Vortex-induced vibration on flexible cylinders

Transcription

1 Chapter 5 Vortex-induced vibration on flexible cylinders J.R. Meneghini 1, F. Saltara 1, R.A. Fregonesi & C.T. Yamamoto 1 NDF, Department of Mechanical Engineering. Department of Naval and Ocean Engineering, Escola Politécnica, University of Sao Paulo, Brazil. Abstract Vortex shedding and vortex-induced vibration (VIV) concepts are shown and discussed in this chapter. The basic knowledge regarding the mechanics of vortex shedding from bluff bodies and its relation to VIV is presented in the first sections. Some of the classical experimental and numerical results are shown, together with applications to marine technology. Special attention is paid to the notion of energy transfer and the differences between forced and free oscillating systems. In the final sections of this chapter, simulations of a marine riser exposed to VIV are shown and the results analysed. 5.1 Introduction In recent years, the need to improve technologies in the field of deep-water oil extraction has led to an increase in studies of the vortex-induced vibration (VIV) phenomenon. Some challenging VIV problems have become common in particular engineering areas. One of those is present in the field of deep-water oil extraction. Recently, offshore oil platforms have been installed in water depths of over 000m. Flexible pipes named risers are used to link the seabed to the 165 doi:10.495/ /05

2 offshore platform for oil production. Risers are cylindrical bodies subjected to vibration due to vortex shedding, and one of the main effects of this vibration is to shorten their durability. Therefore, there is great interest in the development of tools to estimate the characteristics of VIV, and to evaluate its effect on the fatigue of structures and their life expectancy. In the first sections of this chapter, we present some fundamental aspects in connection with VIV and the physics related to the phenomenon. With risers presenting such high aspect ratios and complex flow fields around them, a complete three-dimensional simulation under realistic conditions is unfeasible. With this in mind and aiming at the hydroelastic response of the riser structure, a numerical model in a quasi-three-dimensional fashion is developed in this chapter. By a quasi-three-dimensional approach, we mean that the hydrodynamic forces are evaluated in two-dimensional strips, and the coupling between those strips occurs through a three-dimensional structural calculation. A finite-element structural model based on the Euler Bernoulli beam theory is employed in order to accomplish the task of integrating the hydrodynamic forces calculation and the dynamic response of the cylinder. A general equation of motion is solved through a numerical integration scheme in the time domain. The technique used for evaluating the hydrodynamic forces is the discrete-vortex method (DVM). A practical case of a vertical marine riser is presented. In this case, the results for various uniform and shear currents acting on a single-flexible cylinder, representing a riser with a length equal to 10m with 100m under water, are shown. Envelopes of maximum and minimum in-line and transverse displacements are presented. Our intention is to apply this model to problems occurring in the offshore industry. The risers are slender structures that are subject to shear and oscillatory flows due to currents and waves, flows with a very high degree of complexity, with intensity and direction changing with the depth of water. We believe that the main contribution of this study is to integrate known methods in an innovative way. Building a practical CFD tool to investigate the hydroelastic problems shown in this chapter is of great interest for the oil industry. The life expectancy of the marine risers is significantly affected by vortex-induced vibration. The algorithm developed in this research is especially efficient in cases with very long risers. The results obtained here show the expected behaviour of a riser immersed in a uniform current, and a very good agreement when compared to experimental results of a cantilever flexible cylinder subjected to VIV. Recent investigations have focused on obtaining a reliable procedure to simulate vortex-induced vibration on long cylinders. Wildden and Graham [1] employed a hybrid scheme to examine fluid structure interaction on marine 166

3 risers. The structural part of their scheme is similar to the one employed in our simulations. However, they used a vortex method with a hybrid formulation that required a mesh to evaluate the diffusion of vorticity. Even though their scheme is very accurate, it is more demanding from a computation point of view. A similar approach using a vortex method can be found in the work of Herfjord et al. []. The flow around an elastic cylinder with an aspect ratio of about 60 has also been studied by Wang et al. [3] employing a finite-element method. Other references that should be cited are the extensive work by Evangelinos et al. [4] and Lucor et al. [6]. A spectral element method, as described by Karniadakis and Sherwin [7], has been employed to study the flow structure interaction. They utilized a fully three-dimensional code to investigate vortexinduced vibration. Their simulations were performed at low Reynolds numbers, and showed the applicability of this high-order method for fundamental studies. Patel and Witz [8] and Ferrari [9] developed a structural model for a single riser based on the finite-element technique applied to beam elements. Following Ferrari, a static model is used to obtain the solution for the eigenvalue problem and to evaluate the eigenfrequencies and the eigenmodes. With this approach, the nodes are free to rotate and to displace in the vertical, transverse and in-line directions. The results presented in our investigation are based on the structural model explained briefly above. The hydrodynamic forces are assessed through the DVM, as developed and employed by Spalart et al. [10], Park and Higuchi [11] and Meneghini [1]. This method is a Lagrangian numerical scheme technique for simulating two-dimensional, incompressible and viscous fluid flow. It employs the stream-function-based boundary integral method and incorporates the growing core size or core-spread method in order to model the diffusion of vorticity. An extensive review about vortex methods can be found in Sarpkaya [13]. This chapter is divided into six sections. An introduction to the vortexinduced vibration topic is shown in the first two sections. A description of the structural model employed to evaluate the dynamic response of marine risers is given in the following section. After that explanation, the hydrodynamic model utilized in the calculations is shown and discussed in detail. Calculations for a flexible cantilever model are carried out and compared with the experimental results by Fujarra [14]. Then, the model is applied to a marine riser case. Finally, in the last section, conclusions are drawn based on the results presented. 5. Vortex shedding from bluff bodies A bluff body is defined by Bearman [5] as being one which when placed in a fluid stream, generates separated flow over a substantial proportion of its surface. The occurrence of separation on a two-dimensional bluff-body flow 167

