Keywords: Vortex-Induced Vibrations, flexible cylinder, catenary, experiments, modal analysis.

Transcription

1 EXPERIMENTAL INVESTIGATIONS ON VORTEX-INDUCED VIBRATIONS WITH A LONG FLEXIBLE CYLINDER. PART III: MODAL-AMPLITUDE ANALYSIS WITH A CATENARY CONFIGURATION Felipe Rateiro Celso P. Pesce Rodolfo T. Gonçalves André L. C. Fujarra Department of Mobility Engineering, Federal University of Santa Catarina, Brazil Guilherme R. Franzini Offshore Mechanics Laboratory, Escola Politécnica, University of São Paulo, Brazil Pedro Mendes Petrobras Research Center, (CENPES/PDEP/TDUT), Brazil ABSTRACT The present study provides original results of an experimental approach to understand the VIV phenomenon on long flexible cylinders launched in catenary configuration. The tests were carried out in a towing tank by means of a movable floor attached to the carriage car. Three groups of tests were considered, comprising the incidence of uniforme current profiles in catenary arrangements transverse and longitudinal to the inflow, the second one tested in two different directions. The catenary models were built by means of silicone tubes fulfilled with steel microspheres leading to achieve a mass ratio of m = Displacements along the models were measured by means of an underwater image tracking system and a large number of passive targets along the lines. A Galerkin s modal decomposition scheme is applied to obtain the modal-amplitude time histories and then the root-mean-square amplitudes as functions of the modal reduced velocities. Curiously, the modal results are quite characteristic of simpler systems with low m, particularly in terms of the synchronization behavior observed in VIV reponses of rigid cylinders elastically supported. The modal decomposition procedure, therefore, is presented as a great technique for improvement of the investigations concerning the VIV of long flexible lines, revealing interesting results not yet observed experimentally in a so accurate way. Corresponding author. ceppesce@usp.br Formerly at University of São Paulo. Keywords: Vortex-Induced Vibrations, flexible cylinder, catenary, experiments, modal analysis. INTRODUCTION The investigation of vortex-induced vibrations due to current profiles action on risers is a topic of great relevance for the ocean engineering. In this sense, several numerical and experimental results have been presented, where the understanding of the fundamentals for this fluid-structure interaction problem has been enhanced substantially. Despite the advances obtained, some aspects still deserve consideration, such as VIV experiments with long flexible cylinders in catenary configurations. As already described in Parts I and II, dedicated to VIV of an immersed long flexible cylinder in vertical arrangement, references [1] and [2], the whole experimental investigation is part of a comprehensive project on nonlinear dynamics of risers, carried out at Escola Politécnica of University of São Paulo. In time, some interesting results on parametric resonance responses of vertical configurations to top motion excitations can be found in [3]. Details about the whole research project are presented in [4]. This third part is dedicated to provide results and discussion on VIV experiments with the same long flexible cylinder herein launched in a catenary arrangement. It is worth noting that the difficulty of understanding the VIV

