Vortex-Induced Vibration of Marine Risers: Motion and Force Reconstruction from Field and Experimental Data by Harish Mukundan

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3 Vortex-Induced Vibration of Marine Risers: Motion and Force Reconstruction from Field and Experimental Data by Harish Mukundan Submitted to the Department of Mechanical Engineering on April 5, 28, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Vortex-induced vibration (VIV) of long flexible cylindrical structures enduring ocean currents is ubiquitous in the offshore industry. Though significant effort has gone into understanding this complicated fluid-structure interaction problem, major challenges remain in modeling and predicting the response of such structures. The work presented in this thesis provides a systematic approach to estimate and analyze the vortex-induced motions and forces on a marine riser, and develop suitable methods to improve riser VIV modeling and response prediction. In the first part of the thesis, a systematic framework is developed, which allows reconstruction of the riser motion from a limited number of sensors placed along its length. A perfect reconstruction criterion is developed, which allows us to classify when the measurements from the sensors contain all information pertinent to VIV response, and when they do not, in which case additional, analytical methods must be employed. Reconstruction methods for both scenarios are developed and applied to experimental data. The methods are applied to: develop tools for in-situ estimation of fatigue damage on marine risers; improve understanding of the vortex shedding mechanisms, including the presence of traveling waves and higher-harmonic forces; and estimate the vortex-induced forces on marine risers. In the second part of the thesis, a method is developed to improve the modeling of riser VIV by extracting empirical lift coefficient databases from field riser VIV measurements. The existing experiment-based lift coefficient databases are represented in a flexible parameterized form using a set of carefully chosen parameters. Extraction of the lift coefficient parameters is posed as an optimization problem, where the error between the prediction using a theoretical model and the experimental data is minimized. Predictions using the new databases are found to significantly reduce the error in estimating the riser cross-flow response. Finally, data from a comprehensive experiment is utilized to show that the riser response is resonant in the harmonic component, but non-resonant in the third-harmonic component. It is shown that this happens because the spatial dependence of the third-harmonic fluid force component is dominated by the first-harmonic wavelengths. This finding has significant implications for modeling the higher-harmonic forces and the resulting fatigue damage estimation methodologies. Thesis Supervisor: Michael S. Triantafyllou Title: Professor of Mechanical and Ocean Engineering Thesis Supervisor: Franz S. Hover Title: Assistant Professor of Mechanical and Ocean Engineering

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5 This thesis is dedicated with love and respect to the ever living memories of my elderly.

6 Acknowledgments This PhD thesis is a culmination of five years of my stay here at MIT. Over these years, I have had the opportunity to learn, relearn and I have passed through some of the most difficult and frustrating moments of my life. I recognize that this rigorous learning experience will help me in several more difficult endeavors later in my life. Several people have helped me during this difficult endeavor with their encouraging words, helping hands and material assistance. My advisers during the thesis Prof. Michael Triantafyllou and Prof. Franz Hover have been exemplary, and I thank them for their motivation, mentorship and expert guidance through the labyrinth of research. Prof. Triantafyllou always had an open door and mind for several discussions and I sincerely thank him for his patience. Prof. Hover was always there to consolidate my learning, especially at times when Prof. Triantafyllou was not available. I share a special feeling for my previous advisor Prof. Nicholas Patrikalakis, from whom I have learned a lot and I remain indebted to him. I am grateful to Prof. Kim Vandiver and Prof. Eduardo Kausel who always had time for me in spite of their busy schedule. I also gratefully acknowledge the financial support from British Petroleum and technical support in the form of experimental data from Norwegian Deepwater Programme. I feel fortunate to have been taught by the masters in their corresponding fields at MIT, Professors A. Almazan, T. R. Akylas, K. J. Bathe, J. J. Connor, T. Copeland, T. Chen, D. M. Freeman, D. C. Gossard, V. K. Goyal, S. Hunter, E. Kausel, H. Kite-Powell, P. Koev, H. S. Marcus, D. Margetis, A. V. Oppenheim, A. T. Patera, N. M. Patrikalakis, J. Peraire, R. Stocker, N. P. Suh, J. Sussman, A. Techet, M. S. Triantafyllou, A. Toomre, K. J. Vandiver, D. Veneziano, J. K. White and D. K. P. Yue. I gratefully remember and acknowledge my Professors at IIT-Madras V. G. Idichandy, S. Surendran and R. Natarajan for their recommendation letters without which I would not have been at MIT in the first place. Xiaojing, my good friend has encouraged me in all my efforts and supported me during difficult times at MIT. Yahya and Phil have helped me enormously by proof reading my thesis and presenting several valuable comments. I thank my colleagues Arpit, Costas, Ding, Eric, Jason, Joe, Josh, Matt, Phil, Srini, Tianrun, Vikas, Vivek and Yahya for a pleasant working environment. Thanks are due to past and present administrative staff at Ocean Engineering S. Malley, K. de Zengotita, M. Munger, P. Pickard, J. Lewis, P. Tolan and E. Daniel, and also to the current Mechanical Engineering staff J. Kravit, L. Regan and D. Shea. I also thank the many unknown faces who have helped me in every aspect of my life in these years. I can never forget my Parents T. Mukundan and B. Lali for their unconditional support and encouragement for nearly 28 years without which I cannot even imagine (I don t want to imagine) where I would have been. Finally I thank the Almighty for guiding me through some of the most difficult and confusing times and for keeping up my morale.

7 Contents Abstract 3 Acknowledgments 6 Introduction 27. Background The phenomenon of vortex shedding Vortex-induced vibration of elastically mounted rigid cylinders Vortex-induced vibration of marine risers Modeling vortex-induced vibration of marine risers Research objectives Thesis outline Theoretical background Introduction Equation governing VIV of an elastically mounted rigid cylinder Equation governing VIV of a riser A systematic approach to riser VIV response reconstruction Introduction Problem setting Related literature Overview of the approach Response reconstruction when a combination of acceleration and strain measurements is given A discussion on the reconstruction method and practical issues Full reconstruction criterion (Spatial Nyquist s criterion) Statement of Nyquist s criterion Application to response reconstruction

8 3.5 Applications Application to benchmark data Application to experimental data A study of reconstruction error Identifying sources of uncertainty during reconstruction Uncertainty from the presence of noise in experimental data Uncertainty from the use of finite number of sine and cosine terms Uncertainty from the presence of both acceleration and strain measurements Total uncertainty Concluding remarks Riser VIV modal decomposition and traveling wave identification Introduction Time evolution of VIV measurements using scalograms Extracting a statistically stationary segment of VIV measurement Application of scalograms to NDP data Riser response modes from displacement measurements Definition of response modes and response frequencies Numerical method Application to NDP data and discussion Traveling waves in riser VIV response Method : From magnitude of peak response modes Method 2: From phase angle of peak response modes Method 3: From nodal evolution curves Traveling waves in NDP experimental data Traveling waves in third harmonic of riser VIV response A discussion on traveling waves in riser VIV response Concluding remarks Optimal lift force coefficient databases from riser experiments Introduction A summary of related research Problem statement and solution overview Working hypothesis Solution overview Experimental measurements and theoretical estimates Experimental measurements

9 5.3.2 Theoretical estimates Formulation as an optimization problem Parameterization of lift coefficient databases Nonlinear forward model Error metric (Optimization index) Choice of solution technique and computational framework Universal C lv and C m databases Obtaining candidate databases (candidate parameter-sets) Obtaining the best candidate parameter-set A discussion on the universal C m database A discussion on the universal C lv database A discussion on the optimization method and practical issues Concluding remarks Response reconstruction from few sensor data 7 6. Introduction Problem setting Response reconstruction from few sensor data Solution overview Overall algorithm Illustrative example A discussion on response reconstruction using few sensor data Local lift coefficient database correction Formulation as an optimization problem Application to NDP data Application employing the VIVA generated modes and frequencies Concluding remarks On origin and resonance of third harmonic of VIV on risers Introduction A summary of related research Topics of present study Origin of higher harmonics in vortex-induced fluid force Lissajous figures Application to VIV higher harmonics Inference from NDP data Non-resonant third harmonic in vortex-induced force

10 7.3. Validation based on experimental data from NDP Validation based on theoretical models Comparison with laboratory observations A conceptual model for the third harmonic force Concluding remarks Conclusions 6 8. Summary and principal contributions from each chapter Riser VIV response reconstruction method Riser modal decomposition and traveling wave identification Optimal lift coefficient databases from riser experiments Response reconstruction using data from few sensors Insights on the third harmonic component in riser VIV Recommendations for future research A Nomenclature 67 A. Coordinate system definition A.2 Definition of various coefficients B Available riser VIV experimental datasets 7 B. Norwegian deepwater programme high mode VIV datasets B.2 Lake Seneca experiment datasets B.3 First Gulf-stream experiment datasets C Application of reconstruction method to NDP dataset C. Preparation of the data matrix Ŷ C.2 Preparation of the Φ matrix C.3 Obtaining the ŵ matrix C.4 Results D Implementing scalogram plots in Matlab 87 D. Required data and codes D.2 Procedure E Multi-frequency force coefficients from experimental data 89 E. Multi-frequency lift coefficient decomposition E.. Effective lift coefficient in phase with velocity for a rigid cylinder E..2 Effective lift coefficient in phase with velocity for a flexible cylinder E.2 Multi-frequency power balance

11 F Elastically mounted rigid cylinder theoretical estimates 93 F. Method (Iterative method): F.2 Method 2 (Zero-contour method): G Estimation of riser VIV fatigue life from sensor measurements 97