4 causes the creation of two free shear layers in the rear region of the body. These two layers carry opposite-sign vorticity and they constitute an unstable system when perturbations are present and certain conditions are satisfied. The interactions between the two shear layers are non-linear and represent the essential reason for the vortex-wake formation behind the body. The flow about bluff bodies and its periodic vortex shedding has fascinated researchers since Leonardo da Vinci. The first evidence of vortex shedding can be traced back to the times of the Ancient Greeks who described the Aeolian Tones as a tensioned string vibrated in the wind. However, it was only in 1878 that Strouhal made the first scientific investigation into the subject. He studied a vibrating wire placed in a free-stream of air and concluded that the frequency of the Aeolian tone, f s, resulting from the motion of the wire varied only with the diameter (D) and the velocity (U) of the relative motion (Rayleigh [16]). He also found that when the Aeolian tones coincided with one of the natural tones of the wire, which was free to oscillate, the sound was greatly reinforced. The relation that he found could be expressed by U fs = (5.1) D In 1879 Lord Rayleigh [16], while observing a violin string vibrating due to the flow of air about it, found that the vibration of the string took place in a plane normal to the direction of the flow. Previously, it was thought that the vibration would occur in the plane contained by the direction of the flow. But it was not until 1896 that Lord Rayleigh defined a non-dimensional quantity that became known as the Strouhal number fd s St =. (5.) U The next important historical point about the study of bluff-body flow can be related to the work of Kármán [17], which is summarised in Goldstein [18]. The double vortex rows that form behind a two-dimensional bluff body can be idealised, for an inviscid fluid, so that each vortex is assumed to be concentrated along a line at right angles to the plane of motion, as is presented in Fig Kármán showed that this ideal arrangement for two infinite rows of vortices is stable only if certain conditions are satisfied. The double row of vortices would be stable for any slight displacement of the vortices and then could keep its configuration unchanged, only if a certain value for the relation between the longitudinal and transverse distance of each row is 168

5 satisfied (see Fig. 5.1). Employing stability concepts it is possible to prove that for a stable configuration one must have π hw sinh = 1, (5.3) a which yields the well-known relation w h a w w = (5.4) In addition, the circulation Γ carried by each vortex must satisfy the relation Γ π h = = (5.5) Ua 1 w tanh, v w aw where U v denotes the velocity of the vortices relatively to the undisturbed fluid. In recognition of the importance of Kármán s findings and his work on the subject, a double row of vortices of this kind is usually called a Kármán vortexstreet. However, this analysis was carried out in the absence of a body and no viscous effects or turbulence were considered. Kármán was aware of the shortcomings of the analysis but, at the time, he was only concerned with the stability properties of the wake behind a body and the possibility of representing it by an ideal distribution of vortices. Figure 5.1: The double row of vortices formed behind a bluff body as idealised by Kármán [17] Body geometry and Reynolds-number dependence As has already been said, the occurrence of separation in a bluff-body flow causes the creation of two free shear layers and the interactions between them 169

6 represent the essential reason for the vortex-wake formation. The effect of body geometry becomes important mainly due to the possibility of occurrence of different angles of the free streamline separating from the body. The magnitude of this angle is determined by the nature of the separation process taking place at either a sharp edge, a blunt trailing edge, or on a continuous surface in the presence of an adverse pressure gradient. It is in the latter case that there will be a Reynolds-number dependence, as the separation on a continuous surface having no fixed separation points is highly dependent on it. It must be pointed out, however, that some Reynolds number dependence may be introduced to other body configurations through its influence on shear-layer development. The Reynolds number, Re, is defined as UD Re =, (5.6) ν where U is a characteristic velocity of the flow, D is a characteristic length-scale of the flow (diameter for a circular cylinder) and ν is the kinematic viscosity. The force coefficients are defined as C d = F F x y and C l ρu D = ρu D. (5.7) In this expression, F x and F y are the inline and transverse force, respectively. The effect of body configuration on the streamline separating from a bluff body is shown in Fig. 5.. As can be seen in Roshko [19], there are some qualitative facts that are the main features of the flow about bluff bodies. The first hint given by intuition is that the bluffer the cross section of the body the higher will be the drag coefficient. Secondly, the shedding frequency is inversely related to the width of the wake, so that the bluffer body has the lower Strouhal number. For the bluffer sections the wake is larger, causing a more difficult communication between the two shear layers. For a given cylinder, generally, an increase in base pressure is accompanied by a decrease in shedding frequency. Roshko [19] derived a Strouhal number based on the wake width and velocity at the separation points that is, to a first approximation, universal for many different geometries at different Reynolds numbers. This wake Strouhal number is defined by * fd s St =, (5.8) U s 170

