eq. (3.1) is multiplied by ∂w ∂x and rearranged (Elmore and Heald, 1969; Rowland and Pask, 1999) to the following divergence form that constitutes the differential form of the pseudo-momentum equation: ∂p ( x, t ) ∂T ( x, t ) ∂R ( x, t ) + = , ∂t ∂x ∂x (3.10) ∂w ∂w ⎞ ⎞ ∂w ⎛ ⎛ ∂w p = − ⎜ mp , + mf ⎜ + uf ⎟ t t ∂ ∂ ∂x ⎠ ⎟⎠ ∂x ⎝ ⎝ (3.11) where 2 T = − EI 2 2 ∂w ∂ 3 w 1 ⎛ ∂ 2 w ⎞ 1 ∂w ⎞ ⎛ ∂w ⎞ 1 ⎛ ∂w + EI ⎜ + uf ⎟ + mp ⎜ ⎟ + mf ⎜ ⎟ , t t 2 ∂x ∂x 3 2 ⎝ ∂x 2 ⎠ 2 ∂ ∂ ∂x ⎠ ⎝ ⎠ ⎝ (3.12) and R ( x, t ) is given by Eq. (3.7). In Eq. (3.10), p ( x, t ) is the pseudo-momentum per unit length and T ( x, t ) is the pseudo-momentum flux, both in the axial direction. These are also often referred to as wave momentum and wave pressure (Vesnitskii et al., 1983; Pippard, 1992). From Eq. (3.11), one can see that the pseudo-momentum is nothing else but the transverse momentum of the pipe multiplied by ∂w ∂x .7 Thus, the wave momentum can be thought of as the necessary longitudinal counterpart of the vertical momentum that reflects the fact that each infinitesimal element of the vibrating pipe has a small axial velocity, unless the slope of this element relative to the undeformed axis is zero. Note that the longitudinal pseudo-momentum should not necessarily be equal to the longitudinal momentum of the system (Rowland and Pask, 1999). The pseudo-momentum flux T ( x, t ) is an axial force that exists in every cross-section of the pipe due to small axial movement of the pipe elements. 6 7 Constraints where no energy is lost. Assuming that an element of the pipe conveying fluid moves perpendicular to its local direction, the momentum can be decomposed in a transverse and longitudinal component. Based on geometry, the longitudinal momentum is equal to the transverse momentum of the pipe multiplied by ∂w ∂x , at least in the first-order approximation. Energy considerations for a pipe conveying fluid 33 Using the pseudo-momentum equation, Eq. (3.10), the physical meaning of R ( x, t ) can be identified. Standing on the right-hand side of this equation under the spatial derivative, R ( x, t ) must be an external force. This force must be associated with the fluid flow, since it vanishes with a zero fluid speed, see Eq. (3.7). Therefore, it is logical to conclude that R ( x, t ) is an external force that maintains the given speed of the fluid flow through the pipe. Thus, one may say that the linearized equation of motion, Eq. (3.1), implicitly implies that there should be an external longitudinal force applied at every cross-section of the vibrating system (at some cross-sections and at some time moments this force may be zero) to maintain the given speed of the flow through the pipe. It is the work of this force that explains why Eq. (3.1) predicts divergence and coupled-mode flutter. This is shown in the next section. 3.5 Energy balance in aperiodic motion To obtain the integral form of the energy equation, Eq. (3.4) should be integrated over a time interval t ∈ [ 0, T ] and over a span of the pipe x ∈ [0, L] . This gives T L ∆E = E (T ) − E (0) = − ∫ F 0 dt + ∫ R 0 dx , 0 L T (3.13) 0 L where E (t ) = ∫0 e ( x, t ) dx . The above equation shows that mechanical energy E ( t ) of the pipe can change due to (1) the energy flux through the end cross-sections of the pipe and (2) the work of the external force R ( x, t ) within the pipe span. Let us first assume that the motion is periodic with period T. In this case, the last term in Eq. (3.13) vanishes and this equation, upon substitution of F from Eq. (3.6), reduces to ∆E = − ∫ T 0 ⎡ ⎢m f u f ⎢⎣ L ⎛ ⎛ ∂w ⎞ 2 ∂w ∂w ⎞ ∂w ∂ 3 w ∂2w ∂2w ⎤ − EI ⎥ dt . ⎜⎜ ⎜ ⎟⎟ + EI ⎟ + uf 3 ∂t ∂x ⎠ ∂t ∂x ∂x∂t ∂x 2 ⎥⎦ ⎝ ⎝ ∂t ⎠ 0 (3.14) Equation (3.14) shows that mechanical energy of the system in periodic motion can change owing to the work done at the end cross-sections of the pipe by (1) the fluid (the first term in the integrand), (2) the shear force (the second term), (3) the bending moment (the last term). Obviously, the expression for the work associated with the fluid flow in Eq. (3.14) is the same as given by Eq. (3.3): L ∆W = − m f u f ∫ T 0 ⎛ ⎛ ∂w ⎞ 2 ∂w ∂w ⎞ ⎜⎜ ⎜ ⎟ dt . ⎟ + uf ∂t ∂x ⎟⎠ ⎝ ⎝ ∂t ⎠ 0 (3.15) Equation (3.15) is only valid if the pipe’s motion is periodic and results in ∆W = 0 for a simply-supported pipe. For aperiodic motions, such as divergence or coupled- 34 Chapter 3 mode flutter, with T denoting an arbitrary time function, Eq. (3.13) provides the following expression for the work done by the fluid: L T 2 T ⎛ ⎛ ∂w ⎞ L ∂w ∂w ⎞ ∂w ⎛ ∂w ∂w ⎞ u t m u u ∆W = − m f u f ∫ ⎜ ⎜ + d + + ⎟ f ∫0 f f ∂x ⎜⎝ ∂t f ∂x ⎟⎠ 0 dx . (3.16) 0 ⎜ ∂t ⎟ ∂t ∂x ⎟⎠ ⎠ ⎝⎝ 0 The principal difference between Eqs. (3.15) and (3.16) is that the latter accounts for the work of R ( x, t ) along the pipe length. If the pipe is simply-supported, Eq. (3.16) reduces to T ∆W = m f u f ∫ L 0 2 ⎛ ∂w ∂w ⎛ ∂w ⎞ ⎞ + uf ⎜ ⎜⎜ ⎟ ⎟⎟ dx . ⎝ ∂x ⎠ ⎠ 0 ⎝ ∂t ∂x (3.17) It must be noted that a fuller form of Eq. (3.3) that is applicable to aperiodic motions of the pipe can be found in the paper of Païdoussis and Issid (1974), and in corrected form in Païdoussis (2004, Eq.(7.68)). It has been done by considering a nonlinear statement of the problem that accounts for the axial shortening of the pipe. The implication of this is that the momentum flux of the fluid issuing from the sliding end of the pipe does work on the system.8 To prove that there is no inconsistency anymore between the equation of motion and the energy balance, it is shown in the next section that the pipe-flow system gains energy in case of divergence and coupled-mode flutter. This is in contrast to the commonly accepted thought that ∆W = 0 for a simply-supported pipe conveying fluid. 3.6 Qualitative analysis of energy increase In this section, the possible energy increase of the pipe-fluid system is analyzed qualitatively for three critical velocities. The following velocities are considered: 1. The first critical velocity, u f (1) = π ( EI m f ) L , at which the first eigenfrequency turns to zero (see Fig. 3.1, left), whereas all other eigenfrequencies are real. This velocity corresponds to the onset of the divergence-type instability. The linear problem statement, Eqs. (3.1) and (3.2), predicts a linear growth of the pipe displacement in time in this case. 8 The resulting expression for ∆W as obtained by Païdoussis (2004) reads L T ⎛ ⎛ ∂w ⎞ 2 ∂w ∂w ⎞ 1 L ⎟ dt + ∫0 m f ⎟ + uf ∂t ∂x ⎠ 0 2 ⎝ ⎝ ∂t ⎠ ∆W = − m f u f ∫ ⎜ ⎜ 0 T ⎛ ⎛ ∂w ⎞ 2 ⎛ ∂w ⎞ 2 ⎞ ⎜ −⎜ ⎟ + ⎜uf ⎟ ⎟ dx . ⎝ ⎝ ∂t ⎠ ⎝ ∂x ⎠ ⎠ 0 Similarly to Eq. (3.17), this equation is capable of describing a nonzero energy variation of a simplysupported pipe in case of a non-periodic process. However, the quantitative prediction of the work done by the fluid is quite different. Energy considerations for a pipe conveying fluid 35 2. A velocity u f ( 2 ) (u f ( 2 ) > u f (1) ) at which the first and second eigenfrequencies become equal at the imaginary axis of the complex frequency plane, see Fig. 3.1 (middle). All other eigenfrequencies are real at this velocity. This situation occurs, for example, when m f ( m f + m p ) = 0.1 and u f ( ) ≈ 6.375 EI m f L (Païdoussis, 1998). This velocity marks the onset of 2 one possible type of the post-divergence coupled-mode flutter. The linear theory predicts in this case the displacement to increase according to t exp(at ) , where a is the absolute value of the first (and second) eigenfrequency. 3. A velocity u f ( 3) (u f (3) > u f (1) ) at which the first and second eigenfrequencies become equal at the real axis of the complex plane as shown in Fig. 3.1 (right). All other eigenfrequencies are real at this velocity. This situation occurs, for example, when m f ( m f + m p ) = 0.5 and u f ( 3) ≈ 6.3 EI m f L (Païdoussis and Issid, 1974) and corresponds to another type of the postdivergence coupled-mode flutter. According to the linear problem statement, Eqs. (3.1) and (3.2), the pipe displacement grows in this case proportionally to t cos (ωt + ϕ ) , where ω is the first (and the second) eigenfrequency and ϕ is a real constant that depends on initial conditions. Equations (3.1) and (3.2), supplemented by a set of initial conditions, can be solved using the Laplace integral transform over time (Fuchs et al., 1961-1964). In the Laplace domain, Eq. (3.1) turns to an ordinary differential equation, which can be readily solved subject to the boundary conditions. Applying the inverse Laplace transform to the obtained result, one can express the pipe displacement as: w ( x, t ) = σ + i∞ f ( x, s ) 1 exp ( st ) ds , ∫ 2π i σ − i∞ g ( s ) (3.18) where f ( x, s ) is the non-singular function obtained in the Laplace domain, g ( s ) = 0 is the characteristic equation of the system, σ is a positive real number that is greater than the real parts of all roots of g ( s ) and i = −1 . Im(ω) Im(ω) Im(ω) Mode 1 Coupled-Mode Re(ω) uf(1) Mode 2 Mode 1 Mode 2 Re(ω) Re(ω) Coupled-Mode Fig 3.1 – Divergence and two cases of coupled-mode flutter. 36 Chapter 3 Consider the first critical velocity, u f = u (f1) . In this case, the first two roots of the characteristic equation g ( s ) = 0 are zero: s1 = s2 = 0 , whereas all other roots ( sn , n > 2 ) of this equation are imaginary and simple. Correspondingly, g ( s ) = 0 can be expressed as g ( s ) = s 2 g1 ( s ) , where all roots of g1 ( s ) are purely imaginary. Using the contour integration and the residue theorem (Fuchs et al., 1961-1964), Eq. (3.18) in this case can be reduced to: w ( x, t ) = t ∞ ⎡ f ( x,0 ) ⎡ ∂ f ( x, s ) ⎤ ( s − sn ) ⎤ f x, s exp s t . +⎢ ( n) ( n ) ⎥ + ∑⎢ 2 ⎥ g1 ( 0 ) ⎣ ∂s g1 ( s ) ⎦ s = 0 n = 3 ⎣ s g1 ( s ) ⎦ s = s (3.19) n It is the first term in Eq. (3.19) that describes the linear increase of pipe displacement in time as all other terms are bounded ( sn , n > 2 are all imaginary). Insertion of Eq. (3.19) into Eq. (3.17) will result in a number of bounded terms, a number of terms proportional to t and one term proportional to t 2 . It is the latter term that will prevail as time increases and, therefore, one may say that for sufficiently large t = T , energy variation of the pipe-flow system is given as: ⎛ u (1) ⎞ ∆W ≈ T m f ⎜ f ⎟ ⎜ ⎟ ⎝ g1 ( 0 ) ⎠ 2 2 ∫ L 0 ⎛ ∂ f ( x,0 ) ⎞ ⎜ ⎟ dx > 0 . ∂x ⎝ ⎠ 2 (3.20) The above expression obviously shows that the energy of the pipe-fluid system increases in time in correspondence with the prediction of the eigenfrequency analysis of the system. Consider u f = u (f2 ) , which corresponds to the following roots of the characteristic equation g ( s ) = 0 : s1 = s2 = a, s3 = s4 = − a, a ∈ℜ and sn , n > 4 are all imaginary. 2 2 Consequently, g ( s ) = ( s − a ) ( s + a ) g 2 ( s ) , where all roots of g 2 ( s ) = 0 are imaginary. Using the above and applying the contour integration and the residue theorem, to Eq. (3.18), the following expression is obtained: ⎛ ⎛ ⎡ ⎤⎞⎞ f ( x, s ) ⎜ ⎜ t f ( x, s ) ∂ ⎢ ⎥⎟⎟ + w ( x, t ) = ∑ ⎜ exp ( st ) ⎜ 2 ⎢ ⎥⎟⎟ n 2 ∂ 4 a g s s ( ) n =1 2 s + a ( −1) g 2 ( s ) ⎥ ⎟ ⎟ ⎜ ⎜ ⎢ ⎣ ⎦ ⎠ ⎠ s = a ( −1)n ⎝ ⎝ 2 ( ) (3.21) ∞ ⎡ ⎤ ( s − sn ) +∑ ⎢ f ( x, sn ) exp ( sn t ) ⎥ 2 2 n =5 ⎢ ( s − a ) ( s + a ) g 2 ( s ) ⎥ ⎣ ⎦ s = sn Inserting Eq. (3.21) into Eq. (3.17) and extracting the terms that will prevail at large t, the following expression can be obtained for the energy increase at large t = T : ⎛ ⎞ 1 ∆W ≈ m f T 2 exp ( 2aT ) ⎜⎜ 2 ⎟⎟ ⎝ 4a g 2 ( a ) ⎠ 2 ∫ L 0 ∂f ( x, a ) ⎞ ⎛ ( 2 ) ∂f ( x, a ) ⎞ ⎛ (2) ⎜ u f af ( x, a ) ⎟ + ⎜uf ⎟ dx . (3.22) ∂x ⎠ ⎝ ∂x ⎠ ⎝ 2 Energy considerations for a pipe conveying fluid 37 It can be proven that this expression is positive-definite. Thus, at the onset of coupled-mode flutter that originates from the imaginary axis of the complex frequency plane, the energy increase predicted by the linear theory is proportional to T 2 exp ( 2aT ) . Let us note that the post-divergence coupled-mode flutter does not exist in reality as shown by a superior model of Holmes (1978). Correspondingly, Eq. (3.22) only shows that both the eigenfrequency and energy analyses based on the quadratic energy functional predict the same physically wrong result. Consider, finally, u f = u (f3) , which corresponds to the following roots of the characteristic equation: s1 = s2 = ib, s3 = s4 = −ib, b ∈ℜ and sn , n > 4 are all 2 2 imaginary. Consequently, g ( s ) can be expressed as g ( s ) = ( s − ib ) ( s + ib ) g3 ( s ) with g3 ( s ) = 0 having only imaginary roots. Application of the contour integration and the residue theorem to Eq. (3.18) yields in this case ⎛ ⎛ ⎡ ⎤⎞⎞ f ( x, s ) ⎜ ∂ ⎢ ⎜ t f ( x, s ) ⎥⎟⎟ + w ( x, t ) = ∑ ⎜ exp ( st ) ⎜ 2 2 ⎢ ⎥⎟⎟ n n =1 ⎜ −4b g 3 ( s ) ∂s ⎢ s + ib ( −1) g3 ( s ) ⎥ ⎟ ⎟ ⎜ ⎣ ⎦ ⎝ ⎠ ⎠ s = ib( −1)n ⎝ 2 ( ) (3.23) ∞ ⎡ ⎤ ( s − sn ) +∑ ⎢ ⎥ f ( x, sn ) exp ( sn t ) 2 2 n = 5 ⎢ ( s − ib ) ( s + ib ) g 3 ( s ) ⎥ ⎣ ⎦ s = sn The corresponding energy increase at large t = T is given as ⎛ 1 ⎞ ∆W ≈ m f T 2 ⎜ 2 ⎟ ⎝ 4b ⎠ 2 ∫ L 0 (u ( 3) f ( ) f1 f 2 + u (f ) 3 2 ) f 2 2 dx > 0 , (3.24) where f1 = ib f2 = exp(ibT ) exp( −ibT ) f ( x,ib ) − ib f ( x, −ib ) , g 3 ( ib ) g 3 ( -ib ) exp(ibT ) ∂f ( x, ib ) exp( −ibT ) ∂f ( x, −ib ) + . g3 ( ib ) g 3 ( -ib ) ∂x ∂x This is again in correspondence with the (physically wrong) prediction of the eigenfrequency analysis of the linear problem statement. Hence, it has been shown in this part of the chapter that the inconsistency between the eigenfrequency analysis and the commonly accepted energy equation can be solved by extending the applicability of the energy equation to aperiodic processes. The energy equation and the balance of pseudo-momentum, derived in this chapter, show that the linear statement of the problem implies that an external axial force has to act at every cross section of the pipe to maintain the constant flow speed through the pipe. It is the work of this force that enables energy of the pipe to increase. 38 Chapter 3 In the remaining part of this chapter the scope is broadened to more configurations than only the simply-supported pipe, e.g. a cantilever pipe conveying fluid. As a tool, the travelling wave method is used to analyse the energy exchange at solitary boundaries. 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. 3.7 Introduction to travelling wave method Lee and Mote (1997a,b) investigated the energetics of one-dimensional translating continua, of which the pipe conveying fluid is an example. The main emphasis in the papers of Lee and Mote was placed on the energy exchange at a boundary of the continua. As an approach to analyse this energy exchange, they proposed a very elegant ‘travelling wave method’ based on comparison of the energy of an incident wave to that of the wave reflected by the boundary. Employing this method, a conclusion is drawn on whether the energy is lost or gained at a boundary without explicit identification of the forces acting on this boundary. Studying the wave reflection at a boundary, Lee and Mote considered an incident harmonic wave and stated that ‘the energy ∆E transferred into the continuum span over one period by wave reflection’ (cited from Lee and Mote (1997a), page 724) is given as ∆E = Eλ ,r − Eλ ,i = ( R − 1) Eλ ,i , (3.25) where Eλ ,r and Eλ ,i are the energies contained in one wavelength of the reflected and incident waves, respectively, and R = Eλ ,r Eλ ,i is the ‘energy reflection coefficient’. According to Lee and Mote, if this coefficient is larger than unity then it is concluded that energy is gained at the boundary and vice versa. In the remaining part of the chapter it is shown that the energy exchange at a boundary of a dispersive translating continuum cannot in all cases be analysed using the coefficient introduced by Lee and Mote, but that the following expression for the ‘true’ energy reflection coefficient, Rω , is generally applicable instead: Rω = Eλ ,r kr cgr ,r Eλ ,i ki cgr ,i =R cgr ,r c ph ,i c ph ,r cgr ,i , (3.26) where c ph ,r and cgr ,r are the phase and group velocities of the reflected wave, whereas c ph ,i and cgr ,i are those of the incident wave. The true reflection coefficient represents the ratio of the energy that is contained in a differential frequency bandwidth of the reflected pulse to that contained in the same bandwidth of the incident pulse. The expression for Rω shows, in particular, that the energy reflection coefficient proposed by Lee and Mote does not apply to translating dispersive Energy considerations for a pipe conveying fluid 39 continua. It is applicable, however, if the continuum is not translating ( cgr ,r = cgr ,i and c ph ,r = c ph ,i ) or it is translating but not dispersive ( c ph ,i = cgr ,i and c ph ,r = cgr ,r ). The proof of Eq. (3.26) is given in the following section. 3.8 Energy of a pulse travelling in one-dimensional pipe conveying fluid To obtain the correct expression for the energy reflection coefficient, a pulse of a finite frequency bandwidth is considered. Calculating the total energy of this pulse, an expression is derived for the spectral energy density in the translating continuum. The ratio of this density in the reflected and incident pulses gives the energy reflection coefficient for a differential frequency band of the pulse. This coefficient could be derived by considering reflection of a harmonic wave instead of the pulse. In this case, however, one should additionally account for the difference in the range of wavenumbers, which correspond to the same differential frequency band in the incident and reflected waves. This has not been done by Lee and Mote, which limited the applicability of their results to non-dispersive systems only. It must be noted that the modification of the reflection coefficient proposed in this chapter is not aimed at undermining the usefulness of the ‘travelling wave method’ but at a correct description of the energy transfer at a solitary boundary of a dispersive translating continuum. Consider a generalized, uniform, one-dimensional pretensioned pipe conveying fluid at a constant speed u f , in the positive x-direction. In accordance with Lee and Mote (1997a,b), the total energy per unit length e ( x, t ) (excluding the constant term m f u f 2 2 ) of such a continuum can be written as ⎛ ∂2w ⎞ 1 ⎛ ⎛ ∂w ⎞ ∂w ⎞ ⎛ ∂w ⎛ ∂w ⎞ e ( x, t ) = ⎜ m p ⎜ + mf ⎜ + uf +T ⎜ + EI ⎜ 2 ⎟ ⎟ ⎟ ⎟ 2 ⎜ ⎝ ∂t ⎠ ∂x ⎠ ⎝ ∂t ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ 2 2 2 2 ⎞ ⎟. ⎟ ⎠ (3.27) This is the same equation as Eq. (3.5) except that the energy contribution of the axial pretension T has been added. The equation of the transverse motion of the pretensioned pipe conveying fluid, which corresponds to the energy density given by Eq. (3.27), reads EI ∂4w ∂2w ∂2w ∂2w − ( T − m f u f 2 ) 2 + 2m f u f + (m f + mp ) 2 = 0 . 4 ∂x ∂x ∂x∂t ∂t (3.28) To find the correct expression for the energy transfer at a boundary of the continuum, a propagating pulse with a finite frequency bandwidth is considered in this development. Assuming that this pulse propagates in the positive x-direction, the transverse displacement of the pipe conveying fluid corresponding to this pulse can be represented as 40 Chapter 3 w ( x, t ) = ∞ ∫ w (ω ) exp ( i (ωt − k (ω ) x ) ) dω , (3.29) −∞ where w (ω ) is the complex displacement amplitude of the continuum in the frequency domain, ω is the radial frequency and k (ω ) is the wavenumber, which is assumed real to ensure propagation of all harmonics of the pulse. Substitution of Eq. (3.29) into the equation of motion, Eq. (3.28), yields the following dispersion equation k 4 (ω ) EI + k 2 (ω ) (T − m f u f 2 ) + 2ωk (ω ) m f u f − ω 2 ( m f + m p ) = 0 , (3.30) from which k (ω ) can be derived. Considering an instant when the pulse is located so far from the boundaries that it is not disturbed by their presence, the energy of the pulse E can be computed by integrating the energy density e ( x, t ) over space from minus to plus infinity: ∞ E= ∫ e ( x, t ) dx . (3.31) −∞ By inserting the energy density of the continuum Eq. (3.27) into this expression, making use of the representation of the pulse displacement, Eq. (3.29), and carrying out some mathematical manipulations (see Appendix A) the following expression can be obtained: ∞ ∞ E = 4π ∫ k (ω ) ( T + EIk 2 (ω ) ) cgr (ω ) w (ω ) d ω ≡ ∫ Eω (ω ) d ω , 2 2 0 (3.32) 0 where Eω is the spectral energy-density of the pulse and where the group velocity cgr (ω ) = d ω dk (ω ) is introduced. Since the continuum under consideration is linear, the energy exchange at a boundary during wave reflection can be analysed considering differential bandwidths d ω of the pulse separately. For each bandwidth with the central frequency, ω , the energy transferred into the continuum during wave reflection is given as ∆Wω = Eω ,r − Eω ,i = ( Rω − 1) Eω ,i , (3.33) where Rω = Eω ,r Eω ,i is the ‘true’ energy reflection coefficient. Employing Eq. (3.32), this coefficient can be expressed as Rω = kr2 (T + EI kr2 ) cgr ,r ki2 (T + EI ki2 ) cgr ,i r2 , (3.34) Energy considerations for a pipe conveying fluid 41 where the subscripts r and i stand for the quantities associated with the reflected and incident waves9, respectively and r = w r w i , (3.35) is the amplitude reflection coefficient, which can be found by considering either reflection of a spectral component of the pulse or that of a harmonic wave to give the same result. If the reflection coefficient Rω is larger (smaller) than unity throughout the complete frequency band, then one should conclude that energy is gained (lost) upon reflection of any pulse from the considered boundary. If, on the contrary, Rω − 1 is not a sign-definite function of frequency, the energy transfer at the boundary depends on the amplitude spectrum of a particular pulse. In the papers of Lee and Mote (1997a,b) the energy reflection coefficient R is defined as the ratio of the energy Eλ ,r contained in one wavelength of the reflected wave to the energy Eλ ,i contained in one wavelength of the incident wave. For the continuum, whose energy density is defined by Eq.(3.27), the coefficient R defined by Lee and Mote reads R= k r (T + EI kr2 ) ki (T + EI ki2 ) r2 . (3.36) Comparing Eq. (3.36) to Eq. (3.34), the following relation can be found between Rω and R : Rω = R kr cgr ,r ki cgr ,i =R cgr ,r c ph ,i c ph ,r cgr ,i , (3.37) where c ph ,r = ω k r and c ph ,i = ω ki are the phase velocities of the reflected and incident waves, respectively. The main difference between the method presented here and the method of Lee and Mote, is that we derive the energy reflection coefficient by considering a differential frequency band of the pulse, whereas Lee and Mote derive the energy reflection coefficient by considering a harmonic wave in the wavenumber domain. This would have been correct if they would have accounted for the difference in the range of wavenumbers, which corresponds to the same differential frequency band in the incident and reflected waves. As discussed above, the conclusion as to whether the energy is gained or lost at a boundary is drawn comparing the energy reflection coefficient to unity. Since the factor cgr ,r c ph ,r cgr ,i c ph ,i is not necessarily equal to unity, Eq. (3.37) clearly shows 9 The reflection of a propagating incident wave at a boundary consists of a spatially decaying and a propagating wave. In the considered system, the spatially decaying wave does not propagate energy. 42 Chapter 3 that this conclusion may be wrong if the coefficient R is employed. In the next section the energy exchange for standard boundary conditions are determined based on the correct reflection coefficient, Rω . Note that the travelling wave method is only applicable if propagating waves in the medium are stable. For a tensioned Euler-Bernoulli beam, local unstable waves do not occur if the fluid velocity satisfies the following criterion; u f < T m f . Doaré and de Langre (2006) present a new theoretical method to analyse the stability of a finite length system, considering the role of wave reflections and (unstable) local waves in the medium. They remark correctly that in the original work of Lee and Mote (1997b) the contribution of the bulk of the medium to the instability is neglected. Lee and Mote considered a pipe conveying fluid without pretension, and hence, there exist local unstable waves at any velocity u f > 0 . To avoid the possibility of local unstable waves, in the next section, a tensioned Euler-Bernoulli beam is considered only for fluid velocities u f < T m f . 3.9 Energy exchange at boundaries of a pipe conveying fluid As explained in Chapter 2, the finite length and infinitely long pipe conveying fluid lose stability at different fluid velocities. It was shown that the type of support and the flow direction may have a large impact on the critical velocity. To explain this, the energy transfer at various standard supports is considered, i.e. free, clamped and pinned end. To find the energy reflection coefficients, a harmonic incident wave can be considered of the following form: wi ( x, t ) = wi e ( i ω t − ki x ) , (3.38) where wi is the complex amplitude of this wave and ki is the positive real root of the dispersion equation, Eq. (3.30). Impinging on a downstream support, this wave gives rise to a propagating reflected wave, wrpr ( x, t ) , and to a spatially decaying reflected wave, wrev ( x, t ) , as sketched in Fig. 3.2. Together with the incident wave these waves form the following pattern of the transverse deflection of the pipe: w ( x, t ) = wi ( x, t ) + wrpr ( x, t ) + wrev ( x, t ) = wi e ( i ω t − ki x ) + wrpr e ( i ω t − k rpr x ) + wrev e ( i ωt − k rev x ) , (3.39) where wrpr and wrev are the complex amplitudes of the reflected propagating and reflected spatially decaying waves, k rpr is the negative real root of the dispersion equation, and krev is the complex root of this equation with a positive imaginary part. Energy considerations for a pipe conveying fluid 43 Reflected spatially decaying wave Reflected propagating wave x uf Downstream End Incident propagating wave Fig 3.2 - Reflection of an incident harmonic wave impinging on a downstream boundary. For each support two boundary conditions have to be satisfied: • Clamped end w x =0 = 0 and ∂w =0; ∂x x = 0 (3.40) • Pinned end w x =0 = 0 and ∂2w =0; ∂x 2 x = 0 (3.41) • Free end ∂2w =0 ∂x 2 x = 0 and −T ∂w ∂ 3w + EI 3 =0. ∂x x =0 ∂x x =0 (3.42) Substitution of representation (3.39) into the boundary conditions, Eqs. (3.40)(3.42), results, for each type of support, in a system of two algebraic equations, from which the following expressions for the amplitude displacement reflection10, r, can be found: • Clamped end rc ,d = −k + krev wrpr = pri wi kr − krev • Pinned end rp ,d = − ( ki ) + ( k rev ) wrpr = 2 2 wi ( krpr ) − ( krev ) • Free end rf ,d = ki ( k rev − ki )( − EIkrev ki + T ) wrpr = pr pr wi kr ( kr − krev )( − EIkrev krpr + T ) 2 10 ; (3.43) 2 ; (3.44) . (3.45) For a downstream support, the amplitude reflection coefficient is defined as the ratio between the amplitudes of the wave propagating upstream and downstream. 44 Chapter 3 These amplitude displacement reflection coefficients can be substituted into the expression for the energy reflection coefficient Rω derived in the previous section, Eq. (3.34). The energy exchange at three different supports is shown in Figs. 3.3 and 3.4 for several harmonic waves with different frequencies. Positive and negative fluid velocities are related to, respectively, downstream and upstream support. To plot these figures, the following physical parameters of the system were used: EI = 1.0 ⋅ 109 Nm 2 , T = 1.0 ⋅ 106 N , m f = 1.0 ⋅ 103 kg/m , m p = 1.0 ⋅ 103 kg/m . If the reflection coefficient Rω is larger than unity then there is an energy gain at the support. Obviously, in absence of fluid flow the reflection coefficient is in all cases equal to 1. Two mechanisms are possible if the energy of the reflected wave is larger than the energy of the incident wave: (i) the amplitude of the reflected wave is larger than the amplitude of the incident wave and (ii) the wavelength of the reflected wave is smaller than the one of the incident wave. Since the displacement amplitude of the reflected propagating wave is the same for a pinned and clamped support, the energy reflection coefficients are equal for these two supports (Fig. 3.3). Independent of the frequency, the energy reflection coefficient from a downstream clamped (or pinned) support is always larger than unity (see Fig.3.3). Although the amplitude displacement of the reflected wave reduces, the shortening of the wavelength increases the energy of the reflected wave. For the same fluid velocity, the energy reflection coefficient at an upstream clamped (or pinned) support is the reciprocal of the energy reflection coefficient at a downstream clamped (or pinned) support. Hence, the energy of a clamped-clamped, pinned-pinned or clamped-pinned beam conveying fluid varies periodically in time11; when a wave reflects from a downstream support the energy of the system increases; however, after reflection from the upstream support the total energy of the system regains its original value. In Fig. 3.4 the energy reflection coefficient at a free end12 is shown for different wave frequencies. Energy gain at an upstream free end is related to the increase of the wave displacement amplitude. This amplitude increase dominates the energy decline due to the larger wavelength of the reflected wave. 11 In literature, this is called a gyroscopic conservative system. In this section it is assumed that a free end where fluid is sucked in can be described by the same boundary conditions, Eq.(3.42), as a free end where fluid leaves the pipe. In Chapters 5 and 6 the boundary conditions for a cantilever pipe aspirating fluid are investigated in great detail. 12 Energy considerations for a pipe conveying fluid 45 8 Upstream Downstream 7 Frequency ω = 0.1 rad/s 6 ω = 1.0 rad/s ω = 10 rad/s Rω (-) 5 4 3 2 1 0 -30 -20 -10 0 10 20 30 uf (m/s) Fig 3.3 – Energy reflection coefficient for both clamped and pinned support. Positive fluid velocities represent a downstream support, and negative fluid velocities represent an upstream support. 