COMPARISON BETWEEN FORCE MEASUREMENTS OF ONE AND TWO DEGREES-OF-FREEDOM VIV ON CYLINDER WITH SMALL AND LARGE MASS RATIO

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1 COMPARISON BETWEEN FORCE MEASUREMENTS OF ONE AND TWO DEGREES-OF-FREEDOM VIV ON CYLINDER WITH SMALL AND LARGE MASS RATIO Guilherme R. Franzini Julio R. Meneghini Fluids and Dynamics Research Group - NDF POLI University of São Paulo, Brazil Rodolfo T. Gonçalves André L. C. Fujarra Numerical Offshore Tank - TPN POLI University of São Paulo, Brazil ABSTRACT Experimental results of Vortex-Induced Vibrations (VIV) with one and two degrees-of-freedom obtained (DOF and DOF respectively) with the same apparatus are shown. A rigid cylinder is mounted on a very low damped leaf-spring system and times-series of displacement and forces are acquired. Two values of mass ratio parameter m are tested, m =.6 and m = 8.. The Reynolds number range is < Re <.. The focus is to compare results of forces coefficients, including added mass results, as well as amplitudes and frequency values for both experimental conditions. The results showed a marked difference in behavior between DOF and DOF at the super upper branch and at the in-line synchronization range for m =.6. For m = 8., the difference is constrained to the in-line synchronization range. An interesting feature of the in-line synchronization is the fact that the increase in the mass ratio parameter does not lead to significant differences in the amplification of forces coefficients. Considering the m =.6 DOF experiments, the cross-flow added mass coefficient results differs from the ones obtained with the DOF system only at the super upper branch range. On the other hand, no difference was observed in the added mass plot for m = 8.. Keywords: Vortex-induced vibration, Experiments, One degree-of-freedom, Two degrees-of-freedom, Forces measurements, Added mass coefficient Corresponding author. gfranzini@usp.br Presently at MARIN - Maritime Research Institute Netherlands as visiting researcher. INTRODUCTION The Vortex-Induced Vibrations (VIV) is a selfexcited and self-limited phenomenon that occurs when the vortex-shedding frequency f s is close to one of the natural frequencies of the structure. The phenomenon plays an important role in the behavior of offshore structures, such as risers. Despite maximum amplitudes of oscillation with order of one diameter, VIV can be important in the prediction of fatigue problems. Besides its practical importance, VIV consists on a puzzling fluid-structure resonant problem and has motivated several studies during the last decades. As a result of these studies, there are numerous research works into the subject. The comprehensive papers written by [, ], [], [, ] and [6] are classical examples of surveys regarding the theme. Up to the early s, most of the works, both theoretical and experimental, concerned the problem of a rigid cylinder, free to oscillate only in the cross-flow direction, consisting of an elastic system with one degree-offreedom (DOF). Considering DOF systems, some aspects are extensively studied, such as the effects of structural mass ( [7]), the maximum amplitude of response ( [8]) and the hydrodynamic forces ( [9], [] and []). Motivated by the technological demands, especially those related to the oil and gas exploration in the offshore enviroment, a great amount of effort has been employed on the study of the VIV both on a rigid cylinder with two degrees-of-freedom (DOF) and on a flexible cylinder. Some papers regarding these conditions are those written by [], [], [], [] and [6]. The fundamental studies developed by [7, 8] enlightened some unrevealed aspects concerning the DOF VIV phenomenon and its governing parameters were bet-

2 ter understood according to a enhanced phenomenological fundament. A key-point aspect is the influence of the mass ratio parameter m. Through experiments in water, the authors proposed two dynamical behaviors for the cylinder. One of them was verified for moderate to large mass ratio parameter (m > 6), in which the in-line oscillations are negligible in the cross-flow amplitude of response. The second dynamic behavior, observed for moderate to small mass ratio parameter (m < 6) is characterized by a strong presence of the in-line oscillations. The concomitant presence of in-line and cross-flow oscillations can lead the system to a new stable branch of response, named by the authors as super upper branch and defined by a T pattern of vortex shedding, in which two triplets of vortex are shed in each cycle. At the super upper branch range, the frequency of in-line oscillations are twice the ones observed in the cross-flow ones. Most of the features observed for a rigid cylinder at super upper branch can be extent to a flexible cylinder, as stressed by [9]. Despite the growing number of works discussing amplitude and frequency for DOF VIV, there are a few works that present results about force coefficients. Forces measurements described by [8] reported the presence of a ω component in the lift force at the super upper branch range. The published papers by [] and [] showed experimental DOF VIV results about the mean drag coefficient. In spite of the numerous papers concerning the VIV, the concept of added mass m a and its definition within the phenomenon context still keep some discussion points in Conferences, as can be found in the Preface written by []. There are two ways of defining the added mass; the first one is the term of the hydrodynamic force inphase with the body acceleration and the second one is based on the definition of the kinetic energy of the fluid. The added mass is a key point of hydroelastic problems in which the kinetic energy of the fluid is not negligible compared to the structural kinetic energy, such as the natural frequencies of floating units and