7 where f s is the shedding frequency, d is the wake width and U s is the velocity close to the separation point (see Fig. 5.3). Figure 5.: Different body configurations causing vortex-shedding wakes and their influence on Strouhal number (St) and drag coefficient (C d ). U s U d d Figure 5.3: Parameters used for the definition of a universal Strouhal number, from Roshko [[19]]. Roshko [0] combined Kirchhoff s free streamline theory with Kármán s theory of the vortex street. He allowed some annihilation of the vorticity in the free shear layer and then obtained a solution dependent on only one experimental measurement. Using this theory, the wake Strouhal number can be analytically 171

8 calculated by matching the fraction of the rate of circulation from the cylinder boundary layer surviving vortex formation with Kármán s ideal configuration, which assumes the vortex spacing and strength given by the conditions presented in eqns. (5.4) and (5.5). The wake Strouhal number calculated in this way is 0.164, which agrees well with experiments for a wide range of body geometry and different Reynolds numbers. For a very low Re (Re < 1) the flow around a circular cylinder is nearly symmetric upstream and downstream, with one forward and one rear stagnation point. The flow for this condition is called creeping flow. Stokes in 1851 deduced a formula for the drag on a sphere in creeping flow, which is presented in Milne-Thompson []. This formula estimates the drag accurately for Re < 1. It is interesting to note that Stokes analysis cannot be applied for twodimensional flow, which results in zero velocities everywhere. As is described in Milne-Thompson, Oseen found a more realistic formula for the drag by linearising the inertia terms of the Navier-Stokes equations. His expression was valid for Reynolds number up to 5, and could be used for two-dimensional problems as well. For Re greater than about 5 flow separation occurs at the rear of the cylinder. For this condition, a symmetric pair of standing vortices is formed. It is interesting to note that the separation is non-tangential and on increasing Re the separation points move upstream becoming more tangential with the surface. The wake vortices grow linearly with Re. Above a certain critical value of Re, instabilities of the shear layers cause the onset of oscillations of the wake. This critical value of Re is strongly dependent on the level of external disturbances. For Re greater than a certain critical value around 50 70, the separated shear layers roll up to form vortices. Up to about a value of 150 there is a regular shedding of vortices in the form of a Kármán street. This regime is called laminar vortex shedding. The Strouhal-number variation for this range was studied by Roshko [19] and the so-called Roshko curve is presented in Fig It is interesting to note, that for this range differences in experimental arrangements have led to a large amount of scatter in measured vortex-shedding frequency and to discontinuities in the St Re curve. Williamson [3] showed that the existence of a discontinuity in that curve is due to a transition between different modes of oblique vortex shedding in the spanwise direction. Even when the vortex shedding is laminar, there would be three-dimensionalities in the flow. This aspect is well captured by the flow visualisation presented in Fig Oblique shedding is shown to be caused by the end boundary conditions and by manipulating them, parallel shedding can be induced up to Re (Williamson [3]), which results in a completely continuous St Re curve, as can be seen in Fig Williamson used inclined end plates in order to control 17

9 the boundary conditions and induce parallel vortex shedding and these are shown schematically in Fig The importance of obtaining parallel vortex shedding and defining a continuous S versus Re relationship is that it provides an experimental condition that can be compared directly with two-dimensional computational results. At higher Re, instabilities of the shear layer cause transition to turbulence prior to vortex roll-up, resulting in turbulent fluctuations of wake pressure and shedding frequency and the appearance of large-scale three-dimensional vortexshedding features. Williamson [3] argues that the transition to threedimensionality involves the emergence of vortex loops and causes the discontinuity at about Re 180 in the St versus Re relationship. The discontinuity between Re corresponds with a change to a finer-scale streamwise vortex structure. Williamson called mode A the one corresponding to the first discontinuity. The three-dimensionality is represented by an instability with a wavelength of about 4D in this mode. In the second discontinuity, Re 30 60, the wavelength is about 1D, and Williamson called it mode B of vortex shedding. The flow visualization obtained by Williamson [3] and the numerical simulations by Barkley and Henderson [6] and Siqueira [7] show the existence of these two modes. Figure 5.4: Roshko s curve, from Roshko [19]. 173

10 Figure 5.5: St versus Re over the laminar and transitional regimes, from Williamson [3]. Siqueira [7] employed a finite-element method to simulate the threedimensional flow around a circular cylinder. For Re = 00, Siqueira observed an instability in the wake with a wavelength close to 4D, similar to the one observed experimentally by Williamson [3]. In Figure 5.8(a), we can see the contours of vorticity for Re = 00 and the top view of the wake. The wavelength observed from these calculations is about 4D. The flow visualizations observed by Williamson are shown in Figure 5.9(a) and the wavelength is similar to the one observed in the simulations. Increasing Re to approximately 30, vortices are shed at mode B, with the wavelength close to D. Figure 5.8(b) shows this condition. The experimental results are shown in Figure 5.9(b). In the calculated contours for Re 30 the existence of vortices aligned with the current direction is clear, they look like fingers penetrating into the vortices aligned with the cylinder axe. 5 In the so-called lower subcritical range ( 350 < Re < 10 ), transition waves start to appear in the shear layers springing from the separation points. These oscillate back and forth by a few degrees at about 80. Even the vortices closest to the cylinder are turbulent. The main feature of the flow field is the upstream movement of the point of transition, with increasing Re. There is a laminar separation and the point of transition is close to the point of separation and most of the shear layer is turbulent. 5 5 As Re is increased and the critical regime is entered ( 10 < Re < 7 10 ), the flow field undergoes a dramatic change. The point of separation is very close 174