8 Upstream Downstream 7 Frequency ω = 0.1 rad/s 6 ω = 1 rad/s ω = 10 rad/s Rω (-) 5 4 3 2 1 0 -30 -20 -10 0 10 20 30 uf (m/s) Fig 3.4 – Energy reflection coefficient for a free end. Positive fluid velocities represent a downstream free end, and negative fluid velocities represent an upstream free end. 46 Chapter 3 From Figs. 3.3 and 3.4 it is clear that an undamped cantilever pipe aspirating fluid (downstream clamped and upstream free end) behaves unstable at infinitesimally small flow, since at both ends the total energy of the system is increased. This is in full agreement with the stability analysis of the aspirating cantilever pipe in Chapter 2. Reversing the flow, the system loses energy at both supports. 3.10 Conclusion The energy equation and the balance of pseudo-momentum derived in this chapter show that the linear statement of the transverse vibrations of a pipe conveying fluid implies that an external axial force has to act at every cross section of the pipe to maintain the constant flow speed through the pipe. It is the work of this force that enables energy of the pipe to increase. Only in case of periodic motions this external axial force has a netto zero contribution to the work. It has been shown that the energy exchange at a boundary can be computed using the travelling wave method. The expression for the energy reflection coefficient at a boundary of a pipe conveying fluid, as introduced by Lee and Mote (1997a,b), has been corrected in this chapter. Employing this method, it has been proven that the energy gain of the cantilever pipe aspirating fluid originates from both the upstream free end and the downstream fixed end. CHAPTER 4 Stability of a free hanging riser conveying fluid – stability of the straight configuration In the previous two chapters, the dynamic behaviour of a pipe conveying fluid was examined based on stability analyses and energy considerations. The equation of motion was based on a pretensioned Euler-Bernoulli beam. In order to model more accurately a submerged, free hanging riser aspirating fluid, additional terms are incorporated in the equation of motion. As a first approach, this extended equation is linearized around the straight configuration of the riser. A nonlinear, more realistic, description of the hydrodynamic damping is incorporated in the next chapter. Studying stability of a suspended pipe conveying water, researchers have found a contradiction between theoretical predictions and experiments. Theory predicts instability at small fluid velocities, while experiments did not show such instability. It was commonly accepted that the fluid depressurization at the inlet prohibited the pipe from unstable behaviour. In this chapter, it is shown theoretically that the negative pressurisation influences the stability only slightly and cannot explain the contradiction. What can explain it, is the hydrodynamic drag caused by surrounding water, which is shown to be one of the essential stabilizing factors. The main results of this chapter have been published by Kuiper and Metrikine (2005). 48 Chapter 4 4.1 Introduction Recently, designs have been made for pipes suspended from a floating barge, which will be used for pumping up cooling water (Fig. 4.1). These pipes have an unconstrained tip (lower end) and therefore are referred to as free hanging risers. A question arises whether these risers might become unstable due to conveying fluid. According to existing theory, if fluid is sucked into a cantilever pipe at a free end and no damping is regarded, the energy of the pipe grows ever since it starts moving, so that the cantilever pipe is unstable (see Chapters 2 and 3). However, there is a contradiction between theory and experiments. The theory predicts unstable behaviour for undamped pipes at infinitely small fluid velocity (Païdoussis and Luu, 1985; Sällström and Åkesson, 1990; Kangaspuoskari et al., 1993). On the contrary, in experiments with submerged, cantilevered pipes pumping up water (Hongwu and Junji, 1996; Païdoussis, 1998) no instability has been observed. Later, Païdoussis (1999) explained in a short way why these cantilevered pipes should behave stable. The main reason, according to this explanation, is negative pressurisation of the fluid at the inlet of the riser. 4.2 Assumptions and equation of motion Consider a straight riser that conveys water up to a floating barge as shown in Fig. 4.1. The riser is tubular and submerged fully. It is attached to the barge through a so-called flex-joint connection that fixes the translational motion of the riser top to that of the barge and provides a linear-elastic reaction against a relative angular motion of the barge and the riser top. The tip of the riser is free (not constrained), accordingly the riser is referred to as a free hanging riser. Dynamics of the riser is studied in this chapter under the following assumptions: a) The riser moves in the plane that is depicted in Fig. 4.1. b) The length of the riser and the wavelength of its deformation are large in comparison to the diameter of the riser so that the Euler-Bernoulli theory is applicable for the bending of the riser and the plug-flow model (Païdoussis, 1998) is acceptable. c) Small bending motion of the riser about its equilibrium position is considered. d) The pipe wall behaves elastically and no internal damping is considered. e) The vertical motion of the barge is disregarded. f) The mean flow velocity along the riser is constant. Free hanging riser – stability of the straight configuration 49 w(z,t) Barge z Flex-joint uf L Free hanging riser Fig. 4.1 - Sketch of a free hanging riser conveying fluid. With these assumptions, the equation that governs the horizontal motion of a differential element of the riser can be written as: EI ⎛ 2 ∂2w ∂4w ∂ ⎛ ∂w ⎞ ∂2w ∂2w ⎞ − ⎜ Tr ( z ) − 2u f + ⎟− ⎟ + ρ f Ai ⎜ u f 4 2 ∂z ∂z ⎝ ∂z ⎠ ∂z ∂z∂t ∂t 2 ⎠ ⎝ ∂ ⎛ ∂w ⎞ ∂2w ⎜ ( Ae pe ( z ) − Ai pi ( z ) ) ⎟ + ρ r Ar 2 = f ( z , t ) , ∂z ⎝ ∂z ⎠ ∂t (4.1) where w ( z, t ) is the horizontal riser displacement, z is the co-ordinate along the riser (directed downward), t is the time, EI is the bending stiffness of the riser, Tr ( z ) is the axial tension of the riser, ρr and ρf are the mass density of the riser and the fluid, respectively, Ai and Ae are the internal and external cross-sectional areas of the riser, respectively, Ar = Ae − Ai is the cross-sectional area of the pipe wall, pi ( z ) and pe ( z ) are the water pressure inside and outside the riser, respectively, uf is the velocity of the flow through the riser (directed upward) and f ( z, t ) is the normal dynamic reaction of the surrounding water on the riser element. The first and the last term on the left-hand-side of Eq.(4.1) form the well-known equation for the bending motion of a beam according to the Euler-Bernoulli theory. The second term, ∂ ∂z (Tr ( z ) ∂w ∂z ) , is due to the longitudinal tension in the riser that is caused by gravity and internal fluid resistance. This tension reads Tr ( z ) = ρ r Ar g ( L − z ) − ρ f Ar gL − ρ f Ai 2 f DW u f (L − z) , Di 2 (4.2) 50 Chapter 4 where g is the gravity acceleration, L is the length of the riser (see Fig. 1), Di is the inner diameter of the pipe and fDW is the resistance coefficient of Darcy-Weisbach. Note that the origin of the reference system is fixed to the riser top. At the free end of the riser, i.e. at z = L , only the second term on the right hand side of Eq. (4.2) remains. This term represents the static upward directed buoyancy force. The third term on the left hand side of Eq.(4.1) is the transverse loading per unit length exerted by the internal flow on the pipe (as explained in Chapter 2). The fourth term in Eq.(4.1), ∂ ∂z ( ( Ae pe ( z ) − Ai pi ( z ) ) ∂w ∂z ) , is due to the pressure on the pipe wall. The external hydrostatic pressure pe ( z ) , neglecting the temperature effects (and compressibility, as mentioned above), may be assumed to vary linearly with z , so that pe ( z ) = ρ f gz . (4.3) The description of the internal pressure, which is the key issue of this development, is discussed in detail in the next section. The dynamic reaction of the surrounding water on the riser, f ( z, t ) , is assumed to be a superposition of an inertia force f in ( z, t ) and a drag force f d ( z, t ) . The inertia force depends on the acceleration of the riser and of the surrounding water as f in ( z, t ) = ρ f Ae ( Ca + 1) ∂u ∂2w − ρ f AeCa 2 , ∂t ∂t (4.4) where u ( z, t ) is the horizontal component (located in the plane ( w, z ) ) of the surrounding water velocity and Ca is the added mass coefficient. The drag force depends on the relative motion of the riser and of the surrounding water nonlinearly. Besides, the drag force is influenced by Reynolds number, wall surface roughness, water turbulence, etc. In this Chapter all these effects and the nonlinearity are neglected. The only aim of incorporating the external damping is to investigate whether this might explain the difference between theory and experiments. Hence the following linearised expression for the drag force is used (see Païdoussis, 1998): f d ( z, t ) = 1 ∂w ⎞ ⎛ ρ f DoC d ⎜ u − ⎟ , 2 ∂t ⎠ ⎝ where C d is the adapted drag coefficient, with the dimension of velocity. (4.5) Free hanging riser – stability of the straight configuration 51 4.3 Description of the internal pressure The internal water pressure pi ( z ) in pipes sucking up fluid is a subject of discussion in literature. There exist two different opinions about the description of the internal pressure. Païdoussis and Luu (1985) assumed that the internal pressure does not differ from the external one (at the same depth) and, accordingly, it does not depend on the speed of the internal fluid flow. This approach was also used by Sällström and Åkesson (1990) and by Kangaspuoskari et al. (1993). All these papers reported unstable behaviour of the free hanging, undamped riser sucking up fluid at infinitely small flow velocity. Recently, Païdoussis (1999) suggested that there is a negative pressurisation at the riser tip so that the difference between the external and internal pressure is equal to ρ f u f 2 (see Appendix B). This pressure difference is in agreement with the pressure drop in a so-called Borda's mouthpiece, which is a short tube of the length about equal to the radius that projects into a reservoir. At the inlet the coefficient of contraction for the Borda's mouthpiece is 0.5. By modelling the internal pressure in this way, the pressure drop balances the centrifugal force induced by the internal flow ( ρ f Ai u f 2 ∂ 2 w ∂z 2 ). In this case, the only remaining term in the equation of motion, Eq.(4.1), related to the fluid velocity is the Coriolis force. Païdoussis concluded that no flutter, and hence no instability can occur if only the Coriolis force is present. In this chapter it is shown that this conclusion is incorrect. Modification of the geometry of the inlet can reduce the pressure loss. If the inlet is shaped so that the fluid flow remains fully attached to the wall, no pressure loss occurs at the inlet. In this theoretical case without contraction, the Bernoulli equation for an incompressible flow shows that the difference between the external and internal pressure is equal to ρ f u f 2 2 . In reality the pressure difference depends on the geometry of the inlet and varies between ρ f u f 2 2 and ρ f u f 2 . The effect of the magnitude of the pressure difference has never been studied in the literature, to the authors’ knowledge. Therefore, in what follows, vibrations are studied of the riser for all three aforementioned internal pressure drops and results are compared to each other. Incorporating the pressure loss due to internal wall friction as well, the following expressions for the internal pressure are considered: pi ( z ) = ρ f gz − ∆p ( k ) − ρ f with ∆p (1) = 0, ∆p ( 2) 2 f DW u f (L − z), Di 2 = ρ f u f , ∆p 2 ( 3) k = 1, 2,3 1 = ρ f u f 2. 2 (4.6) 52 Chapter 4 4.4 Dimensionless form of the equation of motion and the boundary conditions Substituting Eqs.(4.2)-(4.6) into the equation of motion Eq.(4.1), the latter can be rewritten in the following form: EI 2 ∂4w ⎛ ∂2w ⎛z ⎞ ⎞ ∂ w T ∂w k + ⎜ m f u f 2 − ∆p ( ) Ai + Ttop ⎜ − 1⎟ ⎟ 2 + top − 2m f u f + 4 ∂z ∂z∂t L ∂z ⎝ L ⎠ ⎠ ∂z ⎝ 1 ∂w ∂2w ∂u 1 + ( mr + ma + m f ) 2 = m f ( Ca + 1) + ρ f DoC d u ρ f DoC d ∂t ∂t ∂t 2 2 (4.7) where mr = ρ r Ar , ma = ρ f Ca Ae , m f = ρ f Ai and Ttop = ( ρ r − ρ f ) Ar gL . The external forces (external flow) do not affect the stability of the system within the accepted linearised model. Consequently, the right-hand-side of the equation of motion is disregarded from hereonwards. The boundary conditions at the ends of the pipe are given as w ( 0, t ) = 0 , ∂2w ∂w = C fl , ∂z 2 z =0 ∂z z = 0 EI EI ∂2w =0, ∂z 2 z = L EI ∂3w = 0, (4.8) ∂z 3 z = L where Cfl is the stiffness of the rotational spring (flex-joint) at the top of the riser. Introducing the following dimensionless variables and parameters: η = w L, τ = t EI ( mr + ma + m f ) L2 , ξ = z L, EI ( mr + ma + m f ) , α = Ttop L2 EI , β = L m f Ttop γ = ρ f DoC d L2 (2 EI ( mr + ma + m f V = u f m f Ttop , ) ) , C ( ) = ∆p ( ) A j j i Ttop , κ = C fl L EI , the statement of the problem Eqs.(4.7)-(4.8) is rewritten as ∂ 4η ∂ 2η ∂ 2η ∂η ∂ 2η ∂η ∂ 2η (k ) 2 α αξ α β γ + − − + + − + + =0 V C 1 2 V ∂ξ 4 ∂ξ 2 ∂ξ 2 ∂ξ ∂ξ∂τ ∂τ ∂τ 2 ( ) η ( 0,τ ) = 0 , ∂ 2η ∂ξ 2 =κ ξ =0 ∂η , ∂ξ ξ = 0 ∂ 2η ∂ξ 2 = 0, ξ =1 ∂ 3η ∂ξ 3 =0 (4.9) (4.10) ξ =1 In Eq.(4.9), coefficient C ( k ) changes for the three descriptions of the pressure difference: ∆p ( ) = 0 corresponds to C ( ) = 0 , 1 1 ∆p ( ) = ρ f u f 2 corresponds to C ( ) = V 2 , 2 ∆p ( 3 ) = 2 1 1 ρ f u f 2 corresponds to C ( 3) = V 2 . 2 2 Free hanging riser – stability of the straight configuration 53 4.5 Characteristic equation To find the eigenvalues of the problem Eqs.(4.9)-(4.10), the displacement η (ξ ,τ ) is to be sought in the following form: η (ξ ,τ ) = W (ξ ) eλτ . (4.11) The pipe is unstable if at least one of the eigenvalues λ has a positive real part. Substituting Eq.(4.11) into the equation of motion Eq.(4.9), the following ordinary differential equation is obtained ( ) d 4W d 2W d 2W dW dW k + α V 2 − C( ) −1 + αξ +α − 2βV λ + γλW + λ 2W = 0 . (4.12) 4 2 dξ dξ dξ 2 dξ dξ This dimensionless ordinary differential equation contains one coefficient that depends on ξ , which implies that the eigenfunctions of this equation are not sinusoidal. A solution to this equation can be sought in the form of a power series expansion (see Huang and Dareing, 1976): ∞ W ( ξ ) = ∑ a nξ n . (4.13) n =0 Substituting Eq.(4.13) into Eq.(4.12), an equation is obtained involving a power series whose sum is equal to zero. Since each term in the series must be equal to zero, the following recurrence relation can be derived: (n ≥ 4) an = An − 2 an − 2 + An − 3an − 3 + An − 4 an − 4 , (4.14) with An − 2 = ( ), −α V 2 − C ( ) − 1 k n ( n − 1) An − 3 = − (α ( n − 3) − 2 β V λ ) n ( n − 1)( n − 2 ) , An − 4 = − (γλ + λ 2 ) n ( n − 1)( n − 2 )( n − 3) . By repeated application of this recurrence relation, starting with n=4, an can be expressed as a linear combination of a0, a1, a2 and a3: an = Fn a0 + Gn a1 + H n a2 + I n a3 , ( n ≥ 0 ) , (4.15) in which F0 = 1, F1 = 0, F2 = 0, F3 = 0, G0 = 0, G1 = 1, G2 = 0, G3 = 0, H 0 = 0, I 0 = 0, H1 = 0, H 2 = 1, I1 = 0, I 2 = 0, H 3 = 0, I 3 = 1. By substituting Eq.(4.15) into Eq.(4.14) the following relations are found: (4.16) 54 Chapter 4 ⎡ Fn ⎤ ⎡ Fn − 2 ⎤ ⎡ Fn − 3 ⎤ ⎡ Fn − 4 ⎤ ⎢G ⎥ ⎢G ⎥ ⎢G ⎥ ⎢G ⎥ ⎢ n ⎥ = An − 2 ⎢ n − 2 ⎥ + An − 3 ⎢ n − 3 ⎥ + An − 4 ⎢ n − 4 ⎥ . ⎢Hn ⎥ ⎢ H n −2 ⎥ ⎢ H n −3 ⎥ ⎢ H n−4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ In ⎦ ⎣ I n−2 ⎦ ⎣ I n −3 ⎦ ⎣ I n −4 ⎦ (4.17) With the starting values given by Eq.(4.16), it is possible to obtain Fn, Gn, Hn and In for every n. Making use of Eq.(4.15), the general solution Eq.(4.13) can then be written as: ∞ ∞ ∞ ∞ n =0 n =0 n =0 n =0 W (ξ ) = a0 ∑ Fnξ n + a1 ∑ Gnξ n + a2 ∑ H nξ n + a3 ∑ I nξ n . (4.18) To find the four unknowns aj, j=0..3, Eq.(4.18) should be substituted into the boundary conditions Eq.(4.10). This yields four linear algebraic equations with respect to a0 – a3. From the first two boundary conditions of Eq.(4.10) it is concluded that: a0 = 0 and a2 = κ 2 a1 . (4.19) Using Eqs.(4.18)-(4.19), the last two boundary conditions of Eq.(4.10) result in the following equations: a1 ( a + κ b ) + a3c = 0 , a1 ( d + κ e ) + a3 f = 0 , (4.20) with ∞ a = ∑ Gn n ( n − 1), b = n=2 ∞ d = ∑ Gn n ( n − 1)( n − 2 ), e = n =3 ∞ 1 ∞ H n n ( n − 1), c = ∑ I n n ( n − 1), ∑ 2 n=2 n=2 ∞ 1 ∞ H n n ( n − 1)( n − 2 ) , f = ∑ I n n ( n − 1)( n − 2 ). ∑ 2 n =3 n =3 The two linear equations Eqs.(4.20) have a non-trivial solution if and only if the following equation is satisfied: ∆ = (a + κ b) f − (d + κ e) c = 0 . (4.21) Eq.(4.21) is the characteristic equation for the problem described by Eqs. (4.9) and (4.10). Free hanging riser – stability of the straight configuration 55 4.6 Argand Diagram Now that the characteristic equation Eq.(4.21) has been obtained, its roots λ have to be analysed to draw a conclusion on the riser stability. To this end, the Argand Diagram (Païdoussis, 1998) is used again. 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. Normally, the flow velocity V is used as such a parameter. In this chapter, along with V, the dimensionless external damping is employed as the parameter to plot the paths of the first four natural frequencies. The natural frequency ω is related to the eigenvalue λ as, ω=iλ. Note that if the imaginary part of a natural frequency ω is smaller than zero, the system is unstable. At the first step of the analysis, both the fluid velocity and the external damping are disregarded. The system possesses in this case only real natural frequencies, which can be easily found numerically. These natural frequencies equal the resonance frequencies of an undamped cantilever beam with varying tension. As the second step, the damping is gradually increased, keeping the velocity zero, and the accompanying complex values of the natural frequencies are computed and plotted in the Argand Diagram. As expected, all complex natural frequencies acquire positive imaginary part, implying that the pipe is stable. Once the damping reaches its ‘true’ (assumed in the present analysis) value, the fluid velocity is gradually increased from zero. As a result, the imaginary part of the complex natural frequencies decreases with increasing fluid velocity. At a certain velocity, the imaginary part of some natural frequencies becomes negative, implying that the pipe becomes unstable. For constructing the Argand Diagram the system parameters are used that are shown in Table 4.1. The selected parameters and the derived dimensionless parameters are shown, respectively, at the right and left side of the table. The internal pressure is taken as independent of the internal fluid velocity, C (1) = 0 . The Argand Diagram is presented schematically in Fig. 4.2. Only the first four natural frequencies are plotted. The paths of the other natural frequencies have a similar shape as those of the second, third and fourth natural frequencies. The smallest fluid velocity for which the natural frequency has a negative imaginary part is denoted as the critical velocity V*. The paths of all the natural frequencies, except the first one, intersect the real ω-axis at almost the same fluid velocity. This implies that the corresponding modes become unstable at the same fluid velocity. 56 Chapter 4 V=0, γ=2.89 Im(ω) V*=0.41 γ=2.89 0 V=0, γ=0 V=0.79, γ=2.89 0 Re(ω) Fig. 4.2 - Argand Diagram for the first four natural frequencies. 2 V=0, γ=2.89 Im(ω) 1 V* 0 V=0 γ=0 -1 V=0.79, γ=2.89 -2 16.4 16.8 V=0.79, γ=2.89 17.2 V=0.79, γ=2.89 17.6 18 18.4 Re(ω) Fig. 4.3. Argand Diagram for the second natural frequency and for three descriptions of the internal pressure. ( C (1) = 0 , C (2) = V 2 , C ( 3) = V 2 2 ). Free hanging riser – stability of the straight configuration 57 Table 4.1 – Parameters used in the calculations Parameter EI mf 1.00*109 Nm2 Dimensionless Parameter L 3 ρf 3 1.00*10 kg/m 100 m 3 3 1.00*10 kg/m α 10.0 β 1.83 ma 1.00*10 kg/m Do 1.00 m γ 2.89 mr 1.00*103 kg/m 1.00*106 Nm/rad κ 0.100 Ttop 1.00*106 N Cfl C d 1.00 m/s Using the Argand Diagram the influence of the description of the internal pressure on the system stability can be analysed. In Fig. 4.3 the second natural frequency of the system is shown for the three different descriptions of the internal pressure as mentioned before. As can be seen, the real part of the complex natural frequency changes significantly by using different descriptions of the internal pressure. However, the critical velocity V* is hardly different for these three descriptions (the deviation is less than 2%). Similar observations can be made for all other natural frequencies, and hence, the influence of the description of the internal pressure on the system stability is marginal. This means that the negative pressurisation cannot be considered as the reason for the stable behaviour of the free hanging riser, which has been found experimentally. What can be a reason for the observed stable behaviour is the external hydrodynamic drag. This drag can greatly influence the stability of the pipe at hand, since it induces energy loss all over the length of the pipe, while energy can be gained at the pipe ends only (Lee and Mote, 1997a,b). In the next section the stabilizing effect of this drag is studied by a D-decomposition method, developed by Neimark (1978) (see also Neimark et al., 2003). The main advantage of this method relative to the Argand Diagram is that it does not require searching for complex zeroes of a complex function. The stability can be analysed using the D-decomposition method by just considering the imaginary axis of the complex λ-plane (the plane of complex eigenvalues). This increases the accuracy of the stability analysis and, in some cases, the calculation speed. As shown in the next sections, the D-decomposition method is especially efficient in combination with the Argand Diagram. 4.7 D-decomposition method The D-decomposition method utilises the fact that the stability of a linear system is fully determined by the sign of the real part of its eigenvalues λ (Eq.(4.11)). The eigenvalues which correspond to unstable vibrations are located in the right halfplane of the complex λ-plane, see Fig. 4.4. Consequently, the imaginary axis of this plane, λ = iΩ, Ω ∈ℜ is the boundary that separates the “stable” and “unstable” 58 Chapter 4 eigenvalues (roots with Re ( λ ) < 0 and Re ( λ ) > 0 , respectively). Assume now that the characteristic equation contains a parameter P that can be expressed explicitly. Such expression can be used as a mapping rule to map the imaginary axis of the λplane onto the complex plane of the parameter P as sketched in Fig. 4.4. To this end, the frequency Ω is varied from minus infinity to plus infinity in the complex P-plane. The resulting mapped line(s), which are referred to as D-decomposition line(s), break the P-plane into domains with different number of “unstable” eigenvalues. Within a domain, this number may not vary. Shading the right side of the imaginary axis of the λ-plane (the side of “unstable” eigenvalues), and keeping the shading at the corresponding side of the Ddecomposition line(s), the information contained in the decomposed P-plane can be enriched. With this shading, it becomes known that passing through a Ddecomposition line in the direction of shading corresponds to the gain of one additional “unstable” eigenvalue by the characteristic equation. Thus, if the number of the “unstable” eigenvalues is known for only one (arbitrary) value of the parameter P, the D-decomposed P-plane allows to draw a conclusion on stability of the system for all admissible values of this parameter at once. P-plane λ-plane Mapping P = F (Ω) Stable Unstable D(N) D(N+1) Fig. 4.4 - Graphical representation of the D-decomposition method. The notation D(N) implies that there are N “unstable” eigenvalues in the domain. Free hanging riser – stability of the straight configuration 59 For the free hanging riser conveying fluid the dimensionless rotational spring stiffness κ can serve as the parameter P, since it can be readily expressed from the characteristic equation (4.21) to give κ= −af + cd . −ce + bf (4.22) The D-decomposition of the κ -plane is depicted in Figs. 4.5 and 4.6 for two velocities of the fluid, namely V=0.32 and V=0.95, respectively. The system parameters used in calculations are shown in Table 4.1. The internal pressure is taken as independent of the internal fluid velocity, C (1) = 0 (the other two descriptions give almost the same result). Note that in both figures only the positive part of the real axis has physical meaning, since κ is the stiffness of the rotational spring. Only the key (low-frequency) part of the D-decomposition curves is plotted in the figures. Taking higher frequencies into account would lead to a larger number of quasi-circles both in the lower and upper half planes. These circles do not change the result of the stability analysis. Although the D-composition curves in Figs. 4.5 and 4.6 have qualitatively the same shapes, these curves run with the increase of Ω in opposite directions. The notation D ( N ) in both figures indicates the number of N “unstable” eigenvalues in every domain. As explained before, passing through a D-decomposition line in the direction of shading corresponds to the gain of one additional “unstable” eigenvalue, i.e. D ( N + 1) . In the figures one should focus on the positive part of the real axis, since κ is the stiffness of the rotational spring and cannot have in reality negative or complex values. In both figures, the D-decomposition curves do not cross the positive part of the real axis. So, in one run with the D-decomposition method it is easily concluded that the riser stability does not depend on the stiffness of the rotational spring. It would be much more laborious to draw the same conclusion by using only the Argand Diagram. The question remains if the pipe is stable or unstable for all real and positive values of the rotational spring. For Fig. 4.6 it is easy to determine that the system is unstable. In Fig. 4.6 there are areas with N-2 “unstable” roots, and consequently, N should be at least 2, since the number of unstable roots cannot become negative. This implies that in the area of our interest, i.e. positive part of the real axis, the number of “unstable” roots is at least 2. The same way of thinking cannot be applied to Fig. 4.5, since the number of unstable roots increases when moving away from the positive part of the real axis. However, to determine the stability in Fig. 4.5, it is sufficient to determine the number N of unstable roots for one particular stiffness of the rotational spring. For 60 Chapter 4 the problem at hand, this can be conveniently done by making use of the Argand Diagram. Note that studying other stability problems, the number of unstable roots for a particular set of parameters can be found using physical considerations (Metrikine and Dieterman, 1997) or the principle of the argument (Verichev and Metrikine, 2002). Im(κ) Ω=0 D(N+2) D(N+1) Ω→ −∞ D(N) Re(κ) D(N+1) Ω→ +∞ D(N+2) Fig. 4.5 - D-decomposition of the κ-plane for V = 0.32 . Im(κ) Ω =0 D(N-2) D(N-1) Ω→ +∞ D(N) Re(κ) D(N-1) D(N-2) Ω→ -∞ Fig. 4.6 - D-decomposition of the κ-plane for V = 0.95 . Free hanging riser – stability of the straight configuration 61 As follows from Fig. 4.2, the pipe is stable if the flow velocity belongs to the interval 0 ≤ V < V * , and is unstable if V > V * , where V * is a critical velocity. For the parameters given in Table 4.1, this critical velocity is V * ≈ 0.41 , which is greater than V = 0.32 (Fig. 4.5) and smaller than V = 0.95 (Fig. 4.6). Thus, combining information containing in Figs. 4.2 and 4.5, it can be concluded that the pipe is stable independently of the flexjoint stiffness if V = 0.32 . On the contrary, as follows from Figs. 4.2 and 4.6, the pipe is unstable if V = 0.95 , again independently of the flexjoint stiffness. These conclusions can be generalised by performing the Ddecomposition of the κ-plane for other velocities V of the flow through the pipe. This decomposition shows that independently of the flexjoint stiffness, the pipe is stable if 0 ≤ V < V * and is unstable if V > V * . Thus, applying the D-decomposition method, the critical velocity V * can be easily found that separates the stable and unstable behaviour of the riser. To this end, it is sufficient to track the half-plane (upper or lower) to which the D-decomposition curve runs as the frequency Ω is increased from zero. If the curve runs to the lower half-plane then the pipe is stable. The flow velocity, at which the curve starts to run to the upper half-plane is the critical velocity V * . This approach is used in the next section to investigate the effect of fluid pressurization at the inlet and of the hydrodynamic drag (external damping) on the pipe stability. 1 0.8 V 0.6 Unstable 0.4 Stable 0.2 0 0 2 4 6 8 γ Fig. 4.7 - The dimensionless critical velocity for three descriptions of the internal pressure as a function of the external damping γ. ( C (1) = 0 , C (2) = V 2 , C ( 3) = V 2 2 ). 62 Chapter 4 4.8 Effect of fluid pressurization and external damping As shown in the previous sections, both the Argand Diagram and the Ddecomposition method can be used to compute the critical velocity beyond which the system becomes unstable. Here we use the D-decomposition method to determine the critical velocity. This velocity depends on all parameters of the system, including the difference between the external and internal hydrostatic pressure and the amount of external damping γ. The dependence of the critical velocity on the external damping breaks the plane (V , γ ) as shown in Fig. 4.7. In this figure the critical velocity is presented for three values of the pressure difference, which are studied in this chapter. This dependence is obtained using the parameter values given in Table 4.1. From Fig. 4.7 the conclusion should be drawn that the effect of the internal fluid pressure is small, especially for low fluid velocities. This means that the negative pressurisation cannot be considered as the reason for the stable behaviour of the free hanging riser, which has been found experimentally. From Fig. 4.7 it is also clear that the external damping has a much more significant influence on the system stability. Without external damping (γ=0) the system is unstable for all fluid velocities irrespectively of the description of the internal pressure. On the contrary, with the external damping, a stable range of velocities is found for which the system behaves stable, as was observed in the experiments. As mentioned in the introduction, the hydrodynamic drag caused by the surrounding water is described in this chapter by a simplistic linearised expression, which does not allow to draw any quantitative conclusions. In the next chapter, the model for the riser is improved by taking into account a nonlinear description of the drag, including the drag dependence on the Reynolds number. In order to investigate whether this damping (drag) is indeed the cause of the discrepancy between theory and experiment, experiments are carried out to check if fluid velocities which should lead to instability, in accordance with the results in Fig. 4.7, agree with measurements (see Chapter 6). 