risers. In numerical studies carried out using the Iwan & Blevins phenomenological model, [] observed that the variation of the added mass coefficient with the reduced velocity allows to capture the lower branch in the amplitude response curve. Another investigation concerning the added mass is the one presented in [], in which the general behavior of the added mass curve was observed for different types of phenomelogical models. In circular cylinders subjected to VIV, the added mass coefficient C a = m a /m d can be significantly different from the potential value C a = as stressed in several works, such as those by [], [] and [6]. The added mass coefficient for DOF VIV can be evaluated from experiments using both the time domain ( [6]) and the frequency domain ( [7]) approaches. In these works, C a curve crosses zero value at reduced velocity equal to 8 monotonically and presents an asymptotic value C a = outside the lock-in region. There are a few works of added mass coefficient for DOF VIV. The paper by [] shows in-line and crossflow added mass coefficients for a large m experiment. For the cross-flow added mass coefficient C a,y, a marked similarity is observed between DOF and DOF experiments, except for an inflection behavior in the range.6 < V R < 6.8, which corresponds to the region with the higher amplitudes of oscillation. The objectives of the present work are to present and discuss aspects about hydrodynamic forces and added mass coefficients for DOF and DOF VIV experiments carried out with the same apparatus, aiming at minimizing behaviors that can be associated to different set-ups. Two values of mass ratio parameter m were chosen, being one of them lower and the other higher than the critical value m c 6 proposed by [8]. It is worth mentioning that the results of added mass for two DOF system, at small value of m, are not found in the literature, at least to the authors knowledge. EXPERIMENTAL ARRANGEMENT AND ANALYSIS METHODOLOGY All the experiments were carried out at the NDF Circulating Water Channel facility of the University of São Paulo (USP). The dimensions of the test section is 7 7 7mm and the flow has low levels of turbulence (less than %). The pump system can operate with good quality with free-stream velocities up to.m/s. Further details concerning the Circulating Water Channel can be found in [8]. The leaf-spring elastic base can be assembled for experiments with DOF or DOF. In the later case, two orthogonal leaf-springs must be employed. The models are made of aluminium with external diameter D =.mm. A schematic representation of the experimental arrangement is shown in Figure.

3 Clamp Lower plate Upper Leaf Springs Lower Leaf Springs Cross-flow direction - y (y(t), C L(t)) In-line direction - x (x(t), C D(t)) Intermediate plate F D (t) = m a,x ẍ c v,x ẋ () F L (t) = m a,y ÿ c v,y ẏ () Flow direction 7 DOF Experiments Lower plate Load Cell Flow direction 7 DOF Experiments Where m a and c v refer to the added mass and the hydrodynamic damping respectively. Applying the Fourier Transform Ϝ on both sides of the Equations and, the following equations are obtained: FIGURE : Experimental Arrangement. Laser position sensors LEUZE model ODSL 8/V were employed to measure the in-line and cross-flow displacements. A load cell ATI, model Mini was used to acquire the lift and drag forces. The sample frequency was Hz and all the data were acquired on a HBM system. Free-decays tests in air allowed identifying a very low structural damping coefficient (ζ s <.%). The aspect ratio (L/D) for all the experiments was kept constant and equal to. The gap between the lower tip of the model and the bottom of the channel was less than D in order to minimize the influence of the flow around this region. No end-plates were employed. Table presents the experimental parameters. TABLE : Experimental parameters. m = m s m ζ ρπd L s [%] ζ = ζ s +m m ζ L/D The oscillation amplitudes were taken by computing the average of the % highest peaks in the time series. The dominant frequency f d refers to the frequency that contains the highest amount energy in the power spectrum density function (PSD). The concept of added mass adopted in this paper is the component of the hydrodynamic force in-phase with the acceleration. Using an analogous approach to the one described in the papers by [7] and [9], the drag and lift forces are decomposed according to the equations: Ϝ[F D ] = m a,x ω ic v,x ω Ϝ[x] () Ϝ[F L ] = m a,y ω ic v,y ω Ϝ[y] () The added mass coefficient for the in-line and crossflow directions can be obtained from Equations and. For each direction, the value of added mass was taken considering the spectral component at the dominant frequency of the displacement spectrum ( f dx or f dy ). RESULTS AND DISCUSSION The experimental results will be presented in the next four subsections. Firstly, the results related to displacement (amplitude and frequency) and to the hydrodynamic forces for m =.6 will be compared to those discussed in the papers by [7] and [8]. Following, a direct comparison between DOF and DOF results will be discussed for both values of the m tested. Finally, the results of cross-flow and in-line added mass coefficients will be presented. COMPARISON WITH PREVIOUS RESULTS - m =.6 Figure presents the comparison between the experimental DOF results from the present work and the results obtained by [7]. One can note that both the amplitude of oscillation (Figure (a)) and the root-mean-square (RMS) lift coefficient (Figure (d)) very well match with the previous results. The frequency response, shown in Figure (b) does not follow either the curve for a fixed cylinder (Strouhal number St.) or the classical lock-in response f dy f N. Figure (c) shows the drag amplification, although there is a difference in the maximum value of C D when compared with the literature.