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13 List of Figures - Depicts our ability to explore and drill in deep waters [75] Vortex shedding behind a 2D fixed cylinder resulting in the famous von Kárman street for a subcritical Reynold s number = 4 [56] Three commonly studied systems undergoing VIV. (a) stationary rigid cylinder under constant flow; (b) elastically mounted rigid cylinder under constant flow; (c) flexible cylinder under a flow profile The dependence of Strouhal number (S t ) on Reynold s number, and the power spectrum of lift force at select Reynold s number regimes. Figure sourced from Pantazopoulos [44] Empirical schemes for predicting the fatigue life of a marine riser. An empirical scheme contains empirical databases and empirical models The extensive C m and C lv database obtained by Gopalkrishnan [4] Depicts the setup of the elastically mounted rigid cylinder under a uniform flow The lift force coefficient databases of C lv (A, V r ) (left) and C m (A, V r ) (right) produced by Gopalkrishnan [4] A riser modeled as a tensioned beam taut along the z direction Theoretical estimates of peak response modes (hydroelastic modes) for a uniform flow profile (top) and a linearly sheared flow profile (bottom) obtained using VIVA Figure depicting the typical arrangement of sensors on a riser Based on the number of sensors available and the bandwidth of the signals from the sensors, we have two separate scenarios of reconstruction. The solution strategy is different for each of the scenarios Flowchart of the algorithm for reconstructing riser VIV displacements from sensor measurements Figure illustrating the spectrum of the η c (t) of a typical benchmark test

14 3-5 Top: comparison of the RMS of the original and reconstructed displacements for the benchmark data using only sine terms; bottom: comparison of the RMS of the original and reconstructed displacements for the benchmark data using both cosine and sine terms Comparison of the CF experimental curvature time series (red), and the reconstructed CF curvature time series (blue) at select strain gage locations for a typical NDP dataset (dataset 24). Note that the reconstructed curvature closely matches the experimental data at least at the sensor locations Comparison of the CF experimental curvature time series (red), and the reconstructed CF curvature time series (blue) at select strain gage locations for a typical NDP dataset (dataset 246). Note that the reconstructed curvature closely matches the experimental data at least at the sensor locations Depicts the displaced shape of the riser at several instances of time for NDP dataset 243. One can clearly observe the presence of disturbances which starts around z = and moves along the riser The span averaged strain signal from NDP dataset 236 is subdivided into three segments. Segments within the solid red vertical lines correspond to zero flow velocity periods, and the segments between the dashed black vertical lines correspond to the period when the riser was towed. The σ n level can be obtained from the ratio of RMS of the zero velocity signal and RMS of the actual VIV signal Variability in reconstruction arising due to the presence of noise in experimental measurements. RMS displacement and error bound assuming σ n = 5% (row I ) and σ n =.5% (row II ) for NDP dataset 236 (U max =.8ms ). RMS displacement and error bound assuming σ n = 5% (row III ) and σ n =.5% (row IV ) for NDP dataset 246 (U max =.8ms ) Uncertainty arising due to the use of finite number of spatial harmonics during reconstruction. Row I : RMS displacement and error bound on noise for NDP dataset 236 (U max =.8ms ); row II : RMS displacement and error bound on noise for NDP dataset 24 (U max =.2ms ); row III : RMS displacement and error bound on noise for NDP dataset 243 (U max =.5ms ); row IV : RMS displacement and error bound on noise for NDP dataset 246 (U max =.8ms ). Note that as the flow velocity increases, the uncertainty in reconstruction increases due to the presence of higher spatial harmonics in the measured riser motions Uncertainty arising due to the use of finite number of spatial harmonics during reconstruction for various linearly sheared flow velocity cases. Note that as the flow velocity increases, the uncertainty in reconstruction increases due to the presence of higher spatial harmonics in the data

15 3-3 Uncertainty arising due to the use of both strain and acceleration signals during reconstruction for NDP datasets 236 (row I ), 24 (row II ), 243 (row III ) and 246 (row IV ). We can observe that with increasing flow velocity, the variability in reconstruction increases Depicts the RMS riser displacements and the bounds y RMS (z) ± σ y (z), which take into account three sources of uncertainty for NDP datasets 236 (row I ), 24 (row II ), 243 (row III ) and 246 (row IV ). We can observe that the total variability in reconstruction increases as the flow velocity increases Three different representations of the CF strain signal measured at z = 8.6m for NDP dataset 2. Top: a plot of the time signal; middle: the spectrum of the signal; bottom: the scalogram of the signal Left: the scalogram of CF response (strain signal) at a given location (z/d = 394) from the Gulf-stream test (B242954). From the scalogram we can observe a multi-frequency response, a sudden event happening around t = 5seconds, and the sporadic appearance and evolution of the third harmonic at reduced frequency f r.6. Right: the scalogram of CF response (acceleration signal) at a given location (z/d = 927) from the Lake Seneca test (47462). We can clearly observe the repeated patterns in the signal, which arise from either a sensor malfunction or a processing error Left: scalograms of select strain signals from NDP experiments [5] (for dataset 2) depicting a typical case of non-statistically stationary behavior; right: scalograms from another set of data (dataset:238) from NDP experiments showing a behavior typical of being statistically stationary Depicts the scalograms of strain signals obtained at 6 representative locations along the riser for NDP dataset 24, and our choice of the statistically stationary segment of data (t stationary = [2.5, 6.5]seconds) shown between the two red lines Depicts the scalograms of strain signals obtained at 6 representative locations along the riser for NDP dataset 24, and our choice of the statistically stationary segment of data (t stationary = [25.5, 29]seconds) shown between the two red lines Magnitude of response of a tensioned string at chosen locations excited by a forcing with a frequency Ω Left: narrow-banded response observed from strain data measured at 3 locations along the riser for NDP dataset 234; right: multi-frequency response observed from strain data measured at 3 locations along the riser for NDP dataset 238. The vertical green lines represent the natural frequencies obtained using the empirical prediction program VIVA

16 4-8 Modal decomposition of the riser response for several linearly sheared flow profile datasets (each row correspond to one dataset) from NDP experiments. Column I : depicts the maximum velocity U max ; column II : depicts the harmonic part of the span averaged displacement spectra for various NDP datasets; column III : depicts the peak response modal magnitudes for various NDP datasets; column IV : depicts the peak response modal phase angles for various NDP datasets Response of a tensioned string under two different damping condition (ζ % ( ) and ζ 35% ( )) acted upon by a force of the form f(z, t) = Re { F (z)e iωt}. Top: depicts the spatial variation F (z) of the applied force; middle: depicts the magnitude of the response; bottom: depicts the phase of the response Figure depicting the nodal evolution plots for 8 different cases of riser displacements of the form given by equation (4.5). Figure 4-(g) depicts a purely standing wave. Figure 4-(h) depicts a purely traveling wave. Figures 4-(a) to 4-(f ) illustrates responses indicating varying presence of standing and traveling waves Contour plots depicting the evolution of the nodes in riser VIV for two linearly sheared flow profiles; Top left: CF displacements for dataset 236 (U max =.8ms ); bottom left: IL displacements for dataset 236 (U max =.8ms ); top right: CF displacements for dataset 242 (U max =.4ms ); bottom right: IL displacements for dataset 242 (U max =.4ms ) Contour plots depicting the evolution of the nodes in riser VIV for two uniform flow profiles; Top left: CF displacements for dataset 26 (U max =.8ms ); bottom left: IL displacements for dataset 26 (U max =.8ms ); top right: CF displacements for dataset 22 (U max =.4ms ); bottom right: IL displacements for dataset 22 (U max =.4ms ) Observed CF wave propagation speed for different linearly sheared velocity profiles Observed CF wave propagation speed for different uniform velocity profiles Left: nodal evolution curves for the harmonic and third harmonic components of riser CF displacements for the NDP dataset 23; right: nodal evolution curves for the harmonic and third harmonic component for the NDP dataset 234. The third harmonic component of the response clearly depict the presence of traveling waves, but not necessarily the same consistency which the harmonic component portray The extensive C m and C lv database obtained by Gopalkrishnan [4] A flow chart describing our method to obtain the peak response frequencies and the peak response modes from experimental data

17 5-3 Parameterization of the C lv database Parameterization of the C m database Left top: nominal C lv database corresponding to the nominal value of parameter p 3 ; right top: nominal C m database corresponding to the nominal value of parameter p ; left bottom: example of a modified C lv database by varying the parameter p 3 from its nominal value 5.4 to 7.3; right bottom: example of a modified C lv database by varying the parameter p from its nominal value 6.2 to Flow chart describing the simulated annealing algorithm to obtain the optimal parameter-set p opt Flow chart illustrating a single evaluation of the optimization index Left: a comparison of the experimental and nominal theoretical prediction; right: a comparison of the experimental and optimal theoretical prediction Application of the optimization method to NDP dataset 243. Left: comparison of the experimental and nominal theoretical prediction; right: comparison of the optimal and experimental prediction. Contour plots below depict the corresponding C lv and C m databases The predictions of Y n (z) and ω n using nominal databases (red dashed) and ( ) ( ) candidate databases C Y (z) lv D, U(z) ωd, p cand j and C Y (z) m D, U(z) ωd, p cand j (green) are compared with the experimentally observed ones (blue) for various datasets from the NDP experiments Flow chart illustrating the process of obtaining the universal parameter-set p univ corresponding to the universal C lvuniv and C muniv databases Left: a comparison of the optimization index J dataset i ( p candj ) for all the candidates isolated; right: depicts the universal error metric J univ ( p candj ) for all the candidates isolated. Note that the use of universal databases result in a 44% reduction in J univ ( p candj ) Strouhal number variation as a function of the Reynold s number, also depicts the Reynold s number corresponding to the Gopalkrishnan database (Reynold s number = ) and the present study (maximum value of Reynold s number in the range 6 to 46). For the range of Reynold s number in the present study, the Strouhal number is expected to be lower than that for Reynold s number around. Figure is adapted from [2, 3, 48, 58] Depicts the universal C muniv database obtained from the optimization code in conjunction with VIVA. A total of nine datasets from NDP experiments (for linearly sheared flow profiles) were used. The maximum value of Reynold s number lies between 6 to