4.9 Conclusion In this chapter, the stability of the straight configuration of a free hanging riser conveying fluid has been considered. It has been proven that the existing contradiction between theory (predicts instability for small fluid velocities) and experiments (show stable behaviour for small fluid velocities) cannot be explained by the negative pressurisation at the riser inlet. The stability analysis that has been carried out in this chapter has shown a marginal effect of this depressurisation. Free hanging riser – stability of the straight configuration 63 The difference between theory and experiment is more likely to be explained by the energy dissipation caused by the hydrodynamic drag, which the riser experiences while moving in water. Experiments are carried out (see Chapter 6) to check if the observed critical fluid velocities agree with the theory proposed in this chapter. CHAPTER 5 Stability of a free hanging riser conveying fluid – steady state vibrations with nonlinear damping In the previous chapter a submerged, free hanging offshore riser aspirating fluid was examined considering the stability of the straight configuration. Using a linearized drag description, it was shown that the hydrodynamic drag might be one of the essential factors to explain the apparent contradiction between theoretical predictions (instability at infinitesimally low fluid velocities) and experiments (no observation of instability). In this chapter a realistic description of hydrodynamic drag is incorporated in the model. In contrast to the previous chapter, here the description of the hydrodynamic drag is based on experimental data available in literature. These data have been obtained by experimenting with submerged rigid cylinders. To make them applicable to flexible pipes, a key step is undertaken to translate the data from the frequency domain to the time domain. Using the time-domain description of the hydrodynamic drag it is shown that the pipe becomes unstable by flutter at a critical velocity which is much higher than that attainable in small scale experiments. For fluid velocities exceeding this critical velocity a stable limit cycle exists (dynamic equilibrium). The main results of this chapter have been published by Kuiper et al. (2007). 66 Chapter 5 5.1 Introduction Studying dynamic stability of a vertically suspended, fully submerged pipe conveying fluid upwards, researchers have found a contradiction between theoretical predictions and experiments. Experiments did not show any instability, while theory predicts instability at infinitesimally low fluid velocities for a pipe. An attempt to explain this contradiction was made by Païdoussis (1999), suggesting that the theoretical prediction is wrong because of an improper description of the negative pressurisation of water at the inlet of the pipe. However, Kuiper and Metrikine 13 (2005) showed that this pressurisation influences the stability only slightly. In addition, they showed, using a linearized drag description, that the external hydrodynamic drag is a major stabilizing factor, which prohibits the pipe instability at low flow speeds. In reaction to this paper, Païdoussis et al. (2005) postulated several new descriptions of the boundary condition for the balance of the transverse forces at the free end of the pipe. Their basic description of the boundary condition assumes that the inflow remains substantially tangential to the undeflected pipe. This is in contrast to the conventional boundary condition, which assumes an inflow direction tangential to the deflected pipe. Using this newly postulated boundary description the pipe is predicted to become unstable by divergence at a high fluid velocity, which has not yet been reached in experiments. Alternative variants of this boundary condition were also discussed and analysed by varying the assumptions concerning depressurization, tension and inflow direction. The main difference between all variants of the boundary condition presented by Païdoussis et al. (2005) and the conventional boundary condition is that the former assume that the fluid momentum changes direction as the fluid enters the pipe, whereas the latter considers this direction to remain unchanged. The correct description of the flow field in the vicinity of the tip is obviously of great importance for the stability of the pipe conveying fluid. In agreement with Païdoussis et al. (2005), it seems plausible that the averaged inflow direction is tangential neither to the deflected pipe end nor to the undeflected pipe end, but that the angle between the pipe and the inflow varies in time between these two bounds. Using the travelling wave method14 (Lee and Mote, 1997) for the situation where the angle of inflow varies between these two bounds it can be proven theoretically that the energy gain at the downstream fixed end is greater than the energy loss at the upstream end. Hence, in the absence of damping, the tensioned cantilever pipe model loses stability at infinitesimal fluid velocities. By adding a linear, external 13 14 Main results of this article have been presented in the previous chapter. This method is explained and improved in Chapter 3. Free hanging riser – steady state vibrations 67 damping Kuiper and Metrikine (2005) showed that using the conventional boundary conditions the critical velocity corresponding to the instability onset is strongly influenced by this damping. Since the damping force employed by Kuiper and Metrikine (2005) was not based on experiments, no quantitative comparison of the critical velocity could be made with measurements. The aim of this chapter is twofold, namely to predict quantitatively (i) the onset of instability and (ii) the amplitude of the steady-state vibrations using physically realistic drag values. The onset of instability corresponds to infinitesimal deflections and velocities of the pipe and, therefore, to the laminar regime of the flow around the pipe. In this regime the drag is proportional to the pipe velocity. The proportionality coefficient is realistically quantified using experimental data available in literature. The steady-state vibrations of the pipe correspond to larger deflections and velocities of the pipe, which cause the flow to separate from the pipe. In this turbulent regime the drag has a quadratic dependence on the transverse velocity of the pipe. This dependence is also quantified in this chapter using experimental data. 5.2 Equation of motion and boundary conditions The same system as in Chapter 4 is considered, viz. an initially straight, submerged, fluid-conveying pipe of finite length as sketched in Fig. 5.1. Applying the same assumptions as summarized in Section 4.2 and disregarding the negative pressurization at the free inlet 15 , the equation of motion of a cantilever pipe conveying fluid can be written as (based on Eq. (4.7)): EI 2 ∂4w ∂ ⎛ ∂w ⎞ ∂2w 1 ∂w ∂2w 2 ∂ w − ⎜ Te ( z ) − 2m f u f + ρ f DoC d + M 2 = 0 , (5.1) ⎟ + mf uf 4 2 ∂z ∂z ⎝ ∂z ⎠ ∂z ∂z∂t 2 ∂t ∂t in which M = mr + m f + ma = ρ r Ar + ρ f Ai + ρ f Ca Ae , Te ( z ) = ( ρ r − ρ f ) Ar g ( L − z ) . In Eq. (5.1) the same symbols are used as in Chapter 4. 15 In the previous chapter it was proven that the negative pressurization at the free inlet has only a minor influence on the stability. 68 Chapter 5 z w(z,t) uf L Fig. 5.1 - Sketch of a submerged riser conveying fluid. The conventional boundary conditions at the ends of the submerged, suspended pipe are given as (Païdoussis, 1998; Païdoussis, 1999; Kuiper and Metrikine, 2005): w ( z, t ) = 0 and ∂ 2 w ( z, t ) =0 ∂z 2 and ∂w ( z, t ) ∂z =0 ∂ 3 w ( z, t ) =0 ∂z 3 at z = 0 , at z = L . (5.2) The boundary conditions at the lower end assume a zero bending moment and a zero shear force. In a recent paper, Païdoussis et al. (2005) postulated a number of new descriptions for the force balance at the free end. We highlight here only their basic description (5.3) and another interesting variant (5.4). Païdoussis et al. (2005) argued that the force balance at the free end should be described not by the last equation of Eq. (5.2), but by one of the following relationships: EI ∂ 3 w ( z, t ) ∂w ( z , t ) ⎞ ⎛ ∂w ( z, t ) − mf uf ⎜ − uf ⎟=0 3 ∂z ∂z ⎠ ⎝ ∂t EI ∂ 3 w ( z, t ) ∂w ( z , t ) − mf uf =0 3 ∂z ∂t at z = L , at z = L . (5.3) (5.4) To explain the physical differences between the descriptions, three sketches are shown in Fig. 5.2. Inside the pipe the internal fluid velocity uf, directed along the pipe, and the transverse velocity ∂w ∂t are indicated by regular arrows. The dotted arrow inside the pipe marks the resultant direction of the flow at the entrance. Obviously, the arrows inside the pipe are the same for all three sketches. The differences originate from the assumptions regarding the flow field just below the entrance. The dotted arrow below the pipe represents the resultant direction of the Free hanging riser – steady state vibrations 69 flow field just below the entrance. The last expression of Eq. (5.2) assumes the inflow direction be tangential to the deflected pipe and the inflow field to move with the same transverse velocity as the tip of the pipe. As a result, the conventional boundary condition (5.2) presumes that the direction of the flow remains unchanged as the fluid enters the pipe. Boundary conditions (5.3) and (5.4) assume that the flow field just below the entrance does not move transversely with the pipe, and hence, the corresponding transverse component is absent in these two sketches. This assumption results in a non-smooth flow field and, as a result, the momentum of the fluid changes as the fluid enters the pipe. The difference between boundary conditions (5.3) and (5.4) is that the former assumes a spatially averaged inflow direction tangential to the undeflected pipe, while the latter assumes the inflow direction tangential to the deflected pipe. Further to this description of the balance of transverse forces, Païdoussis and coworkers (2005) discussed several other variants accounting for various depressurization, tensioning effects and direction of inflow. However, in the mathematical formulation all their variants contain the term m f u f ( ∂w ∂t ) , which is absent in the conventional boundary condition. This term cancels the effect of the Coriolis force in the equation of motion, insofar as the calculation of the energy gain at the boundaries is concerned. Using their basic description, boundary condition (5.3), Païdoussis et al. (2005) explain that the pipe cannot become unstable at small fluid velocities since the total (over a period of vibrations) averaged amount of energy of the system remains constant. In this chapter only the two extreme boundary conditions for the balance of the transverse forces, (5.2) and (5.3), are addressed. In the next chapter the alternative description, Eq. (5.4), is considered as well. (5.2) (5.3) (5.4) uf uf uf ∂w ∂t ∂w ∂t ∂w ∂t Fig. 5.2 - Three sketches of inflow conditions corresponding to boundary conditions (5.2), (5.3) and (5.4). 70 Chapter 5 Introducing the following dimensionless variables16 and parameters: η = w L , ξ = z L , τ = t EI M L2 , α = Ar g ( ρ r − ρ f ) L3 EI , β = m f M , U = u f L m f EI , γ * = ρ f DoC d L2 (2 ) MEI , the equation of pipe motion, Eq. (5.1), is rewritten as 2 ∂ 4η ∂ ⎛ ∂η ⎞ ∂ 2η ∂ 2η ∂η 2 ∂ η α ξ β − 1 − − 2 U + U + +γ * =0. ( ) ⎜ ⎟ 4 2 2 ∂ξ ∂ξ ⎝ ∂ξ ⎠ ∂ξ∂τ ∂ξ ∂τ ∂τ (5.5) The four conventional boundary conditions (5.2) and the postulated basic description of the force balance at the free end (5.3) (Païdoussis et al., 2005) are written in dimensionless form as, respectively: η ( ξ ,τ ) = 0 ∂ 2η (ξ ,τ ) =0 ∂ξ 2 and and ∂η (ξ ,τ ) =0 ∂ξ at ξ = 0 , ∂ 3η (ξ ,τ ) =0 ∂ξ 3 ∂ 3η (ξ ,τ ) ∂η (ξ ,τ ) ⎞ ⎛ ∂η (ξ ,τ ) −U ⎜ β −U ⎟=0 3 ∂ξ ∂τ ∂ξ ⎠ ⎝ at ξ = 1 , at ξ = 1 . (5.6) (5.7) 5.3 Stability of a pipe conveying fluid in the linear approximation The dynamic stability of a linear system is determined by its eigenfrequencies. To find these, the displacement of the pipe can be sought for in the following form: η (ξ ,τ ) = W (ξ ) e iωτ , (5.8) where ω is the dimensionless, generally complex frequency. In accordance with Eq. (5.8), the system is stable if the imaginary part of all eigenfrequencies is positive, and unstable if at least one eigenfrequency has a negative imaginary part. An Argand diagram is used to analyse the eigenfrequencies (see Chapters 2 and 4). In this diagram the real and imaginary parts of the natural frequencies are plotted parametrically, as they depend on some system parameters. In this section the dimensionless flow velocity and the coefficient γ * of the hydrodynamic drag are employed as such parameters. 16 To be consistent with most literature about pipes conveying fluid, the dimensionless internal fluid velocity is defined differently than in Chapter 4. Free hanging riser – steady state vibrations 71 U = 2.5, γ*= 12.0 Im(ω) U = 0.0, γ*= 12.0 U = 0.0, γ*= 12.0 U = 0.0, γ*= 0.0 0 U = 0.0, γ*= 0.0 U = 1.59, γ*= 12.0 0 Re(ω) Fig.5.3 - Argand diagram with the first two loci using the parameters of Table 5.1 and the newly proposed boundary condition, Eq.(5.7). Pipe becomes unstable by divergence at U=1.59. For the equation of motion of the free-hanging pipe, Eq. (5.5), with the conventional boundary conditions (5.6), the analysis and the results have been thoroughly discussed in the previous chapter. It was found that the stability of the pipe in this case depends strongly on the hydrodynamic drag. In the absence of the linearized drag the system becomes unstable at infinitesimal fluid velocity. By increasing the adapted drag coefficient γ * , the fluid velocity for which the transition occurs from stable to unstable behaviour is increased significantly. The pipe becomes unstable by flutter (stable focus to unstable focus bifurcation). A completely different scenario occurs for the equation of motion of the freehanging pipe, Eq. (5.5), in combination with the new boundary condition (5.7) (Païdoussis et al., 2005). The corresponding Argand diagram is shown in Fig. 5.3. Only the first two eigenfrequencies are plotted using the parameters given in Table 5.1. These parameters simulate a flexible, plastic pipe of 2.0 m length fully submerged in water. As a first step the adapted drag coefficient is gradually increased from zero while keeping the fluid velocity zero. The accompanying complex values of the natural frequencies are computed and plotted in the Argand diagram (see Fig. 5.3). As expected, all complex natural frequencies acquire positive imaginary part, implying that the pipe is stable. Once the adapted drag coefficient reaches the value for which the stability is studied (corresponds to C d = 1.0 m/s ), the first natural frequency has a pure imaginary value. 72 Chapter 5 Table 5.1 – Parameters used as base case Parameter E 1.00·109 N·m-2 ρf 1.00·103 kg·m-3 L 2.00 m ρr 1.20·103 kg·m-3 Do 3.00·10-2 m Ca 1.00 ν 1.00·10-6 m2·s-1 Di -2 2.60·10 m After gradually increasing the fluid velocity, the imaginary part of the first natural frequency decreases. At a certain velocity the path of the first natural frequency intersects first with the real ω − axis . Since this path crosses the real axis at the origin of the frequency plane the pipe becomes unstable by divergence (stable node to saddle bifurcation). For a system which becomes unstable by divergence the amount of hydrodynamic drag has no influence on the critical fluid velocity. This means that even in the absence of drag a relatively high fluid velocity should be achieved to get the pipe unstable. For the parameters in Table 5.1 the dimensionless critical velocity is equal to U crit = 1.59 , corresponding to u f ,crit = 4.53 m/s . For the cantilever pipe, divergence can only occur if the spatially averaged fluid inflow direction is tangential to the undeflected pipe. However, as noted in the introduction, the inflow direction at the free end is probably tangential neither to the deflected nor to the undeflected pipe. This means that divergence type of instability is unlikely and, therefore, the amount of hydrodynamic drag is of crucial importance for stability predictions. To assess the onset of instability and the amplitude of the pipe motion after the onset, more realistic drag description based on experiments should be incorporated into the model. 5.4 Hydrodynamic drag on flexible cylinders 5.4.1 Available experimental data In order to obtain reliable predictions of the dynamic behaviour of submerged pipes, a drag description based on previous experiments is used. For measuring hydrodynamic drag acting on a pipe section in still water, a straight rigid cylinder is used conventionally. In literature, three test set-ups have been described: • A water tank where the cylinder is subjected to forced oscillations at a constant frequency (e.g. Chaplin and Subbiah., 1998), • A water tank where a cylinder undergoes decaying oscillations following an initial displacement (e.g. Bearman and Russell, 1996), Free hanging riser – steady state vibrations • 73 A U-shaped water tunnel where the flow oscillates about a cylinder at a constant frequency (e.g. Sarpkaya, 1986). The drag on a rigid cylinder oscillating in still water is characterised by the Stokes parameter β and the Keulegan-Carpenter number KC defined as, respectively, β= Do 2 νT and KC = 2π xˆ , Do (5.9) where x̂ is the amplitude of the cylinder displacement, T is the period of oscillation and ν is the kinematic viscosity of the water. At low values of KC ( KC < 0.1) , i.e. small displacement-to-diameter ratios, the flow remains predominantly attached to the cylinder, and the drag is due to viscous forces in a thin boundary layer attached to the body. For a circular cylinder oscillating harmonically at small amplitudes in water otherwise at rest, the damping is conveniently expressed as proposed by Stokes (1851) and later extended by Wang (1968). For large values of β ( β > 105 ) , this damping is approximately proportional to the velocity of the cylinder, dx dt . Its value per unit length is given as: FStokes = 2π 3 2 ρ f ν β ρ D ν dx dx = 2π 3 2 f o , dt dt T (5.10) At larger values of KC ( KC > 0.1) , the flow separates from the pipe, forming a turbulent wake behind the body. In this turbulent regime the drag has a quadratic dependence on the transverse velocity of the pipe. In this case the instantaneous force acting on the cylinder per unit length is normally expressed as the often-used Morison damping FMorison = 1 dx dx ρ f DoCd . 2 dt dt (5.11) Conventionally, even the linear Stokes damping is expressed in terms of a quadratic dependence on velocity, although the instantaneous force is not proportional to the square of the instantaneous velocity, as the use of the Morison expression normally suggests. In order to do this, the nonlinear drag force is rewritten as a Fourier series, under the assumption that the velocity has a sinusoidal character, 1 1 1 2 2⎛ 8 ⎞ ρ f DoCd ( xˆω ) cos (ωt ) cos (ωt ) ≈ ρ f DoCd ( xˆω ) ⎜ cos (ωt ) + cos ( 2ωt ) + .... ⎟ , 2 2 2 ⎝ 3π ⎠ where ω = 2π T . Taking only the first Fourier term into account, i.e. neglecting the higher harmonics, the hydrodynamic drag coefficient Cd of the Morison damping at low KC is then given by 74 Chapter 5 Cd = 3π 3 1 , 2 π KC β (5.12) The inverse dependence of Cd on KC compensates for expressing a linear drag through a quadratic dependence of velocity. Experimental studies of hydrodynamic drag on a cylinder oscillating in still water at small amplitudes and large Stokes parameters ( β > 6 ⋅ 104 ) have been carried out among others by Bearman and Russell (1996), Chaplin and Subbiah (1998) and Chaplin (2000). These experiments for low Keulegan-Carpenter numbers show a near doubling of the Cd value in comparison with the value predicted by the StokesWang theoretical analysis, Eq. (5.12). Sarpkaya (2001) attempted to find a reason for the deviation of the measured values from the Stokes-Wang laminar flow theory, however he found no good explanation for the doubling of the drag. In the turbulent regime, in which the displacement amplitudes are relatively large, KC > 0.1 , Bearman and Russell (1996) argue that a contribution to Cd of 0.08KC can be expected. They proposed a semi-empirical formula for the total drag coefficient of the form Cd = ζ1 KC β + ζ 2 KC = ζ1 Re KC + ζ 2 KC (5.13) with ζ 1 = 55 and ζ 2 = 0.08 . Especially for very small KC numbers this formula predicts the drag coefficient in good agreement with the experiments. Expression (5.13) is only valid for smooth cylinders and is measured in the abovementioned experiments to a KC-number of 0.3. For smaller values of the Stokes parameter ( β < 3 ⋅104 ) experiments have been conducted for higher Keulegan-Carpenter numbers up to 12 (Justesen, 1988; Rodenbusch, 1986). Although the same trend for Cd is observed in all the results, there is an appreciable scatter in the data for the higher Keulegan-Carpenter numbers. Sarpkaya (2001) observed that with increasing Keulegan-Carpenter number regular (Honji-instabilities) and irregular mushroomshaped vortices arise first, followed by turbulence, separation and vortex shedding. Due to the stochastic behaviour of vortices a large scatter was observed in the experimental results for larger KC-values. Nevertheless, we assume that formula (5.13) is valid for Keulegan-Carpenter numbers up to 8. The semi-empirical formula, Eq. (5.13), has been plotted in Fig. 5.4 for two values of the Stokes parameter. For rough cylinders the drag coefficient should be increased, depending on the roughness (see e.g., Bearman and Mackwood, 1992; Chaplin and Subbiah, 1998). Free hanging riser – steady state vibrations 75 2.5 2 Cd 1.5 1 0.5 0 0 2 4 6 8 KC Fig. 5.4 - Drag coefficient according to Eq. (5.13) versus Keulegan-Carpenter number for β =670.000 (regular line) and β =1.277.000 (bold line). 5.4.2 Experimental drag force applicable to flexible pipes In order to use Eq. (5.13) for prediction of the dynamic behaviour of a flexible pipe in the time domain, an iterative procedure is required, because through KC the drag depends on the unknown displacement amplitude and period. Such procedure is straightforward if the pipe vibrates in a single-mode regime. However, flexible pipes often exhibit multi-mode vibrations. In this case, the motion of each section of the pipe is approximately decomposable into a number of harmonic vibrations that have different amplitudes. This makes the iterative procedure cumbersome and not necessarily convergent. Therefore, in this chapter, an alternative approach is proposed for description of the drag coefficient. Instead of using the frequency domain representation of the drag coefficient, Eq. (5.13), a time domain expression for the drag is proposed. Using the time domain expression, vibrations of a flexible pipe can be studied by any conventional time domain method (FEM, finite difference, Galerkin Method, etc.) without employing an iterative procedure. The idea is to reconstruct a time domain drag coefficient on the basis of Eq. (5.13), which should result in a similar dynamic pipe response. A rigid, elastically mounted cylinder is externally excited at the resonance frequency ωe . The resonance excitation is assumed because it is most sensitive to damping. The predictions obtained using the frequency and time domain descriptions are compared to each 76 Chapter 5 other requiring that the displacement amplitudes are the same for the same external force. The following equation of motion describes oscillations of this cylinder in water provided that the Morison’s formula is applicable: d 2 x ρ f DlCd dx dx F + + ωe 2 x = sin (ωe t ) , dt 2 2m dt dt m (5.14) where x is the displacement of the cylinder, l the length of the rigid cylinder, m the mass of the cylinder plus the added mass and F the amplitude of the external force. The Stokes parameter β and the resonance frequency ωe are related to each other, as indicated by Eq. (5.9): β= Do2ωe . 2πυ (5.15) Owing to the oscillatory force, the cylinder reaches a periodic motion, and its amplitude can be characterized by the Keulegan-Carpenter number. Starting with the frequency domain description of the drag coefficient, Eq. (5.13), the drag force reads: Fdrag = ρ f Do l ⎛ ζ 1 ν T 2π ⎞ dx dx +ζ2 . xˆ ⎟ ⎜ Do ⎠ dt dt 2 ⎝ 2π xˆ (5.16) Using this drag force, the steady state solution can be computed (using an iterative procedure) for a given external oscillatory force. This computation has been repeated for 20 amplitudes of the external force, covering the range of KeuleganCarpenter numbers of our interest ( 0.01 < KC < 8 ) . Besides, three natural frequencies are considered, by using different values for the Stokes parameter ( β1 = 6.700 ⋅105 , β 2 = 1.025 ⋅106 , β3 = 1.277 ⋅106 ) according to Chaplin’s experiments (2000). The time-domain description of the drag coefficient should result in the same dynamic response of the rigid cylinder as it is loaded by the same force. As explained before, the hydrodynamic damping for small displacements ( KC < 0.1) is caused by viscous effects apparent in a thin boundary layer attached to the pipe. This force is proportional to the velocity of the pipe. For larger displacements and velocities the flow separates from the pipe, forming a turbulent wake behind the body and the drag is proportional to the square of the pipe velocity. A superposition of these two contributions is proposed for the description of the drag force in the time-domain: ⎛ µ dx 1 dx dx ⎞ Fdrag ,new = Do l ⎜ A1 + ρ f A2 ⎟, dt dt ⎠ ⎝ Do dt 2 (5.17) Free hanging riser – steady state vibrations 77 where A1 and A2 are two unknown dimensionless constants and µ is the dynamic viscosity of the fluid (ν = µ ρ f ) . According to Eq. (5.13) the drag coefficient increases with amplitude (for KC < 8 ), while Eq. (5.17) has a constant coefficient in front of the velocity squared term. This suggests a cubic velocity dependence to capture this difference. However, it is not easy to get the dimensions of an additional cubic term correctly. Nor is it straightforward to link this term to the underlying physics. Speculating about this, we think that the cubic term should depend on the vortex shedding frequency. Unfortunately, the current available experiments do not give information about this. Therefore, we do not pursue this cubic velocity dependence of the drag. Note that this cubic term does not have an effect on the onset of instability, but only on the steady-state amplitude. To find the unknown constants A1 and A2 , Eq. (5.14) is rewritten using Fdrag,new: ⎛ µ d 2x dx 1 dx dx ⎞ F 2 A1 + ρ f A2 + Do l ⎜ ⎟ + ωe x = sin (ωe t ) . 2 dt m dt dt ⎠ ⎝ Do dt 2 (5.18) The steady-state solution of this equation is computed for the same 20 amplitudes and 3 frequencies of the external force as in the previous case. The two unknowns in Eq. (5.17), A1 and A2, should be chosen such that the differences between the results for the dynamic response of the cylinder corresponding to the descriptions of the drag coefficient in the frequency domain and in the time domain are minimal in some sense. To achieve this, a procedure was applied to minimize the difference in KC-value (for each given F and β ) in the steady state motion for both drag descriptions by varying the unknowns A1 and A2. It was found that the smallest error in KC is obtained by choosing A1 = 27 ⋅103 and A2 = 0.24 . The KC-F dependence obtained with these values of A1 and A2 is shown in Fig. 5.5. The same parameters as in the experiments performed by Chaplin (2000) were used: m = 4.08 ⋅ 103 kg , Do = 0.75 m , l = 4.29 m , ρ f = 1.00 ⋅ 103 kg ⋅ m -3 , ν = 1.14 ⋅ 10−6 m 2 ⋅ s-1 . 78 Chapter 5 10 β = 670,000 5 2 1 0.5 β = 1,025,000 KC [-] 0.2 0.1 0.05 0.02 0.01 β = 1,277,000 0.005 0.002 0.001 0.1 1 10 100 1000 10000 100000 F [N] Fig. 5.5 - Keulegan-Carpenter number versus the amplitude of the external force for different Stokes parameters. Bold line: Eq. (5.16); regular line Eq. (5.17). 0.4 0.3 0.2 x [m] 0.1 0 -0.1 -0.2 -0.3 -0.4 0 2 4 6 8 10 t [s] Fig 5.6 - Time series of a damped cylinder displacement subjected to an external oscillatory force using different drag coefficients, based on Eq. (5.16) (bold line) and based on Eq. (5.17) (regular line). Free hanging riser – steady state vibrations 79 The Keulegan-Carpenter number in the steady state regime is plotted in Fig. 5.5 as a function of the amplitude of the external oscillatory force. The bold lines correspond to the drag according to Eq. (5.16), while the regular lines correspond to the new description of the drag, Eq. (5.17). As an example, the displacement in the time domain of a cylinder subject to the external oscillatory force with F = 1.00 ⋅ 105 N and β = 1.277 ⋅ 106 is given in Fig. 5.6. In this figure the two lines are plotted using the two different descriptions of the hydrodynamic drag. As can be concluded from Figs. 5.5 and 5.6, the time domain description can be considered to reproduce the hydrodynamic drag in reasonable accordance with the experiments in the range of KC up to 8. In Fig. 5.5 a difference in slope is observed for values of KC above 0.2. The reason for this deviation is that in expression (5.16) Cd has a cubic dependence on KC while in expression (5.17) this dependence is quadratic. As noted, this difference does not affect the calculated onset of instability. 