11 to the point of transition but still precedes it. There is a tendency for the shear layer to reattach itself to the surface. The region between laminar separation and turbulent reattachment is termed a separation-reattachment bubble, Roshko [0]. The flow is very sensitive to flow disturbances like surface roughness or free-stream turbulence. In the critical regime the stability of the separation bubble is very weak and a bubble can appear simultaneously on each side of the cylinder or only at one side, which results in a steady lift (Bearman [8]). On increasing Re even further, the point of transition moves upstream until it reaches the point of separation and then the bubble disappears. Separation is delayed to about 10 since a turbulent boundary layer can better withstand an adverse pressure gradient. However, with increased Re, the point of transition and the separation point move upstream on the cylinder surface. As transition precedes separation, the postcritical regime is reached. The wake is then widened and the state of the shear layers becomes such that they cause regular vortex shedding from the cylinder and the separation occurs at about Figure 5.6: Visualisation of the different modes of oblique vortex shedding: (a) without control of end conditions, and (b) with control of end condition by using inclined end plates, from Williamson [3]. 175

12 Figure 5.7: Schematic view of different oblique vortex patterns and control of end conditions by using inclined end plates, from Williamson [3]. The different regimes cause pronounced changes in the drag coefficient (C d ) versus Re and S versus Re curves. In Fig. 5.10, C d versus Re curve is shown, in which it is possible to see clearly the effect of the different flow conditions for each regime on C d. In the laminar regime, C d initially decreases linearly for increasing Re. In the subcritical regime C d changes slightly until the critical regime is reached when there is a sudden drop in C d. For higher values of Re, C d initially increases and then an almost constant value is reached. The S versus Re curve, Fig. 5.11, presents a similar trend if compared with C d -1 versus Re. For the laminar regime, S increases linearly for increasing Re, then there is a discontinuity, already presented in detail in the beginning of this section, and for the subcritical regime S remains fairly constant. When the critical regime is approached, S increases dramatically and then there is a sudden drop when the postcritical regime is reached. 176

13 z z A B x y x y x y x y Figure 5.8: Contours of vorticity in the spanwise direction. a) Re = 00; b) Re = 50, reproduced from Siqueira [7]. A B Figure 5.9: Flow visualizations of the flow around a circular cylinder: Mode A of vortex shedding ( Re 00 ); b) Mode B of vortex shedding ( Re 30 ), reproduced from Williamson [3]. 177

14 Figure 5.10: The drag coefficient versus Re curve, reproduced from Schlichting [4]. Figure 5.11: The Strouhal number versus Re curve, reproduced from Norberg [5]. 178

15 5.. Mechanism of vortex formation A physical description of the mechanics of the vortex-formation region has been given by Gerrard [9]. He suggests that the mutual interaction between the two separating shear layers is a key factor in the formation of a vortex street. Gerrard postulated that a vortex grows by gaining circulation from its connected separated shear layer. At a certain instant, the growing vortex is strong enough to draw the opposing shear layer across the near wake. The approach of oppositely signed vorticity, in sufficient concentration, cuts off further supply of circulation to the growing vortex, which is then shed and moves downstream. Gerrard s vortex-formation model is illustrated in Fig. 5.1, where an instantaneous filament lines pattern is shown. He postulates that the fluid particles of the opposite shear layer drawn across the wake can follow one of the routes: a) they can be entrained into the growing vortex thus reducing its strength, b) they can find their way into the shear layer with vorticity of opposite sign to theirs, and c) they can be fed back into the near-wake region. The balance of the amount of fluid that follows these routes controls the shedding frequency, the strength of the vortices shed and the base pressure. The entrainment of opposite vorticity into the shear layer is a very important stabilising mechanism. If, for instance, the base pressure falls, the separating shear layer will become stronger, the growing vortex will also become stronger, more of the opposite shear layer will be drawn across the wake, more of it will be entrained into the shear layer and hence the reduction of the shear-layer circulation will reduce the strength of the growing vortex, thus acting as a stabilising mechanism. Considering that vortex formation involves the mixing of flows of oppositely signed vorticity, the strengths of individual wake vortices will be less than the total circulation shed from one side of a bluff body during a shedding cycle. Assuming the pressure remains constant through a shear layer, as can be seen in Bearman [5], the velocity U s at the edge of the boundary layer at separation is given by U = U (1 C ), (5.9) s where U is the free-stream velocity and C pb is the mean base pressure coefficient. The rate of shedding of circulation at a separation point is equal to the flux of vorticity through a plane normal section of a shear layer, and is given by y y1 pb dγ U = ωudy (1 Cpb ) dt =, (5.10) 179

16 in which y 1 and y are two points representing the edges of the shear layer, uy ( 1) = 0 and uy ( ) = Us and expression (5.9) has been used to evaluate U s. Using these arguments, it is possible to deduce that the fraction of the original circulation that survives vortex formation is given by the expression S Γv αv = (5.11) UD(1 C ) where Γ v is the strength of a wake vortex. Roshko [19], and Bloor and Gerrard [30] have calculated α v using Γ v obtained by matching a measured vortex convection speed to a predicted value evaluated from a suitable array of idealised point vortices. Fage and Johansen found α v = 0.60 for a flat plate and Roshko [19] found α v = 0.43, as the average value for a circular cylinder, 90 wedge, and a flat plate. For a circular cylinder Bloor and Gerrard [30] found α v to be between 0. and 0.3, depending on Re. pb Figure 5.1: Gerrard s vortex formation model showing entrainment flows, from Gerrard [9]. 5.3 Vortex-induced vibration Vortex shedding can be dramatically changed when a bluff body is oscillating in a fluid stream, or when it is exposed to an imposed oscillatory stream. In certain ranges of amplitude and frequency of oscillation, the body motion or the oscillatory stream can take control of the instability mechanism that leads to vortex shedding. 180