18 5-5 Depicts the universal C lvuniv database obtained from the optimization code in conjunction with VIVA. A total of nine datasets from NDP experiments (for linearly sheared flow profiles) were used. The maximum value of Reynold s number lies between 6 to VIV oscillation amplitude of an elastically mounted freely oscillating cylinder in uniform flow. Left: experiments conducted by Smogeli [53] for various values of reduced velocity V r ; right: experiments conducted by Vikestad [27] for various values of nominal reduced velocity V rn. Note that the amplitude ratio A takes values as high as A snapshot of the results from the use of the universal database with VIVA (for linearly sheared flow profiles). The figure compares experimental with the theoretical prediction using nominal (+ sign) and universal ( sign) lift coefficient databases for various linearly sheared flow profile datasets from NDP experiments. The use of universal databases improves our estimate of both the peak response frequency and the peak response modal magnitude. The agreement between the experiments and the predictions using the universal databases result in 44% reduction in error Figure depicting the typical arrangement of N s sensors at locations z s on a riser The spectrum of signal ŷ(z s, ω) from a sensor is subdivided by bandpass filtering them around each of the peak response frequencies ω n to obtain N m separate spectra ŷ n filt (z s, ω) Overview of the algorithm for reconstructing riser response using measurements from few sensors Left: the span averaged spectrum of the benchmark data; right: the RMS of the displacements measured along the riser Left: the span averaged spectra of the benchmark data. The vertical lines represents the VIVA predicted ω n. Right: the spectra of sensor signals ŷ(z s, ω) are subdivided by bandpass filtering them around each ω n to obtain N m separate spectra ŷ n filt (z s, ω) Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 2 are illustrated

19 6-7 Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 3 are illustrated Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 4 are illustrated Left: depicts the original and reconstructed signals over 5 seconds for a case when N s = 3; right: depicts the original and reconstructed signals over 5 seconds for a case when N s = Depicts the average and maximum reconstruction error as a function of the error in the flow profile Umax U max used to obtain Y n (z) and ω n for various N s cases. We can observe that the accuracy of reconstructed response is highly sensitive to the error in the flow profile used during reconstruction Overview of the method used for local lift coefficient database correction Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s locations. = 2 for various cases of sensor A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s locations. = 3 for various cases of sensor A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s locations. = 4 for various cases of sensor A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals (using nominal lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 2 for various cases of sensor locations. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals (using nominal lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 4 for various cases of sensor locations. The locations of the sensors are marked by the sign

20 6-7 Application of the local lift coefficient extraction method to NDP dataset 243. Left: comparison of the experimental and nominal theoretical prediction; right: comparison of the optimal and experimental prediction Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 2 for various cases of sensor locations. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 3 for various cases of sensor locations. The locations of the sensors are marked by the sign Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 4 for various cases of sensor locations. The locations of the sensors are marked by the sign The average error during reconstruction for all possible combinations using data from the 8 accelerometers for NDP dataset Different Lissajous figures obtained for different values of frequency ratio ( ωx ω y ) and (ψ x ψ y ). A x = A y is assumed to be constant Sample phase-plots from NDP experiment (dataset: 234) depicting the phaselocked behavior of CF fpeak (t) with CF f3peak (t), and CF fpeak (t) with CF f5peak (t). The figure illustrates the remarkable periodicity between the harmonic and the higher harmonic components of the response Sample phase-plots from NDP experiment (dataset: 243) depicting the phaselocked behavior of CF fpeak (t) with CF f3peak (t), and CF fpeak (t) with CF f5peak (t). The figure illustrates the remarkable periodicity between the harmonic and the higher harmonic components of the response D iso-contours of instantaneous vorticity obtained from DNS simulations by Dahl et al. []. We can observe the presence of spatial persistence of vortex triplets being shed during every half cycle Depicts span averaged spectra for various NDP datasets corresponding to different linearly sheared flow profiles. Left: the maximum velocity U max ; middle: the span averaged spectra of CF strain; right: the span averaged spectra of CF acceleration.47 2

21 7-6 Top: peak response frequency obtained for various NDP datasets corresponding to the maximum velocity U max. Bottom: the C m (ω ) for the harmonic component obtained for different datasets corresponding to the maximum velocity U max ; ( ) corresponds to estimate obtained by evaluating the force in phase with acceleration; (+) corresponds to estimate obtained from equation (7.8). Note the remarkable agreement between the C m (ω ) estimated using the two methods Comparison of harmonic (+) and third harmonic ( ) components of the stiffness and inertia forces for different flow velocities. Top left: RMS of the harmonic and third harmonic components of stiffness force; top right: RMS of the harmonic and third harmonic components of net inertia force; bottom left: the RMS of the harmonic and third harmonic components of residual excitation force (inertia force minus stiffness force); bottom right: the ratio of RMS of the harmonic and third harmonic components of inertial force by the RMS of stiffness force. Note that for the harmonic component, the inertia forces nearly matches the stiffness force. However for the third harmonic component the inertia force dominate over the stiffness force Magnitude of the transfer function of a single degree of freedom system. At resonant frequency the inertia force balances the stiffness force, at high frequencies we observe an inertia dominated system and at low frequencies we have a stiffness dominated system Displacement magnitudes of a tensioned string (similar properties as the NDP riser) for three cases of forcing of the form f(z, t) = Re { sin(kz)e iωt}. Note that Case I (top), and the Case III (bottom) will result in increased displacement magnitude (resonance) much higher than the displacement magnitude in Case II (middle) Depicts the harmonic component F (z) and third harmonic component F 3 (z) of the fluid force obtained using lift coefficient database from Dahl et al., for various cases from NDP experimental data. We can observe that the dominant wavelength of F 3 (z) closely follows the dominant wavelength of F (z) (a) Typical span averaged spectrum of CF fluid force. (b) Typical spatial variation of the harmonic part of the CF fluid force. (c) Proposed concepts for the spatial variation of the third harmonic of the CF fluid force A- Definition of coordinate system used throughout the thesis B- Setup of NDP experiment as reproduced from Trim et al. [6, 5] B-2 Setup of the Lake Seneca experiment as reproduced from Vandiver et al. [7]

22 B-3 Left: first Gulf Stream experimental setup as reproduced from Vandiver et al. [72]; right: cross-sectional view of the riser depicting placement of the optical strain gage fibers. There are four sets of fiber couples (Q, Q2, Q3, Q4), each containing 7 strain locations C- A glance at the NDP data 235 depicts the following. (a) Magnitude of the span averaged acceleration spectrum with red dashed lines depicting the band pass filter used; (b) Magnitude of the span averaged curvature spectrum with red dashed lines depicting the band pass filter used; (c) Magnitude of the span averaged displacement spectrum (obtained from the acceleration spectrum), from which we can observe that the higher harmonic components do not contribute much to displacement; (d) Magnitude of the span averaged acceleration spatial spectrum obtained using the dispersion relation of a tensioned string and assuming a variability in C m [.5, 3]. The magenta dashed line depicts the maximum number of spatial harmonics which we can use during reconstruction C-2 RMS of the spatial Fourier coefficients using the sine terms from n = 3 to n = 4 and cosine terms from n = to n = 4. Though the magnitudes of the spatial Fourier coefficients are large, but an extensive investigation shows that there are cancellations happening between the sine and cosine terms allowing adequate representation of complicated riser motions C-3 Comparison of the RMS of the original (at accelerometer locations) and reconstructed displacements for NDP dataset 235 using both cosine (n = to n = 4) and sine (n = 3 to n = 4) terms C-4 Comparison of the experimental curvature time series (red), and the reconstructed curvature time series (blue) at few strain gage locations. Note that the reconstructed curvature closely matches the experimental data at least at the sensor locations C-5 A study of the uncertainty during reconstruction. Row I : RMS displacement and error bound due to the presence of noise in experimental measurements assuming σ n =.5%; row II : uncertainty arising due to the use of a finite number of sine and cosine terms during reconstruction based on an ensemble containing 8 members; row III : uncertainty arising due to the use of both strain and acceleration signals during reconstruction for β [., ]; row IV : the RMS riser displacements and the bounds y RMS (z) ± σ y (z), which take into account three sources of uncertainty.86 22

23 F- Top: equation (F.) has zeros which in the V r, A space represented by the contour labeled with ; middle: equation (F.2) has zeros which in the V r, A space represented by the contour labeled with ; bottom: the intersection of both contours represent the solution to the system of equations (F.) and (F.2) F-2 The equations (F.) and (F.2) can have no zeros, single zeros or multiple zeros depending on the input flow velocity G- Overview of the rainflow method for stress cycle counting

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25 List of Tables 5. A comparison of the typical input and observables of a VIV problem Definition of the parameters p to p 9 corresponding to the C lv database. Please refer to Figure 5-3 for a depiction of these parameters Nominal values of the parameters used for representing C lv database Definition of the parameters p to p 2 corresponding to the C m database Nominal values of the parameters used for representing C m database Two desktop PCs with the above specifications were employed to run the optimization codes B. Important properties of riser model used in NDP experiment as reproduced from Trim et al. [6, 5] B.2 Important properties of pipe used in Lake Seneca experiment as reproduced from Vandiver et al. [7] B.3 An overview of the various test runs conducted during the Lake Seneca experiment as reproduced from [69, 7] B.4 Important properties of pipe used in First Gulf Stream experiment as reproduced from Vandiver et al. [7, 72, 7] B.5 An overview of the various test runs conducted during the First Gulf Stream experiment as reproduced from [7] C. Depicts the assembled Φ matrix accurate up to 3 decimal places. The shaded region of the table corresponds to acceleration, while the rest of the table corresponds to curvature. The part of the table which is slanted corresponds to cosine terms while the rest of the table corresponds to sine terms