5.5 Stability of a pipe conveying fluid with nonlinear drag In this section the influence of the hydrodynamic drag on stability of a free-hanging pipe conveying fluid is investigated by using a nonlinear, time-domain description of the hydrodynamic drag, as derived in the previous section. At the free end the conventional boundary conditions are used, Eq. (5.6). Only for these boundary conditions does the hydrodynamic drag have a significant effect on the onset of instability of the pipe. For the boundary conditions proposed by Païdoussis et al. (2005), Eq. (5.7), the results of section 5.3 are applicable for prediction of the onset, as this is not affected by the hydrodynamic drag. The equation of motion of the submerged pipe subject to the nonlinear hydrodynamic drag reads EI 2 ∂4w ∂ ⎛ ∂w ⎞ ∂2w 2 ∂ w T z m u m u − − 2 + + ( ) f f f f ⎜ e ⎟ ∂z 4 ∂z ⎝ ∂z ⎠ ∂z∂t ∂z 2 ⎛ µ ∂2w ∂w 1 ∂w ∂w ⎞ M 2 + Do ⎜ A1 + ρ f A2 ⎟ = 0. ∂t ∂t 2 ∂t ∂t ⎠ ⎝ Do (5.19) Using dimensionless variables and parameters η = w L , ξ = z L , τ = t EI M L2 , α = Ar g ( ρ r − ρ f ) L3 EI , β = m f M , U = u f L m f EI , ς = µ L2 MEI , γ = ρ f Do L ( 2 M ) , the equation of motion, Eq. (5.19), is rewritten as ∂ 4η ∂ ⎛ ∂η ⎞ ∂ 2η ∂ 2η ∂ 2η ∂η ∂η ∂η −α + U 2 2 + 2 +ς A1 + γ A2 = 0. ⎜ (1 − ξ ) ⎟ − 2βU 4 ∂ξ ∂ξ ⎝ ∂ξ ⎠ ∂ξ∂τ ∂ξ ∂τ ∂τ ∂τ ∂τ (5.20) 80 Chapter 5 Note that the critical fluid velocity does not depend on the nonlinear term in Eq. (5.20). Therefore, the critical velocities predicted by Eq. (5.5) and Eq. (5.20) may differ only due to a difference between γ * and ς A1 . The advantage of using ς A1 is that the numerical value of this coefficient is based on experimental data. 5.5.1 Galerkin method Different methods (FEM, finite difference, etc.) are applicable for solving nonlinear partial differential equation (5.20). In this chapter, we have chosen the Galerkin method, since it provides better insight in the physics of the dynamic stability of the pipe than the other time domain numerical methods. The Galerkin procedure is used to approximate the nonlinear partial differential equation by a finite set of coupled ordinary differential equations, with the solution expressed as ∞ η (ξ ,τ ) = ∑ φm (ξ ) qm (τ ) , (5.21) m =1 where qm (τ ) are unknown time-dependent functions and φm (ξ ) are spacedependent shape functions forming a complete set. The φm (ξ ) are chosen as the eigenfunctions of the following operator, Lξ = d4 d ⎛ d ⎞ −α ⎜ (1 − ξ ) ⎟ , 4 dξ dξ ⎝ dξ ⎠ (5.22) satisfying the conventional boundary conditions, Eq. (5.6). The corresponding eigenvalues λm are all imaginary and in the following are represented by the realvalued eigenfrequencies Ω m = iλm . The functions φm (ξ ) form a complete orthogonal set. The orthogonality condition is used to obtain the functions qn (τ ) . Substituting Eq. (5.21) into Eq. (5.20), multiplying through by φn , and integrating over ξ ∈ ( 0,1) , it is found that ∞ ∞ ∞ ∞ d 2 qn dq dq dq dqm − 2 β U ∑ Bmn m + Ω n 2 qn + U 2 ∑ Cmn qm + ς A1 n + γ A2 ∑∑ Dlmn l = 0 (5.23) 2 dτ d d dτ dτ τ τ m =1 m =1 m =1 l =1 where d φm φn d ξ dξ 0 1 Bmn = ∫ 1 d 2φm φn d ξ dξ 2 0 1 2 ∫ φn dξ , Cmn = ∫ 0 1 1 2 ∫ φn dξ , Dlmn = ∫ φl φmφn dξ 0 0 1 ∫φ n 0 2 dξ . Free hanging riser – steady state vibrations 81 5.5.2 One-mode approximation First, the results of the one-mode Galerkin approximation are shown (only the first mode is accounted for). This approximation is presented to explain the basic instability mechanisms of a cantilever pipe conveying fluid. In the next two sections higher modes are taken into account to achieve higher accuracy. For the one-mode Galerkin approximation (n=1) Eq. (5.23) can be written as two coupled first-order differential equations: 0 ⎧ q1 ⎫ ⎡ ⎨ ⎬=⎢ 2 2 ⎩ p 1 ⎭ ⎣ −Ω1 − U C11 1 ⎤ ⎧ q1 ⎫ ⎨ ⎬ + f ( p1 ) , 2 β UB11 − A1ς ⎥⎦ ⎩ p1 ⎭ (5.24) where f ( p1 ) is a vector containing the nonlinear terms. This system has only one equilibrium point given by q1 = p1 = 0 . The type of this equilibrium is found by linearizing Eq. (5.24) in the vicinity of this point. The complex eigenvalues λ are shown in the Argand diagram in Fig.5.7 for varying flow velocity U. Note that the axes of this diagram represent the real and imaginary values of the eigenvalues λ instead of the eigenfrequencies ω , which was the case in Section 5.3 ( λ = iω ). This change is made since it is more common in the field of nonlinear dynamics to work with eigenvalues. 12 Ucrit=8.74 8 Im(λ) 4 0 U=0 U=0 -4 -8 Ucrit=8.74 -12 -25 -20 -15 -10 -5 0 5 Re(λ) Fig. 5.7 - Argand diagram of the complex eigenvalues for a one-mode Galerkin approximation. The numerical value of U crit is based on the parameters of Table 5.1. 82 Chapter 5 The criterion for the instability to occur is that at least one of the eigenvalues has a positive real part. The one-mode system loses stability through the Hopf bifurcation, i.e. stable focus to unstable focus bifurcation, at a critical fluid velocity of U crit = A1ς . 2 β B11 (5.25) Using the base case parameters as shown in Table 5.1, the dimensionless critical velocity of the simulated pipe is found as U crit = 8.74 . It is interesting to investigate which terms in the equation of motion (5.20) are responsible for the energy input. At the critical velocity, the energy of the system considered over one period remains constant since both eigenvalues λ are imaginary. For the energy analysis the equation of motion in this case can be written as: η (ξ ,τ ) = φ1 (ξ ) q1 (τ ) = φ1 (ξ ) qˆ1 cos (ω1τ ) , (5.26) where q̂1 is the amplitude of the harmonic motion and ω1 = Ω12 + U crit 2C11 . The energy variation over one period of the motion is caused by the centrifugal force, the Coriolis force and the linear part of the hydrodynamic drag. The contributions of these forces to the energy loss read: T 1 ∆ECent . = ∫ ∫ FCent . 0 0 T 1 ∆ECor . = ∫ ∫ FCor. 0 0 d 2φ1 dq ∂η φ q 1 d ξ dτ = 0 , d ξ dτ = U crit 2 ∫ ∫ 2 1 1 dξ dτ ∂τ 0 0 T 1 2 (5.27) ∂η d φ ⎛ dq ⎞ dφ d ξ dτ = −2 β U crit ∫ ∫ 1 φ1 ⎜ 1 ⎟ d ξ dτ = −2 β U crit q12πω ∫ 1 φ1d ξ ,(5.28) d d dξ ξ τ ∂τ ⎝ ⎠ 0 0 0 T 1 T 1 ∆E DragL. = ∫ ∫ FDragL. 0 0 1 2 ∂η ⎛ dq ⎞ d ξ dτ = ς A1 ∫ ∫ φ12 ⎜ 1 ⎟ d ξ dτ = ς A1q12πω ∫ φ12 d ξ . ∂τ ⎝ dτ ⎠ 0 0 0 T 1 1 (5.29) The remaining terms in the equation of motion, Eq. (5.20), do not contribute to the energy variation over one period. As can be seen from Eq. (5.27) the centrifugal force does not contribute to the energy variation in the first mode approximation. By substituting Eq. (5.25) into Eq. (5.28), it can be seen that the energy loss due to the linearized hydrodynamic drag (5.29) is exactly equal to the energy gain due to the Coriolis force (5.28) at the critical velocity. In order to investigate the behaviour of the cantilever pipe conveying fluid after the onset of flutter, the nonlinear terms, as indicated in Eq. (5.24), have to be taken into account. The phase-plane for this system is shown in Fig. 5.8 for a fluid velocity exceeding the critical one, Eq. (5.25). One can see that the motion grows from sensibly zero to a steady oscillation of finite amplitude, i.e. to a stable limit cycle. Free hanging riser – steady state vibrations 83 0.3 0.2 q1 0.1 • 0 -0.1 -0.2 -0.3 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 q1 Fig. 5.8 - Phase-plane for a fluid velocity exceeding the critical fluid velocity, showing a stable limit cycle. 5.5.3 Two-mode approximation A key difference between a one-mode and a two-mode Galerkin approximation is that in the latter case the possibility of mode interaction is accounted for. For the two-mode Galerkin approximation (n=2) Eq. (5.23) can be rewritten in a system of four coupled first-order ODEs: 0 ⎧ q1 ⎫ ⎡ ⎪ p ⎪ ⎢ −Ω 2 − U 2C ⎪ 1⎪ ⎢ 1 11 ⎨ ⎬= ⎢ 0 q ⎪ 2⎪ ⎪⎩ p 2 ⎪⎭ ⎢⎣ −U 2C12 1 2 β UB11 − A1ς 0 2 β UB12 0 −U 2C21 0 2 −Ω 2 − U 2C22 0 ⎤ ⎧ q1 ⎫ ⎥ ⎪p ⎪ 2 β UB21 ⎥ ⎪⎨ 1 ⎪⎬ + f ( p1 , p2 ) (5.30) ⎥ ⎪ q2 ⎪ 1 ⎥ 2 β UB22 − A1ς ⎦ ⎪⎩ p2 ⎪⎭ where f ( p1 , p2 ) is a vector containing the nonlinear terms. The complex eigenvalues of the system linearized about the equilibrium (there is only one, in the origin of the phase space) are plotted in Fig. 5.9 for increasing values of U. Similar to the one-mode approximation, the instability occurs through the Hopf bifurcation, leading to flutter. Using again the base case parameters (Table 1), the critical velocity for the two-mode approximation is equal to U crit = 4.45 , which is approximately half of the value for the one-mode approximation. 84 Chapter 5 To understand this significant reduction in critical flow velocity, an energy analysis is performed for the two-mode approximation. At the critical velocity the solution of the equation of motion, using the two-mode approximation, can be written as: 2 η (ξ ,τ ) = ∑ φm (ξ ) qm (τ ) = φ1 (ξ ) qˆ1 cos (ω2τ ) + φ2 (ξ ) qˆ2 cos (ω2τ − α ) (5.31) m =1 where ω2 is the dimensionless real frequency with which modes 1 and 2 vibrate and α is the phase lag between mode 1 and 2. In principle, the time-dependent functions q1 and q2 are a summation of four exponential functions with complex arguments (Fig. 5.9). However, only the pair of pure imaginary eigenvalues is taken into account, since the contribution of the other two complex eigenvalues vanishes quickly with time. The pair of pure imaginary eigenvalues results in one real frequency for the time-dependent functions. Using the two-mode solution, Eq. (5.31), the contributions to the energy loss over one period given by the centrifugal force, the Coriolis force and the linear part of the hydrodynamic drag are now equal to, respectively: T 1 ∆ECent . = ∫ ∫ FCent . 0 0 T 1 ⎛ d 2φ d 2φ2 ⎞ ⎛ dq1 dq ⎞ ∂η + φ2 2 ⎟ d ξ dτ d ξ dτ = U crit 2 ∫ ∫ ⎜ 21 q1 + q2 ⎟ ⎜ φ1 2 ∂τ ξ ξ τ d d d dτ ⎠ ⎠⎝ 0 0⎝ ⎛ d 2φ d 2φ ⎞ π qˆ1qˆ2 sin (α ) ∫ ⎜ 21 φ2 − 22 φ1 ⎟ d ξ , dξ dξ ⎠ 0⎝ 1 = U crit T 1 ∆ECor . = ∫ ∫ FCor . 0 0 (5.32) 2 ⎛ d φ dq d φ dq ⎞ ⎛ dq dq ⎞ ∂η d ξ dτ = −2 β U crit ∫ ∫ ⎜ 1 1 + 2 2 ⎟ ⎜ φ1 1 + φ2 2 ⎟ d ξ dτ ∂τ d ξ dτ d ξ dτ ⎠ ⎝ dτ dτ ⎠ 0 0⎝ T 1 ⎛ dφ ⎞ ⎛ dφ dφ ⎞ dφ = −2 β U critπω ∫ ⎜ 1 φ1qˆ12 + ⎜ 1 φ2 + 2 φ1 ⎟ qˆ1qˆ2 cos (α ) + 2 φ2 qˆ2 2 ⎟ d ξ , dξ dξ ⎠ dξ ⎝ dξ ⎠ 0⎝ 1 T 1 ∆E DragL. = ∫ ∫ FDragL. 0 0 2 ∂η dq ⎞ ⎛ dq d ξ dτ = A1ς ∫ ∫ ⎜ φ1 1 + φ2 2 ⎟ d ξ dτ ∂τ dτ dτ ⎠ 0 0⎝ T 1 1 = A1ςπω ∫ (φ q + φ2 q2 ) d ξ . 2 2 1 1 2 (5.33) (5.34) 2 0 Using Eq. (5.22), it can be shown that the remaining terms in the equation of motion, Eq. (5.20), do not contribute to the energy variation over one period. In contrast to the one-mode approximation, the centrifugal force contributes to the energy variation in the two-mode approximation. Whether this force leads to energy loss or energy gain depends on the value of the phase shift α between the two modes. Using the parameters in Table 5.1, the energy expression for the centrifugal force, Eq. (5.32), can be found to be negative, implying an energy gain. The energy gain caused by the centrifugal force is of the same order as caused by the Coriolis force. Free hanging riser – steady state vibrations 85 20 15 10 Ucrit=4.45 Im(λ) 5 0 -5 Ucrit=4.45 -10 -15 -20 -25 -20 -15 -10 -5 0 5 Re(λ) Fig. 5.9 - Argand diagram of the complex eigenvalues for a two-mode Galerkin decomposition. The additional contribution of the centrifugal force results in a reduction of the critical velocity with almost a factor two with respect to the one-mode approximation. Using the two-mode approximation, the behaviour of the cantilever pipe after the onset of flutter has been analysed by incorporating the full nonlinear equation, Eq. (5.30). As expected, for a fluid velocity exceeding the critical one, a stable limit cycle arises in the four-dimensional phase space (not shown). 5.5.4 Multi-mode approximation In order to achieve a sufficient accuracy with the Galerkin method more than two modes should be taken into account. It appears that for this system twelve modes are sufficient, since by incorporating more modes the response of the pipe hardly changes. For the pipe of 2 m length and the parameters given in Table 5.1 the dimensionless critical velocity is 4.40, which is only slightly less than computed with the two-mode approximation. The twelve-mode approximation is compared with a numerical method employing a central, explicit finite difference scheme. This comparison is shown in Fig. 5.10. The displacement of the tip of the pipe is plotted in the time domain after the system is released from an initial deflection for 86 Chapter 5 an internal fluid velocity just exceeding the critical one. Figure 5.10 shows good correspondence between predictions of the limit state by the two methods. The differences in the transient regime are caused by slightly different initial conditions used in the computations. After analyzing the one-, two- and twelve-mode approximations it can be concluded that: • A one-mode approximation describes the physical behaviour of the system incorrectly since in this approximation the centrifugal force does not contribute to the energy variation in the system. • A two-mode approximation describes the dynamic behaviour of the system qualitatively correctly. Even in a quantitative sense this is a reasonable approximation. • Both the centrifugal and Coriolis forces contribute to the energy gain of the free hanging pipe conveying fluid. 0.2 η 0.1 0 -0.1 -0.2 0 5 10 15 20 25 τ Fig 5.10 - Comparison of tip deflections using 12 mode approximation (regular line) and finite difference method (bold line). Free hanging riser – steady state vibrations 87 5.6. Conclusion Experiments with cantilever pipes aspirating water up to now never showed any instability. Using the new tip boundary conditions proposed by Païdoussis et al. (2005), Eq. (5.3), the simulated pipe of 2 m length is predicted to become unstable by divergence at a dimensionless flow velocity U crit = 1.59 . This corresponds to a fluid velocity of u f ,crit = 4.53 m/s . It is far from easy to achieve such high internal fluid speeds in a pipe of 0.03 m diameter, and hence, it is not surprising that the experiments did not show any instability. Using the conventional boundary condition, which assumes that the momentum of the fluid does not change direction as the fluid enters the pipe, in combination with a realistic description of the hydrodynamic drag, the cantilever pipe is predicted to become unstable at U crit = 4.40 , corresponding to a fluid speed of u f ,crit = 12.6 m/s . It is impossible to achieve such high internal fluid speeds in a 0.03 m diameter pipe. Hence, in our view theory and experiments do not show any contradiction. Probably, the truth will not be described by just one of the two discussed theories. We expect that an intermediate approximation is more likely. Both the correct description of the flow field in the vicinity of the tip and the external hydrodynamic drag will be of great importance for the prediction of the stability of the pipe conveying fluid. For this reason we initiated new experiments using longer pipes, so that the critical velocity becomes attainable. The results of these new experiments are described in the next chapter. CHAPTER 6 Experimental investigation of a cantilever pipe conveying fluid As reported in the previous chapters, there is a contradiction between theoretical predictions and experiments as to the dynamic stability of a free hanging pipe aspirating fluid. Experiments conducted in the past did not show any instability, while theory predicts instability beyond a critical fluid velocity. All existing theories predict instability, either oscillatory or static. A new test set-up was designed to investigate the hypothesis that previous experimental set-ups could not allow observations of pipe instability or the pipe aspirating water is unconditionally stable. In this new test set-up, the fluid velocity could exceed the theoretically predicted critical velocities. A cantilever pipe of about 5 m length was partly submerged in water. The free open end of the pipe was in the water, whereas the fixed end was above the waterline. The experiments clearly showed that the cantilever pipe aspirating water is unstable beyond a critical velocity of water convection through the pipe. Below this velocity the pipe is stable, whereas above it the pipe shows a complex motion that consists of two alternating phases. The first phase is a nearly periodic orbital motion with the amplitude of a few pipe diameters, whereas the second one is a noise-like vibration with very small deflections. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion, but does not change the pipe motion qualitatively. The main results of this chapter have been published in Journal of Fluids and Structures (Kuiper and Metrikine, 2008). 90 Chapter 6 6.1 Introduction As explained in the previous chapters, there is no consensus among researchers as to the mechanism of instability and the magnitude of the fluid velocity beyond which the instability occurs. 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. In contrast to this, the early theoretical models, neglecting energy loss on the pipe motion in a surrounding fluid, predicted instability at infinitesimal fluid velocities (Païdoussis and Luu, 1985). In attempts to explain this contradiction, several papers appeared recently, improving the theoretical modelling of a cantilever pipe aspirating fluid: 1. Païdoussis (1999) suggested an improved description of the negative pressurisation of water at the inlet of the pipe. However, this appeared to have a small effect on the stability of the cantilever pipe. 2. Kuiper and Metrikine (2005) and Kuiper et al. (2007) showed that the external hydrodynamic drag is a major stabilizing factor, which is likely to prohibit the pipe instability at the speeds reached in the experiments. 3. Païdoussis et al. (2005) postulated several new descriptions of the boundary condition for the balance of transverse forces at the free end of the pipe. The basic description of this boundary condition in this paper assumes that the inflow remains nearly tangential to the undeflected pipe. Using this boundary condition, the pipe is predicted to lose stability by divergence at a non-zero fluid speed. Despite some differences, all improved theories still suggest that the pipe should become unstable, in either oscillatory or static manner, at a certain fluid speed. Thus, it remained unclear whether the previous experimental set-ups did not allow to observe the instability, or whether the pipe aspirating water is unconditionally stable and all existing theories have to be improved further. No instability was observed in any experiment we are aware of. In the experiment of Hongwu and Junji (1996) a 1 m length pipe with an internal diameter of 8.2 mm was used. The velocity of fluid flow through the pipe was relatively low due to the small diameter of the pipe (high internal wall friction) and the capacity of the pump. For the maximum attainable flow rate, no flutter was observed. In the book of Païdoussis (1998) it is mentioned that the author conducted small-scale experiments with an elastomer pipe; however, exact dimensions of the test set-up are not given. Flutter was not observed at small flow velocities. After increasing the flow rate, local shell-type collapse of the pipe near the support was observed owing to the underpressure in the pipe. Hence, in all experiments known no sign of global pipe instability was observed. Experimental investigation of a cantilever pipe conveying fluid 91 The aim of this chapter is to investigate whether the instability of a pipe aspirating fluid can be observed experimentally, and if so, to compare the observational results to theoretical predictions. To this end, we have built a test set-up in which the fluid velocity through the pipe could exceed all predicted critical velocities of the above mentioned theories. A cantilever pipe of about 5 m long with a diameter of about 0.1 m was partly submerged in water (to reduce the hydrodynamic drag). The internal fluid velocity could reach a value of nearly 7 m/s. 6.2 Critical velocities predicted by different theories In this section several theoretical models of an initially straight, submerged pipe aspirating fluid are briefly summarized. The models differ mainly in the description of the boundary condition at the free inlet. Using the Argand diagram (Païdoussis, 1998), the critical velocities and the type of instability are examined. The design of the experimental set-up should be such that these velocities can be achieved. For modelling the cantilever pipe the same assumptions are used as summarized in Chapter 4. In this section, the effective tension, the material damping and the effect of depressurization at the inlet are disregarded as having minor effect on the pipe stability (the tension can be neglected because an almost neutrally buoyant pipe is used in the experiments). A novel element with respect to the two previous chapters is that we account here for the two transverse directions of the pipe. The reason is that the drag couples the motion of these two directions. The two coupled equations of transverse motion of the submerged pipe subjected to a nonlinear hydrodynamic drag read (for the drag description see Chapter 5): ∂4v ∂2v ∂2v ∂ 2v − 2m f u f + mf uf 2 2 + M 2 4 ∂z ∂z∂t ∂z ∂t 2 2 ⎛ µ ∂v 1 ∂v ⎛ ∂v ⎞ ⎛ ∂w ⎞ ⎞ ⎜ + Do A1 + ρ f A2 ⎜ ⎟ +⎜ ⎟ ⎟ = 0, ⎜ Do ∂t 2 ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎟ ⎝ ⎠ (6.1) ∂4w ∂2w ∂2w ∂2w − 2m f u f + mf uf 2 2 + M 2 + 4 ∂z ∂z∂t ∂z ∂t 2 2 ⎛ µ ∂w 1 ∂w ⎛ ∂v ⎞ ⎛ ∂w ⎞ ⎞ A1 + ρ f A2 Do ⎜ + ⎜ ⎟ ⎜ ⎟ ⎟ = 0, ⎜ Do ∂t 2 ∂t ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎟ ⎝ ⎠ (6.2) EI EI The two terms in the brackets represent the linear and quadratic damping caused by the surrounding quiescent fluid (see Chapter 5). The onset of instability is governed by the linearized equations of motion. Linearization of Eqs. (6.1) and (6.2) results in two identical uncoupled equations for 92 Chapter 6 v ( z, t ) and w( z , t ) . Hence, for analyzing the onset of instability it is sufficient to consider only the equation of motion in one plane: EI 2 ∂4w ∂2w ∂2w ∂w 2 ∂ w − 2 + + + µ A1 =0. m u m u M f f f f 4 2 2 ∂z ∂z∂t ∂z ∂t ∂t (6.3) The difference between several theoretical models is related to the balance of transverse forces at the free inlet. All models use a zero bending moment at this inlet. The conventional boundary condition assumes a zero shear force at the free end (Païdoussis, 1998; Païdoussis, 1999; Kuiper and Metrikine, 2005): EI ∂3w =0 ∂z 3 at z=L. (6.4) In a recent paper, Païdoussis et al. (2005) postulated new descriptions for the transverse force balance at the free end. In the previous chapter we highlighted two of these descriptions: EI ∂ 3w ∂w ⎞ ⎛ ∂w − mf uf ⎜ − uf ⎟=0 t ∂z 3 ∂ ∂z ⎠ ⎝ EI ∂3w ∂w − mf uf =0 ∂z 3 ∂t at at z=L, z=L. (6.5) (6.6) The physical differences between the three descriptions for the balance of forces at the free end have been explained in the previous chapter. They are related to the direction of inflow (tangential to the undeflected or deflected pipe) and whether the flow field just below the entrance moves transversely with the pipe or not. As a result, the momentum of fluid remains unchanged in direction (Eq. (6.4)) or changes direction (Eqs. (6.5) and (6.6)) as the fluid enters the pipe. The dynamic stability of a linear system is determined by its eigenfrequencies. To find these, the displacement of the pipe can be sought for in the following form: w ( z , t ) = W ( z ) e iω t , (6.7) where ω is the complex frequency. The system is stable if the imaginary parts of all eigenfrequencies are positive, and unstable if at least one eigenfrequency has a negative imaginary part. Like in the previous chapters, the eigenfrequencies are analysed using an Argand diagram. The natural frequency ω are plotted parametrically, as they depend on the flow velocity uf and the linear drag coefficient A1 . At the first step of the analysis, both the fluid velocity and the external damping are disregarded. The system possesses in this case only real natural frequencies, which can be easily found numerically. As the second step, the damping is gradually increased, keeping the velocity zero, and the accompanying complex values of the natural frequencies are computed and plotted in the Argand diagram. Experimental investigation of a cantilever pipe conveying fluid 93 As expected, all complex natural frequencies acquire positive imaginary part, implying that the pipe is stable. Once the linear coefficient A1 reaches the value for which the stability is studied, the fluid velocity is gradually increased from zero, keeping the value A1 constant. As a result, the imaginary part of the complex natural frequencies decreases with increasing fluid velocity. At a certain velocity, the imaginary part of some natural frequencies becomes negative, implying that the pipe becomes unstable. For the three above mentioned descriptions of the force balance at the inlet, Eqs. (6.4)-(6.6), the Argand diagrams have been plotted in Fig. 6.1 using the parameters given in Table 6.1. These parameters correspond to a flexible plastic pipe of 4.75 m length fully submerged in water and are close to the dimensions of the actual experimental pipe. In Fig. 6.1 only the evolution of the first natural frequency is shown, since this frequency is the first to enter the lower half plane. As can be seen from these diagrams, the three boundary conditions result in different critical fluid velocities and different types of instability: • Eq. (6.4) results in a critical velocity of 7.86 m/s and predicts flutter beyond this velocity; • Eq. (6.5) results in a critical velocity of 5.05 m/s and predicts divergence beyond this velocity; • Eq. (6.6) results in a critical velocity of 10.6 m/s and predicts flutter beyond this velocity. To check which of the three predictions resembles reality, the experimental set-up should allow achieving the highest critical velocity predicted by Eq. (6.6). The difficulty in doing so was that it is impossible to steadily pump water through a pipe of 0.075 m diameter with a velocity higher than 7 m/s. This limit is set by wall friction and cavitation. A solution was found in submerging only a part of the pipe. This reduced the hydrodynamic drag, and consequently, the critical velocities in the cases of flutter, predicted by Eqs. (6.4) and (6.6). Table 6.1 - Parameters used for obtaining the Argand Diagrams in Fig. 6.1. Parameter 9 -2 2.40·10 N·m Di L 4.75 m µ Do 0.075 m Ca E ρf 1.00·103 kg·m-3 1.00·10 kg·m ·s ρp 1.06·103 kg·m-3 1.00 A1, max 27.0·103 0.070 m -3 -1 -1 94 Chapter 6 2 2 uf =0, A1=27.000 uf =0, A1=27.000 1.5 1.5 1 uf =7.86 m/s, A1=27.000 1 Im(ω) Im(ω) 0.5 0.5 0 uf =0, A1=0 uf =5.05 m/s, uf =0, A1=0 A1=27.000 0 -0.5 uf =0, A1=0 -0.5 0 0.4 0.8 1.2 1.6 2 -1 -2 -1 0 1 2 Re(ω) Re(ω) 2 uf =0, A1=27.000 1.5 1 Im(ω) 0.5 0 uf =0, A1=0 uf =10.6 m/s, A1=27.000 -0.5 0 0.4 0.8 1.2 1.6 2 Re(ω) Fig. 6.1 - Argand diagrams for a fully submerged pipe subject to (left) boundary condition (6.4), (right) boundary condition (6.5) and (bottom) boundary condition (6.6). In view of the diagram symmetry with respect to the vertical axis, only the right half-plane is shown in the left and bottom diagrams. 6.3 Experimental set-up and instrumentation The experimental set-up, built at WL Delft Hydraulics in the Netherlands, consisted of a plastic pipe partly submerged in a tank of 6 m diameter filled with water, as shown in Figs. 6.2 and 6.3 (side view and top view). The free inlet of the pipe was 1.25 m above the bottom of the tank, so that the flow near the entrance was insignificantly affected by the bottom. The depth of water in the tank was 3.2 m. If needed, the water level inside the tank could be lowered, but we first focused on this level. In this situation the length of the submerged part of the pipe was 1.95 m. The top of the pipe was rigidly fixed to a stiff steel pillar mounted to a concrete balcony. Experimental investigation of a cantilever pipe conveying fluid 95 Fig 6.2 - Side view of the experimental set-up. (1) Partly submerged cantilever pipe, (2) Rigid connection to stiff steel pipe, (3) Centrifugal pump, (4) Return pipe, (5) Concrete balcony. Table 6.2 - Parameters of the experimental pipes and the fluid. Parameter Smaller pipe Larger pipe Parameter Smaller pipe Larger pipe 3 E 2.70·10 N·m 2.20·10 N·m ρp 1.06·10 kg·m 1.06·103 kg·m-3 L 4.75 m 4.66 m Ca 1.6 1.4 Ldry 2.80 m 2.89 m γe 0.5 9 -2 9 -2 -3 0.5 3 Lsubm 1.95 m 1.77 m A1 2.4·10 Do 0.075 m 0.110 m A2 0.24 2.4·103 0.24 Di 0.070 m 0.104 m α 1.62 kg·m·s 9.53 kg·m·s-1 ρf 1.00·103 1.00·103 µ 1.00·10-3 1.00·10-3 kg·m-3 kg·m-3 kg·m-1·s-1 kg·m-1·s-1 -1 96 Chapter 6 Above this fixation the pipe widened to reduce the internal wall friction. A centrifugal pump was located outside the tank and the flow re-entered the tank through a return pipe at the other side. Two pipes of different diameter were tested. Their main parameters and the water parameters are given in Table 6.2. To give a better impression of the experimental set-up, a photograph of the 4.75 m long pipe in the dry tank is shown in Fig. 6.4. For the smaller diameter pipe, the maximum internal fluid velocity we could reach was nearly 7.0 m/s. For this speed one could clearly hear that cavitation started to occur. And even worse, the under-pressure in the plastic pipe was so high that the pipe buckled locally at the top; the circular shape of the cross-section became unstable. To avoid the latter, the top part of the pipe was stiffened on the outside by metallic rings. For the larger diameter pipe we could reach a maximum fluid velocity of 5.3 m/s, which was close to the discharge capacity of the pump. For this pipe, cavitation and local buckling was not an issue. Fig. 6.3 - Top view of the experimental set-up. (1) Cantilever pipe aspirating fluid, (2) Centrifugal pump, (3) Return Pipe, (4) Tank wall. Experimental investigation of a cantilever pipe conveying fluid 97 Fig. 6.4 - Cantilever plastic pipe when the tank was empty. Although not desirable, the pipe axis was only 25 cm away from the wall of the tank. The practical reason was that in this case the top of the pipe could easily be fixed to the steel pillar mounted to the concrete balcony. Although the pipe was positioned at a distance of only five times its radius from the wall, it seemed that the wall influence was not pronounced. During several experimental runs we injected a blue ink to visualize the flow near the pipe entrance and near the wall. The ink clearly indicated that there were no wall-induced flow patterns between the pipe and the wall. Nevertheless, in a second set of experiments, as described in Section 6.6, the pipe was located almost in the middle of the tank to reduce possible wall influences to a minimum. Two displacement transducers were installed to measure the transverse deflections of the pipe in two perpendicular directions. Since these transducers were noncontact (induction-based), they did not affect the pipe motion. They were installed at the balcony level and recorded the motion with a sampling frequency of 100 Hz. An underwater camera recorded the motion of the pipe near the tip. The water discharge was electromagnetically measured just downstream of the centrifugal pump with a sampling frequency of 100 Hz. This signal showed a steady fluid discharge during the experiments. From the discharge we could easily calculate the fluid velocity through the cantilever pipe, assuming an incompressible fluid. The 98 Chapter 6 water pressure was measured downstream of the cantilever pipe (close to the pipe top), just after the pipe widened. 