17 Essentially, there are two ways of investigating the influence of the vibration on vortex shedding. In the first, the effect is analysed by the application of forced oscillations on a cylinder mounted in a water or wind tunnel. In the second, the interaction between the shedding of vortices and the body oscillation is investigated directly by mounting the cylinder on an elastic base. An adjustable spring and damping is employed in this base. In fact, this last choice is a method that tries to investigate the phenomenon directly. On the other hand, the first method is an indirect way of analysing the interaction. There are pros and cons in each of these approaches. The use of a cylinder mounted on an elastic base shows directly the evidence of the non-linear interaction between the excitation and the system response. However, the number of parameters to be measured and analysed is augmented significantly, implying a difficulty in interpreting the experimental results. When forced oscillations are employed, the number of parameters is inferior, but some of the characteristics observed in practical problems of vortex-induced vibrations may be not reproduced. The question that may arise is: under which conditions one experiment with a cylinder forced to oscillate is equivalent to an experiment where the cylinder is mounted on an elastic base. Alternatively, under which conditions the forced oscillations could occur if the cylinder were free to oscillate. To answer those questions, one has to investigate the direction of the energy transfer involved in the fluid oscillation interaction: if it occurs from the fluid to the body or vice versa. In the next section, it will be shown that the energy transfer is related to the phase angle between the force and the body displacement. When the energy transfer in a forced-oscillation experiment occurs from the fluid to the body, then, probably, the body would oscillate if it were mounted on an elastic base with a natural frequency close to that of the forced oscillation. When we are carrying out experiments with a cylinder mounted on an elastic base, oscillations occur only when the energy transfer is from the fluid to the body. The usual convention employed for studies of VIV is to consider the energy transfer positive when it occurs from the fluid to the body. Logically, in the dynamical system analysis we must consider the net energy transfer evaluated by the integration of the hydrodynamic force and displacement in many cycles of oscillation, i.e. we must calculate the average energy transferred from the fluid to the body. On the other hand, when we carry out forcedoscillation experiments, the energy transferred is not restricted to positive values, and depending on the frequency and amplitude of oscillation, negative or positive values can be obtained. 181

18 5.3.1 Energy transfer The study of the phase angle in cases where the body is forced to oscillate allows direct comparisons with the free-oscillating body case. The importance of the phase angle (φ) lies in the fact that the energy transferred from the fluid to a freeoscillating body is proportional to sin(φ). To prove this affirmation, we consider the energy transferred per cycle as T E = F () t dy, (5.1) 0 y where F y is the transverse force per unit length in the spanwise direction. This force leads the body displacement y b by the phase angle φ and is given by Fy U DCl cos ft where C l is the amplitude of the lift coefficient and b 1 = ρ ( π + φ), (5.13) ( π ) y = Acos ft. (5.14) b With expressions (5.13) and (5.14), the energy transferred to the body by the fluid per cycle of oscillation can be written as π 1 E = U DCl A cos( + ) sin d 0 where τ = πft. This expression yields ρ τ φ τ τ, (5.15) π E = ρ U DCl Asin φ. (5.16) The non-dimensional energy is which is proportional to sin(φ). E A = = sinφ, (5.17) ρu D E πcl 1 D 18

19 5.3. Forced oscillations Bishop and Hassan [31] were among the first researchers to investigate the influence of body oscillation on vortex shedding. They studied how the forces on a circular cylinder are influenced by forcing the body to oscillate transversally perpendicular to the free stream. Bishop and Hassan found that when the forcing frequency, f, of the cylinder approached the Strouhal frequency, f s, the oscillation frequency and the measured lift-force frequency become synchronised. In other words, the oscillation was able to change the shedding frequency to its own value. As has already been pointed out in the first section of this chapter, this synchronisation is usually called lock-in. Bishop and Hassan noted that the synchronisation persisted over a range of frequencies for a given amplitude of oscillation. Some years after Bishop and Hassan s experiments, Koopman [3] also investigated the effect of forced transverse oscillation of a cylinder on vortex shedding. His main concern was to determine the synchronisation region, or lock-in boundaries. He found that lock-in occurred only above a threshold amplitude of oscillation. Figure 5.13: Boundary of lock-in, reproduced from Koopman [3]. The lock-in boundaries, as determined by Koopman, are shown in Fig It is possible to see that the upper and lower range of frequencies for which the 183