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27 Chapter Introduction. Background Vortex-induced vibration (VIV) of ocean structures is a major factor affecting all stages of development of offshore structures (conceptualization, design, analysis, construction and monitoring) and governs the arrangement of risers, details during fabrication, method of installation and instrumentation and operation. Advances to deeper waters in search for petroleum resources have resulted in multi-billion dollar offshore projects off the Gulf of Mexico, for example: Cheyenne (2747), Independence Hub (244m), Atlantis (245m), Nakika (2m), Thunder Horse (9m) to name a few (refer Figure -). In such water depths, long flexible cylinders are increasingly required (umbilics, risers, conductor tubes, pipeline spans), and prediction of VIV response has become increasingly important. A recent estimate by British Petroleum puts the estimated cost of countering VIV to approximately % of the project cost itself. Figure -: Depicts our ability to explore and drill in deep waters [75]. 27

28 .. The phenomenon of vortex shedding The presence of a 2D cylinder (bluff object) held static in a current results in separated flow (consisting of two layers of separation) in the cylinder wake. Two layers of fluids moving with different velocity leads to natural instability resulting in the formation of vortices along each of the mixing layers. Over time the vorticity is concentrated at downcrossing/upcrossing points of the perturbed upper/lower shear layers emanating from the cylinder, resulting in a staggered array of vortices leading to the famous von Kárman street depicted in Figure -2. For a stable staggered von Kárman street behind a stationary cylinder (Figure -3(a)), it was found that the frequency of vortex shedding (f s ) is directly proportional to the current velocity (U) and inversely proportional to the diameter of the cylinder (D), or f s U D [4]. The constant of proportionality is called the Strouhal number (S t ), and it was found out to be dependent on the Reynold s number (RE) as depicted in Figure -4 [44, 52]. The Strouhal number was found to be approximately within the range.7 to.2 in the subcritical Reynold s number regime. In the critical and the supercitical Reynold s number regimes the Strouhal number was found to be quite different, taking values up to.4 in the critical and around.26 in the supercritical regime. Thus the vortex shedding frequency could be estimated from the flow speed U as: f s S t D. The vortices hence shed are low pressure regions, resulting in an alternating force driven at the frequency of shedding. The alternating forcing has two components, a cross-flow (CF) component (lift force) which oscillates about zero mean and an in-line (IL) component which oscillates about a non-zero mean (drag force). Figure -2: Vortex shedding behind a 2D fixed cylinder resulting in the famous von Kárman street for a subcritical Reynold s number = 4 [56]...2 Vortex-induced vibration of elastically mounted rigid cylinders An important phenomenon of lockin occurs if the cylinder is elastically mounted (refer Figure -3(b)) and is free to move under the condition that the frequency bandwidth of forces induced by vortex shedding includes the natural frequency of the system. Under the above conditions, the natural frequency of the system f n which depends on its inertia (mass+added mass) changes due to the formation of vortices, and the vortex shedding frequency f s changes due to the oscillation of the cylinder. A feedback mechanism occurs such that both the natural frequency of the 28

29 system and the vortex shedding frequency coalesce, leading to amplification of the oscillation due to resonance. The oscillations are such that the motion in the CF direction is several times the motion in the IL direction. (a) (b) (c) U U U (z) Figure -3: Three commonly studied systems undergoing VIV. (a) stationary rigid cylinder under constant flow; (b) elastically mounted rigid cylinder under constant flow; (c) flexible cylinder under a flow profile. S RE = t S t RE =. 9 6 S t RE = S t RE =. 3 5 S t S t RE = RE S t RE = 7. 6 Figure -4: The dependence of Strouhal number (S t ) on Reynold s number, and the power spectrum of lift force at select Reynold s number regimes. Figure sourced from Pantazopoulos [44]...3 Vortex-induced vibration of marine risers Marine risers are long flexible cylinders with spatially varying material and geometric properties leading to variation in their stiffness, inertia and damping characteristics. Long flexible cylinders (depicted in Figure -3(c)) behave in a more complicated fashion due to the spatially 29

30 varying magnitude and direction of the fluid flow. This results in vortices being shed at different frequencies at different locations along the riser. As a consequence the vortex-induced forcing tends to have a complicated spatial and temporal variation. Under such circumstances a nonlinear dynamic equilibrium is reached. This occurs as a result of a resonant matching between fluid excitation and small amplitude response of the riser. The frequency content of the response depends on the modal density, an intrinsic property of the riser, and the vortex shedding bandwidth corresponding to the flow. Vortex-induced vibration of marine risers is driven at relatively high frequencies leading to significant damage from fatigue. Since the CF amplitudes are typically larger than the IL motions by at least a factor of two, the fatigue damage is primarily due to bending stresses arising from motions in the CF direction. Thus the underlying aim of any scheme for predicting riser VIV is to estimate the riser fatigue life corresponding to a given flow profile...4 Modeling vortex-induced vibration of marine risers The ideal approach for solving the complicated problem of VIV (fluid-structure interaction) would have been the use of computational fluid dynamics (CFD) techniques [22, 5]. CFD methods when used with the appropriate mesh size can take into account the complicated wake around the riser. However, a fully 3D simulation of a riser at realistic Reynold s numbers is prohibitive due to the intense computational requirements. At present, the most widely used methods for predicting the riser VIV fatigue life are the empirical prediction schemes like VIVA [6, 59], SHEAR7 [66, 6] or VIVANA [27]. A typical such empirical scheme consists of two parts: a) the empirical models, and b) the empirical databases as depicted in Figure -5. Empirical scheme for VIV fatigue prediction Empirical models Hydrodynamic model Structural dynamic model Fatigue prediction model Empirical databases C lv (lift coefficient in phase with velocity) C m (added mass coefficient) C d (drag coefficient) Figure -5: Empirical schemes for predicting the fatigue life of a marine riser. An empirical scheme contains empirical databases and empirical models. The important empirical models are the hydrodynamic models, the structural dynamic models, and the fatigue damage models. The empirical databases on the other hand primarily contain 3

31 hydrodynamic information in the form of the lift force coefficients (lift coefficient in phase with velocity C lv (V r, A ) and added mass coefficient C m (V r, A )) and drag coefficients (C d (V r, A )). These lift force coefficients are usually obtained from extensive laboratory experiments and are functions of the non-dimensional amplitude A = A D (A is the amplitude of oscillation) and nondimensional frequency of oscillation V r = U fd (f is the frequency of oscillation) of the response in the CF direction. Figure -6 depicts the most extensive such lift coefficient databases obtained by Gopalkrishnan [4]. A more complete description of the theoretical model for riser response is given in Chapter Figure -6: The extensive C m and C lv database obtained by Gopalkrishnan [4]. Another way of looking at the problem at hand is to view it from a systems perspective. The problem of obtaining the vortex-induced CF response of a riser can also be subdivided into a hydrodynamics subproblem which quantifies the action of the fluid (surroundings) on the structure (system), and secondly a structural dynamics subproblem, which predicts the response of the system given excitation from the fluid. From a hydrodynamic view point, long flexible cylinders encounter significant change in both magnitude and direction of the current along its span. This leads to significant variation in the magnitude, direction and frequency of vortex-induced forcing along the structure. It is a common practice [59, 66] to use a strip theory based (quasi-uniform) approach to subdivide the riser and estimate the vortex-induced forces on each segment which acts like an elastically mounted rigid cylinder. The harmonic part of the vortex-induced forces on a cylinder corresponding to a local flow velocity U and density ρ f boils down to estimating its two components; the excitation force in phase with velocity and the excitation force in phase with acceleration [4, 5, 59] for each segment of the riser. Estimating these forces requires empirical lift coefficient databases of added mass coefficient C m (V r, A ) and lift coefficient in phase with velocity C lv (V r, A ) mentioned earlier [57, 4]. 3

32 From a structural dynamics view point, a riser is adequately modeled as a tensioned beam with the appropriate boundary conditions, acted upon by the external hydrodynamic force. Since VIV is a self limited process with small amplitudes of motions, the riser structural dynamic model is assumed to the linear. This linear structural dynamics model is used to obtain the riser response once we quantify the forces acting from the fluid on the structure. The hydrodynamic force however depends on the riser motions themselves and thus the above two problems are coupled. So an iterative scheme going back and forth between the hydrodynamics and structural dynamics subproblems is needed to evaluate the riser response..2 Research objectives The overall research objective is to better understand the VIV of risers, and significantly improve the present empirical schemes for predicting the riser fatigue damage. For this thesis, we will focus on answering two well defined questions: ) How can we use data measured using a limited number of sensors placed along a riser and recreate the VIV response of the entire riser? A systematic approach to this problem is necessary. It starts with defining the conditions under which a set of measurements contain adequate information about the riser VIV. Methods to recreate (reconstruct) the riser VIV response, when adequate number of sensors are available, and compensation for the limitations arising from the use of very few sensors will be presented. The applications of such a response reconstruction method include developing tools for in-situ estimation of fatigue damage due to VIV on a riser in the field (fatigue damage monitoring), and improving our understanding of vortex shedding mechanism and vortex-induced forces on marine risers. 2) How can we improve the present empirical schemes for predicting riser VIV cross-flow response? We aim at significantly improving the schemes for predicting the riser VIV crossflow response by improving the empirical lift coefficient databases. The empirical C lv and C m databases have certain limitations. These C lv and C m databases were obtained by Gopalkrishnan, Smogeli et al. [4, 53] from simplified experiments at laboratory (MIT towing tank) where a cylinder was forced to oscillate at prescribed trajectories at select Reynold s numbers for specific frequencies [8, 4, 65]. The databases were obtained under certain conditions: ) databases are available for limited Reynold s number regimes; 2) cylinders undergo simplified motion tests at single frequency; 3) cylinder motions are restricted to CF direction instead of both CF and IL motions. In addition, the applicability of the C lv and C m databases obtained from forced vibration experiments to predict free vibrations also poses an issue [8]. In real marine risers, the Reynold s number at different locations along the riser may belong to the flow regimes of sub-critical, critical or super-critical simultaneously. Risers under real conditions are 32