6.4 Observations – qualitative description During the experiments the internal fluid velocity was gradually increased from zero to a desired velocity. For about 5 minutes the fluid velocity was kept constant before measurements were started. From the observations, two regimes could be distinguished. Below a certain critical fluid velocity the pipe did not move. Exceeding this critical velocity, the pipe motion was observed in the form of a complex motion that consisted of two alternating phases: a nearly periodic orbital motion and a noise-like vibration with very small amplitudes. Usually noise-like vibrations were first observed during a few minutes before the pipe started to orbit. Then, the orbital motion started and lasted for about a few minutes. Thereafter, the orbital motion ceased to exist and noise-like vibrations were observed again, etc. (it should be emphasized that during an experimental run all parameters were kept constant). This alternating behaviour was observed for both pipes. There was no qualitative difference of the pipe behaviour observed for fluid velocities just above critical or for much larger flow velocities. In both cases the motion consisted of the two alternating phases: small, noise-like vibrations and orbital motion with displacements with order of magnitude of the pipe diameter. However, both the frequency and the amplitude of the orbital motion increased with fluid velocity. For the maximal attainable fluid velocity of 7.0 m/s, the orbital motion of the tip of the smaller diameter pipe had a radius of more than 1.5 times the pipe diameter. It was also observed that the pipe orbited always in the same direction. This may be due to a number of unknown asymmetries of the experimental set-up with respect to the pipe axis. We never observed a pure planar motion. 6.5 Observations – quantitative description As a first step, the first natural bending frequency of the partly submerged pipes filled with still (not flowing) water was determined. The first natural frequency was identified for the situation that the lower 1.95 m of the pipe was submerged. The pipe was given an initial deflection and then released. From the record of the pipe displacements, the period of free decaying vibrations was extracted and the natural frequencies were calculated. The first bending natural frequency for the smaller and larger diameter pipes filled with still water was 1.5 rad/s and 2.1 rad/s, respectively. It is not easy to accurately determine the critical fluid velocity beyond which the pipe is unstable. The problem is that the motion that corresponds to the near-critical Experimental investigation of a cantilever pipe conveying fluid 99 velocities is very small. Hence, the period of measurements needs to be long enough to determine whether the pipe exhibits orbital motion during some time. Looking at the data one could estimate that the smaller pipe (Do=0.075 m) exhibited orbital response starting from a velocity of 2.1 - 2.4 m/s. The larger pipe (Do=0.11 m) showed this behaviour from a velocity of 1.9 - 2.2 m/s. For the smaller pipe, the frequency of the orbital motion for fluid velocities just exceeding the critical velocity was around 1.6 rad/s, which is only slightly larger than the first natural frequency of the pipe filled with still water (1.5 rad/s for u f = 0.0 m/s ). Lateral displacement of the pipe at balcony level (cm) The most inexplicable observation for us was the alternating pipe motion for the fluid velocities in the post-critical regimes. As an example of the alternating pipe motion, a measured transverse displacement of the smaller diameter pipe is shown in Fig. 6.5. This displacement was measured at the balcony level, at a height of 3.17 m above the pipe tip, for a fluid velocity of 6.2 m/s. A blow-up of the displacement of the orbital motion (350-520 s) and of the noise-like vibrations (550-850 s) are shown at the top left and top right of Fig. 6.6, respectively. The spectra of both displacement records are shown below in the same figure. It is interesting to see that in the spectrum of the orbital motion (bottom left) there is a clear peak around the first natural frequency of the pipe. 1 0.5 0 -0.5 -1 0 200 400 600 800 1000 Time (s) Fig. 6.5 - Y-displacement of the smaller diameter pipe measured by the displacement transducer for a fluid velocity of 6.2 m/s. 100 Chapter 6 1 Lateral displacement (cm) Lateral displacement (cm) 1 0.5 0 -0.5 -1 0.5 0 -0.5 -1 360 400 440 480 520 550 600 650 10 10 1 1 0.1 0.01 0.001 0.0001 1E-005 0.0001 1E-005 1E-007 8 Radial frequency (rad/s) 850 0.001 1E-007 4 800 0.01 1E-006 0 750 0.1 1E-006 1E-008 700 Time (s) Spectral density (cm2/(rad/s)) Spectral density (cm2/(rad/s)) Time (s) 12 16 1E-008 0 4 8 12 16 Radial frequency (rad/s) Fig. 6.6 – (Left) A blow-up of Figure 6.5 (350 to 520 s) and below the accompanying Ydisplacement spectrum. (Right) A blow-up of Figure 6.5 (550 to 850 s) and below the accompanying Y-displacement spectrum. In comparison with the situation when the water in the pipe is not flowing, the natural frequency of the first bending mode increases from 1.5 rad/s ( u f = 0.0 m/s ) to 1.6 rad/s ( u f = 2.2 m/s ) and then to around 2.2 rad/s ( u f = 6.2 m/s ). In the regime of noise-like vibrations the peak around the first natural frequency is completely absent (see bottom right plot in Fig. 6.6). It was checked whether the spectra contained energy around the second or higher natural frequencies. This was hardly the case. The small amplification around 12 rad/s in both spectra (see Fig. 6.6) is related to the frequency of the second bending mode. Hence, the orbital motion originates mainly from the first bending mode instability. The smaller peak in the spectra of the orbital motion (bottom left plot in Fig. 6.6) around 4.8 rad/s is explained in a subsequent section. Experimental investigation of a cantilever pipe conveying fluid 101 6.6 Second set of experiments The unexplainable complex motion of the cantilever pipe was the key reason to conduct a second set of experiments. In particular, we wanted to check whether the internal fluid flow was indeed the cause of the observed instability. There were still some suspicions about other mechanisms for the pipe’s orbital motion, like a pumpinduced swirling motion inside the pipe, a (sheared) flow in the tank, a boundary effect of the tank wall, etc. In order to invalidate these suspicions we rebuilt the original experimental set-up. The main modifications with respect to the first set of experiments are listed below. • The pipe was positioned further away from the wall of the tank to eliminate possible boundary effects. In the original experimental set-up the centre of the pipe was about 0.25 m away from the tank wall, whereas in the new setup this distance was increased to 1.75 m. • The flow from the return cylinder, i.e. the flow back into the tank, was modified by installing a perforated cage around the return cylinder. The main thought behind this idea was to get a more symmetric flow in the tank and to break the direct flow from the return cylinder to the cantilever pipe. • Additionally, several inlet shapes (see Fig. 6.7) were fabricated, which could be manually connected to the tip of the pipe. In this way it could be checked whether a different inflow field would result in a change of the dynamic stability of the pipe. Besides, an anti-swirling device (see Fig. 6.8) could be internally installed near the tip of the pipe. This would reduce a possible internal swirling fluid motion induced by the centrifugal pump. Fig 6.7 – Three inlet pieces (standing up-side-down); (left) inlet piece with holes, the fluid can only enter the pipe in radial direction, (middle) inlet piece slanting at 45 degrees, (right) bell-mouth piece. 102 Chapter 6 Fig 6.8 – Anti-swirling device, which could be internally installed near the pipe tip. In the new experimental set-up, the motion of the bare pipe, i.e. without additional inlet piece and without anti-swirling device, consisted again of the two alternating phases for internal fluid velocities exceeding a critical value. However, two main differences were observed with respect to the first set of experiments. Firstly, the orbital motion occurred not in all runs. Secondly, the maximum displacement amplitude of the orbital motion was much smaller compared to the first set of experiments (0.25 versus 1.5 times the pipe diameter). Some other interesting observations are summarized below. • During one run, although with a small amplitude, the pipe orbited first anticlockwise and later clockwise. This suggests that a possible internal swirl motion, generated by the centrifugal pump, cannot be considered as the main driver of the orbital motion. • This thought has been strengthened after conducting some runs with the anti-swirling device installed. Even with this device installed, the pipe motion alternated between the two phases. • Experiments with the different inlet pieces (shown in Fig. 6.7) did not show qualitatively different pipe behaviour than without an inlet piece, i.e. the pipe motion still consisted of the two alternating phases. It could be, however, that due to the small pipe motions, differences were not clearly visible. Experimental investigation of a cantilever pipe conveying fluid 103 Fig 6.9 – Cantilever pipe with a nearby wooden plate (in an empty tank). The reason for the reduction in amplitude and accidental occurrence (not in all runs) of the orbital motion, especially in case of the bare pipe, was unclear. Two obvious differences with respect to the first set of experiments were the distance from the cantilever pipe to the tank wall and the distortion of the direct flow from the return cylinder to the test pipe. The influence of the latter could easily be investigated by removing the perforated cage around the return cylinder. This appeared to have no significant influence on the pipe motion. To check the influence of a nearby wall on the pipe motion a wooden plate (see Fig. 6.9) was installed at several distances from the cantilever pipe, ranging between 0.23 m and 0.48 m (distance from the centre of the pipe to the wooden plate). As expected, when the pipe was close to the wooden plate and the discharge was sufficiently high the pipe was sucked to the wooden plate and stuck to it. However, we never observed oscillations with similar amplitude as in the first set of experiments. 6.7 Modelling the instability In this section the experimental observations are compared with predictions of different theories. All current theories in the literature, known to us, are not able to predict a complex motion that consists of two alternating phases: a nearly periodic orbital motion and a noise-like vibration. For comparison between the experiment and theories, we first focus on the critical velocity and the frequency of the orbital motion at near-critical fluid velocities. To enable a meaningful comparison, all material and fluid parameters had to be carefully identified. In order to do so a few additional tests were carried out. 104 Chapter 6 First, the elasticity modulus and the material damping in the pipe were identified. To this end, the dry pipe, i.e. hanging in air and without fluid inside, was given an initial deflection and then released. From the record of the pipe displacements, the period and decrements of free decaying vibrations were extracted. For this particular case, the motion of the free-hanging pipe in a plane is assumed to be described by the following equation of motion: EI ∂4w ∂ ⎛ ∂w ⎞ ∂3w ∂2w − ρ − − α + =0, A g L z m ( ) p ⎜ p p ⎟ ∂z 4 ∂z ⎝ ∂z ⎠ ∂z 2∂t ∂t 2 (6.8) in which ρ p is the density of the pipe material, Ap is the cross-sectional area of the pipe wall, L is the length of the pipe, α is the material damping coefficient and mp is the mass of the pipe per unit length. The material damping is assumed proportional to the pipe curvature. As a second step, the same free vibration tests were carried out with the cantilever pipes partly submerged in water and completely filled with still water. The added mass coefficient was determined from the measured period of vibration. From the decrement of vibrations, making use of the previously determined material damping, the hydrodynamic drag coefficients were calculated. Neglecting the internal underpressure in the pipe, the equations of motion in one plane, applicable to above-water and submerged parts, are given as: ∂4w ∂ ⎛ ∂w ⎞ − ⎜ ρ p Ap g ( L − z ) − ρ f Ap g ( L − Ldry ) − ρ f Ai g ( z − Ldry ) ⎟ ∂z 4 ∂z ⎝ ∂z ⎠ ∂3w ∂2w 0 ≤ z ≤ Ldry , −α 2 + ( m p + m f ) 2 = 0, ∂z ∂t ∂t EI EI { } ∂4w ∂ ⎛ ∂w ⎞ ∂3w − ⎜ ( ρ p − ρ f ) Ap g ( L − z ) ⎟ −α 2 4 ∂z ∂z ⎝ ∂z ⎠ ∂z ∂t + ( m p + m f + ma ) ⎛ µ ∂2w ∂w 1 ∂w ∂w ⎞ + D0 ⎜ A1 + ρ f A2 ⎟ = 0, ∂t 2 ∂ ∂t ∂t ⎠ 2 D t ⎝ 0 (6.9) (6.10) Ldry < z ≤ L, in which Ldry is the length of the pipe in air, ρ f is the density of the fluid, Ai is the internal cross-sectional area of the pipe, mf is the mass of the internal fluid per unit length, and ma is the added mass per unit length defined as ma = Ca ρ f ( Ap + Ai ) , with Ca the added mass coefficient. From these auxiliary experiments without internal flow, the added mass coefficient Ca and the hydrodynamic drag coefficient A1 were identified. The value obtained for the coefficient A1, indicating the amount of linear viscous fluid damping, was smaller than predicted by Kuiper et al. (2007). The reason is that the parameter A1 was estimated in that paper for a Stokes Experimental investigation of a cantilever pipe conveying fluid 105 parameter β = D02 (υ T ) ≈ 106 , while in our experiments the Stokes parameter was about 103 . In the above expression, T is the period of oscillation and υ is the kinematic viscosity of the water. The parameter A2, related to the external drag, cannot be estimated from the auxiliary free vibration experiments described above. Therefore, since this coefficient does not depend on the Stokes parameter, the value proposed by Kuiper et al. (2007) was used. Obviously, the latter parameter has no influence on the onset of instability. For both pipes, all measured and calculated parameters are summarized in Table 6.2. The measured Y-displacement at the balcony level after releasing the pipe from an initial deflection and the corresponding theoretical prediction (with fitted parameters) based on Eqs. (6.9) and (6.10) are shown in Fig. 6.10 as an example. After identifying all parameters, the onset of instability could be theoretically calculated. To this end, the equations of motion for the pipe without fluid flow, Eqs. (6.9) and (6.10), were extended with fluid flow dependent terms: EI ∂4w ∂ − ∂z 4 ∂z ({ρ A g ( L − z ) − ρ p p f Ap g ( L − Ldry ) − ρ f Ai g ( z − Ldry ) + 1 ∂2w ∂ 3w ∂2w ⎫ ∂w ⎞ ρ f Ai u 2f ( −1 + γ e ) ⎬ ⎟ − 2m f u f − α 2 + ( m p + m f ) 2 = 0, 2 ∂z∂t ∂z ∂t ∂t ⎭ ∂z ⎠ EI (6.11) 0 ≤ z ≤ Ldry , ∂4w ∂ ⎛ ⎧ 1 ∂3w ⎫ ∂w ⎞ 2 ρ ρ ρ γ α − − A g L − z + A u − 1 + − ( ) ( ) ( ) ⎨ p f p f i f e ⎬ ⎟ ∂z 4 ∂z ⎝⎜ ⎩ 2 ∂z 2∂t ⎭ ∂z ⎠ ⎛ µ ∂2w ∂2w ∂w 1 ∂w ∂w ⎞ −2 m f u f + ( m p + m f + ma ) 2 + D0 ⎜ A1 + ρ f A2 ⎟ = 0, Ldry < z ≤ L. ∂z∂t ∂t ∂t 2 ∂t ∂t ⎠ ⎝ D0 (6.12) The coefficient γ e indicates the pressure loss at the entrance, varying between 0 for a smooth bell mouth inlet and 1 for a square-cut inlet. In order to calculate the onset of instability, the boundary conditions should be known. At the top, the pipe is assumed to be fixed. As mentioned before, at the free entrance the bending moment is zero and for the balance of transverse forces two options are analysed: EI EI ∂3w ∂2w −α =0 3 ∂z ∂z∂t ∂3w ∂2w ∂w − − mf uf =0 α 3 ∂z ∂z∂t ∂t at z = L , at z = L . (6.13) (6.14) 106 Chapter 6 1 Displacement at balcony level (cm) 0.5 0 -0.5 -1 -1.5 -2 0 20 40 60 80 Time (s) Fig. 6.10 - Measured (bold line) and theoretical prediction (thin line) of Y-displacement of the pipe in the free vibration auxiliary test. These two boundary conditions are similar to Eqs. (6.4) and (6.6), except that the effect of the material damping is included. The third option for the balance of transverse forces, Eq. (6.5), is not considered, since this, in contrast to the experiments, would predict divergence but not flutter (divergence was not observed in our experiments). Before the nonlinear equations, Eqs. (6.11) and (6.12), are solved in the time domain, the onset of instability predicted by the linear equations is determined by using the Argand Diagram. Terms related to the nonlinear part of the hydrodynamic drag and the effective tension of the riser are disregarded in Eqs. (6.11) and (6.12). The key difference with Section 6.2 is that in this Section the partly submerged pipe is considered instead of the fully submerged pipe. To obtain the Argand Diagrams, the values corresponding to the smaller pipe diameter (see Table 6.2) are used. In Fig. 6.11 the paths of the first natural frequency are plotted as they depend on the linear damping coefficient and the fluid velocity. The left plot is obtained using the conventional boundary condition at the free inlet, Eq. (6.13), while the right one is based on boundary condition (6.14). As explained above, the linear part of the hydrodynamic damping is not so easy to determine since it depends on the Stokes number. To investigate the sensitivity of the critical velocity to the linear damping, the paths of the first natural frequency have been plotted for both the measured linear damping coefficient ( A1 = 2400 ) and a three times larger value ( A1 = 7200 ) . Obviously, a larger damping coefficient results in a larger critical velocity. Experimental investigation of a cantilever pipe conveying fluid 0.5 0.4 0.5 0.4 uf =0, A1=7200 0.3 0.3 uf =2.0, A1=7200 0.2 0.1 Im(ω) uf =0, A1=2400 0 0.1 Im(ω) uf =0, A1=0 -0.2 -0.2 uf =0.73, A1=2400 -0.4 1.5 1.51 1.52 Re(ω) 1.53 uf =9.3, A1=7200 uf =0, A1=2400 0 -0.1 -0.3 uf =0, A1=7200 0.2 -0.1 -0.5 1.49 107 uf =0, A1=0 uf =6.8, A1=2400 -0.3 -0.4 1.54 -0.5 1.48 1.52 1.56 1.6 Re(ω) Fig. 6.11 - Argand diagrams for partly submerged pipe, (left) subject to boundary condition (6.13) and (right) subject to boundary condition (6.14) . The regular and dotted lines represent A1 = 2400 and A1 = 7200 , respectively. For the conventional boundary condition at the free inlet, Eq. (6.13), and using the parameters of the smaller pipe, the theoretical critical velocity is 0.73 m/s, while the experiments showed an onset of instability at velocities between 2.1 and 2.4 m/s. The experimentally observed critical velocity would only be predicted using a much higher linear damping coefficient than measured (see the dotted line in the left plot in Fig. 6.11). Besides, it is interesting to check whether the theoretical model captures the frequency at near-critical velocities. In the experiments the first natural frequency increased with fluid velocity from 1.5 rad/s at zero fluid velocity to 1.6 rad/s at near-critical fluid velocity. With measured frequency at near-critical velocities we mean the dominating frequency during orbital motion. This increase of frequency is in qualitative agreement with the left plot in Fig. 6.11 but not quantitatively. Using the boundary condition proposed by Païdoussis and co-authors, Eq. (6.14), the onset of flutter for the smaller diameter pipe is at 6.8 m/s (right plot in Fig. 6.11), which is much larger than observed in the experiments. Also in this case the theory predicts an increase in the real part of the frequency with increasing fluid velocity from zero to near-critical. Again this is in qualitative agreement with the observations. For the larger diameter pipe the same trend is observed. The experiments showed an onset of instability at velocities between 1.9 and 2.2 m/s. The conventional boundary condition predicts an onset at 0.56 m/s, whereas the boundary condition proposed by Païdoussis and co-authors predict an onset at 7.1 m/s. 108 Chapter 6 In order to analyse the post-critical regime the nonlinear equations, Eqs. (6.11) and (6.12), were solved in the time domain. In these equations only the nonlinear hydrodynamic drag was accounted for to limit the structural vibrations. In the experiments it was observed that the displacements were relatively small, and hence, geometrical nonlinearities were neglected. The equations of motion, Eqs. (6.11) and (6.12), and the boundary conditions are discretized using a second-order central difference approximation. The length of the riser L is divided into N segments, all having a length h = L N . The first node is located at x = 0 and the (N+1)th node at x = L. This yields a system of (N+1) second-order differential equations, resulting in a system of 2(N+1) first-order ordinary differential equations. In the calculations the riser is divided into 100 segments. In the post-critical regime both boundary conditions, Eqs. (6.13) and (6.14), predict finite amplitude oscillations, i.e. a limit cycle, due to the nonlinear drag. Two spectra of the calculated pipe displacement in the post-critical regime are shown in Fig. 6.12. The purpose of calculating these spectra is to check whether the predicted dominant frequencies coincide with the measured ones. Since it is much more difficult to correctly predict the amplitude of motion, we skipped the values on the vertical axis (the displacement transducers show maximum amplitudes in the order of 1 cm, while for the same location theory predicts maximum amplitude in the order of 10 cm). The left plot is obtained using the conventional boundary condition, Eq. (6.13), and an arbitrarily chosen internal fluid velocity of u f = 3.6 m/s . The spectrum is determined for the pipe displacement at the balcony level in order to compare calculations (left plot in Fig. 6.12) with measurements (bottom left plot in Fig. 6.6). Obviously, the comparison is limited to the situation when the pipe orbits. The locations of the calculated and measured peaks agree quite well. According to the model, both the first and second mode are unstable, whereas the measurements show only first mode instability. Due to the nonlinear drag, peaks in the calculated spectrum arise additionally at 3ω1 , ω2 − 2ω1 and ω2 + 2ω1 . Especially, the location of the measured peak around 4.8 rad/s (bottom left plot in Fig. 6.6) coincides nicely with the prediction ( 3ω1 ). A similar spectrum is plotted at the right side of Fig. 6.12 for the boundary condition proposed by Païdoussis and co-authors, Eq. (6.14). To be in the postcritical regime a much higher internal fluid velocity was needed. To this end u f = 9.0 m/s was chosen in computations. For this fluid velocity the second mode was not yet unstable. In these computations, additional peaks were observed as well, originating from the nonlinear hydrodynamic damping. Again, the frequencies of the calculated and measured peaks agree quite well. Experimental investigation of a cantilever pipe conveying fluid 109 ω1 Spectrum Spectrum ω1 3ω 1 3ω 1 ω 2−2ω 1 ω2 ω 2−2ω 1 ω 2+2ω 1 0 4 8 Radial frequency (rad/s) 12 16 0 4 ω2 8 12 16 Radial frequency (rad/s) Fig. 6.12 - Predicted displacement spectrum; (left) pipe is subject to boundary condition (6.13) and u f = 3.6 m/s , (right) pipe is subject to boundary condition (6.14) and u f = 9.0 m/s . By using two coupled equations in the two transverse directions, similar to Eqs. (6.1) and (6.2), one can successfully find an orbital motion with a finite amplitude. In the experiments, however, the instability was observed in the form of a complex motion that consists of two alternating parts: nearly periodic orbital motion with maximum amplitude of 1.5 times the pipe diameter and noise-like vibration with small deflections. Therefore, it can be concluded that there are still discrepancies between theory and experiments for both the onset of instability and the behaviour in the post-critical regime. In the next section some possibilities are examined which might explain these discrepancies. 6.8 Possible explanations for the experimentally observed pipe behaviour Existing theories do not predict: 1. The correct magnitude of the critical fluid velocity; 2. The relatively long period that the pipe needs to start orbiting; 3. The unsteady unstable motion that consists of an orbital motion alternating with a noise-like vibration; 4. The amplitude of the steady-state orbital motion. We attempted to improve existing models, to simulate the observed behaviour of the experimental pipe. Although a great deal of ideas was investigated, only three modelling attempts are addressed below. These attempts should be considered as thought experiments, rather than as quantitative, realistic models. First, in all the abovementioned models the fluid flow velocity in the pipe was assumed to be constant. However, at the high fluid velocities attained in the 110 Chapter 6 experiments, the flow might have a pulsating character. This might result in an additional vibration in the longitudinal direction of the system, which may affect the pipe motion in the transverse direction through parametric resonance. This could explain why the pipe needs so much time to start oscillating in the unstable regime. For a parametrically excited system, the first-order instability zone is located around Ω = 2ω0 , where Ω is the frequency of the parametric excitation and ω0 is an eigenfrequency of the system. From the measured data we observed that the pipe vibrates only at the first natural frequency, which is about 2 rad/s. The parametric excitation, i.e. the pulsating fluid flow, to be able to perceptibly affect the transverse motion, should show a peak in the fluid velocity spectrum around 4 rad/s. The measurements do not show any peak around this frequency. The same holds for lower excitation frequencies which could lead to higher-order parametric instability zones. Hence, parametric resonance is not likely to occur in our experimental setup. As explained in section 6.2, the correct description of the flow field in the vicinity of the tip is of great importance for the stability of the pipe conveying fluid. The complex inflow is largely simplified by the boundary condition at the tip of the pipe. To improve the boundary condition at the free inlet, a model was developed in which the angle at the free end of the pipe ( ∂w ∂x x = L ) relative to the angle of inflow ( φ ) was not prescribed but introduced as a degree of freedom. This was done by introducing a “pipe tail” representing the part of the inflow below the inlet, as shown in the Fig. 6.13. This part lags behind the inlet due to interaction with the surrounding water. The boundary condition at the free end for this situation is: EI ∂3w ∂2w ⎛ ∂w ⎞ − α − mf uf 2 ⎜ − +φ ⎟ = 0 3 ∂z ∂t∂z ⎝ ∂z ⎠ at z = L (6.15) An additional pendulum equation was added to describe the motion of the “tail”. After analyzing this system, it became clear that the term m f u f ( ∂w ∂t ) in the boundary condition, which is absent in Eq. (6.15), is the dominating factor for stability rather than the angle of inflow. Hence, we thought of another simplified model to describe the main characteristics of the flow field near the entrance. If one would imagine a model in which the boundary condition for the transverse force switches in time between the descriptions given by Eqs. (6.13) (excluding the term m f u f ( ∂w ∂t ) ) and (6.14) (including the term m f u f ( ∂w ∂t ) ), one would come remarkably close to the experimental observations, since the pipe response would consist of sustained vibrations alternated with a non-moving phase. Experimental investigation of a cantilever pipe conveying fluid 111 w z=0 uf EI, m f z=L z φ lt , m t Fig. 6.13 - Sketch of a cantilever pipe with a free angle of inflow. The boundary condition postulated by Païdoussis and co-authors (2005), Eq. (6.14), seems to be quite reasonable if the pipe is standing almost still. However, when the pipe starts to orbit, the fluid beneath the pipe might start to rotate with it, and in this case this boundary condition is not valid anymore. When the fluid beneath the pipe moves with the same speed as the tip of the pipe, expression (6.13) seems to be more realistic. To catch these effects in a simplified model, an artificial fluid disk has been introduced beneath the pipe tip, with the centre of the disk located at the z-axis (see Fig. 6.14). The artificial fluid disk rotates with an unknown rotational speed Ω . To start simple, the diameter and the height of the artificial fluid disk are taken as fixed fractions of the diameter of the pipe. The fluid which is sucked into the pipe originates from the rotating disk. The change of momentum while the fluid enters the pipe depends in this model on the rotational speed of the disk. The boundary conditions for the balance of transverse forces at the free tip in the two perpendicular directions read: EI ∂3w ∂2w ⎛ ∂w ⎞ − α − mf uf ⎜ + Ωv ⎟ = 0 , 3 ∂z ∂z∂t ⎝ ∂t ⎠ (6.16) EI ∂ 3v ∂ 2v ⎛ ∂v ⎞ − α − m f u f ⎜ − Ωw ⎟ = 0 . 