20 shedding frequency is controlled by the driving frequency of the cylinder oscillation is mainly amplitude dependent and only slightly Reynolds-number dependent. The lock-in boundary determined by Koopman has the shape of what is called an Arnold s tongue in non-linear system mechanics and chaos theory. Griffin et al. [33] performed experiments for a circular cylinder that was free to oscillate. Their experiments were in the range of Reynolds numbers from 350 to 900. They studied the phase difference between body displacement and lift coefficient and the energy transfer from the fluid to the cylinder. The phase jump that occurs close to the resonance frequency has also been studied by Bearman and Currie [34]. They compared results from experiments where the body was forced to oscillate with results from experiments by Feng [35] in which the body was mounted in an elastic system and free to oscillate. The results for both cases can be seen in Fig The behaviour of the phase angle for each case is similar, but the forced-oscillation experiments have a practical advantage in that the amplitude and frequency of the motion can be varied at will. The importance of the phase angle (φ) values lies in the fact that the energy transferred from the fluid to the oscillating body is proportional to sin(φ). The explanation for this can be found, for example, in Griffin and Ramberg [36]. When 0 < φ < 180, the energy transferred is positive. Then, if the body is free to oscillate, it is in this range that vortex shedding can induce body oscillation. Besides the frequency synchronisation that occurs over a certain range of amplitude and frequency of oscillation, there is a large increase in correlation length of surface pressure along the span of a bluff body forced to oscillate. Bearman and Obasaju [37] measured the correlation of surface pressures along the span of a square cylinder forced to oscillate. Their results show that in the lock-in region the flow is more two-dimensional. This seems to cause an increase in strength of the vortices being shed because there is a perfect coherence along the span. 184

21 Figure 5.14: Comparisons between forced vibrations at y/d=0.11, by Bearman and Currie [34], and free-vibrating circular cylinder (Feng [35]). Figure 5.15: Measurements of the phase angle between suction and cylinder displacement versus reduced velocity. o, y/d=0.05; x, y/d=0.1;,y/d=0.5, from Bearman and Obasaju [37]. 185

22 Bearman and Davies [38] showed that the after-body shape plays an important role in the phase shift. Bodies with a large after-body, such as a square or rectangular cylinder, experience a phase shift that is most negative for frequencies of oscillation around the shedding frequency. Bodies with a small after-body are not subjected to VIV under normal conditions. In Fig. 5.15, the phase shift that occurs for a square cylinder forced to oscillate is shown, and in this figure, the frequency of oscillation is considered in terms of the reduced velocity V r. The resonance occurs for V r = Analysing these results, vortex shedding for a square cylinder has mainly a negative energy transfer. This energy transfer is positive only for a small range of frequency of oscillation, and if the body is free to oscillate, it is only in this range that oscillation can be induced by vortex shedding. The work of Zdravkovich [39], who analysed the flow-visualisation studies of Den-Hartog [40], Meir-Windhorst [41], and Griffin and Ramberg [36], shows that the phase jump can be explained by a change of the timing of the vortices being shed. Öngoren and Rockwell [4] also drew the same conclusion by analysing results using modern techniques of flow visualisation. They noticed that for the frequencies of oscillation above the natural shedding frequency, the vortex formed on one side of the cylinder was shed when the cylinder was near the maximum amplitude on the opposite side. On decreasing the frequency of oscillation the timing changes suddenly, such that a vortex is shed when the cylinder is at its maximum amplitude on the same side. Williamson and Roshko [43] presented in a paper one of the few attempts to carry out experiments for large amplitudes of oscillation. They found a series of synchronisation regions, which have been classified in terms of the number of vortices shed per cycle of oscillation. They classified as fundamental lock-in, the region where the experiments by Koopman [3] were carried out. In this region, two vortices are shed with opposite circulation per cycle of oscillation. For higher amplitudes, regions were found where two pairs of vortices are shed per cycle. Their experiments were carried out in a X-Y towing tank, and a flowvisualisation technique that utilised aluminium particles on the fluid surface was used Free oscillations Vibrations induced by vortex shedding have been a main subject of research in the field of fluid structure interactions. These vibrations are modelled through a spring mass damper system. If the cylinder is free to vibrate only in the transverse direction in relation to the flow stream, as in Figure 5.16, the movement of the cylinder is given by the equation: 186

23 c k y+ y + y = Fy, (5.18) m m in which m is the cylinder mass, c is the damping coefficient, k is the spring stiffness, and F y is the lift force acting on the cylinder, calculated through the integration of pressure and viscous stress on the cylinder surface. The mass parameter m*, the non dimensional amplitude A*, the non dimensional frequency f*, the damping parameter ζ, and the reduced velocity V r, are defined as m 4m m* =, m = π D lρ (5.19) d n A A* =, (5.0) D 1 π k Tn = = = π /, (5.1) f ω m n foscillation f * =, (5.) f n c ζ =, (5.3) km V r U =. (5.4) fd n If we employ the definitions shown above and the definition of C l, eqn (5.18) can be rewritten as y+ ζωny + ωny = U C * l. (5.5) π Dm 187

24 Figure 5.16: Cylinder free to vibrate in the transverse direction. The non-dimensional time can be defined as 1 τ, (5.6) and the derivatives of eqn (5.5) are given by D D y = DY, y = Y, y = Y. (5.7) T T T n n With all these definitions, eqn (5.5) is written as D D Y + ζω n Y + ωndy = U C * l (5.8) Tn Tn π Dm or Y + 4πζ Y + 4π Y = V * r Cl. (5.9) π m This is the non-dimensional oscillator equation. The non-linearity is expressed by the lift coefficient and its components in phase with acceleration and velocity. The effect of the added mass is contained in this coefficient. A similar equation is shown in the review by Parkinson [45]. The amplitude of vibration A/D is a function of the mass parameter m*, the damping ζ and the natural frequency ω n of the cylinder. In general, experiments n 188