33 almost never restricted to move only in the CF direction, and allowing the IL motions has a significant effect on the motions in the CF direction [9,, 9]. To address the issues mentioned previously, we need a method which allows us to extract the C m and C lv information from the realistic conditions of field experiments (e.g.: Norwegian Deepwater Programme (NDP) tests, Lake Seneca test, Gulf Stream test) [5, 7, 7]. These new databases when used should produce better theoretical estimates of the CF response than the present estimates. Improving the present empirical schemes also require improvement in modeling the 3 rd harmonic component of the forces. This thesis seek to answer questions regarding the spatial characteristics of the 3 rd harmonic component of the vortex-induced force on a riser. For modeling the 3 rd harmonic of fluid forces, there is a clear need to understand the complicated coupling between the hydrodynamic forces and riser motions, leading to the phenomena such as resonance. This thesis also provides significant insights into such phenomena and produces a conceptual model for the higher harmonic force..3 Thesis outline This thesis is structured as follows. Chapter presents a brief introduction on the topic and describes the issues that will be addressed in the thesis. The theoretical background necessary for predicting the VIV response of two systems: ) an elastically mounted rigid cylinder, and 2) a flexible cylinder (riser) are presented in Chapter 2. Chapter 3 presents a method to recreate the riser motions using data measured from a limited number of sensors. The approach is based on Fourier decomposition in both time and space, and is independent of predicting the riser response a priori. Using this Fourier decomposition framework we develop a full reconstruction criterion to classify when the measurements from the sensors contain all information pertinent to VIV response and when it does not. This will be followed by an attempt to quantify the uncertainty during the reconstruction. In Chapter 4 we develop techniques critical to understand and quantify the important characteristics of riser VIV from experimental measurements. For this three aspects of riser VIV are studied: ) methods to extract a statistically stationary segment of a measured signal, 2) modal decomposition of the observed riser response, and 3) methods to identify and characterize traveling waves in experimental measurements. Chapter 5 formulates the problem of extracting the lift force coefficient information from experimental data as an optimization problem. This optimization problem is solved using data from NDP experiments and consequently obtain the universal lift force coefficient databases which will minimize the error in theoretical prediction of riser VIV response. Chapter 6 considers the response reconstruction problem when data from very few sensors 33

34 are available. We develop a method based on the modal decomposition approach which utilizes the riser VIV prediction capabilities. A method to correct the lift force coefficients at the measurement locations is also developed to improve both the prediction and subsequently the reconstruction. In Chapter 7 we utilize data from a comprehensive experiment to identify some important characteristics of the higher-harmonic components in riser VIV. The origin of higher harmonics in riser response, and the resonance of a riser in the harmonic and in the third harmonic components are studied. A conceptual model for evaluating the third harmonic component of vortex-induced force is also presented. Chapter 8 concludes the thesis by summarizing and presenting the principal contributions from the thesis. This is followed by recommendations for future research. 34

35 Chapter 2 Theoretical background 2. Introduction This chapter will focus on modeling the dynamics of two systems undergoing VIV: ) an elastically mounted rigid cylinder, and 2) a flexible cylinder (riser). In the first part, the equation governing the transverse oscillations of an elastically mounted rigid cylinder under a constant current U in the in-line direction is obtained. This governing equation will be used to predict the frequency of peak response and the corresponding amplitude of peak response. This will be followed by modeling the transverse oscillation of a riser (flexible cylinder) under a steady constant flow profile U(z) in the in-line direction. We will obtain the equation governing the transverse oscillation of the flexible cylinder, which will allow us to predict the set of peak response frequencies and the corresponding peak response modes. 2.2 Equation governing VIV of an elastically mounted rigid cylinder The ordinary differential equation (ODE) governing the motion of a single degree of freedom (SDOF) system like a rigid cylinder (depicted in Figure 2-) of mass m, mounted on springs with spring constant k and dashpots with damping constant b, under a uniform current of velocity U can be written as: my tt + by t + ky = F fluid (t). (2.) where, F fluid (t) represents the time varying external force acting from the fluid on the rigid cylinder. Equation (2.) represents a linear time invariant (LTI) system. For a harmonic excitation at a frequency ω, for an LTI system we can assume a solution of the form: y(t) = Re { Ae iωt}. (2.2) 35

36 Figure 2-: Depicts the setup of the elastically mounted rigid cylinder under a uniform flow. where, Re {x} means the real part of x. We are interested in obtaining the amplitude A of the oscillation and the frequency ω of the oscillation. The harmonic part of the forcing can then be written with one term in phase with velocity and another term in phase with acceleration as: {[ F fluid (t) = Re m a Aω 2 + i ( ρf U 2 2 ) DLC lv ] e iωt }. (2.3) where, m a = ρ f πd 2 L 4 C m. Note that the above expansion of the fluid force F fluid (t) (in equation (2.3)) incorporates the definition of both C lv and C m, the lift force coefficient in phase with velocity and the added mass coefficient respectively. Both C lv and C m are found to depend on the non-dimensional frequency of oscillation (reduced velocity) V r = 2πU ωd amplitude (amplitude ratio) A = A D [5, 49, 76]. That is: and non-dimensional C lv = C lv (A, V r ) & C m = C m (A, V r ). (2.4) Now we can substitute F fluid (t) from equation (2.3) and y(t) from equation (2.2) into equation (2.) to obtain: ( (m + m a )Aω 2 ρf U 2 ) cos(ωt) baω sin(ωt) + ka cos(ωt) = DLC lv sin(ωt). (2.5) 2 We can group terms in phase with displacement (cos(ωt)) and terms in phase with velocity (sin(ωt)). obtain: If we equate the terms in phase with displacement (cos(ωt)) in equation (2.5) we [ ] k m + C m (A, V r ) πd2 Lρ f ω 2 =. (2.6) 4 36

37 The frequency ω is non-dimensionalized by substituting reduced velocity V r = 2πU ωd (2.6) to obtain: [ k m + C m (A, V r ) πd2 Lρ f 4 ] [ ] 2πU 2 D Vr 2 Similarly equating the terms in phase with velocity (sin(ωt)) in equation (2.5): in equation =. (2.7) ( ρf U 2 ) baω DL C lv (A, V r ) =. (2.8) 2 Rewrite equation (2.8) in terms of reduced velocity V r we obtain: ( ) A ρf UDL C lv (A, V r ) V r =. (2.9) 4πb We can non-dimensionalize flow speed U, as V rn = U f, where f nd n is the nominal natural frequency (obtained by assuming an added mass coefficient equal to unity). The set of equations (2.7) and (2.9) can then be written in non-dimensional form as: [m V r (V rn ) = V + C m (A ], V r ) rn m, (2.) + A (V rn ) = C lv (A, V r ) V r 2 4π 3 [m + C m (A, V r )] ζ, (2.) where, ζ is the damping ratio, m is the mass ratio and are defined in Appendix A.2 in equations (A.) and (A.) respectively. Given the empirical C m (A, V r ) and C lv (A, V r ) databases, we can in principle solve for the unknowns A and V r which satisfy the two equations (2.) and (2.). Experimental lift force coefficient database The experimental lift force coefficient database consists of the lift coefficient in phase with velocity C lv (A, V r ) and the added mass coefficient C m (A, V r ) both as functions of A and V r. The most extensive such databases were obtained by Gopalkrishan by conducting forced motion experiments on a rigid cylinder at the MIT towing tank [4] and are depicted in Figure 2.2. These databases were obtained by forcing a rigid cylinder to oscillate at prescribed trajectories and frequencies under different flow speeds. The flow speeds were such that the maximum Reynold s number achieved is. As observed from Figure 2.2, the C m (A, V r ) database depicts one major peak and a valley (transition) parallel to the A axis at V r close to 6. This transition has important physical significance and is directly related to the free vibration frequency. The C lv (A, V r ) database on the other hand depicts the presence of two resonant lobes. However, the feature most important 37

38 to free vibration of an elastically mounted rigid cylinder is the C lv = contour. It was found that the free vibration solution of A and V r for various flow velocities V rn closely follow the C lv = contour [8]. In addition, the C lv = contour reaches a peak A value close to V r Figure 2-2: The lift force coefficient databases of C lv (A, V r ) (left) and C m (A, V r ) (right) produced by Gopalkrishnan [4]. A less extensive database is also obtained by Smogeli et al. [53] for higher Reynold s numbers. 2.3 Equation governing VIV of a riser We will model a riser (a flexible cylinder) of length L, mass per unit length m, structural damping b and bending stiffness EI as a tensioned beam, acted upon by an external fluid force f fluid (z, t) (refer Figure 2-3). Since the vortex-induced motions have amplitudes of the order of f fluid ( z, t) f T T z z = z = L Figure 2-3: A riser modeled as a tensioned beam taut along the z direction. the riser diameter, the structural dynamics can be described using a linear model. The equation governing the transverse oscillation of such a beam taut along z-axis between z = and z = L under a tension T, acted upon by a time-varying fluid force f fluid (z, t) in the y direction (lift force) is given by: m 2 y t 2 + b y t ( T y ) ( ) + 2 z z z 2 EI 2 y z 2 = f fluid (z, t). (2.2) 38