3 ∂z ∂z∂t ⎝ ∂t ⎠ (6.17) 112 Chapter 6 v(z,t) w(z,t) z=0 uf EI, mf z=L Ω z Fig. 6.14 - Sketch of a rotating fluid disk below the free hanging pipe. These boundary conditions are coupled and nonlinear. When the fluid disk is not rotating, Ω = 0 , expressions (6.16) and (6.17) are similar to expression (6.14), as proposed by Païdoussis and co-authors (2005). When the fluid disk rotates with the same speed as the pipe, ∂w ∂t = −Ωv and ∂v ∂t = Ωw , the expressions are similar to the conventional boundary condition, Eq. (6.13). To compute the rotational speed of the disk an extra equation of motion for the disk has been introduced: J dΩ ⎛ ∂w ⎞ ⎛ ∂v ⎞ + mf uf ⎜ + Ωv ⎟ v − m f u f ⎜ − Ωw ⎟ w + Crot Ω = 0 , dt ⎝ ∂t ⎠ ⎝ ∂t ⎠ (6.18) in which J is the rotary inertia of the fluid disk and Crot is a damping parameter in the rotational direction (in Nms). Releasing this system from relatively small initial conditions, the “critical velocity” for this system is of the same order as the critical velocity for the partly submerged pipe subject to boundary condition (6.14), u f , crit = 6.8 m/s . However, in principle it is possible to have sustained oscillations for fluid velocities close to the critical velocity for the partly submerged pipe subject to boundary condition (6.13), u f , crit = 0.73 m/s . This is only possible if the initial conditions are such that the pipe and the fluid disk rotate already with almost the same speed. Hence, depending on the initial conditions, the pipe starts to rotate or not. Similarly, external random forces can change the type of pipe vibration from a non-moving phase to an orbital motion. As an example, two plots of the pipe motion subject to an impulse loading (around 12 s) on the disk are shown in Fig. 6.15. Both plots are obtained for an Experimental investigation of a cantilever pipe conveying fluid 113 internal fluid speed u f = 4 m/s . The left plot shows no pipe motion, after the disk is subjected to a pulse loading. In contrast to this, the right plot shows an orbital pipe motion with the same angular velocity as the fluid disk, after it is loaded with a slightly larger pulse. From this figure it is clear that for the same internal fluid velocity two types of pipe response can occur. This phenomenon was also observed in the experiments. In Fig. 6.14 the basic model has been shown. However, other variants were tested as well. For example, in one variant the centre of the disk was not fixed at the z-axis, but it could also move in the two horizontal directions. In another variant the diameter of the disk was adjustable and dependent on the position of the pipe tip. Although these models resulted in quantitatively different results, in all models both types of pipe response, a non-moving pipe response and an orbital pipe motion, could still occur for the same internal fluid velocity. This is qualitatively in agreement with the experiments, and hence, more research will be done in this direction. Especially, the research should be related to the transition from the orbital pipe motion back to the static equilibrium position. In the current model it is difficult to disturb the energy transfer from the fluid to the pipe during orbital pipe motion, and hence, to bring the pipe back to the non-moving phase. 3 Tip displacement (m), Angular velocity (rad/s) Tip displacement (m), Angular velocity (rad/s) 1.2 0.8 0.4 0 -0.4 -0.8 2 1 0 -1 0 10 20 30 Time (s) 40 50 60 0 10 20 30 40 50 60 Time (s) Fig. 6.15 - Displacement of the tip of the pipe w(z=L) (regular line), v(z=L) (bold line) and the angular velocity of the disk (dashed line). (Left) non-moving pipe response after a relatively small pulse loading (torque) on the disk. (Right) orbital pipe motion after a relatively large pulse loading (torque) on the disk. 114 Chapter 6 6.9 Conclusion For the first time, it was experimentally proven that a partly submerged cantilever pipe aspirating water becomes unstable beyond a critical internal fluid velocity. Below this velocity the system is stable, whereas above it the system shows a complex motion that consists of two alternating phases: nearly periodic orbital motion and noise-like vibration with very small amplitudes. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion as well as in a larger amplitude of the noise-like vibration, albeit to a smaller extent. In these experiments the maximum displacement amplitude was a few pipe diameters. Existing theories predict the dominant frequencies quite well. However, they do not correctly predict the critical velocity and the pipe behaviour in the unstable regime. It seems that the flow field in the vicinity of the tip is of great importance for the correct prediction of the dynamic behaviour of the cantilever pipe conveying fluid. As a consequence, it is recommended that future research, either experimentally, theoretically or numerically, should be concentrated on this key topic. CHAPTER 7 Destabilization of deep-water risers by a heaving platform The main focus in the previous chapters was on axial flow-induced instability of a free hanging riser aspirating fluid. In this chapter a straight top-tensioned riser in deep water is considered. In contrast to the free hanging riser, this riser is connected to both the sea bottom (via the wellhead) and the floating platform. Obviously, there is also an internal fluid flow (oil and gas) through these top-tensioned risers, which in principle can destabilize the system. Owing to the low fluid speed and the absence of a free end, the internal fluid flow cannot destabilize these top-tensioned risers in practice. There are other mechanisms possible that could possibly lead to instability of toptensioned risers. One of them is the heave (vertical motion) of a floating platform that induces a fluctuation in time of the axial tension of the riser. A possible and undesirable phenomenon is the excitation of a transverse riser vibration caused by this fluctuation. Although this is a different topic than discussed in the previous chapters, related solution techniques are used to find the instability zones. There are several original components in this work. In finding the instability zones, viscous hydrodynamic damping, depth-dependent axial tension and a high modal density of deep-water risers are accounted for, so that conventional analytical methods are practically impossible. Two qualitatively different mechanisms of stability loss are distinguished, discussed and exemplified. The first is classical parametric resonance that occurs solely due to periodic time variation of the axial tension. The second mechanism occurs if the amplitude of vibration of the platform is large enough to change tension into compression in a segment of the riser for a part of the vibration cycle. The main contents of this chapter have been published in Journal of Sound and Vibration (Kuiper et al., 2008). 116 Chapter 7 7.1 Introduction Exploitation of offshore oil and gas fields is moving into deeper waters of more than 2000 meters. In order to convey the hydrocarbon to the sea level, a steel pipe, conventionally referred to as a riser, is installed between wellhead at the sea bed and floating platform. A straight, top tensioned riser or a catenary-shape riser may be used, depending on the floating unit from which the riser is suspended. Despite a number of advantages of the straight top-tensioned riser over the catenary riser, straight production risers have not been used yet in water depths over 2000 meters. There are two specific concerns that stop the offshore industry from using top-tensioned straight risers in deep waters. The first concern is associated with fluctuation of the axial tension in the riser that is caused by vertical motion (heave) of the 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 straight configuration of the riser and cause it to vibrate at a dangerously high level. The second concern is that the stroke of the heave compensator cannot always be made sufficiently large. This, however, can be dealt with by properly tuning the stiffness of the heave compensator to the environmental conditions. This chapter addresses the first above-mentioned concern, which is the possible destabilization of the riser due to fluctuation of the axial tension. The primary aim of the chapter is to find the range of practically relevant amplitudes and frequencies of the vertical displacement of the platform which may destabilize the riser and cause significant transverse vibrations. The second aim is to discover possible mechanisms of the destabilization and to distinguish those mechanisms which may lead to dangerously high dynamic amplification of the stresses in the riser. Instability of a riser that is caused by fluctuation of its tension in time belongs to the class of parametric instabilities. This type of instability of offshore cables, risers and tethers received close attention of researchers in the past. Hsu (1975) was one of the first who analysed parametric resonance for offshore cable applications. Patel and Park (1991, 1995) studied dynamic behaviour of buoyant platform tethers assuming a constant pre-tension over the height and considering combined lateral and axial excitation at the top. The partial differential equation, governing the transverse motion of a tether, was reduced by them to a set of Mathieu equations with the help of the Galerkin method. These equations were then solved numerically taking into account the first four modes. In the first step of the analysis, determining the stability of the equilibrium, they did not consider the coupling between the modes. For a submerged cable of 1000 m length, parametrically excited at the top, Chatjigeorgiou and Mavrakos (2002) studied numerically the transverse motion of the cable based on the first four modes taking the coupling between these modes Destabilization of deep-water risers by a heaving platform 117 into account. Suzuki et al. (2004) performed a small-scale test in air with a Teflon tube of 11 m long that was parametrically excited at the top. They simulated dynamic behaviour of a 3000 m long free hanging riser, demonstrating the possible danger of parametric instability. Chatjigeorgiou and Mavrakos (2005) found after extensive mathematical manipulations a closed-form solution for a parametrically excited riser based on the first two modes. These analytical limitations were the reason for Park and Jung (2002) to use a finite element method for investigating long slender marine structures. Parametric instability of straight deep-water cables, tethers and risers should be studied in two steps. First, stability of the straight configuration should be analyzed by linearizing the equations of motion in the vicinity of this equilibrium. This allows to determine the amplitude and frequency of parametric excitation which makes the straight configuration unstable. Then, the dynamic equilibrium should be studied, which the structure would reach if its straight configuration is unstable. At this stage, at least one of the two major nonlinearities that are capable of limiting structural vibrations should be accounted for. These nonlinearities are the geometric nonlinearity of the structure and hydrodynamic drag. Once the dynamic equilibrium is found, the corresponding displacements of and stresses in the structure can be calculated. On that basis a conclusion can be drawn as to whether the parametric instability may cause a dangerously high level of vibration or not. The first step of analysis, i.e. determination of stability of the straight configuration, is of major importance. It allows to determine the parameter domain, in which a nonlinear analysis of the dynamic equilibrium is necessary. A notable problem at this stage is linearization of hydrodynamic damping, which is conventionally described as a quadratic function of the velocity of structural motion through surrounding water. Because of its quadratic character, in all above-mentioned papers the damping was not accounted for in the linear analysis of stability of the straight configuartion. In this chapter, based on recent results of Kuiper et al. (2007), an improved description of hydrodynamic drag is used, which contains a linear term. Taking this term into account, the domain of parameters corresponding to unstable straight configuration is significantly reduced, which makes the nonlinear analysis of dynamic equilibrium significantly quicker. There are two more factors which have to be accounted for during stability analysis of the straight configuration. These are the depth-dependent axial tension and a high modal density of deep-water risers. Both these factors make analytical analysis of stability of the straight configuration practically impossible. Therefore, instead of simplified analytical estimation of the equilibrium stability, which was used in the vast majority of papers on the subject, a numerical technique is applied here. 118 Chapter 7 Another original component of this chapter is an attempt to distinguish a number of qualitatively different mechanisms of stability loss by the riser. The difference between these mechanisms originates from the depth-dependent axial tension of the riser, which decreases significantly with elevation. This enables a so-called local dynamic buckling, which occurs if tension changes into compression in a segment of the riser during a part of the vibration cycle. 7.2 Assumptions and equation of motion The top-tensioned riser under consideration is an initially straight, tensioned pipe of a finite length, as sketched in Fig. 7.1. It is assumed that the riser moves only in the plane that is depicted in this figure. The transverse displacement of the pipe is considered to be small and of a long wavelength relative to the diameter of the pipe, so that the Euler-Bernoulli theory is applicable for description of the pipe dynamic bending. The top of the riser is connected to the floating platform by means of a heave compensator, which has two functions. First, this device provides a large static tensile force at the top of the riser to avoid buckling. Secondly, it reduces the longitudinal stress variation induced by the relative vertical motion of the platform and the riser. As commonly done, the heave compensator is modelled in this chapter as a pretensioned vertical spring with a stiffness that is much lower than the axial stiffness of the riser. At the bottom, the connection between the riser and the wellhead is modelled by a hinge. k w(z,t) z L EI, M Fig. 7.1 - Sketch of a deep-water riser connected to a floating platform. Destabilization of deep-water risers by a heaving platform 119 Given the low stiffness of the heave compensator, the extensibility of the riser plays a minor role in its dynamics. Therefore, the riser is assumed to be inextensible so that any variation of the tension at the top of the riser, imposed by the heave compensator, causes the corresponding instant change of tension all over the riser length. With these assumptions, the equation of motion governing the transverse displacement w ( z, t ) of the riser from its straight vertical configuration as a function of elevation z and time t can be written as EI ∂4w ∂ ⎛ ∂w ⎞ ∂2w ∂w 1 ∂w ∂w − ⎜ Te ( z, t ) + ρ f Do A2 = 0, ⎟ + M 2 + µ A1 4 ∂z ∂z ⎝ ∂z ⎠ ∂t ∂t 2 ∂t ∂t (7.1) where EI is the bending stiffness of the pipe, Te is the effective tension in the riser, M is the mass per unit length of the system (riser, internal fluid and added masses), µ is the dynamic viscosity of water around the riser, ρf is the water density, Do is the outer diameter of the riser and A1 and A2 are two dimensionless constants. The last two terms on the left-hand side of Eq. (7.1) represent the hydrodynamic damping due to the surrounding water as explained in Kuiper 17 et al. (2007). The term proportional to the transverse velocity of the pipe represents the viscous effects apparent in a thin boundary layer attached to the pipe. This term is important for small ( w Do < 0.02 ) displacements of the riser, and hence, for the onset of instability. For larger displacements and velocities the flow separates from the pipe, forming a turbulent wake behind it. In this regime the drag is proportional to the square of the pipe velocity. The constants A1 and A2 can be derived from experiments as shown in Kuiper et al. (2007), assuming that the riser is surrounded by quiescent water. Conventionally, the measured hydrodynamic drag is expressed as a function of displacement amplitude and period of vibration. The idea is to reconstruct a time domain drag expression on the basis of the measured drag. To this end, the predictions obtained using the frequency and time domain descriptions are compared to each other requiring that displacement amplitudes are equal for the same external force. The coefficients A1 and A2 are identified by minimizing the differences between the predictions. The additional forces associated with a flow through the riser 18 are neglected assuming that the flow is sufficiently slow. The effective tension in the riser has static and dynamic components. The static component of the tension results from the pretension imposed by the heave compensator and submerged weight. In the offshore industry it is common practice 17 See for full explanation Chapter 5 of the thesis. In Chapters 2 and 3 of the thesis it is proven that the internal flow-induced instability for a pinnedpinned beam can only occur for relatively high speeds. 18 120 Chapter 7 to use a pretension that is 1.3 times the submerged weight of the riser. The dynamic component of the tension is caused by the heaving platform. Owing to the assumption of inextensibility, this tension component depends only on time. Assuming that the platform vibrates harmonically, the effective tension can be expressed as: Te ( z , t ) = Wa ( − z + fL ) − ka sin ( Ωt ) , (7.2) where Wa is the submerged weight of the riser per unit length, f is a dimensionless pretension factor (in this chapter, f =1.3), L is the length of the riser, k is the stiffness of the heave compensator, a and Ω are the amplitude and frequency of platform heave. The boundary conditions at the ends of the riser, assuming that the connection of the heave compensator to the riser can be modelled as a hinge, are given as ∂ 2 w ( z, t ) ∂ 2 w ( z, t ) = = 0. ∂z 2 ∂z 2 z =0 z=L w ( 0, t ) = w ( L, t ) = 0, (7.3) Zero transverse motion at the top of the riser assumes no surge (horizontal motion) of the platform. Note that a small surge would not affect the results of linear stability analysis and only marginally the nonlinear one. A large surge would cause an additional variation of the riser tension and, consequently, would alter the results of both linear and nonlinear analyses. Using the following dimensionless variables and parameters: = Ω M EI L2 , α = W L3 / EI , η = w L , ξ = z L , τ = t EI M L2 , Ω a κ = kaL2 EI , ς = µ L2 MEI , γ = ρ f Do L ( 2 M ) , the equation of motion (upon substitution of Eq. (7.2)) and the boundary conditions are rewritten as: 2 2 ∂ 4η ∂ 2η τ ) ∂ η + α ∂η + ∂ η + ς A ∂η + γ A ∂η ∂η = 0, (7.4) − α ( f − ξ ) 2 + κ sin ( Ω 1 2 4 2 ∂ξ ∂ξ ∂ξ ∂ξ ∂τ 2 ∂τ ∂τ ∂τ η ( 0,τ ) = η (1,τ ) = 0, ∂ 2η (ξ ,τ ) ∂ 2η (ξ ,τ ) = = 0. ∂ξ 2 ξ = 0 ∂ξ 2 ξ =1 (7.5) Analysis of Eqs. (7.4) and (7.5) is subdivided into two parts. In the next section, the nonlinearity in Eq. (7.4) is disregarded to determine the domain of system parameters which correspond to instability of the straight shape of the riser. Then, in Section 7.4, the development of this instability is studied taken the nonlinearity into account. Destabilization of deep-water risers by a heaving platform 121 7.3 Stability of the straight configuration Stability of the straight shape of the riser is determined by the linearized ( A2 = 0 ) equation of motion. To find the instability domain in the space of system parameters, it is customary to employ the Galerkin method. This method allows to decompose the partial differential equation of motion of the riser into an infinite set of ordinary differential equations with respect to time. Owing to the third term in Eq. (7.4), the latter equations will have coefficients that depend on time periodically. This fact calls for application of the Floquet theory (Nayfeh and Mook; 1970), which is a convenient method for determining system parameters that break the parameter space into “stability” and “instability” domains. If the system parameters belong to the former domain, the static (and the only) equilibrium of the riser is stable. On the contrary, if the system parameters belong to the “instability domain”, the straight configuration is unstable and, given an initial perturbation, the energy of vibration of the riser would grow in time until the hydrodynamic nonlinearity would bring the energy input and the energy loss in balance. This section is structured as follows. First, the linearized equation of motion is decomposed into a set of linear differential equations with periodic in time coefficients. Then, the Floquet theory is shortly described and applied numerically. Finally, the resulting stability charts are plotted and discussed focusing on the underlying physical mechanisms of instability. 7.3.1 Galerkin decomposition of linearized equation of motion According to the Galerkin method, solution to the linearized Eq. (7.4) is sought for in the form: ∞ η (ξ ,τ ) = ∑ φm (ξ ) qm (τ ) , (7.6) m =1 where qm (τ ) are unknown time-dependent functions and φm (ξ ) are spacedependent shape functions forming a complete orthogonal set. In this Chapter, φm (ξ ) are chosen as the eigenfunctions of the following linear differential operator: Lξ = d4 d2 d −α ( f −ξ ) 2 +α , 4 dξ dξ dξ (7.7) satisfying the boundary conditions, Eq. (7.5). The corresponding eigenvalues λm are all imaginary and in the following are represented by the real-valued eigenfrequencies Ω m = iλm . The eigenfrequencies and eigenfunctions of Lξ can be 122 Chapter 7 found numerically or in a closed form using a power series expansion19 (Kuiper and Metrikine; 2005). Substituting Eq. (7.6) into the linearized Eq. (7.4), multiplying through by φn (ξ ) , and integrating over ξ ∈ ( 0,1) , it is found that ∞ d 2 qn dqn 2 τ ) B q = 0, A q + ς + Ω + κ sin Ω ( ∑ n n mn m 1 dτ 2 dτ m =1 d 2φm =∫ φn d ξ dξ 2 0 1 Bmn 1 ∫φ n 2 dξ , (7.8) n, m = 1, 2,..., ∞. 0 This infinite set of ordinary differential equations can be approximated by a finite system of 2N first order coupled equations: ⎧ dyn ⎪⎪ dτ = y N + n ⎨ ∞ τ) B y , ⎪ dy N + n = −ς A1 y N + n − Ω n 2 yn − κ sin ( Ω ∑ mn m m =1 ⎩⎪ dτ (7.9) n, m = 1, 2,..., N , where n and N stand for, respectively, the mode number and the total number of modes considered. In matrix notation, Eq. (7.9) can be written as: y = A (τ ) y , (7.10) where y = [ y1 , y2 ,..., y2 N ] , the overdot means ordinary derivative with respect to τ , and A (τ ) is a 2N by 2N matrix, which is periodic in time, i.e. A (τ ) = A (τ + 2π Ω ) . Stability of the equilibrium of Eq. (7.10) can be studied employing the Floquet theory, which is briefly outlined in Appendix C. A rigorous description of this theory can be found in Nayfeh and Mook (1970). T In short, the theory states that for a system of first order linear ordinary differential equations with periodic coefficients, like Eq. (7.10), the stability of the system can be determined by solving the following matrix differential equation: (τ ,0 ) = A (τ ) ⋅ Φ (τ , 0 ) , Φ (7.11) with initial conditions Φ ( 0, 0 ) = I , where I is the identity matrix. Stable behaviour of the system is guaranteed if all eigenvalues λi of the state transition matrix Φ (τ ,0 ) at the end of one period T satisfy the following stability criterion: λi ≤ 1, 19 See for more details Chapter 4 of the thesis. i = 1, 2,..., 2 N . (7.12) Destabilization of deep-water risers by a heaving platform 123 This criterion is applied in the next sub-section to find the instability zones of the riser on the basis of Eq. (7.9). 7.3.2. Determination of instability zones Stability of the straight configuration of the riser is studied in the plane of the frequency and displacement amplitude of the harmonic heave motion of the platform. The frequency Ω is varied between 0.1 rad/s and 2.2 rad/s, whereas the amplitude a is varied between 0.0 m and 5.0 m. In the chosen frequency range, sea waves contain the most energy. The other parameters of the system are chosen as follows. The stiffness k of the heave compensator in practice is commonly tuned to compensate for the submerged weight of the riser in case that the platform heaves with a given critical amplitude ac, usually set at 10 meters. This means that a realistic value of k is given as k = LWa ac . (7.13) In this section the stiffness of the heave compensator will be varied to investigate its effect on stability of the riser. Kuiper et al. (2007) estimated the coefficient A1 for the amount of linear viscous fluid damping at 27.0·103. This estimation was based on experiments with the Stokes parameter being around 106 . The Stokes parameter is defined as β = Do2 (υ T ) , where T is the period of oscillation and υ is the kinematic viscosity of water. Since the Stokes parameter for the deep-water riser that is considered in this Chapter is in the order of 103 − 104 , depending on the mode of vibration, a lower value of A1 is to be expected. Besides, the outer wall roughness of the riser influences the value of A1 . To cope with these uncertainties, three values of A1 are considered, namely 6.75·103, 13.5·103 and 27.0·103. The values of the other parameters are shown in Table 7.1. Table 7.1 - Values of the system parameters used in calculations E 2.1·1011 N·m-2 ρf 1.025·103 kg·m-3 L 2000 m µ 1.00·10-3 kg·m-1·s-1 Do 0.25 m M 136.7 kg·m-1 Di 0.22 m Wa 735.3 N·m-1 f 1.3 ac 10 m 124 Chapter 7 (1) (2) 5 5 4 4 2ω 1 3 a (m) a (m) 3 2 2 2ω 2 2ω 3 1 1 2ω 4 2ω 5 0 0 0.5 1 2ω 6 2ω 7 2ω 8 1.5 0 2 0 0.5 Ω (rad/s) 1 1.5 2 Ω (rad/s) (3) 5 4 a (m) 3 2 1 0 0 0.5 1 1.5 2 Ω (rad/s) Fig. 7.2 - Stability charts obtained from the linearized equation of motion with k = LWa ac ; (1) A1 = 6.75 ⋅ 103 , (2) A1 = 13.5 ⋅ 103 and (3) A1 = 27.0 ⋅ 103 . Fig. 7.2 shows stability charts for the three damping coefficients A1. The dotted domains correspond to unstable behaviour of the riser. The stability charts have been computed by considering the first 10 modes ( N = 10 ) . With the chosen parameters (see Table 7.1), the first natural frequency is 0.137 rad/s, whereas the tenth is 1.38 rad/s. The relatively wide distinct instability zones in Fig. 7.2 correspond to the first order parametric resonance, for which the relation Ω = 2ωn holds ( ωn is the n-th natural frequency of the riser). The positions of these zones are Destabilization of deep-water risers by a heaving platform 125 specified in the top left chart of Fig. 7.2. The narrower zones are related to either combined or higher order parametric resonance conditions. Comparing the three stability charts in Fig. 7.2, it can be concluded that the instability zones shrink, if the linear viscous damping is increased. The distinct first order parametric zones disappear for sufficiently large linear viscous damping. In the absence of damping the instability zones would begin at the horizontal axis. Besides, many more very narrow instability zones associated with higher order and combined parametric resonance would appear. (1) (2) 5 4 4 3 3 a (m) a (m) 5 2 2 1 1 0 0 0.5 1 1.5 0 2 Ω (rad/s) 0 0.5 1 1.5 2 Ω (rad/s) (3) 5 4 a (m) 3 2 1 0 0 0.5 1 1.5 2 Ω (rad/s) Fig. 7.3 - Stability charts obtained from the linearized equation of motion for different spring constants of the heave compensator; (1) k = LWa ( 2ac ) , (2) k = LWa ac and (3) k = 2 LWa ac for a linear damping coefficient of A1 = 13.5 ⋅ 103 . 126 Chapter 7 From a practical viewpoint it is interesting to investigate the effect of the stiffness of the heave compensator on the riser stability. Fig. 7.3 presents three stability charts obtained for three values of this stiffness. This figure shows that the stiffness of the heave compensator influences stability of the riser dramatically. The higher the stiffness, the larger the instability zone in the chosen practically relevant parameter domain. Therefore, though a stiffer heave compensator is much easier to construct in practice, relatively soft heave compensators are recommended to ensure stability of the riser. 7.3.3 Preliminary discussion of the mechanisms of instability Though the Floquet theory is a convenient tool for studying stability of the straight configuration of the riser, this theory does not enlighten in detail the physical mechanisms of this instability. It is clear, for example, that instability of the riser is not necessarily related to parametric resonance but also might be caused by local buckling of the riser. Indeed, the mean value of the tension of the riser decreases with the distance from the sea surface. Therefore, the platform, if heaving with sufficiently large amplitude, can change the tension into compression near the sea bottom. If the local compression cannot be counteracted by the restoring forces, the riser will lose stability. In order to check whether local buckling near the sea bottom might occur, a static analysis is carried out to compute the maximum platform set down beyond which the riser buckles. Neglecting the time dependent terms in Eq. (7.4) and considering the floating platform in the lowest position, i.e. when the riser top tension has its lowest value, the following equation governing static equilibrium of the riser can be deduced: ∂ 4η ∂ 2η ∂η − (α ( f − ξ ) − κ ) 2 + α =0. 4 ∂ξ ∂ξ ∂ξ (7.14) The general solution of this equation can be written in terms of the Airy functions of the first, Ai (ψ ) , and the second kind, Bi (ψ ) : ξ ξ 0 0 η (ξ ) = C1 + C2 ∫ Bi (ψ 1 ) d ξ1 + C3 ∫ Ai (ψ 1 ) d ξ1 ξ1 ξ1 ⎛ ⎞ + C4 ∫ ⎜ -Ai (ψ 1 ) Bi (ψ 1 ) ∫ Bi (ψ 2 ) d ξ2 ∫ Ai (ψ 2 ) d ξ2 ⎟ d ξ1 , ⎜ ⎟ 0⎝ 0 0 ⎠ ξ (7.15) where ψ 1 = (α ( f − ξ1 ) − κ ) α 2 3 and ψ 2 = (α ( f − ξ2 ) − κ ) α 2 3 . Substituting Eq. (7.15) into the boundary conditions, Eq.(7.5), a system of four linear algebraic equations can be obtained with respect to the unknown constants C1-C4. Static instability occurs if the determinant of this system of equations equals zero. This determinant Destabilization of deep-water risers by a heaving platform 127 is studied numerically by gradually increasing the value of κ. The calculation shows that buckling will occur if κ=14.3, which corresponds to a vertical set down of 3.14 m using parameters in Table 7.1 and k = LWa ac . At first glance, it might seem remarkable that in Fig. 7.2 some areas are stable for amplitudes exceeding this value. However, it is known that parametric excitation can stabilize dynamical systems. For frequencies tending to zero, the instability boundary originates from the ‘static buckling amplitude’ of 3.14 m. It is not visible in Fig. 7.2 because the lowest frequency considered is 0.1 rad/s. The effect of local buckling is investigated in more detail in the next section. 