25 for given values of ζ are carried out, and curves of A/D as a function of the reduced velocity V r are plotted. Feng [35], and Brika and Laneville [46] carried out experiments for high mass parameters. The main difference between the two experiments was that Feng used a rigid cylinder free to vibrate in only one direction, while Brika and Laneville employed a flexible cylinder. Consequently, in the latter, the cylinder could vibrate in both x and y directions. In both experiments, a hysteresis in the behaviour of the y amplitude of vibration was observed, dependent on the procedure adopted in the experiment. It was noted that, when the experiment was carried out with small increments of the reduced velocity, the amplitude of vibration followed what was called the upper branch of vibration, reaching a peak amplitude and dropping suddenly to what was called the lower branch through path (a) (Figure 5.17). When large increments of reduced velocity were used, the amplitude soon detached from the upper branch at a low reduced velocity and, following path (b), reached the lower branch without developing the peak amplitude. When the experiment was initiated with large reduced velocity and the velocity was decreased, independent of having large or small decrements, the amplitude followed the lower branch and, at a low reduced velocity, detached and through path (c) reached the upper branch without developing the peak amplitude. Mass parameter m* was around for Feng and around 80 for Brika and Laneville. The peak amplitude A/D obtained by Feng was around 0.4 for a reduced velocity V r 6. 0 and damping ζ = Brika and Laneville obtained a peak amplitude around 0.54 for a reduced velocity slightly higher than 6.0 and variable damping one order of magnitude smaller than the value of Feng. Khalak and Williamson [44] carried out experiments for very low mass parameters, with m* in the range between.0 and 10.0 and damping so that m*ζ = No observations were made about hysteresis, but their results show clearly the two branches of vibration, with peak amplitudes A/D closer to 1.0. For m*=3.3 an amplitude of 1.1 was reached. As the values of mass parameter of Khalak and Williamson are closer to those characteristic of marine risers, we decided to use their results to make comparisons with our numerical calculations. 189

26 Figure 5.17: General plot of the amplitude of vortex-induced vibration as a function of reduced velocity. 5.4 Hydrodynamic model The discrete-vortex method (DVM) is a Lagrangian numerical scheme technique for simulating two-dimensional, incompressible and viscous fluid flow. The method employs the stream-function-based boundary-integral method and incorporates the growing core size or core-spread method in order to model the diffusion of vorticity. The body wall is discretized in N w panels, and N w discrete vortices with circulation Γ i are created from a certain distance from the wall, one for each panel. These vortices are convected and their velocities are assessed through the sum of the free-stream velocity and the induced velocity from the other vortices. The induced velocities are calculated through the Biot Savart law. Since N w vortices are created at each time step, the number of vortices N v in the computation grows rapidly. Then, it is necessary to use some procedure to keep the number of vortices under control. Otherwise, a longer simulation would become unfeasible, mainly due to the number of calculations to evaluate the induced velocities through the Biot Savart law. The scheme employed is an amalgamation procedure. The vorticity transport equation for an incompressible Newtonian fluid with an isotropic viscosity, and the stream-function vorticity equation can be written as, 190

27 Dω Dt = ν ω, (5.30) ψ = ω. (5.31) Equation (5.31) can be written for each discrete vortex as, ψ i = ωi. (5.3) The induced velocity of each vortex can be obtained from the Biot Savart law. To avoid the discontinuous distribution of induced velocity by a point vortex, Spalart et al. [10] employed a rigid vortex particle, or blob vortex, with a radius σ. The velocity induced by the vortex is given by U θ Γ = π r r + σ. (5.33) The vortices are created at a distance σ 0 from the body surface. This parameter is considered as the initial vortex core radius. From potential theory and considering expression (5.33), it is known that the stream function at a wall point i on the body wall due to a vortex distribution with N w vortices each with strength Γ j, a free-vortex distribution with N v vortices each with strength Γ k (representing the wake), and free-stream velocity components U and V, is given by Nw 1 ψ i = Im Zw ( U iv) ln i j Zw Z i c σ Γ + j 0 4π j = 1 (5.34) Nv 1 Γk ln ( Zw Z ), i k + σ 4π k = 1 for 1 i N, where w Z w i represents the complex coordinate of the control point i on the wall; Z c i is the coordinate of the j vortex created around the body; and Z k is the coordinate of the k vortex located in the wake. Similarly, the stream function at the wall point i+1 is obtained by just replacing the subscript i by i+1, and since the body is a streamline, we can write ψ i 1 ψ = + i Vbody. n S, (5.35) 191

28 where V body denotes the velocity and S the size of the panel. Expression (5.35) yields a set of linear equations in the form, in which A Γ = b, (5.36) cj cj 1 Zw Z + σ i+ 1 A aij = ln (5.37) 4π Z Z + σ wi k = 1 (( )( )) i+ 1 i b b = Im Z Z ( U U ) i( V V ) 1 4π i w w BODY BODY Nv Z Γk ln Z wi + 1 wi k k Z + σ Z + σ (5.38) and A is determined by the body geometry and vortex-creation points, Γ is the vector containing values of the circulation of the bound vortices (solution) and b is known from the free-stream velocity, circulations and positions of free vortices, and is recalculated at each time step. With the purpose of conserving the total circulation, Kelvin s theorem is enforced on the linear system, expression (5.36). This is equivalent to imposing Nw Nvlost Γ i = Γk. (5.39) i= 1 k= 1 When a vortex hits the body, it is removed from the computation. The righthand side of eqn (5.39) is the same as the sum of lost circulation, thus the lost vortices are recreated by Kelvin s theorem. When more than one body is in the fluid, this equation is written for each body separately. The core-spread method uses the vortex core σ to model the diffusion of vorticity. The core size in eqn (5.33) is substituted by a new core size at each time step. The rate at which the core size grows can be calculated as follows. First, the analytical solution of a diffusing vortex core is considered U θ Γ = 1 e π r r 4ν t. (5.40) 19