2 in catenary risers is intrinsic to its geometry, affecting the hydrodynamic interaction along the length. In fact, even in a constant current profile, different excitation conditions may appear due to the variation of the current incidence angle, depending on the position along the line. This could cause multimodal responses, with or without intermittency, travelling waves, mode switches, etc, setting a very complicated analysis scenario; like those described in [5] and [6], as well as in the pioneer work [7]. The Galerkin s projection scheme of analysis is used again to construct modal-amplitude time-histories. At this time precise modal shapes obtained numerically, through Poliflex 3D, a in-house riser dynamics code, were used to compose the projection functions space. With such a technique, quite interesting dynamic behaviours are revealed, towards an increasing understanding of the VIV of catenary lines. EXPERIMENTAL ARRANGEMENT AND ANALYSIS METHODOLOGY The experiments were carried out at IPT (Institute for Technological Research of the State of São Paulo) towing tank, with a moveable floor connected to the carriage at 2.5m depth. A long flexible cylinder was built by means of a silicon tube with external diameter of D = 22.2mm filled with iron micro-spheres, arrangement that results in a mass ratio parameter of m = 3.72; additional details can be found in [8]. Table 1 presents the main properties of the long flexible cylinder tested. Two catenary arrangements were considered: the first acted on by a constant current profile in the plane of launching (longitudinal catenary arrangement) and the second one with the current profile perpendicular to the launching plane (transverse catenary arrangement). The lengths measured from the top connections to the touchdown points on the moveable floor were respectively 4.15m and 4.8m. In the first arrangement, two current directions were considered, comprising concave and convex incidences, the second one corresponding to towing velocities from left to right; see Fig. 1(a). As can be observed for the transverse catenary arrangement, Fig. 1(b), both towing directions represent the same out-of-plane incidence of current. Despite that, for the sake of accuracy, only the towing direction having the underwater cameras downstream the catenary plane were considered herein. The Reynolds numbers ranged from to , based on the minimum and maximum towing TABLE 1: Main properties of the long flexible cylinder tested in catenary arrangement. External diameter, D Axial rigidity, EA Cylinder properties 22.2 mm 1.2 kn Bending stiffness, EI.56 Nm 2 Linear mass, m l Immersed weight, γ Longitudinal catenary arrangement Water depth, h Top angle with the horizontal, θ L Vertical position of top connection, H Length from top to the touchdown, L Total length, L t Water depth, h Transverse catenary arrangement Top angle with the horizontal, θ L Vertical position of top connection, H Length from top to the touchdown, L Total length, L t 1.19 kg/m 7.88 N/m 25 mm mm 415 mm 665 mm 25 mm mm 48 mm 58 mm carriage speeds, respectively.3m/s and.26m/s. The dynamic behavior of the riser model was monitored and recorded by means of passive targets (44 submerged and 5 emerged in the longitudinal catenary, as well as 29 and 5 in the transverse one). Six submerged and two emerged cameras were used to track and record the dynamics of the long flexible cylinders in catenary configuration. It is important to highlight, however, that a relatively large effort was needed for positioning, calibrating and synchronizing the monitoring system, what resulted in a good portion of the line monitored by the cameras. In fact, that is a great advantage of the instrumentation presented herein, which provides direct measurements of the displacements in a larger number of predefined positions along the flexible lines, differently from others based on indirect measurements, usually through accelerometers installed inside the lines in a smaller number of pre-defined positions and/or with strain measure-

3 the flexible cylinder, at every instant t j, onto a space composed by modal functions ψ(z). Such a technique has been used before in [3], and at references [1] and [2], those two presented at this same conference as Parts I and II. The respective modal amplitudes corresponding to a given n mode were then written by: (a) Longitudinal arrangement ã x n(t j ) = ã y n(t j ) = ã z n(t j ) = X (s,t j )ψ n (s)ds (ψ n(s)) 2 ds Y (s,t j )ψ n (s)ds (ψ n(s)) 2 ds (1) (2) Z (s,t j )ψ n (s)ds (ψ, (3) n(s)) 2 ds (b) Transverse arrangement FIGURE 1: Catenary model in the longitudinal and transverse arrangements. ments systems. Chronologically, some good examples of conventional instrumentation for VIV experiments can be found in: [7] and [9] through strain measurements, [1] through acceleration measurements, [5] and [11] through strain and acceleration measurements, [12] and [13] through strain measurements and some verification by means of displacements directly measured by means of a laser device in few positions along the line. None of those experiments, however, addressed the VIV phenomenon on immersed catenary lines. The large number of monitored targets, however, generated a huge amount of displacement time-histories, making the procedure of analysis extremely laborious, aiming at observing the catenary dynamics from a global point of view. In order to cope with this task, each timehistory was observed as a summation of many vibration mode contributions, obtained by mean of a Galerkin s scheme, i.e., by projecting the deformed configuration of where X (s,t j ), Y (s,t j ) and Z (s,t j ) are respectively the in-line, cross-wise and vertical measured displacements normalized by the diameter D with respect to the towing carriage reference frame at the point of spanwise coordinate s at an instant t j. The projection space was constructed with quasiexact eigenfunctions, taking into account geometric and axial rigidity. Such functions were determined with the already mentioned in house software Poliflex3D. More details about this software can be found in [14]. As stated by Eqns. (1) to (3), the procedure must be applied at every instant of time, for all the natural modes, resulting in the respective modal-amplitude timehistories, as exemplified in Fig. 2 for one of the tests with the longitudinal catenary arrangement. Time-histories in Fig. 2(a) are for the concave incidence at V R,1 = 18.4, while the ones in Fig. 2(b) are for the convex incidence at a similar reduced velocity. Note that the distinct modal contributions vary over time, reconstructing the overall dynamics of the catenary regardless the incidence condition. Still in Fig. 2(a), it possible to observe switch over responses, particularly between modes third and fifth (n = 3 and 5), which is not observed in the case of the longitudinal catenary with convex incidence, Fig. 2(b). The effectiveness of this method was evaluated by means of the reconstruction of experimental results for a given instant of time and then its comparison with the data obtained directly from the monitoring system. Figure 3 shows an example of experimental reconstruction after applying the Galerkin s scheme. In the graphs, the blue circles represent coordinates experimentally measured and red lines are the results obtained through the modal contributions overlay, according to