7.4 Nonlinear analysis of instability development If the straight shape of the riser is stable, any perturbation of this shape would slowly decay and the riser would return to its straight configuration. On the contrary, if the straight shape is unstable, any perturbation would lead to a temporary growth of the energy of transverse vibration of the riser. This growth would proceed until the energy loss caused by the nonlinear hydrodynamic damping would balance the energy input by the heaving platform, leading to a dynamic equilibrium. In this section, this dynamic equilibrium is studied numerically and conclusions are drawn as to different scenarios leading to this equilibrium. 7.4.1 Nonlinear time domain analysis of the dynamic equilibrium To study the dynamic equilibrium that the riser reaches if its straight configuration is unstable, the original nonlinear equation of motion, Eq.(7.4), with the boundary conditions, Eq.(7.5), are discretized in space using a second-order central difference approximation. The riser is given an initial sinusoidal deflection: η (ξ , 0 ) = ηˆ sin (ξπ ) and ∂η (ξ ,τ ) =0, ∂τ τ =0 (7.16) where ηˆ = 0.01 . The length of the riser L is divided into S segments, all having a length of h = L S . The first node is at z = 0 and the (S+1)th node is at z = L. To incorporate the boundary conditions, at each end one dummy node is added. This discretization gives a system of (S+1) second-order nonlinear ordinary differential equations with respect to time, or, equivalently, a system of 2(S+1) first-order ordinary differential equations. In the calculations the riser is divided into 100 segments. This latter system is solved numerically by using the Runga-Kutta-Verner fifth order method. 128 Chapter 7 Before giving results of numerical analysis, it is important to discuss the major physical limitation of Eq. (7.4). Though this equation is nonlinear, the only nonlinearity taken into account in this equation is related to the hydrodynamic drag. No geometrical nonlinearities are accounted for, which limits applicability of Eq. (7.4) to relatively small rotations and curvatures of the riser. Since the bending stiffness has minor effect on dynamics of deep water risers, the major limitation to be concerned with is the rotation of the riser. The geometric nonlinearity related to 2 large rotations can be neglected if ( ∂w ∂z ) << 1 , which is the classical condition of small transverse vibrations of taut strings. In order to distinguish the domain in the parameter space within which the condition of small rotations is violated, the limit 2 of ( ∂w ∂z ) = 0.2 was chosen. If during a numerical run this limit was exceeded, the run was interrupted and the resulting riser amplitude was set to 7.5 m. No physical significance is attached to this value, which is chosen merely to achieve good graphical representation of the results. The parameter domain that is distinguished by this value of riser amplitude needs additional analysis which takes geometric nonlinearities of the riser into account. Employing the above described numerical approach, three-dimensional charts are obtained that are shown in Fig. 7.4 (a-c). The horizontal axes in these figures show the frequency and heave amplitude of the platform, whereas the vertical axis gives the maximum deflection along the riser in the dynamic equilibrium. These charts are obtained using the same parameters as used in the three charts in Fig. 7.2. Fig. 7.4.a - Maximum deflection of the riser versus the amplitude and frequency of the heaving platform for A1 = 6.75 ⋅ 103. Destabilization of deep-water risers by a heaving platform 129 Fig. 7.4.b - Maximum deflection of the riser versus the amplitude and frequency of the heaving platform for A1 = 13.5 ⋅ 103. Fig. 7.4.c - Maximum deflection of the riser versus the amplitude and frequency of the heaving platform for A1 = 27.0 ⋅ 103. 130 Chapter 7 2 4.5 3 4 1 3.5 3 2.5 2 1.5 1 0.5 0.5 1 1.5 2 Ω (rad/s) Fig. 7.5 - Three areas of qualitatively different destabilization mechanisms of the riser ( A1 = 13.5 ⋅ 103 , k = LWa ac ). An in-depth study of Fig. 7.4 and corresponding dynamic deflections of the riser allows to visually distinguish three qualitatively different scenarios leading to the dynamic equilibrium of the riser (the corresponding frequencies and amplitudes of the platform are indicated in Fig. 7.5). These scenarios can be referred to as • Classical parametric resonance (area 1 in Fig. 7.5 ); • Sub-critical local dynamic buckling (area 2 in Fig. 7.5); • Super-critical local dynamic buckling (area 3 in Fig. 7.5). In the following sub-sections the specific features of each of these scenarios are discussed in detail. 7.4.2. Classical parametric resonance The frequency and amplitude of the platform that correspond to destabilization of the riser via classical parametric resonance correspond to area 1 in Fig. 7.5. Subject to these frequencies and amplitudes of excitation, the riser gains energy (owing to simultaneous variation of tension) along its entire length and loses stability at a specific mode or a combination of modes. Destabilization of deep-water risers by a heaving platform 131 0.8 0.6 Deflection w (m) 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 200 240 280 320 360 Time (s) Fig. 7.6 - Transverse deflection of the midpoint of the riser versus time for a = 3.0 m, Ω = 1.6 rad/s ( A1 = 13.5 ⋅ 103 , k = LWa ac ). 0 Depth (m) -400 -800 -1200 -1600 -2000 -1.5 -1 -0.5 0 0.5 1 1.5 Deflection w (m) Fig. 7.7 - Shape of the riser in the dynamic equilibrium for a = 3.0 m, Ω = 1.6 rad/s ( A1 = 13.5 ⋅ 103 , k = LWa ac ). As an example, a deep-water riser subjected to a platform that heaves with amplitude of 3.0 m and frequency of 1.6 rad/s is considered. This frequency can be shown to be about twice the sixth natural frequency of small vibrations of the riser. The deflection of the midpoint of the riser is depicted as a function of time in Fig. 132 Chapter 7 7.6 starting from a moment when the influence of initial conditions has almost disappeared. This figure shows that, in accordance with the condition of the main zone of classical parametric resonance, the dominant frequency of vibration equals about half the excitation frequency. Correspondingly, in the dynamic equilibrium the riser vibrates primarily in its sixth mode, as can be seen in Fig. 7.7. Note that in this figure the influence of spatially varying tension is clearly visible. Due to the tension decrease with the distance from the sea surface, the amplitude of the riser increases with this distance, whereas the internodal spacing decreases. 7.4.3. Sub-critical local dynamic buckling For frequencies and heave amplitudes of the platform corresponding to area 2 in Fig. 7.5, the riser experiences what can be called ‘sub-critical local dynamic buckling’. This type of instability occurs if the amplitude of the heaving platform is sufficiently large to change tension into compression at a segment of the riser during a part of the vibration period. The resulting compressive force should be large enough to overcome restoring forces and cause local buckling of the riser. This buckling occurs near the sea bottom because the tension of the riser is the smaller the closer the riser cross-section is to the sea bottom. The time interval within which the buckling occurs is limited to a part of the vibration cycle that corresponds to a compressive force in the riser. During this time interval, the riser deflection grows locally and starts propagating along the riser. Then, the riser becomes tensioned again and the buckling-induced local disturbance propagates along the riser gaining no energy. In the so-called ‘sub-critical local dynamic buckling’ regime, this disturbance is not amplified by the next local buckling that happens after one period of vibration but travels along the riser until it vanishes due to the hydrodynamic damping. Thus, in the regime under consideration a sequence of distinct deflection pulses travels along the riser without overlap. Obviously, in the case that the excitation frequency is low enough ( Ω ≤ 0.5 rad/s ), these pulses do not amplify each other so that the buckling occurs not too often. Therefore, the area in Fig. 7.5 that corresponds to this regime occupies the region of relatively high amplitudes and low frequencies. To exemplify the riser vibration in the case of sub-critical local dynamic buckling, the deflection of a riser point close to the sea bottom (at z = 0.9L) is shown in Fig. 7.8 as a function of time. The amplitude and the frequency of the platform heave are, respectively, 4.5 m and 0.2 rad/s. Fig. 7.8 clearly shows a sequence of short pulses that pass through the chosen cross-section. These pulses are very similar to each other and occur periodically. One can see that between the pulses the riser deflection is nearly zero, which implies that the pulses do not amplify each other. Destabilization of deep-water risers by a heaving platform 133 2 1.5 Deflection w (m) 1 0.5 0 -0.5 -1 -1.5 -2 200 240 280 320 360 Time (s) Fig. 7.8 - Transverse deflection of a point of the riser at z = 0.9 L versus time for a = 4.5 m, Ω = 0.2 rad/s ( A1 = 13.5 ⋅ 103 , k = LWa ac ). Initiation and propagation of the pulses that result from sub-critical local dynamic buckling is illustrated in Fig. 7.9 that shows the shape of the riser at three successive instants. Fig. 7.9(a) shows local buckling near the sea bottom that results in a relatively large deflection of the riser. Figs. 7.9(b) and 7.9(c) show two successive instants when the riser is everywhere in tension and the deflection pulse excited by the buckling travels upward. While propagating, this pulse widens owing to dispersion and loses energy because of the hydrodynamic damping. Upon reflection from the upper end of the riser, this pulse would travel downward but by the time it would reach the sea bottom its energy would be negligible and the contribution of this pulse to the riser deflection in the dynamic equilibrium would vanish. 7.4.4. Super-critical local dynamic buckling The main difference between the sub-critical and super-critical local dynamic buckling is the frequency of the buckling events. In the latter regime this frequency is larger so that the buckling-generated pulses are not separated in space as in the sub-critical case. In the regime of super-critical local dynamic buckling, the amplitude and frequency of the heave of the platform correspond to area 3 in Fig. 7.5. Within this area both the amplitude and frequency are relatively high. 134 Chapter 7 (b) 0 0 -400 -400 -800 -800 Depth (m) Depth (m) (a) -1200 -1200 -1600 -1600 -2000 -6 -4 -2 0 2 4 -2000 6 -4 -6 Deflection w (m) -2 0 2 Deflection w (m) 4 6 (c) 0 Depth (m) -400 -800 -1200 -1600 -2000 -6 -4 -2 0 2 4 6 Deflection w (m) Fig. 7.9 - Shape of the riser at three successive time moments for a = 4.5 m, Ω = 0.2 rad/s ( A1 = 13.5 ⋅ 103 , k = LWa ac ). To exemplify this instability regime, the typical shape of the riser in dynamic equilibrium is shown in Fig. 7.10 for a = 4.5 m and Ω = 1.5 rad/s. Obviously, in contrast to the sub-critical regime, one can not distinguish separate bucklinginduced pulses. Instead, the riser vibrates in a certain combination of its modes. This vibration is reached, however, not via classical parametric resonance but via periodic generation of buckling-induced pulses near the bottom that is followed by decomposition of these pulses (due to dispersion) into a combination of riser modes. Destabilization of deep-water risers by a heaving platform 135 0 Depth (m) -400 -800 -1200 -1600 -2000 -0.8 -0.4 0 0.4 0.8 Deflection w (m) Fig. 7.10 - Shape of the riser in the dynamic equilibrium for a = 4.5 m, Ω = 1.5 rad/s ( A1 = 13.5 ⋅ 103 , k = LWa ac ). 4 3 3 2 Velocity ∂w/∂t (m/s) Velocity ∂w/∂t (m/s) 2 1 0 -1 1 0 -1 -2 -2 -3 -4 -2 -1.5 -1 -0.5 0 0.5 Deflection w (m) 1 1.5 2 -3 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 Deflection w (m) Fig. 7.11 - Phase portraits of the riser motion at z = 0.9L in case of sub-critical (left) and super-critical (right) local dynamic buckling ( A1 = 13.5 ⋅ 103 , k = LWa ac ). The difference between the sub- and super-critical local dynamic buckling can also be clearly illustrated by phase plots, as shown in Fig. 7.11 for a point close to the sea bottom, z = 0.9L. The frequencies of vibration of the platform in the left and right plots are 0.2 rad/s and 1.5 rad/s, respectively. In both cases the platform heave amplitude is 4.5 m. For sub-critical local dynamic buckling (the left figure) the 136 Chapter 7 phase trajectories first run away from the origin and then, quickly, return back to the vicinity of the origin where they stay for some time. This behaviour of the trajectories corresponds to generation of the pulse, its propagation away from the generation area and the “silent time” before the next buckling occurs. In the regime of super-critical local dynamic buckling (the right figure) the phase trajectories in the dynamic equilibrium do not approach the origin. This is because each new buckling-induced pulse is generated before the previous ones are dissipated by the hydrodynamic damping. 7.5 Bending stresses in the riser From the viewpoint of design of deep-water risers, it is more important to calculate stresses in the riser than its deflections. A heaving platform induces bending, axial and radial stresses in the riser. Here only the maximum bending stress in dynamic equilibrium is considered and compared to the yield stress of steel, which is around 3.8·108 N·m-2 (=380 MPa). The bending stress at the outer diameter is calculated using the following equation: σb (z) = E ∂ 2 w ( z ) Do , ∂z 2 2 (7.17) where σ b ( z ) is the bending stress and E is elasticity modulus of steel. The same range of amplitudes and frequencies of vibration of the platform is considered as in the previous sections, i.e. 0.1 rad/s ≤ Ω ≤ 2.2 rad/s and 0 ≤ a ≤ 5.0 m. As in the 2 previous section, the calculations are interrupted if ( ∂w ∂z ) > 0.2 and in this case the maximum riser bending stress is put to 5.0·108 N·m-2 (=500 MPa) (no physical significance is attached to this value). The bending stress is computed using the parameters in Table 7.1, k = LWa ac , A1=13.5·103 and A2=0.24. The maximum bending stress along the riser, subjected to the chosen range of platform motions, is shown in Fig. 7.12. From this figure it can be concluded that classical parametric resonance (area 1 in Fig. 7.5) of the riser corresponds to insignificant bending stress, while local dynamic buckling (areas 2 and 3 in Fig. 7.5) is a real threat for the riser, since the bending stress exceeds the yield stress. Destabilization of deep-water risers by a heaving platform 137 Fig. 7.12 - Maximum bending stress in the riser versus the amplitude and frequency of the heaving platform ( A1 = 13.5 ⋅ 103 , k = LWa ac ). 7.6 Conclusion In this chapter, stability of a straight deep-water riser connected to a heaving floating platform has been considered. It is shown that a vertical harmonic motion of the platform can result in a loss of stability of the riser. Three distinct mechanisms of stability loss may occur, namely classical parametric resonance, sub-critical local dynamic buckling and super-critical local dynamic buckling. The two latter mechanisms are the most dangerous for the riser since they result in very high bending stress in the riser, which might even exceed the yield stress. A necessary condition for the local dynamic buckling is that the riser is statically unstable at the moment of the maximum set down of the platform. CHAPTER 8 Main results, practical relevance and recommendations 8.1 Main results The main goal of the present work is to improve the understanding of the dynamic stability of submerged cantilever pipes aspirating fluid. For pipes conveying fluid there are two possible sources of instability. The first possibility is that unstable waves are generated in the medium. This kind of instability occurs at relatively high fluid velocities. A second possible source of instability are wave reflections at the boundaries of the pipe-flow system. 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, like for the cantilever pipe aspirating fluid. The energetics of a pipe conveying fluid has attracted considerable attention of researchers because of the profound effect the fluid flow can have. To understand the energy exchange at the boundaries of a pipe conveying fluid the travelling wave method has been used in literature. The expression for the energy reflection coefficient at a boundary of a pipe conveying fluid, as introduced by Lee and Mote (1997a,b) has been corrected in this thesis to make it applicable to dispersive translating continua. Employing this method, it has been proven that the energy gain of the cantilever pipe aspirating fluid originates from both the upstream free end and the downstream fixed end. For even the simplest pipe-flow systems, like the simply-supported pipe conveying fluid, there did not exist in literature a generalized energy expression which could predict an energy increase for aperiodic processes. This is remarkable since the primary instability mechanism of the simply-supported pipe conveying fluid is divergence, which is an aperiodic process. In this thesis, the energy equation has been derived that is in one-to-one correspondence with the linearized equation of motion. It is shown that the linear statement of the problem implies that an external force has to be applied at every cross section of the pipe to maintain the constant 140 Chapter 8 flow speed through the vibrating pipe. In systems where wave reflections at the boundaries do not contribute to the net energy exchange of the system, e.g. for a simply-supported pipe conveying fluid, it is the work of this effective force that enables energy of the pipe to increase. Until 2004, the accepted theory predicted unstable behaviour for slightly damped cantilever pipes at small fluid flow velocity, whereas experiments did not show any instability. It was commonly thought that this contradiction is due to negative pressurization of the fluid at the inlet of the pipe that was not accounted for in the theory. In this thesis it is pointed out that this explanation was at best incomplete. Even when the negative pressurization is accounted for the theory still predicts instability. This finding initiated a re-thinking of this contradiction and initiated a few new explanations. In this thesis it is proven that the external hydrodynamic drag caused by the surrounding fluid is a major stabilizing factor. In literature, data of the hydrodynamic drag have been presented based on experiments with submerged rigid cylinders. To make them applicable to flexible pipes, a key step in this thesis is undertaken to translate the data from the frequency domain to the time domain. Using the time-domain description of the hydrodynamic drag, it is shown that the drag caused by the surrounding fluid prevents instability of the cantilever pipe at low fluid flow velocities. Another explanation for the existing contradiction between theory and experiments was given by Païdoussis et al. (2005). They proved that the correct description of the flow field in the vicinity of the tip is of major importance for the stability analysis of the pipe conveying fluid. We fully support this finding. In order to verify the new theories a test set-up was built in which the fluid velocity through the pipe could exceed the critical velocities predicted by different theories. Our experiment showed that the pipe is stable below a critical velocity of water convection through the pipe. Contrary to all existing theories, for velocities above this critical value the pipe shows a complex motion that consists of two alternating phases. One phase is a nearly periodic orbital motion with displacement amplitude of a few pipe diameters, whereas the second one is a noise-like vibration with very small deflections. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion as well as in a larger amplitude of the noise-like vibration, albeit to a smaller extent. The unexplainable complex motion of the cantilever pipe was the key reason conduct a second set of experiments. In particular, the aim of the second set experiments was to check whether the internal fluid flow was indeed the cause the observed instability. The main modifications with respect to the first set to of of of Main results, practical relevance and recommendations 141 experiments were (i) the larger distance between the pipe and the tank wall and (ii) the avoidance of the direct flow from the return cylinder to the entrance of the cantilever pipe. In the new test set-up, the motion of the pipe consisted again of the two alternating phases for internal fluid velocities exceeding a critical value. However, the main difference with respect to the first set of experiments was the much smaller amplitude of the orbital motion in the second set of experiments. The reason for the reduction in amplitude of the orbital motion was and is still unclear. Existing theories predict the dominating frequencies quite well. However, they do not correctly predict the critical velocity and the pipe behaviour in the unstable regime. It seems that the flow field in the vicinity of the inlet is of great importance for the correct prediction of the dynamic behaviour of the cantilever pipe conveying fluid. In the past it was common to assume a zero shear force as a boundary condition at the free inlet. Now it is clear that this is an oversimplification of reality. Assuming a zero shear force at the free inlet predicts lower critical fluid velocity than observed in the experiments. Several variants for the balance of shear forces at the free end were proposed by Païdoussis et al. (2005). One variant assumes that the flow field just below the entrance does not move transversely with the pipe and assumes a spatially averaged inflow direction tangential to the deflected pipe. We consider this variant as interesting, since it also predicts instability through flutter. However, this boundary condition predicts a higher critical fluid velocity than observed in the experiments. We attempted to improve the existing models, to simulate the observed intermittent behaviour of the experimental pipe. In one hypothetical concept, the motion of a part of the fluid beneath the pipe was incorporated in a simplistic way. The orbital motion of the pipe tip might induce a rotational motion of a part of the fluid beneath the pipe. In that case, the inflow of the fluid into the pipe is affected by this rotational fluid motion, and hence, the boundary conditions should depend on this rotational motion. This effect was modeled by using an artificial disk with rotational inertia. For some of these simplistic concepts, the solution is bivalent, i.e. for the same internal fluid velocity in one case the pipe does not move perceptibly, while in the other case the pipe response consists of an orbital motion. This is new in comparison to all existing models. This thesis gives rise to a number of follow-up questions. It is proven experimentally that a free hanging pipe aspirating fluid can become unstable. Several theories are developed in this thesis, however there is still a gap between theory and experiments. Based on the gathered knowledge, it seems that the description of the fluid flow at the intake is of crucial importance for correct stability predictions. A new attack is necessary to solve this problem, which turns out to be more complicated than originally thought. 142 Chapter 8 8.2 Practical relevance For the offshore industry it would be valuable if one could predict whether a free hanging water intake riser behaves stable at a certain operating fluid velocity. Since there is still a mismatch between small-scale experiments and theory, it is difficult to translate the obtained knowledge to stability predictions of a real-scale offshore water intake riser. As a consequence, the discussion in this section should be used with caution until further research is carried out. For design purpose it would be interesting to indicate a maximum operating fluid velocity below which no flow-induced instability occurs. In this regime flowinduced fatigue damage in the riser material could be disregarded. The critical fluid velocity, indicating the transition from stable to unstable behavior, might be considered as the maximum operating fluid velocity. However, theory and smallscale experiments disagree about the value of the critical fluid velocity. The critical velocity based on a simple model of the fully submerged free hanging riser in combination with the inflow at the free end modeled by the conventional boundary condition (like the description in Eq. (6.13)) would lead to a fairly conservative approach for the maximum operating fluid velocity. The reason to consider this as a conservative approach is that the critical velocity predicted by this theory was much lower than observed during experiments. For a fully submerged water intake riser of steel with a length of 150 m, an outer diameter of 1.07 m (42.0 inch) and a wall thickness of 0.0254 m (1.00 inch) this conservative approach would lead to a maximum operating velocity of only 0.5 m/s. However, the current concepts of a floating LNG plant require large quantities of cooling water with a maximum required rate of water intake in the order of 75,000 m3/hour. Consequently, large diameter risers in combination with a flow velocity around 2-3 m/s are needed to limit the number of water intake risers to less than ten. Limited by the current knowledge, one can only speculate what happens with the free hanging riser for these internal fluid velocities. If the critical fluid velocity for the 150 m long riser is based on a theoretical model with a boundary condition as proposed by Païdoussis et al. (2005) (like the description in Eq. (6.14)) one would find a value of 17 m/s. Hence, the preferred fluid velocity around 2-3 m/s lies between the two critical fluid velocities of 0.5 and 17 m/s predicted by theory using, respectively, the conventional boundary condition and the boundary condition proposed by Païdoussis et al. This would imply that the 150 m long riser with an internal fluid velocity around 2-3 m/s may show similar instability behaviour as observed in the tank, i.e. a complex motion that consists of an orbital motion alternated by noise-like vibrations. Without any doubt, the influence of other external loading mechanisms, like wave loading, currents and ship motions, on the riser stability should be investigated. Main results, practical relevance and recommendations 143 8.3 Recommendations It is quite remarkable that such a fundamental problem as the stability of a cantilever pipe aspirating fluid is not understood anno 2008. To bridge the gap between theory and experiments it is inevitable to work on both. In this section some thoughts about possible approaches are suggested. First, some challenges related to future experiments are discussed. It would be quite interesting to see what happens if the ratio of the maximum attainable fluid velocity in small-scale experiments to the critical fluid velocity is increased further. Several burning questions are still unanswered: • When the fluid velocity is increased further, do the two alternating phases (periodic flutter and noise-like vibration) still appear or does the pipe “choose” for one of these two phases? • It was shown that an increase of internal fluid velocity results in a larger amplitude of the orbital motion. Does this trend still exist for larger fluid velocities than currently tested or is there some kind of self-limited mechanism as observed in e.g. vortex-induced-vibrations of cylindrical structures? Based on theory, it seems that the modeling of the flow field in the vicinity of the tip is of great importance for the correct prediction of the dynamic behaviour of the cantilever pipe conveying fluid. It is not straightforward to visualize the flow field near the pipe entrance in the experiments, although this would yield important insight for correct modeling of the boundary conditions. One way to visualize the flow field near the entrance is by using particle image velocimetry (PIV). PIV is an optical measurement technique based on the determination of the displacement of tracer particles inserted into the flow (see e.g. Raffel et al., 1998). It consists in illuminating a slice of the flow field (using laser light sources) and in recording two or more images of the tracer particles at successive instants (using digital cameras). The displacement of the tracers can be determined through an analysis of the particle images. Assuming that the particles are reliable flow tracers, the velocity of the fluid at the location of a particle is equal to the velocity of this particle. In addition to the suggested experimental improvements, some theoretical challenges are discussed below. To help decide about the correct assumptions regarding the flow field in the vicinity of the inlet, computational fluid dynamics (CFD) might be used. Païdoussis et al. (2005) state that they would like to advance in this direction by initiating a study using ANSYS (finite element analysis). Due to the high Reynolds number this is not a simple task. To our knowledge there is no paper where the dynamic stability of a 144 Chapter 8 submerged cantilever pipe, either aspirating or discharging fluid from the free end, has been modeled using CFD. In a private communication Païdoussis explained that even in the most simple case where the fluid is discharged from free end, there was no pipe motion observed, though experiments and analytical methods clearly show flutter type of instability for internal fluid velocities exceeding a critical value. Hence, although CFD seems to be an ideal tool to gain insight in this problem, some problems still have to be tackled. Based on the results of this thesis, it is clear that the analytical models have to be improved further. We strongly believe that the description of the boundary conditions at the free end is currently the weakest part of the analytical models. It is very likely that a nonlinear description is needed, like the model with the artificial fluid disk below the pipe entrance. In the most ideal situation, a relatively simple model would be developed which describes the dynamic stability of the pipe aspirating fluid and explains the physical mechanisms of the two alternating phases, (a) periodic flutter and (b) noise-like vibration, as well as the transitions between these. References Anagnostopoulos, A. (2002) Flow-induced vibrations in engineering practice. WitPress, Southampton. Ashley, H. and Haviland, G. (1950) Bending vibrations of a pipe line containing flowing fluid. Journal of Applied Mechanics 17, 229-232. Bearman, P.W. and Mackwood, P.R. (1992) Measurements of the hydrodynamic damping of circular cylinders. Proceedings of the 6th International Conference on the Behavior of Offshore Structures, London, 405-414. Bearman, P.W. and Russell, M.P. (1996) Measurements of the hydrodynamic damping of bluff bodies with application to the prediction of viscous damping of TLP hulls. Proceedings of the 21st Symposium on Naval Hydrodynamcis, Washington D.C., U.S.A., 622-634. Benjamin, I. (1961) Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proceedings of the Royal Society A 61, 457-486. Bourrières, F.