29 Due to computational reasons, instead of using eqn (5.40) Park and Higuchi [07] used a growing core size where the core radius was the position of the maximum value of velocity calculated by this expression, which yielded This equation can be written as σ =.4 ν t. (5.41) o σ = σ + ν t, (5.4) n+ 1 n ( ) ( ) where n denotes the time step, and ν is the kinematic viscosity of the fluid. Thus, the induced velocity is assessed with a growing vortex core, which models the diffusion of vorticity. Note that when a discrete vortex is moving away from the body, the core grows, and the induced velocity around it decreases. Vortices are convected by the sum of the free-stream and the induced velocity by all other vortices. The new positions of the already existing vortices are calculated employing a Lagragian scheme, X + = X + U t. (5.43) n 1 n n i i i Since N w vortices are created at each time step, the total number of vortices increases rapidly. As previously mentioned, the presence of too many vortices would make the computation infeasible. With the intention of avoiding this occurrence, Spalart et al. [10] employed a merging scheme if two vortices satisfy the following condition ΓΓ i i j i j Γ +Γ j Z Z ( D + d ) ( D + d ) i 0 j < V, (5.44) 0 D 0 and V 0 are the merging and tolerance parameters. D 0 is used to control resolution near the body. The merging tolerance, V 0, preserves the desirable total number of vortices, N v. The resulting amalgamated vortex is located at a position that satisfies the first moment of the distribution, i.e. at with a vortex strength Z Γ + Z Γ Z = Γ +Γ i i j j i j, (5.45) Γ =Γ +Γ. (5.46) i j 193

30 Drag and lift forces are calculated from direct integration of the pressure field and skin friction around the body. The total force, F total, is given by F = F + F total pressure skin. (5.47) A particular form of Kelvin s theorem, valid for viscous flow, is employed with the purpose of evaluating the pressure contribution to the total force. If one considers the definition of circulation, and a Lagrangian approach for the acceleration of material particles, we find d Γ d dv = Vdr. = dr dt dt. (5.48) dt C The Navier Stokes equations are evoked for the total velocity derivative, yielding d Γ = p dr + ν V dr = ν V dr dt. (5.49) C C S In this expression, the pressure integral is zero as it is evaluated in a closed contour around the body. The last term of eqn (5.49) can be expressed in terms of the normal derivative of the vorticity at the wall, yielding Kelvin s theorem for viscous flow dγ V V ν ds dt s s n n S s s = + = Vn Vs ω ν + ds = ν ds. S n s n S n This expression is rewritten in a discrete form as C wall (5.50) Γ ω = ν S, (5.51) t n in which Γ is the net circulation generated in a given panel. The relationship between the circulation and the pressure is obtained by employing once again the Navier-Stokes equations. At the wall, for a moving cylinder p ω ρabody es = ρν, (5.5) s n 194

31 where abody es is the body acceleration projected in the wall-tangent direction (s). Rearranging this expression, and employing eqn (5.49) we have p Γ 1 = ρ ρabody es. (5.53) s t S Now, the contribution to the pressure distribution from the circulation is referred A B to as p and from the body acceleration as p, the pressure is evaluated in a given panel i by the following equation p A i i ρ = p Γ A 0 j. (5.54) t j= 1 A In this expression, p0 is a reference pressure and can be made equal to zero. The term due to the body acceleration can also be integrated along the cylinder wall. In doing so, the total pressure force is then calculated by A B F pressure = pends = ( p + p ) ends = Nw p S e + ρπ R ( a e + a e ). i= 1 S A i i ni x x y y S (5.55) Finally, the skin-friction contribution is obtained from the definition of shear stress at the wall, and the definition of vorticity, which yields Vs τ w µ = e s, (5.56) n where τ w is the shear stress at the wall, and µ is the dynamic viscosity. In order to evaluate the normal derivative, a linear variation of the velocity is assumed close to a vortex panel yielding V τ s w = µ e s. (5.57) σ In this expression, σ 0 is the initial core, which corresponds to the normal distance from the newly created vortices to the cylinder wall. The skin-friction contribution to the total force is obtained by integrating eqn (5.57), 0 195

32 V F ds S e. (5.58) Nw si skin = τ w = µ i si i 1 S = σ 0i After considering the contributions from skin-friction and pressure, the force components are resolved in the two directions x,y, yielding F x and F y. These forces are then non-dimensionalised by expressions (5.7)a and (5.7)b. 5.5 Structural model A finite-element structural model based on the Euler Bernoulli beam theory is employed to calculate the dynamic response of the cylinder. A general equation of motion is solved through a numerical integration scheme in the time domain. This numerical structural model is based on the studies of Patel and Witz [8] and Ferrari [9]. First, we find a static solution for the riser. Then, in the dynamic analysis, the stiffness matrix obtained from the static analysis is used as an average approximation. A lumped approach is employed. A mass-lumped matrix is constructed and the damping matrix is evaluated in a global manner. Figure 5.18: Riser model, where SWL stands for still-water level. The Euler Bernoulli beam equation gives the transverse and rotational degrees of freedom. Figure 5.18 shows the model of riser. As can be seen in 196