39 The time varying force f fluid (z, t) is hydrodynamic in nature and it is a function of the motion of the structure. However, when the response is temporally stationary (more detail given in Section 4.2) and further monochromatic, the oscillation of the structure is synchronized with the vortex shedding forces. The monochromatic steady state response of a riser can be written as: y(z, t) = Re { Y (z) e iωt}, (2.3) Corresponding to the above steady state response (2.3), we can write the fluid force on the riser using a quasi-uniform approximation (or strip theory) where segments of the riser are assumed to behave like a rigid cylinders. Using the quasi-uniform approximation, the harmonic part of the fluid force is written in two parts: the lift force in phase with velocity and the lift force in phase with acceleration (the added mass force) as: {[ ( Y (z) f fluid (z, t) = Re C m D, V r(z) ) (ρ f πd 2 4 ( Y (z) +ic lv D, V r(z) ) (Y (z) ω 2 ) ) ( ρf U 2 ) Y (z) 2 D Y (z) ] e iωt }. (2.4) where, U(z) is the local flow velocity which is assumed to be time invariant, and ρ f is the fluid density. The experimentally obtained added mass coefficient C m and lift coefficient in phase with velocity C lv databases, such as the ones obtained by Gopalkrishnan [4] (Figure 2.2) can be used to estimate the excitation force from fluid to cylinder. Note that these lift coefficients ( ) ( ) C Y (z) lv D, V r(z) and C Y (z) m D, V Y (z) r(z) are now functions of the local amplitude ratio D and local non-dimensional frequency V r (z) = 2πU(z) ωd which vary along the length of the riser. However for simplicity we use the notation C lv and C m respectively. Substitution of equations (2.3) and (2.4) into equation (2.2) gives: [ [ ω 2 m + ibω ] Y ( T Y ) + 2 z z z 2 πd = [C m (ρ 2 f 4 which can be rewritten as: { ω 2 [m + C m (ρ f πd 2 4 ( EI 2 Y z 2 )] e iωt ) (Y (z) ω 2 ) + ic lv ( ρf U 2 2 D ) Y (z) Y (z) ] e iωt, (2.5) )] } + iωb Y ( T Y ) ( ) + 2 z z z 2 EI 2 Y z 2 ( ρf U 2 ) Y = ic lv 2 D Y. (2.6) The above equation (2.6) with the appropriate boundary conditions represents the nonlinear eigenvalue problem. The nonlinearity arises from the presence of terms like Y (z) Y (z), C m and C lv 39

40 which depend on Y and ω. However, solving the nonlinear eigenvalue problem in the above form is very difficult. Instead, if we multiply each side of the equation (2.6) with the complex conjugate of Y n (z) and perform an integration by parts (assuming zero boundary conditions), we obtain the weak form of the nonlinear eigenvalue problem as: L z= ( T dy dz 2 + EI d 2 Y dz 2 = 2 + i bω Y 2 ) dz L z= ρ f πd ([m 2 ] + C m Y 2 ρ f U 2 ) D dz + i C lv Y dz. (2.7) 4 2 The real and imaginary part of the integral relation (2.7) can be separately written as: L z= T dy dz 2 + EI d 2 Y dz 2 L z= 2 dz = ω 2 bω Y 2 dz = L z= L z= [m + C m ρ f πd 2 4 ] Y 2 dz, (2.8) ρ f U 2 D C lv Y dz. (2.9) 2 The above equations (2.8) and (2.9) have physical significance from an energy conservation view point. Equation (2.8) can be viewed as the potential energy - kinetic energy balance, while the equation (2.9) represents the balance between the power coming in from fluid to riser and power dissipated through fluid and structural damping (power in - power out). An iterative scheme can be used to obtain the characteristic shapes (hydroelastic modes or peak response modes) Y n (z) and coupled frequencies (peak response frequencies) ω n using semi-empirical program like VIVA [57, 6, 59]. The peak response modes of such a coupled fluid-structure system are complex functions, and unlike free vibration modes from a linear eigenvalue problem are not in general orthogonal to each other. Figure 2-4 depicts the peak response modes of a riser for two cases of flow profiles. Figure 2-4(top) depicts the peak response modes obtained from VIVA for a uniform flow profile, and Figure 2-4(bottom) depicts the peak response modes obtained from VIVA for a linearly sheared velocity profile. 4

41 Figure 2-4: Theoretical estimates of peak response modes (hydroelastic modes) for a uniform flow profile (top) and a linearly sheared flow profile (bottom) obtained using VIVA. 4

42 42

43 Chapter 3 A systematic approach to riser VIV response reconstruction 3. Introduction Understanding the complicated fluid-structure interaction problem of riser VIV requires careful observation of riser motions. Knowing the riser VIV motions allows us to estimate the vortex-induced forces, and predict the vortex-induced stresses and stress hot-spots on the riser, which in-turn allows us to predict the fatigue damage on the riser. Ideally, we need to obtain the measurements at all points along the length of the riser. However, in reality riser motions are measured using sensors placed at select locations along its span. These sensor measurements have the following characteristics:. High temporal sampling rate: Each of the sensors measures the signal with high sampling rate in time. These samples are usually uniform and these measurements are obtained for a sufficiently long period of time. 2. Limited number of sensors: Measurements of VIV responses are typically limited to a relatively small number of sensors located along the riser length. A large number of sensors in field data is very expensive due to installation, instrumentation and upkeep costs, especially since these sensors must be light and of small volume so they do not affect the properties of the riser; also several sensors may fail during long testing periods. In addition, the sensors are often unevenly placed or become uneven due to the failure of some of them. 3. Indirect measurement of displacement: Presently, direct measurement of riser displacements are not feasible. Vortex-induced motions along a riser are typically measured as accelerations (using accelerometers) or strains (using strain gages) at select locations along 43

44 the riser. A typical experimental measurement would consist of data from both accelerometers and strain gages. In addition, the riser motions due VIV has the following characteristics:. Narrow banded in time: Several experiments conducted in the laboratory and field suggests that the riser displacements due to vortex shedding occurs around a preferred frequency band around the local Strouhal frequency. 2. Narrow banded in space: In addition, the temporal frequency ω and spatial frequency (wave number) k are related through the dispersion relationship [7]. Since the displacement spectra are band limited in ω, the displacement spectra are band limited in k although the spread of the band could be affected due to the variability in effective added mass along the riser. 3.. Problem setting Consider a flexible riser of length L taut along the z-axis as shown in Figure 3-. Assume we have N s sensors placed along the riser at locations z, z 2,..., z s,..., z Ns. These sensors may measure displacements y(z s, t), accelerations a(z s, t) or axial strains due to bending ɛ zz (z s, t). Since the sampling in time is very dense, for the purpose of explaining the concept, we can consider t as a continuous variable. We define response reconstruction as the process of using the data from these sensors (y(z s, t), a(z s, t), ɛ zz (z s, t)) and obtaining the displacements y(z, t) at any arbitrary location z along the riser. While considering the solution strategy, two separate Figure 3-: Figure depicting the typical arrangement of sensors on a riser. scenarios (refer Figure 3-2) emerge based on the available number of sensors: Scenario I: When complete information of the riser motion is present in the data obtained from the N s sensors. Scenario II: The number of sensors are so few that the data obtained from these sensors alone is not enough to completely describe the riser motions. 44

45 This chapter will focus on the first scenario where the number of sensors are high enough to completely describe the riser motions. For this, we will pose the reconstruction problem from the perspective of a Fourier decomposition of the displaced shape of the riser. Next, we will obtain the criterion for full reconstruction to determine when we do have enough number of sensors or when we do not have enough sensors. Finally, we will attempt to quantify the error during reconstruction. Scenario II will be discussed in much greater depth in Chapter 6. For the second scenario where we need to complete the riser motions, we require our prior understanding of the vortex-induced motions to reduce the set of plausible solutions. Full reconstruction criterion (Spatial Nyquist criterion) Scenario I: Satisfy reconstruction criterion (Measurements contain all information about riser VIV response) Scenario II: Does not satisfy reconstruction criterion (Measurements do not contain all information about riser VIV response) Figure 3-2: Based on the number of sensors available and the bandwidth of the signals from the sensors, we have two separate scenarios of reconstruction. The solution strategy is different for each of the scenarios Related literature Several researchers have considered the response reconstruction problem; the relevant references include Kaasen et al. [2], Trim et al. [6], Lie and Kassen [29]. These methods are based on linear structural dynamics techniques combined with the mode superposition principle, where the displacements are written as a linear combination of the riser free vibration modes weighted by the modal response factors. The free vibration modes are used as a complete basis to project the resulting vibration. The present methodology differs with respect to prior publications in that the spatial responses at each frequency are allowed to be different from the free vibration modes; this results in economy of representation, since the response is caused by vortex shedding which may not be of uniform phase, hence traveling waves are possible [35] the need to transfer energy in sheared current cases from a lockin region to a damping region makes this argument even stronger. Also, the large variability in the added mass along the riser may affect the patterns of response. In addition, the problem is that of a forced vibration rather than a free vibration, involving complicated fluid forces, possibly of a traveling wave type. Even using the riser response modes from semi-empirical prediction codes like VIVA [59] and SHEAR7 [66] may be inadequate due to the differences between the prediction and the actual response. Moreover, the complicated riser response features the presence of boundary layers at the riser ends, resulting in large jumps in strain near the boundary. This further complicates the 45

46 reconstruction when we are faced with strain signals and derivatives of numerically evaluated response modes. What is needed is a consistent and systematic approach to the reconstruction problem which is independent of predicting the riser response modes and can use measured displacement derivatives (acceleration and curvature/strain). The criterion for full reconstruction [29, 68, 6] is posed from a matrix inversion point of view, based on the number of modes needed in the modal analysis and number of sensors available. This criterion is hence sensitive to the choice of response modes. We develop a criterion for full reconstruction which is independent of the choice of the response modes. 3.2 Overview of the approach For demonstrating the overall approach, we will first consider the hypothetical case when we have sensors measuring only displacements. In the following section we will generalize the approach when we are given a combination of displacements, accelerations and strain measurements. The underlying idea is to write the displaced shape of the riser at any instance of time (say t = t ) as a Fourier series of the form: y(z, t ) = Y (z) = a + n= [ ( nπ ) ( nπ )] a n cos L z + b n sin L z. (3.) The series converges (under certain conditions [26]) even if we use a finite number of terms as: Y (z) a + N m n= [ ( nπ ) ( nπ )] a n cos L z + b n sin L z, (3.2) where, N m is the maximum number of spatial harmonics that is required to represent the displaced shape of the riser. The underlying idea in this approach is to solve for the Fourier coefficients a n and b n from the measured data y(z s, t ). Obtaining Fourier coefficients a n and b n can be posed in a matrix form: cos ` π y(z, t ) L z y(z 2, t ) cos ` π L z 2 y(z 3, t ) cos ` π L z 3. =.. y(z s, t ) cos ` π zs L y(z Ns, t ) cos ` π {z } L z N s Y... cos Nmπ L z sin ` π L z... sin Nmπ L z 3... cos Nmπ L z 2 sin ` π L z 2... sin Nmπ L z 2... cos Nmπ L z 3 sin ` π L z 3... sin Nmπ L z cos Nmπ L zs sin ` π zs... sin Nmπ L L zs cos Nmπ L z N s sin ` π L z 5 N s... sin Nmπ L z N s {z } Φ a a a 2. a Nm b b 2.. b Nm {z } w (3.3) 46