4 x/d. Modal Amplitude (a/d) y/d t(s) z/d. 16 (a) Concave incidence with V R,1 = Archlength Coordinate s (mm) Modal Amplitude (a/d) t(s) (b) Convex incidence with V R,1 = 17.3 FIGURE 2: Example of modal-amplitude time-histories for the longitudinal catenary. In these graphs: a n (t j ) = a/d, where n = 1 to 15. Eqn. (4) given just for the x coordinate (considering that similar equations apply to y and z coordinates): x(t,s)/d = N n=1 ã x n(t)ψ n (s) (4) According to Fig. 3, a good adherence between the directly measured coordinates and the reconstructed ones is observed. Additionally, it is clear that the aerial cameras played an important role in the reconstruction process and consequently in the modal amplitude evaluation procedure; see the group of blue circles on the right side of the graphics. These points were obtained by the two aerial cameras sketched in Fig. 1. FIGURE 3: Snapshots of the experimental results and modal reconstruction for the longitudinal catenary at V R,1 = It is important to highlight that fifteen modal functions were enough for a good representation of all the measured data, n = 1 to 15, although in the cases under low towing velocity, where only the low-order modes are excited, the same analysis could be conducted with only a few. Unfortunately, similarly to what happens in experiments through conventional instrumentation, the monitored part of the transverse catenary arrangement was smaller than that originally planned, as a consequence of restrictions imposed to the assembly of this experimental arrangement. In order to keep the accuracy of the results, a complementary procedure of analysis was adopted as follows. Figure 4 compares two evolutions in time of the nondimensional displacements in the normal direction along the transverse catenary at the same reduced velocity. In the upper graph, the results were obtained by means of the modal reconstruction technique, as described in Fig. 3. In this case, note that the prediction of the unmonitored region (between dashed lines) was included. The lower graph, on the other hand, shows the evolution in time based only in the displacements measured in the predefined positions with the targets, which makes necessary to interpolate the values within the unmonitored region. By comparing the graphs in Fig. 4, one can conclude that the evolution in time of the nondimensional displacements become more accurate when based on the recon-