4 .8.6 S t =,.. S t =,.8.6. f = fdy/fn...8 f = fdy/fn.6... Present work Khalak & Williamson (999) (a) Amplitude of cross-flow oscillation = A y /D. (b) Frequency of cross-flow oscillation f dy. A x.... fdx/fdy.. CD.... C L.. Present work Jauvtis & Williamson () (a) Amplitude of oscillation = A y /D.. (b) Frequency of oscillation f dy. Present work Khalak & Williamson (997) (c) Mean drag coefficient C D. Present work Khalak & Williamson (999) (d) Root-mean-square lift coefficient C L. CD... C L.. FIGURE : Experimental Results DOF - m =.6. The comparison between the present results from DOF experiments and those published in [8] are shown in Figure. The amplitude of oscillation matches with the previous results very well, including the in-line resonance observed in < V R <. The frequency plot (Figure (b)) shows the well known result regarding the twice in-line frequency of oscillation. The results for force coefficients are presented in Figures (c) and (d). The good agreement with previous results for the RMS lift coefficient can be observed. The mean drag coefficient follows the same trend, in spite of a slightly higher value obtained in the present study. Present work Jauvtis & Williamson () (c) Mean drag coefficient C D. Present work Jauvtis & Williamson () (d) Root-mean-square lift coefficient C L. FIGURE : Experimental Results DOF - m =.6. monotonically crescent and higher than the DOF results. Figures (c) and (d) present the results for the mean drag coefficient and RMS lift coefficient respectively. Notice, also, the asymptotic value C D. at V R, corresponding to the mean drag coefficient observed in a stationary circular cylinder. COMPARISON BETWEEN DOF and DOF RESULTS - m =.6 Figure presents the direct comparison between DOF and DOF results. For the values of reduced velocities V R <., i.e., before the range characterized by the upper branch (DOF) and the super upper branch (DOF), the amplitude of oscillation, shown in Figure (a), is independent of the in-line degree-of-freedom. Considering the results in the lower branch range, the amplitudes for the DOF system are slightly higher than those for the DOF case. Notice that the onset of desynchronization occurs for the same reduced velocity for both conditions V R. Besides the higher amplitudes of oscillation at the lower branch, the frequency of oscillation shows no distinction between DOF and DOF experiments at this range, as shown in Figure (b). Considering the range < V R < 8 in the frequency plot, the DOF curve is. DOF DOF (a) Amplitude of cross-flow oscillation. CD DOF DOF (c) Mean drag coefficient C D. f = fdy/fn. S t =, DOF DOF (b) Frequency of cross-flow oscillation f dy. C L.. DOF DOF (d) Root-mean-square lift coefficient C L. FIGURE : Comparison between DOF and DOF results - m =.6.