47 The above equation (3.3) can be written in a compact form as: Y = Φ w. (3.4) Where Y is the vector containing the displaced shape measured by the sensors, w is the vector of unknown Fourier coefficients, and Φ is a function of only the sensor locations z s, our choice of N m and is independent of time (choice of N m is described in detail in Section 3.4.). We can solve for the unknown Fourier coefficient vector w by taking the pseudo-inverse Φ + of the matrix Φ as: w = Φ + Y. (3.5) In a similar fashion, reconstructing the entire signal (for every instance of time) can be posed as: y(z, t ) y(z, t 2 )... y(z, t Nt ) y(z 2, t ) y(z 2, t 2 )... y(z 2, t Nt ) y(z 3, t ) y(z 3, t 2 )... y(z 3, t Nt ) y(z s, t ) y(z s, t 2 )... y(z s, t Nt ) } y(z Ns, t ) y(z Ns, t 2 ) {{... y(z Ns, t Nt ) } Y = Φ The equation (3.6) can be written in a compact form as: a (t ) a (t 2 )... a (t Nt ) a (t ) a (t 2 )... a (t Nt ) a 2 (t ) a 2 (t 2 )... a 2 (t Nt ) a Nm (t ) a Nm (t 2 )... a Nm (N t ). (3.6) b (t ) b (t 2 )... b (t Nt ) b 2 (t ) b 2 (t 2 )... b 2 (t Nt ) } b Nm (t ) b Nm (t 2 ) {{... b Nm (N t ) } w Y = Φ w. (3.7) where, Y = Y (z s, t), and w = w(t) and Φ is the same matrix as in equation (3.6). Equation (3.7) can be solved by taking the pseudo-inverse Φ + as: w = Φ + Y. (3.8) Once we solve for w, we can obtain the displacements y(z, t) at any arbitrary location as: y(z, t) = a (t) + N m n= [ ( nπ ) ( nπ )] a n (t) cos L z + b n (t) sin L z. (3.9) However, the process of taking the pseudo-inverse breaks down when the Φ matrix is rankdeficient. That is the number of linearly independent rows of Φ matrix is less than the number 47

48 of unknowns. In Section 3.4., we will discuss this issue in greater detail from both a matrix inversion perspective and from a signal processing perspective. 3.3 Response reconstruction when a combination of acceleration and strain measurements is given Presently we do not have the luxury of measuring the displacements explicitly. More often than not, we have a combination of sensors measuring accelerations, strains or displacements. In such a case where we need to use information from both strain gages and accelerometers, we formulate the problem in a slightly different way. As the first step we relate the bending strains ɛ zz (z, t) measured on the surface of the riser to the local curvature κ(z, t) as: κ(z, t) = ɛ zz(z, t) R, (3.) where R the radius of the riser. Thus we are presented with curvature κ(z sk, t), acceleration a(z sa, t) and displacement y(z sy, t) measurements at select locations along the riser. Next, we represent the displaced shape of a riser using Fourier expansion as given in equation (3.2). Let us say we use φ n (z) as a proxy for the sine and cosine terms and w n (t) the proxy corresponding to the Fourier coefficients a n (t) and b n (t) as: y(z, t) a (t) + N m n= [ ( nπ ) ( nπ )] 2N a n (t) cos L z + b n (t) sin L z m+ = w n (t)φ n (z) (3.) Such an expansion allows us to write the curvature and acceleration of the riser under bending as: κ(z, t) y zz (z, t) = a(z, t) = y tt (z, t) = 2N m+ n= 2N m+ n= n= w n (t) φ nzz (z), (3.2) w ntt (t) φ n (z). (3.3) The idea again is to solve for the Fourier coefficients w n (t) at every instance of time. However, posing the equations (3.), (3.2) and (3.3) in a matrix form to solve for w n (t) presents an additional difficulty due to the presence of w ntt (t) in addition to unknown w n (t). To mitigate this issue we pose the problem in the frequency domain. Since each of the sensors measures the signals at high time sampling rates over several cycles of the data we can take their Fourier transforms in time. If we denote the following Fourier transforms: F{w n (t)} by ŵ n (ω), 48

49 F{y(z, t)} by ŷ(z, ω), F{a(z, t)} by â(z, ω), and F{κ(z, t)} by ˆκ(z, ω) we can obtain: ŷ(z, ω) = 2N m+ n= 2N m+ â(z, ω) = ω 2 ŷ(z, ω) = ω 2 ˆκ(z, ω) = 2N m+ n= ŵ n (ω) φ n (z), (3.4) n= ŵ n (ω) φ n (z), (3.5) ŵ n (ω) φ nzz (z). (3.6) Posing the problem in frequency domain allows us to explicitly solve for ŵ n (ω) by putting together equations (3.4), (3.5) and (3.6) in a matrix form. Also accounting for the discrete nature of the frequencies we obtain the corresponding matrix equation as: 2 3 φ (z ) φ 2 (z )... φ N (z ) φ (z 2 ) φ 2 (z 2 )... φ N (z 2 ) φ (z 3 ) φ 2 (z 3 )... φ N (z 3 ) φ (z My ) φ 2 (z My )... φ N (z My ) φ (z ) φ 2 (z )... φ N (z ) 2 φ (z 2 ) φ 2 (z 2 )... φ N (z 2 ) φ (z 3 ) φ 2 (z 3 )... φ N (z 3 ) φ (z Ma ) φ 2 (z Ma )... φ N (z Ma ) φ (z ) φ 2 (z )... φ N (z ) φ (z 2) φ 2 (z 2)... φ N (z 2) φ (z 3) φ 2 (z 3)... φ N (z 3) φ (z Mk) φ 2 (z Mk)... φ N (z Mk) {z } Φ wˆ (ω ) wˆ 2 (ω ) wˆ 3 (ω ). wˆ (ω 2 )... wˆ 2 (ω Λ ) wˆ 2 (ω 2 )... wˆ 2 (ω Λ ) wˆ 3 (ω 2 )... wˆ 3 (ω Λ ) wˆ N (ω ) wˆ N (ω 2 )... wˆ N (ω Λ ) {z } ŵ 2 3 = ŷ(z, ω ) ŷ(z, ω 2 )... ŷ(z, ω Λ ) ŷ(z 2, ω ) ŷ(z 2, ω 2 )... ŷ(z 2, ω Λ ) ŷ(z 3, ω ) ŷ(z 3, ω 2 )... ŷ(z 3, ω Λ ) ŷ(z My, ω ) ŷ(z My, ω 2 )... ŷ(z My, ω Λ ) â(z,ω ) ω 2 â(z 2,ω ) ω 2 â(z 3,ω ) ω 2. â(z Ma,ω ) ω 2 â(z,ω 2 ) ω 2 2 â(z 2,ω 2 ) ω 2 2 â(z 3,ω 2 ) ω 2 2. â(z Ma,ω 2 ) ω â(z,ω Λ ) ω Λ 2 â(z 2,ω Λ ) ω Λ 2 â(z 3,ω Λ ) ω Λ â(z Ma,ω Λ ) ω 2 Λ ˆκ(z, ω ) ˆκ(z, ω 2 )... ˆκ(z, ω Λ ) ˆκ(z 2, ω ) ˆκ(z 2, ω 2 )... ˆκ(z 2, ω Λ ) ˆκ(z 3, ω ) ˆκ(z 3, ω 2 )... ˆκ(z 3, ω Λ ) ˆκ(z Mk, ω ) ˆκ(z Mk, ω 2 )... ˆκ(z Mk, ω Λ ) {z } Ŷ where, N = 2N m +. The above equation (3.7) in the frequency domain can be written in a compact form as: Φ ŵ = Ŷ, (3.8) where, Φ is the matrix containing the sine and cosine terms and their spatial derivatives evaluated at the sensor locations, and Φ is independent of the temporal frequency ω. Ŷ is the Fourier transform of the measured experimental data (data matrix). ŵ is the matrix containing the unknown Fourier coefficients of the sine and cosine terms in the frequency domain. The solution for each ω can be obtained by taking a pseudo-inverse Φ + of Φ. But instead of repeating the procedure for each time step, we repeat the procedure for each frequency ω as: 3 (3.7) 7 5 ŵ = Φ + Ŷ. (3.9) 49

50 After solving for the Fourier coefficient matrix in the temporal frequency domain, ŵ, one can easily obtain the Fourier coefficient matrix in the time domain w by a simple inverse Fourier transform IF{}. The displacements can then be reconstructed as: y(z, t) = a (t) + N m n= [ ( nπ ) ( nπ )] a n (t) cos L z + b n (t) sin L z. (3.2) The overall solution methodology is described as a flow chart in Figure 3-3. Response reconstruction algorithm Given Data: y( z, t ), sy j a( z, t ) sa ε( zsk, t j ) κ ( zsk, t j ) = R j Compute Fourier transform: ) Y z, ω ) = Y( z, t ) F { } ( s l s j Assemble: Φ( z s ) Reconstruction in frequency domain: Φ wˆ ( ω ) = Yˆ ( ω ) wˆ ( ω ) = Φ l l + l Yˆ ( ω ) l Obtain displacement at any location y( z, t j ) = Φ( z) w( t j ) Compute inverse Fourier transform: ) wn ( t j) n l =IF { w ( ω )} Figure 3-3: Flowchart of the algorithm for reconstructing riser VIV displacements from sensor measurements A discussion on the reconstruction method and practical issues During the actual implementation we need to obtain the pseudo-inverse Φ + only once resulting in increased speed of the reconstruction algorithm. In addition, we do not need to consider all frequencies while solving for ŵ. If the signals are band limited in the temporal frequency, then the signals with only the non-zero Fourier coefficients need to be evaluated. Since the matrix Φ has a size N s (2N m + ), obtaining the pseudo-inverse requires N s > (2N m + ). In practice, the error grows as the number of sensors comes close to the prefect reconstruction criterion. As a result we have to require that N m to be slightly lower than the critical value of N s /2. The reconstruction criterion is discussed in greater detail in Section 3.4. and a systematic study of reconstruction error is discussed in Section 3.6. Few lower order spatial harmonics have to be removed while assembling the Φ matrix because of their extreme sensitivity to noise which is present in the real data. While preparing the data 5