5 TABLE 2: Natural periods of the catenaries. FIGURE 4: Evolution in time of the nondimensional displacements in the normal direction along the transverse catenary at V R,1 = 8.2. Upper graph from modal reconstruction and lower one from directly measured displacements. struction of the time histories after the modal decomposition, allowing a clearer and neater identification of the dominant eigenmode, as well as to observe eventual occurrences of traveling waves. RESULTS AND DISCUSSIONS In addition to the numerical analysis with Poliflex3D, some decay tests were performed to evaluate the natural periods of the catenary models as built. Table 2 compares the natural periods obtained numerically with those respectively obtained through the decay tests (in fact, mean values among at least three repetitions). The decay tests consisted in disturbing the catenary model from its equilibrium position, verifying its natural periods through Fourier analysis on its motion time series. The orientations of the vibration modes are also described in Tab. 2, noting that an axial mode could not be experimentally assessed. Regardless the catenary arrangement, longitudinal or transversal, there is an excellent agreement between numerical and experimental results, which points out to accurate results based on the modal decomposition analysis. These natural frequencies were used to plot the modal-amplitude (rms - root mean square of the timehistories as those in Fig. 2) as function of the modal reduced velocity, calculated for each natural mode, V R,n = U / f n D, following [1]. No considerations on the inclination of the line are here applied. Longitudinal Arrangement Periods T n (s) Mode Orientation Numerical Experimental 1 out-of-plane in-plane out-of-plane in-plane out-of-plane axial.67 7 in-plane out-of-plane in-plane out-of-plane in-plane out-of-plane in-plane out-of-plane in-plane Transverse Arrangement Periods T n (s) Mode Orientation Numerical Experimental 1 out-of-plane in-plane out-of-plane in-plane out-of-plane axial.7 7 in-plane out-of-plane in-plane out-of-plane in-plane out-of-plane in-plane out-of-plane in-plane In fact, this last procedure was applied for all different velocities, resulting in the amplitude-reduced velocity curve for each vibration mode, as presented in Fig. 5(a) and (b), for the longitudinal catenary under concave and convex current incidence, respectively. According to Fig. 5(a), the synchronization region overlap is clearly observed for the first three natural out-

6 a rms /D Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 1 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 a rms /D Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 1 Mode 11 Mode 12 Mode 13 Mode 14 Mode a rms /D Vr n (a) Concave incidence Vr n (b) Convex incidence Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 1 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 FIGURE 5: VIV response amplitudes of the longitudinal catenary. out-of- and in-plane modes of oscillation are respectively represented by ( ) and ( ) markers. of-plane modes, typically between 4 V R,n < 1. However, for the first mode this synchronization range is wider, extending to V R,1 13. In fact, this result is quite characteristic of simpler systems with low reduced mass value, m < 6, displaying a lower response branch for some natural modes, as discussed in reference [15]. Also in terms of the synchronization, the same behavior can be observed for the longitudinal catenary under convex incidence of current. By comparing the concave- and convex-incidence VIV results, it is possible to observe that the concave VIV amplitudes are higher than the convex ones. This conclusion is in accordance with the behavior observed in reference [16] through experiments with a rigid cylinder curved upstream and downstream. The same procedure of analysis can be applied for Vr n FIGURE 6: VIV response amplitudes of the transverse catenary. out-of- and in-plane modes of oscillation are respectively represented by ( ) and ( ) markers. the tests with the transverse catenary, resulting the plots of modal-amplitudes as function of the modal reduced velocity presented in Fig. 6. In this case, as should be expected and conversely to what happened with the longitudinal catenary, in the transverse arrangement most of the VIV response is due to the in-plane natural modes; see previous theoretical considerations on that, as well as consequences of that, in [17]. By means of the modal decomposition regardless the catenary arrangement, it is clear that the multimodal VIV response follows the same well-known behavior from experiments with rigid cylinders elastically supported. Although widely considered by theoretical approaches, up to our knowledge such a VIV behavior for the eigenmodes of long flexible cylinders in catenary configuration is an original offering of the present work, which may turn to be an important contribution for validation of numerical investigations on VIV of offshore lines and for further experimental works on this issue. FINAL REMARKS There is no doubt about how important is to understand precisely the VIV behavior on long flexible cylinders, usually found in the offshore scenario of oil and gas production. Considerably complex by the possible multimodal response, the understanding of the VIV behavior on catenaries become a more difficult task because of the arrangement itself, which is responsible for increasing the coexistence of several natural modes inside of the global VIV response observed.