-J. (1939) Sur un phénomène d’oscillation auto-entretenue en mécanique des fluides réels. Publications Scientifiques et Technique du Ministère de l’Air, No.147. Chaplin, J.R. and Subbiah, K. (1998) Hydrodynamic damping of a cylinder in still water and in a transverse current. Applied Ocean Research 20, 251-259. Chaplin, J.R. (2000) Hydrodynamic damping of a cylinder at β ≈106. Journal of Fluids and Structures 14, 1101-1117. Chatjigeorgiou, I.K. and Mavrakos, S.A. (2002) Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications. Applied Ocean Research 24, 341-354. Chatjigeorgiou, I.K. and Mavrakos, S.A. (2005) Nonlinear resonances of parametrically excited risers – numerical and analytical investigation for Ω = 2ω1 Computers and Structures 83, 560 – 573. 146 References Chen, S.S. and Rosenberg, G.S. (1971) Vibrations and stability of a tube conveying fluid. Argonne National Laboratory Report ANL-7762, Argonne, Illinois, USA. Chernov, L.A. (1946) Flow and density of acoustical energy in a moving medium (in Russian). Zhurnal Tekhnicheskoi Fiziki 16, 733-736. Translated as TG 230-T291 (1962), APL Library Bulletin, The Johns Hopkins University, Applied Physics Laboratory, Silver Spring Maryland. Chung, J.S., Whitney, A.K. and Loden, W.A. (1981) Nonlinear transient motion of deep ocean mining pipe. ASME Journal of Energy Resources Technology 103, 2-10. De Langre, E. and Ouvrard, A.E. (1999) Absolute and convective bending instabilities in fluid-conveying pipes. Journal of Fluids and Structures 13, 663-680. Doaré, O. and De Langre, E. (2006) The role of boundary conditions in the instability of one-dimensional systems. European Journal of Mechanics B Fluids 25, 948-959. Done, G.T.S. and Simpson, A. (1977) Dynamic stability of certain conservative and nonconservative systems. I. Mech. E. Journal of Mechanical Engineering Science 19, 251263. Elmore, W.C. and Heald, M.A. (1969) Physics of Waves, McGraw-Hill, New York. Feodos’ev, V.P. (1951) Vibrations and stability of a pipe when liquid flows through it. Inzhenernyi Sbornik 10, 169-170. Floquet, G. (1883) Sur les équations différentielles linéaires à coefficients périodiques, Annales scientifiques de l’École Normale Supérieure Sér. 2. 12, 47-88. Forrester, A.T. (1986) Inverse sprinklers: A lesson in the use of a conservation principle. American Journal of Physics 54, 798-799. Fuchs, B.A., Shabat, B.V. and Berry, J. (1961-1964) Functions of a complex variable and some of their applications, Pergamon, Oxford. Ginzburg, V.L. and Tsytovich, V.N. (1990) Transition Radiation and Transition Scattering, Hilger, Bristol. Gregory, R.W. and Païdoussis, M.P. (1966a) Unstable oscillation of tubular cantilevers conveying fluid. I. Theory. Proceedings of the Royal Society A 293, 512-527. Gregory, R.W. and Païdoussis, M.P. (1966b) Unstable oscillation of tubular cantilevers conveying fluid. II. Experiments. Proceedings of the Royal Society A 293, 528-542. Goldstein, H. (1980) Classical Mechanics, 2nd edition, Addison-Wesley, Massachusetts. Holmes, P.J. (1977) Bifurcations to divergence and flutter in flow-induced oscillations: a finite-dimensional analysis. Journal of Sound and Vibration 53, 471-503. References 147 Holmes, P.J. (1978) Pipes supported at both ends cannot flutter. Journal of Applied Mechanics 45, 619-622. Hongwu, C. and Junji, T. (1996) Effect of boundary conditions on the stability of a cantilever pipe discharging and aspirating fluid. JSME International Journal Series C, 39 (1), pp. 20-24. Housner, G.W. (1952) Bending vibrations of a pipe line containing flowing fluid. Journal of Applied Mechanics 19, 205-208. Hsu, C.S. (1975) The response of a parametrically excited hanging string in fluid. Journal of Sound and Vibration 39, 305-316. Hsu, L. (1988) Inverse sprinklers: Two simple experiments and the resolution of the Feynman-Forrester conflict. American Journal of Physics 56, 307-308. Huang, T. and Dareing, D.W. (1976) Natural frequencies of marine drilling risers. Journal of Petroleum Technology, 813-818. Justesen, P. (1988) Hydrodynamic forces on large cylinders in high Reynolds number oscillatory flow. BOSS conference, 805-819. Kangaspuoskari, M., Laukkanen, J. and Pramila, A. (1993) The effect of feedback control on critical velocity of cantilevered pipes aspirating fluid. Journal of Fluids and Structures 7, 707-715. Korn, G.A. and Korn, T.M. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York. 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. (2004) On stability of a clamped-pinned pipe conveying fluid. Heron 49, 211-232. 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. 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). Kuiper, G.L., Metrikine, A.V. and Battjes, J.A. (2007) A new time-domain drag description and it influence on the dynamic behaviour of a cantilever pipe conveying fluid. Journal of Fluids and Structures 23, 429-445. Lee, S.-Y. and Mote, C.D. (1997a) A generalized treatment of the energetics of translating continua, part I: strings and tensioned pipes. Journal of Sound and Vibration 204, 717-734. 148 References Lee, S.-Y. and Mote, C.D. (1997b) A generalized treatment of the energetics of translating continua, part II: beams and fluid conveying pipes. Journal of Sound and Vibration 204, 735-753. McIver, D.B. (1973) Hamilton’s principle for systems of changing mass. Journal of Engineering Mathematics 7, 249-261. 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. Metrikine, A.V. and Dieterman, H.A. (1997) Instability of vibrations of a mass moving uniformly along an axially compressed beam on a visco-elastic foundation, Journal of Sound and Vibration 201, 567-576. Möhring, W. (1978) Acoustic energy flux in nonhomogeneous ducts. Journal of the Acoustical Society of America 64, 1186-1189. Morfey, C.L. (1971) Acoustic energy in non-uniform flows. Journal of Sound and Vibration 14, 159-170. Naudascher, E. and Rockwell, D. (1994) Flow-induced vibrations; an engineering guide. A.A. Balkema, Rotterdam. Nayfeh, A.H. and Mook, D.T. (1970) Nonlinear Oscillations, Jon Wiley and Sons, New York. Neimark, Y.I. (1978) Dynamic systems and controllable processes, Nauka, Moscow (in Russian). Neimark, Y.I., Golumov, V.I. and Kogan, M.M. (2003) Mathematical models in natural science and engineering,. Springer, Berlin. Niordson, F.I. (1953) Vibrations of a cylindrical tube containing flowing fluid. Kungliga Tekniska Hogskolans Handlingar (Stockholm) No.73. Païdoussis, M.P. (1970) Dynamics of tubular cantilevers conveying fluid. Journal Mechanical Engineering Science 12, 85-103. Païdoussis, M.P. (1998) Fluid-Structure Interactions: Slender Structures and Axial Flow, Vol. 1. Academic Press, London. Païdoussis, M.P. (1999) Aspirating pipes do not flutter at infinitesimally small flow. Journal of Fluids and Structures 13, 419-425. Païdoussis, M.P. (2004) Fluid-Structure Interactions: Slender Structures and Axial Flow, Vol. 2. Academic Press, London. References 149 Païdoussis, M.P. (2005) Some unresolved issues in fluid-structure interactions. Journal of Fluids and Structures 20, 871-890. Païdoussis, M.P. (2008) The canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across Applied Mechanics. Journal of Sound and Vibration 310, 462-492. Païdoussis, M.P. and Issid, N.T. (1974) Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration 33, 267-294. Païdoussis, M.P. and Luu, T.P. (1985) Dynamics of a pipe aspirating fluid, such as might be used in ocean mining. ASME Journal of Energy Resources Technology 107, 250-255. Païdoussis, M.P., Semler, C., Wadham-Gagnon, M. (2005) A reappraisal of why aspirating pipes do not flutter at infinitesimal flow. Journal of Fluids and Structures 20, 147-156. Park, H.I. and Jung, D.H. (2002) A finite element method for dynamic analysis of long slender marine structures under combined parametric and forcing excitations. Ocean Engineering 29, 1313-1325. Patel, M.H. and Park, H.I. (1991) Dynamics of tension leg platform tethers at low tension. Part I – Mathieu stability at large parameters. Marine Structures 4, 257-273. Patel, M.H. and Park, H.I. (1995) Combined axial and lateral responses of tensioned buoyant platform tethers. Engineering Structures 17, 687-695. Pippard, A.B. (1992) Momentum and pseudo-momentum: 1. Classical pseudo-momentum and wave pressure. European Journal of Physics 13, 82-87. Raffel, M., Willert, C. and Kompenhans, J. (1998) Particle Image Velocimetry – A Practical Guide, Springer-Verlag, Berlin. Rayleigh, Lord (1902) On the pressure vibrations. Philosophical Magazine 3, 338-346. Rodenbusch, G. and Kallstrom, C. (1986) Forces on a large cylinder in random twodimensional flow. OTC, Houston. Roth, W. (1964) Instabilität durchströmter Rohre. Ingenieur-Archiv 33, 236-263. Rowland, D.R. and Pask, C. (1999) The missing wave momentum mystery. American Journal of Physics 67, 378-388. Sällström, J.H. and Åkesson, B.Å. (1990) Fluid-conveying damped Rayleigh-Timoshenko beams in transverse vibration analyzed by use of an exact finite element. Part I: theory. Journal of Fluids and Structures 4, 561-572. Sarpkaya, T. (1986) Force on circular cylinder in viscous oscillatory flow at low KeuleganCarpenter numbers. Journal of Fluid Mechanics 165, 61-71. 150 References Sarpkaya, T. (2001) Hydrodynamic damping and quasi-coherent structures at large Stokes numbers. Journal of Fluids and Structures 15, 909-928. Stein, R.A. and Tobriner, W.M. (1970) Vibrations of pipes containing flowing fluids. Journal of Applied Mechanics 37, 906-916. Stokes, G.G. (1851) On the effect of the internal friction fluids on the motion of pendulums. Transactions of Cambridge Philosophical Society 9, 8-106. Suzuki, H., Takano, K., Enomoto, K. and Oka, N. (2004) Axial and lateral response of a deepsea riser for scientific drilling, Proceedings of Twenty Third International Conference on Offshore Mechanics and Arctic Engineering, Vancouver, British Columbia, Canada, OMAE2004-51094. Verichev, S.N. and Metrikine, A.V. (2002) Instability of a moving bogie on a flexibly supported Timoshenko beam, Journal of Sound and Vibration 253, 653-668. Vesnitskii, A.I., Kaplan, L.E. and Utkin G.A. (1983) The laws of variation of energy and momentum for one-dimensional systems with moving mountings and loads. PMM Journal of Applied Mathematics and Mechanics 47, 692-695. Vesnitskii, A.I. and Lisenkova, E.E. (2002) Transformation of elastic wave energy to the energy of motion of bodies, Acoustical Physics 48, 34-38. Wang, C-Y (1968) On high frequency oscillating viscous flows. Journal of Fluid Mechanics 32, 55-68. Appendix A Spectral energy density of a pulse Considering a time moment when the pulse is located so far from the boundaries that it is not disturbed by their presence, the energy of the pulse E can be computed by integrating the energy density e ( x, t ) over space from minus to plus infinity: ∞ E= ∫ e ( x, t ) dx . (A.1) −∞ By inserting the energy density of the pipe conveying fluid, ⎛ ∂2w ⎞ 1 ⎛ ⎛ ∂w ⎞ ∂w ⎞ ⎛ ∂w ⎛ ∂w ⎞ e ( x, t ) = ⎜ m p ⎜ + mf ⎜ + uf +T ⎜ + EI ⎜ 2 ⎟ ⎟ ⎟ ⎟ 2 ⎜ ⎝ ∂t ⎠ ∂x ⎠ ⎝ ∂t ⎝ ∂x ⎠ ⎝ ∂x ⎠ ⎝ 2 2 2 2 ⎞ ⎟, ⎟ ⎠ (A.2) into this expression, and taking into account that this displacement can also be represented by its complex conjugate: w ( x, t ) = ∞ ∫ ( ) w (ω ) exp i (ω t − k (ω ) x ) d ω = −∞ ∞ ∫ w (ω ) exp ( −i (ω t − k (ω ) x )) dω * 1 1 1 1 , (A.3) −∞ the following expression can be obtained: ∞ ∞ ∞ E= { 1 2 ∫ −∞∫ −∞∫ ( m f + m p ) ω ω1 + (T + m f u f ) k (ω ) k (ω1 ) − 2m f u f ω k (ω1 ) 2 −∞ } + EI k (ω ) k (ω 1 ) w (ω ) e 2 2 1 i (ωt − k (ω ) x ) w (ω 1 ) e * ( ( )) − i ω 1t − k ω 1 x (A.4) d ωd ω1dx. The integral over x in Eq. (A.4) can be evaluated using the following representation of the Dirac delta-function (Korn and Korn, 1961): ∞ ∫ exp ( ±iα x ) dx = 2 π δ ( ±α ) , (A.5) −∞ to give E =π ∞ ∞ ∫ ∫ {( m f + m p ) ω ω 1 + ( T + m f u f 2 ) k (ω ) k ( ω 1 ) − 2m f u f ω k (ω 1 ) −∞ −∞ } + EI k (ω ) k (ω 1 ) w (ω ) w (ω 1 ) e 2 2 1 * ( i ω t −ω 1t ) ( ) δ k (ω ) − k (ω 1 ) d ω d ω 1 . (A.6) 152 Appendix A To calculate the integral over ω1 in Eq. (A.6), the following property of the deltafunction can be used (Ginzburg and Tsytovich, 1990): ∞ ∫ δ ( k (ω ) − k (ω ) ) f (ω )d ω 1 1 −∞ 1 = f (ω1 ) dk (ω 1 ) d ω1 . (A.7) ( ) k ω 1 = k (ω ) Employing Eq. (A.7), the expression for the energy of the pulse can be reduced to E =π ∞ ∫ {( m f + m p ) ω 2 + ( T + m f u f 2 ) k 2 (ω ) −∞ −2m f u f ω k (ω ) + EI k (ω )} cgr (ω ) w (ω ) d ω , (A.8) 2 4 where the group velocity cgr (ω ) = d ω dk (ω ) is introduced. Equation (A.8) can be simplified further by making use of the dispersion equation and by noting that the resulting integrand is an even function of the frequency. This simplification yields ∞ ∞ E = 4π ∫ k (ω ) ( T + EIk 2 (ω ) ) cgr (ω ) w (ω ) d ω = ∫ Eω (ω ) d ω , 2 0 where Eω is the spectral energy-density of the pulse. 2 0 (A.9) Appendix B Depressurization at the inlet Consider the streamlines at the intake of a pipe as shown in Fig. B.1. Far away from the pipe the fluid velocity is assumed to be zero. The fluid pressure is considered with respect to the hydrostatic pressure. Applying the Bernoulli equation along a streamline from an undisturbed area to cross section 1 results in: p0 + 1 1 ρU 02 = p1 + ρU12 2 2 ⇒ 1 p1 = − ρU12 , 2 (B.1) in which p0 and U 0 are the undisturbed fluid pressure and undisturbed fluid velocity, respectively, and p1 and U1 are the fluid pressure and fluid velocity at cross section 1, respectively. From Eq. (B.1) it is clear that due to the difference in fluid velocities there is a pressure drop at the intake. 1 2 p0=0, U0=0 Fig. B.1 – Sketch of streamlines at the intake of a pipe. A few pipe diameters downstream of cross section 1 we introduce cross section 2, where all streamlines are parallel. In order to establish a relation between the contraction coefficient µ at cross section 1 and the pressure at cross section 2, p2 , a momentum balance between cross sections 1 and 2 is applied, p1 A + ρU12 µ A = p2 A + ρU 22 A , (B.2) in which A is internal pipe area. Taking into account a volume balance between cross sections 1 and 2, µ AU1 = AU 2 , and substituting Eq. (B.1) into Eq. (B.2) the pressure inside the pipe is equal to: 154 Appendix B ⎛ 1 1 ⎞ p2 = ρU 22 ⎜ −1 + − 2 ⎟ . 2 µ µ ⎝ ⎠ (B.3) For a pipe as sketched in Fig. B.1 the contract coefficient µ is close to ½. In this case the pressure drop is equal to p2 = − ρU 22 . (B.4) Modification of the geometry of the inlet can reduce the pressure loss. If the inlet is shaped so that the fluid flow remains fully attached to the wall, e.g. using a bellmouth, the contraction at the inlet is avoided, µ = 1 . In this case the pressure drop inside the pipe is reduced to 1 p2 = − ρU 22 . 2 (B.5) Appendix C Floquet Theory In this appendix a short explanation of the Floquet theory is given. A detailed description and examples can be found in the book of Nayfeh and Mook (1970). The method is applicable to determine the stability of linear ordinary differential equations with periodic coefficients. Any linear ordinary differential equation with periodic coefficients can be written as a system of first order linear differential equations: y = A (τ ) y , (C.1) where A (τ ) is periodic in time, i.e. A (τ ) = A (τ + T ) . The fundamental matrix solution U (τ ) is defined as the matrix with linearly independent solutions to Eq. (C.1): U (τ ) = ⎡⎣ x1 (τ ) , x 2 (τ ) ,.., x N (τ )⎤⎦ (C.2) where x i (τ ) is a vector containing the i-th independent solution to Eq. (C.1). The Floquet theorem (Floquet; 1883) states that the fundamental matrix solution U (τ ) of Eq. (C.1) is given as U (τ ) = P (τ ) exp (τ F ) , (C.3) and F is where P (τ ) is a periodic N × N matrix function with the period T = 2π Ω a N × N matrix (possibly complex). As follows from Eq. (C.3), the fundamental matrix solution U (τ ) grows in time if and only if at least one of the eigenvalues of F has a positive real part. To find the eigenvalues of F numerically, it is convenient to introduce the state transition matrix Φ (τ ,τ 0 ) : Φ (τ ,τ 0 ) = U (τ ) U −1 (τ 0 ) . (C.4) Substituting the fundamental matrix solution into this expression and setting τ = T ,τ 0 = 0 , one obtains Φ ( T , 0 ) = P ( T ) exp ( T F ) P −1 ( 0 ) = P ( 0 ) exp (T F ) P −1 ( 0 ) (C.5) 156 Appendix C Stability of linear systems and, in particular, matrix F is independent of initial conditions. Therefore, any initial conditions can be employed in numerical evaluation of F . As can be seen from Eq. (C.5), it is convenient to choose initial conditions such that U ( 0 ) = P ( 0 ) = I , where I is the identity matrix. In this case, Eq. (C.5) reduces to Φ (T , 0 ) = exp (T F ) (C.6) Therefore, with the initial conditions U ( 0 ) = P ( 0 ) = I, matrix F can be expressed through the state transition matrix over the period T : F = T −1 ln ( Φ ( T ,0 ) ) (C.7) The state transition matrix Φ ( T ,0 ) can be found by solving numerically Eq. (C.1) in the time interval 0 ≤ τ ≤ T . Eq. (C.1) has to be solved N times, each time giving one of the components of the vector y the unit initial value, while keeping all other initial values zero. For example, to obtain the first row of the fundamental matrix solution, one should take y1 ( 0 ) = 1 and yi ( 0 ) = 0, i = 2, 3,..., N ; the second row is obtained by taking y2 ( 0 ) = 1 and yi ( 0 ) = 0, i = 1, 3, 4,..., N and so forth. Once the state transition matrix Φ ( T ,0 ) is composed, the eigenvalues λi , i = 1, 2,..., N of this matrix have to be found. According to Eqs. (C.6) and (C.7), if the absolute value of λi is larger than one, the corresponding eigenvalue of matrix F has a positive real part. Therefore, a system governed by Eq. (C.1) is stable if the absolute value of all eigenvalues of Φ ( T , 0 ) is not larger than one: λi ≤ 1, i = 1, 2,..., N . (C.8) Correspondingly, at the boundary of the stability zone in the space of the system parameters the absolute value of at least one of these eigenvalues equals one, whereas the absolute values of all other eigenvalues are not larger than one. Summary Stability of Offshore Risers Conveying Fluid The offshore industry develops more often large gas fields far away from the shore. To eliminate the cost of expensive pipelines, a new concept has been developed in which the gas is liquefied offshore on a barge. To be able to liquefy gas, large quantities of cooling water are required. Depending on the location, to be sufficiently cold, this water has to be pumped up from depths ranging from 150 to 500 meters below sea level. The pumping takes place through a set of steel waterintake risers, whose top is attached to the barge, whereas the lower end is free. Until now, this type of risers, conveying a large amount of water and having an unconstrained end, has not been used in the offshore industry and, accordingly, there is no experience with the dynamic behaviour of such risers. In addition to the conventional issues associated with the dynamics of deep-water risers, such as vortex- and wave-induced vibrations, free-hanging water intake risers are sensitive to a less-known problem of the dynamic instability due to this intake. It is well known that a cantilever pipe, which conveys fluid from the fixed end to the free end, becomes unstable when the fluid velocity exceeds a critical value. A well-known example is the unstable behaviour of a garden hose. However, for the reversed flow direction, i.e. from the free end towards the fixed end, there is no consensus among researchers as to occurrence or the mechanism of instability and the water velocity at which the instability may occur. For the offshore industry the latter situation is of clear relevance, since this describes the free-hanging water intake risers. The main objective of this thesis is to improve the understanding of the dynamic stability of submerged cantilever pipes aspirating fluid. Until 2004, the accepted theory predicted unstable behaviour for slightly damped cantilever pipes at small fluid velocity, whereas experiments did not show any instability. It was hypothesized that this contradiction was due to depressurization of the fluid at the inlet of the pipe, a phenomenon that was not accounted for in the theory. In this thesis it is pointed out that this explanation is at best incomplete. Even when the depressurization is accounted for, the theory still predicts instability. This finding stimulated us to reconsider the apparent contradiction, leading to a few more adequate explanations. In this thesis it is proven that the external hydrodynamic drag caused by the surrounding fluid is a major stabilizing factor, which prohibits the pipe instability at low flow speeds. In addition, a correct description of the flow field in the vicinity of the tip is shown to be of great 158 Summary importance for an appropriate stability analysis of the pipe conveying fluid. In the past it was common to assume a zero shear force as a boundary condition at the free inlet. Now it is clear that this is an oversimplification of reality. In this thesis it is shown that the stability predictions depend strongly on whether the flow field just below the entrance is assumed to move transversely with the pipe or not. In order to verify the new explanations an experimental set-up has been built which allows the fluid velocity through the pipe to be increased over the critical velocities as predicted by different theories. The experiments clearly show that the pipe is stable below a critical fluid velocity. Contrary to all existing theories, for velocities above this critical value the pipe shows a complex motion that consists of two alternating phases. One phase is a nearly periodic orbital motion with a displacement amplitude at the tip of a few pipe diameters, whereas the other one is a noise-like vibration with very small deflections. Increasing the internal fluid velocity results in a larger amplitude of the orbital motion as well as in a larger amplitude of the noise-like vibration, albeit to a smaller extent. Since none of the existing theories predict this kind of behaviour, we have attempted to adapt existing models. To simulate the observed intermittent behaviour of the experimental pipe, several hypothetical concepts have been developed. For some of these concepts, the solution is bivalent, i.e. for the same internal fluid velocity in one case the pipe does not move perceptibly, while in the other case the pipe response consists of an orbital motion. This is new in comparison to all existing models. However, more research is necessary to fully explain this phenomenon, which turns out to be more complicated than originally thought. Guido Kuiper Samenvatting Stabiliteit van stijgbuizen voor vloeistoffen zoals gebruikt in de offshore industrie De offshore-industrie ontwikkelt steeds vaker grote gasvelden ver uit de kust. Om kosten van dure pijpleidingen te elimineren is een nieuw concept ontwikkeld, waarbij het gas offshore op een schip wordt gecondenseerd. Hiervoor is een grote hoeveelheid koelwater nodig. Afhankelijk van de locatie moet het water opgepompt worden van een diepte tussen de 150 en 500 meter om voldoende koud te zijn. Het water zal worden aangevoerd door vrijhangende (vanaf het platform) stalen stijgbuizen. Dit type stijgbuizen, die een grote hoeveelheid water doorvoeren en een vrij uiteinde hebben, zijn tot nu toe nog niet toegepast in de offshore industrie, zodat nog geen ervaring is opgedaan met het dynamisch gedrag ervan. Behalve de bekende problemen van stijgbuizen in diep water, zoals trillingen opgewekt door wervels en golven, blijkt de vrijhangende stijgbuis gevoelig te zijn voor een minder bekend dynamisch instabiliteitsverschijnsel veroorzaakt door de inname van een grote hoeveelheid water. Het is bekend dat een éénzijdig ingeklemde buis, waardoor vloeistof stroomt van de ingeklemde zijde naar het vrije uiteinde, instabiel wordt wanneer de vloeistofsnelheid een kritieke waarde overschrijdt. Een bekend voorbeeld hiervan is het slingergedrag van een losse tuinslang. Echter, in geval van de tegenovergestelde stromingsrichting, dus als de vloeistof bij het vrije uiteinde de buis wordt ingezogen, zijn de onderzoekers het nog niet eens over het optreden en het mechanisme van de instabiliteit noch over de vloeistofsnelheid waarboven instabiliteit op zou kunnen treden. Voor de offshoreindustrie is deze laatste situatie zeer relevant, omdat die overeenkomt met de vrijhangende buizen die koelwater moeten innemen. Het belangrijkste doel van deze dissertatie is beter begrip te krijgen van het dynamisch gedrag van een ondergedompelde vrijhangende buis die water opzuigt. Tot 2004 voorspelde de gangbare theorie instabiel gedrag voor lichtgedempte vrijhangende buizen bij lage vloeistofsnelheden, terwijl experimenten geen enkel teken van instabiliteit toonden. Deze tegenstrijdigheid werd verondersteld een gevolg te zijn van het feit dat de verlaging van de waterdruk bij het vrije uiteinde niet in de theorie in rekening werd gebracht. In deze dissertatie wordt aangetoond dat deze verklaring op z’n minst onvolledig is. Zelfs als de negatieve waterdruk in rekening wordt gebracht voorspelt de theorie nog steeds instabiliteit. Deze uitkomst heeft tot heroverweging van de genoemde tegenstrijdigheid geleid en vervolgens tot een paar nieuwe verklaringen. In deze dissertatie wordt bewezen dat 160 Samenvatting de externe hydrodynamische demping, veroorzaakt door het omringende water, een belangrijke stabiliserende factor is, die instabiliteit van de buis voor lage vloeistofsnelheden voorkomt. Ook is daarbij gebleken dat een goede beschrijving van de vloeistofstroming in de buurt van het vrije uiteinde van groot belang is voor een juiste stabilteitsanalyse van de buis. In het verleden was het vanzelfsprekend om als randvoorwaarde de dwarskracht aan het vrije uiteinde gelijk te stellen aan nul. Het blijkt nu dat dit een te simpele beschrijving van de werkelijkheid is. In dit proefschrift is aangetoond dat de stabiliteitsvoorspellingen sterk afhangen van de veronderstelling of de vloeistofstroming onder het vrije einde transversaal met de buis meebeweegt of niet. Om de nieuwe verklaringen te verifiëren is een testopstelling gebouwd, waarin de kritieke vloeistofsnelheid door de buis, zoals voorspeld door de verschillende theorieën, kon worden overschreden. De experimenten lieten duidelijk zien dat de buis stabiel is beneden een kritieke vloeistofsnelheid. In tegenstelling tot alle bestaande theorieën vertoonde de buis boven deze kritieke vloeistofsnelheid een complexe beweging, die uit twee afwisselende fasen bestond. Eén fase is een bijna periodieke, orbitale beweging met een verplaatsingsamplitude van het einde van de pijp van een paar keer de buisdiameter, terwijl de andere fase bestaat uit een ruisachtige trilling met erg kleine verplaatsingsamplitudes. Een toenemende interne vloeistofsnelheid resulteert in een grotere amplitude van de orbitale beweging en, zij het in mindere mate, ook in een grotere amplitude van de ruis-achtige trilling. Omdat geen enkele van de bestaande theorieën dit gedrag voorspelt, is een poging gedaan de bestaande theorieën te verbeteren. Om het afwisselende gedrag van de experimentele buis te simuleren zijn verschillende hypothetische concepten ontwikkeld. Voor enkele concepten is de oplossing tweewaardig. Bij eenzelfde interne vloeistofsnelheid beweegt in het éne geval de buis niet merkbaar, terwijl die in het andere geval een orbitale beweging uitvoert. Dit is nieuw in vergelijking met alle bestaande modellen. Meer onderzoek is noodzakelijk om dit verschijnsel, dat gecompliceerder is dan aanvankelijk gedacht, volledig te kunnen verklaren. Guido Kuiper Curriculum Vitae Guido Kuiper was born in Maarheeze, the Netherlands, on 30 November 1977. After having graduated from Philips van Horne Scholengemeenschap in Weert, he attended in 1996 the Faculty of Civil Engineering and Geosciences of Delft University of Technology. In the beginning of 2002, he received Master of Science in Civil Engineering (cum laude), specializing in applied mechanics with emphasis on structural dynamics. He received an award for the best student graduated in 2001-2002 from the Faculty of Civil Engineering by the Universiteitsfonds Delft. During his study he was a student-assistant for the second year course on general dynamics of systems at the Faculty of Civil Engineering. After his graduation he worked one year fulltime as an offshore structures engineer in the Civil/Marine group in Shell. Most projects were related to the design of deepwater riser systems. This year was also used to write and submit a research proposal to Technologiestichting STW. This research proposal was granted and in 2003 he started to work on his doctoral research in Wave Mechanics Group of the Faculty of Civil Engineering and Geosciences. The PhD study was a co-operation between Delft University of Technology and Shell. Three days a week were spent at the University on the doctoral thesis, whereas the other two days were spent in Shell on closely related projects. Doing both theory and practice at the same time resulted in a fruitful bridge between the academic world and the offshore industry.

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