5 Some interesting aspects should be highlighted in the force coefficients plots. The first one is the marked amplification in both C D and C L observed in the in-line resonance, despite the small cross-flow amplitudes.. This fact demonstrates a dependence on the hydrodynamic forces not only with the amplitude, but also with the presence of synchronization. The second aspect that must be emphasized is the similar maximum value of C L, although the maximum cross-flow amplitude for DOF case is % higher than for DOF system. Another feature regarding the DOF results is the broader range of synchronization at the super upper branch range. Additionally to the classical curves of response discussed above, the Lissajous figures shown in Figure were plotted aiming at enhancing the comparative analysis. A first aspect that can be better understood with the Lissajous plot is the increase in the forces coefficients, for DOF system, at the in-line synchronization range,characterized by a symmetric vortex-shedding pattern (see [8]). At V R =., the well defined eightshapped xy Lissajous plot indicates a highly correlated wake along the span, and thus higher hydrodynamic forces can be expected in comparison with the case in which there is no spanwise wake correlation. At the same reduced velocity, the DOF system presents amplitude of oscillation lower than D, which is an indicative of lower correlation length. Another aspect that can be discussed through the Lissajous plot is related to the spectral distribution of both the lift force and the cross-flow oscillations. As pointed out by [8], the super upper branch range is characterized by the T vortex shedding pattern and by the presence of a third subharmonic (ω) component on the lift force coefficient. Analyzing reduced velocities V R =.6 and V R = 6., the xy plot presents the classical 8-shape and the C L (t) clearly reveal the presence of the referred subharmonic. Focusing on the reduced velocity immediately before the jump to the lower branch (V R = 7.9), the in-line amplitude is A x. although the xy figure does not follow the classical 8 shape pattern, whereas the C L (t) clearly does not presents the ω harmonic. Therefore, the ω component on the lift force is related to the twice inline frequency of oscillation, despite the non negligible in-line amplitude. COMPARISON BETWEEN DOF and DOF RESULTS - m = 8. The results for the condition with moderate to large mass ratio is presented in Figure 6. As mentioned in pre- V R Re DOF x (t) C L (t) x (t) x (t) x (t) x (t) CL(t) CL(t) CL(t) CL(t) FIGURE : Lissajous plot - m =.6. vious works, the amplitude of cross-flow oscillations for the DOF system very well agrees with that observed in the DOF experiments, even though some minor differences are shown in the upper branch, as presented in Figure 6(a). The review by [] reported the narrowing of the lower branch for DOF systems, fact in total agreement with the present DOF results. The non-dimensional frequency of response, shown in Figure 6(b), is closer to the unity than the results for the experiments with m =.6. This feature was also discussed, for DOF experiments in [7]. The analysis of the forces coefficients results deserves a more detailed discussion. Considering the range V R >, both the C L (Figure 6(d)) and the C D (Figure 6(c)) plots show a very good agreement DOF and DOF experiments. This aspect extents, to the forces coefficients, the assertion that for moderate to large mass ratio parameter, the in-line oscillations are negligible; there is thus no difference between one or two degrees-of-freedom systems.

6 . DOF DOF (a) Amplitude of cross-flow oscillation. CD. f = fdy/fn. S t =, DOF DOF (b) Frequency of cross-flow oscillation f dy. C L.. Cay Cax (a) Added mass coefficient. DOF and m =.6. Cay Cax (b) Added mass coefficient. DOF and m = 8.. DOF DOF DOF DOF (c) Mean drag coefficient (d) Root-mean-square lift C D. coefficient C L 6 8. Cay Cay A x.... (e) Amplitude of in-line oscillation DOF DOF DOF DOF (c) Comparison between (d) Comparison between DOF and DOF experiments. m =.6. ments. m = DOF and DOF experi- 8.. FIGURE 7: Added mass coefficient. FIGURE 6: Comparison between DOF and DOF results - m = 8.. However, for the range in which the in-line synchronization is observed ( < V R < ), the marked increase in both forces coefficients is observed. Moreover, the maximum values in this range very well agree with those observed for moderate to small mass ratio parameter experiments; therefore, the effects of the larger values m are more evident in the super upper branch range than in the in-line synchronization range. ADDED MASS COEFFICIENT The added mass coefficient results for both mass ratio parameters tested are presented in Figure 7. Considering m =.6 and DOF, the results for in-line, as well as cross-flow added mass (Ca x and Ca y respectively) are found in Figure 7(a). In the Ca x plot, the zero-crossing result occurs at V R, which corresponds to the end of the in-line synchronization. Analyzing the super upper branch range of reduced velocities, both Ca x and Ca y show an inflection behavior at V R 6., although the decreasing rate of the cross-flow results are more noticeable in the Ca y plot. Figure shows that, at V R = 6.9 the DOF Lissajous plots are more pronouncedly coupled, in agreement with the results described by [] for moderate to large m ex- periments. In [], the authors pointed out that the inflection in Ca y plot could be an effect of the in-line oscillations, and consequently to change in the vortex shedding pattern. At the range < V R < 7, the Ca y is constant and the ω component is present in the lift force spectrum; hence the T vortex shedding pattern is observed. The inflection occurs when the ω component is no longer observed, implying a change in the vortex shedding pattern, as previously pointed out, with no significant change in the in-line amplitude of oscillation, though. The present results show the same trend in spite of the higher value of in-line oscillations compared to those presented in the cited work (A x. in present work, A x. in []). Also notice the marked similarity of the values of Ca y at the lower branch for both DOF and DOF systems (see Figure 7(c)). Conversely to the m =.6 results, the cross-flow added mass coefficient for DOF very well agrees with those from DOF experiments for the whole range of reduced velocities, as presented in Figure 7(d). Because of the very small oscillations, specially at the lower branch, the in-line oscillations, as well as the corresponding accelerations, are not well defined thus, through Equation, the in-line added mass is poorly defined for moderate to large m.