7 As a consequence, just few researches are found in literature about the VIV on catenaries, particularly if one looks for experimental investigations. In VIV experiments with catenary arrangements, the simple attempt of monitoring the displacements along the line is a challenge, usually solved by means of indirect measurements through strain-gages and/or accelerometers positioned at just a few points inside the line. In this scenario, the present work seeks to contribute decisively to the understanding of the VIV phenomenon on catenary arrangements subjected to uniform in-plane and out-of-plane current profiles, bringing original results of modal amplitude obtained by directly monitoring the displacements in several points along the length. Besides of bringing precise results, the monitoring of several points simultaneously made possible to apply a modal decomposition analysis technique. Moreover, through subsequent modal reconstructions, such a technique was useful for estimating displacements in unmonitored positions along the catenary, which by itself may be seen as a major experimental analysis achievement. Regardless the catenary arrangement, longitudinal or transverse to the current, this procedure makes possible to analyze each modal response separately, revealing characteristics inaccessible through standard experimental techniques. In general, by isolating the modal responses it is possible to show that the synchronization behaviors are quite similar to those observed in VIV experiments with rigid cylinders elastically supported. Typically, the synchronization regimes are found in reduced velocities from 4 to 14 and some differences are found with respect to the modal amplitudes at the higher part of that interval, particularly for the longitudinal catenaries compared to the transverse one. In terms of maximum modal amplitudes, values of.7d have been observed in the lower-order natural modes: out-of-plane modes number 1 and 3 for the longitudinal catenary with concave current incidence, and in-plane modes number 2 and 4 for the transverse catenary. For those two catenary arrangements, the higherorder natural modes presented maximum modal amplitudes of approximately.6d (for natual mode n = 5 in the longitudinal catenary under concave incidence) and.5d (for natual mode n = 6 in the transverse catenary). Besides, maxima of.6d were observed for the first three excited natural modes in the case of convex incidence on the longitudinal catenary. As a final remark it is important to highlight that the first results presented herein are just part of a comprehensive campaign of analysis aiming at a deeper understanding concerning VIV of catenary risers. Experiments with the same catenary configurations, driven by top motion excitations, in both, still water and under concomitant current action, have been carried out as well and will be presented in further papers. Such a investigation effort eventually aims at providing a comprehensive data set for theoretical models benchmarking. ACKNOWLEDGMENT The results presented in the paper were obtained from a comprehensive research project on non-linear dynamics of risers sponsored by Petrobras and carried out in Fujarra, Pesce and Franzini acknowledge CNPq (National Council for Scientific and Technological Development, Brazil) for the financial support, grants 39591/213-9, 3899/214-5 and 31595/215-. Gonçalves and Franzini acknowledge financial support from FAPESP for their post-doctoral scholarships, grants 214/243-1 and 213/ Special thanks to the IPT towing tank technical staff. REFERENCES [1] Franzini, G. R., Pesce, C. P., Gonçalves, R. T., Fujarra, A. L. C., and Mendes, P., 216. Experimental investigations on vortex-induced vibrations with a long flexible cylinder. Part I: modal-amplitude analysis with a vertical configuration. In Proceedings of the 11th International Conference on Flow-Induced Vibration - FIV216. [2] Franzini, G. R., Pesce, C. P., Gonçalves, R. T., Fujarra, A. L. C., and Mendes, P., 216. Experimental investigations on vortex-induced vibrations with a long flexible cylinder. Part II: effect of axial motion excitation in a vertical configuration. In Proceedings of the 11th International Conference on Flow- Induced Vibration - FIV216. [3] Franzini, G. R., Pesce, C. P., Salles, R., Gonçalves, R. T., Fujarra, A. L. C., and Mendes, P., 215. Experimental investigation with a vertical and flexible cylinder in water: response to top motion excitation and parametric resonance. Journal of Vibration and Acoustics, 137 (3), pp [4] Pesce, C. P., 213. Riser dynamics: experiments with small scale models. LabOceano - Ten-Years Anniversary Celebration Workshop, April [5] Vandiver, J. K., Marcolo, H., Swithenbank, S., and Jhingran, V., 25. High mode number vortexinduced vibration field experiments. In Proceedings of Offshore Technology Conference, Houston. [6] Vandiver, J. K., Jaiswal, V., and Jhingran, V., 29. Insights on vortex-induced, traveling waves on

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