7 FINAL REMARKS Experiments of vortex-induced vibrations (VIV) were carried out with a circular cylinder free to oscillate in one or two directions using the same apparatus. Two conditions of structural mass were tested, allowing studying either the effects of moderate to small or moderate to large mass ratio parameter. A direct comparison of results was discussed, enabling to identify differences in amplitudes of oscillation, forces and added mass coefficients between DOF and DOF for both values of mass ratio parameter. For moderate to small mass ratio parameter m =.6, the results showed that at the lower branch, there are no differences between DOF or DOF systems, except for slightly higher amplitudes for DOF. At the super upper branch the in-line degree of freedom introduces a marked difference in the plots. It is worth mentioning the fact that the maximum value of RMS lift force coefficient is similar for both DOF and DOF, even though the maximum cross-flow amplitude differs by %. Considering moderate to large values of m, the results for amplitude, frequency and forces coefficients show very good agreement between DOF or DOF systems for both the upper branch and lower branch ranges. At the range in which the in-line synchronization is observed <V R <, both the DOF RMS lift force and mean drag coefficients showed a marked increase if compared with the DOF results. Moreover, both the crossflow and the forces magnification in this range were less influenced by the effects of the mass ratio parameter. Results of in-line and cross-flow added mass coefficient were also presented. The discontinuities at the m =.6 curves are associated to the end of the in-line synchronization for the Ca x and to the end of the super upper branch and the onset of the lower branch for cross-flow added mass coefficient. The Ca y plot for m = 8. very well agreed of the results for DOF system. For this value of mass ratio parameter, the in-line added mass is poorly defined. Further works will include the study of force and added coefficients for different ratio of in-line and crossflow stiffness. ACKNOWLEDGMENT The authors acknowledge FAPESP (São Paulo State Research Foundation), CNPq (National Council of Research), Petrobras (Brazilian company of oil and gas) and FINEP (National agency for research sponsoring) for the financial suport on Vortex-Induced Vibrations (VIV) projects at USP. G.R.F. is also grateful to FAPESP for his PhD scholarship, process 8/ Special thanks to César Monzu Freire, Ivan Korkischko, Douglas Silva and Reinaldo Marcondes Orselli for their help with the experiments. Prof. Fujarra is grateful to the Brazilian Navy and Maritime Research Institute Netherlands for all support provided during his sabbatical period,. REFERENCES [] Bearman, P. W., 98. Vortex shedding from oscillating bluff bodies. Annual Review of Fluids Mechanics, 6, Jan, pp. 9. [] Bearman, P. W.,. Circular cylinders wake and vortex-induced vibrations. Journal of Fluids and Structures, 7, pp [] Sarpkaya, T.,. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures, 9, pp [] Williamson, C. H. K., and Govardhan, R. N.,. Vortex-induced vibrations. Annual Review of Fluids Mechanics, 6, pp.. [] Williamson, C. H. K., and Govardhan, R. N., 8. A brief review of recents results in vortex-induced vibrations. Journal of Wind Engineering and Industrial Aerodynamics, 96, pp [6] Gabbai, R., and Benaroya, H.,. An overview of modeling and experiments of vortex-induced vibration of circular cylinders. Journal of Fluids and Structure, 8, pp [7] Khalak, A., and Williamson, C. H. K., 999. Motions, forces and modes transitions in vortexinduced vibration at low reynolds number. Journal of Fluids and Structures,, pp [8] Govardhan, R. N., and Williamson, C. H. K., 6. Defining the modified griffin plot in vortexinduced vibration: reavealing the effect of reynolds number using controlled damping. Journal of Fluid Mechanics, 6, pp [9] Hover, F. S., Miller, S. N., and Triantafyllou, M. S., 997. Vortex-induced vibration of marine cables: Experiments using force feedback. Journal of Fluids and Structures,, pp [] Khalak, A., and Williamson, C. H. K., 997. Fluid forces and dynamics of a hydroelastic structure with very low mass and damping. Journal of Fluids and Structures,, pp [] Sarpkaya, T., 99. Hydrodynamic damping, flowinduced oscillations and biharmonic response. Journal of Offshore Mechanics and Artic Engineering, 7, pp. 8.

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