51 matrix Ŷ and the Φ matrix, the boundary conditions have to be enforced as extra constraints on the problem and in fact they serve as additional sensors placed at the riser ends. Need for both sine and cosine terms: The reconstruction using sine and cosine terms assumes a periodic extension in the period z = [ L, L], where we have sensors placed only between z = [, L]. It is also possible to assume an odd periodic extension in the period z = [ L, L] by placing hypothetical sensors in the region z = [ L, ]. However, placing hypothetical sensors is an extra constraint which will result in major deviations observed in the reconstructed signal as will be shown in extensive benchmark tests given in Section Using sine and cosine terms produces excellent accuracy otherwise impossible with only sine terms. The justification for using cosine terms is that, the available datasets consists of both accelerations and curvatures. These displacement derivatives and their temporal Fourier transforms could have boundary layers at the ends due to the complicated riser motions. Accurate representation of such a displaced shape requires enough flexibility for the interpolating curve. Using the sine and cosine terms allows the interpolating curve more freedom and hence better accuracy during reconstruction. 3.4 Full reconstruction criterion (Spatial Nyquist s criterion) The criterion for full reconstruction of a signal from its samples was first described by Nyquist [4]. Our reconstruction methodology employs a Fourier decomposition of the displaced shape of the riser using both sine and cosine terms. Since we posed the question of reconstruction from the fundamental principle of Fourier decomposition, we can evaluate this criterion in terms of the number of spatial harmonics required N m and the available number of sensors N s Statement of Nyquist s criterion The Nyquists criterion (sampling theorem) [43, 42] states that if we have a signal y(z) with its Fourier transform ŷ(k), such that: ŷ(k) =, for k > k m, (3.2) that is y(z) is band limited in spatial frequency k with a cutoff spatial frequency equal to k m, then y(z) is uniquely determined by its samples y(n z ) if: k s = 2π z > 2k m. (3.22) where, z is the spacing between the adjacent samples. In other words Nyquist s criterion states that an exact reconstruction of a signal from its samples is possible if the signal is band limited and the sampling frequency k s is greater than two times the bandwidth of the signal k m. Though, 5

52 the above statement of Nyquist s criterion assumes a uniform sampling, this requirement can be relaxed for uneven sampling as in the original criterion published by Nyquist [4]. However in practice a more uniform distribution of sensors is found to yield better reconstruction accuracy than the presence of several sensors which are clustered. It is also interesting to note that the Nyquist s criterion is posed as a sufficient condition and is not a necessary condition Application to response reconstruction The reconstruction methodology we have discussed assumes a periodic extension of the signal with a period z = [ L, L]. Based on this we can evaluate the Nyquist criterion by assuming N s sensors placed (for simplicity we can assume uniformly placed) over a period 2L resulting in a spatial sampling frequency: k s = 2π 2L N s = π L N s. (3.23) If we assume that the displaced shape of the riser is such that the spatial frequency k is band limited, then the highest non-zero harmonic present in the Fourier decomposition N m is related to the spatial band width as: The Nyquist criterion then becomes: [ nπ ] k m = max = π L L N m. (3.24) k s > 2k m, or, π ( π L N s > 2 m) L N, or, N s > 2N m. (3.25) However if we also include the constant term (a (t)), we obtain the criterion as: N s 2N m +. (3.26) In other words the number of sensors N s should be greater than two times the number of spatial harmonics N m present in the function y(z) plus one. criterion we obtained based on obtaining the pseudo-inverse of Φ. Note that this criterion matches the Instead of using both sine and cosine terms, if we assume that the riser displacements are restricted by assuming an odd periodic extension using only sine terms, the full reconstruction criterion is obtained as N s > N m. However, this assumption artificially introduces certain characteristics for riser VIV response which may or may not be valid. Violating the above full reconstruction criterion will lead to spatial aliasing which is observed as linearly dependent rows of the Φ matrix. The spatial frequency (wave number) k is related to the temporal frequency ω through the dispersion relationship. Since the displacement spectra is band limited in ω, displacement spectra is band limited in k although the spread of the band could be affected due to the variability in effective added mass over the riser. 52

53 3.5 Applications To understand and verify the accuracy, speed and limitations of the method, we first apply the method to benchmark data where the displacements are known a priori. This will be followed by the application of our methodology to experimental data from NDP high mode VIV tests [5] to obtain both cross-flow (CF) and in-line (IL) displacements Application to benchmark data The benchmark data was created for a riser with properties similar to that of the NDP riser model (NDP experimental setup is described in detail in Appendix B.). Displacements are simulated by choosing one of the complex peak response modes Y c (z) obtained for a linearly sheared profile from VIVA, and using the formula: y(z, t) = Re {Y c (z) η c (t)}, (3.27) where, Y c (z) is the chosen complex VIVA mode and η c (t) is a narrow banded function comprising of 5 frequencies centered around the modal frequency f c corresponding to Y c (z). η c (t) is obtained as: η c (t) = where, a j, f j and φ j are given as: 5 a j e i(2πf jt+φ j ) (3.28) j= a j = [ ] (3.29) f j = [ ] f c (3.3) φ j = random (3.3) The spectrum of the η c (t) is depicted in Figure 3-4. Since the idea is to simulate a typical NDP Spectrum of η c (t ) ˆ ( f ) η c frequency ( f ) [ Hz] Figure 3-4: Figure illustrating the spectrum of the η c (t) of a typical benchmark test. 53

54 dataset, from displacements we extract accelerations at the NDP accelerometer locations (8 in number) and strains at the NDP strain gage locations (24 in number). Thus the benchmark data consisted of 24 strain signals and 8 accelerometer signals all sampled at 2Hz (corresponding to sampling frequency in NDP data). ' &% $ # "!.4.3 &% ' ( $ # "!.4.3 ( Figure 3-5: Top: comparison of the RMS of the original and reconstructed displacements for the benchmark data using only sine terms; bottom: comparison of the RMS of the original and reconstructed displacements for the benchmark data using both cosine and sine terms. We apply the reconstruction methodology to the benchmark data, and compare the root mean square (RMS) of the original and the reconstructed displacements. To illustrate the importance of the cosine terms we perform two separate reconstructions:. Using only sine terms: As depicted in Figure 3-5(top), there is significant deviation between the RMS of the original signal and the RMS of the reconstructed displacements over a large part of the riser. Use of increased number of spatial harmonics (increased number N m ) does not solve this problem. The reconstruction using increased number of sine terms converges but converges to the wrong solution. 2. Using both cosine and sine terms: As depicted in Figure 3-5(bottom), the reconstruction closely matches the original over the entire riser. Similar results were observed for several benchmark datasets and this illustrates the importance of using both sine and cosine terms for accurately reconstructing the riser motions. As mentioned previously, this is due to the fact that the riser motions arising from VIV consist of traveling 54

55 waves arising from nonlinear interaction between the riser and the fluid. This will result in boundary layers close to the riser ends yielding sharp jumps in curvature. This introduces problems during decomposition using sines alone due to interpolation of the curvatures but produces better results when we use both sine and cosine terms Application to experimental data The application of reconstruction to several benchmark tests gives us confidence to apply the method to NDP experimental data (a detailed description of the experiment is provided in section B.). The number of sensors employed in NDP experiment (32 in CF and 48 in IL) is expected to allow the reconstruction of the harmonic part of the CF motion and the second harmonic part (twice the Strouhal frequency) of the IL motion. A large portion of this thesis will rely on using the reconstructed data. For a detailed step-by-step example illustrating the reconstruction methodology refer to Appendix C. Figures 3-6 and 3-7 depicts the typical comparison of the original and reconstructed curvature signals at few chosen locations along the riser for datasets 24 and 246 respectively. These figures clearly illustrate how the reconstruction closely resembles the experimental data at least at the sensor locations. Another good way to observe the CF and IL motions are to view it as a 3D animation. Since an animation cannot be incorporated into the thesis, we present several frames depicting motion of the riser in Figure 3-8. One significant finding is the presence of traveling waves present in the riser response. Linearly sheared flows clearly depict the presence of disturbances (pulses) in CF displacements which travel along the entire length of the riser with directional consistency. As flow velocity increases, the presence of traveling waves become more apparent due to increased number of pulses generated in each second. The IL motions are more irregular depicting the presence of standing waves and traveling waves over a part of the riser. CF displacements for uniform flow on the other hand depict disturbances which propagate over a part of the riser length with no directional consistency. Also observed is the presence of standing waves whose shape differ significantly from the purely sinusoidal free vibration modes predicted by prediction programs. In fact this could be a case where the given standing wave may be combination of several purely sinusoidal modes with their corresponding modal frequency converging to a single value (frequency coalescence). Another significant observation is the interaction of CF and IL motions during the propagation of each pulse. Allowing IL motions has the effect of stabilizing or enabling such wave propagation. A detailed description and review of traveling wave identification procedures are presented in Section

56 543 / 543 / ,..5.5 : : , , !"#$ &%('(") *& +'(* " !"#$% '&)()#*!+'!,()+!# Figure 3-6: Comparison of the CF experimental curvature time series (red), and the reconstructed CF curvature time series (blue) at select strain gage locations for a typical NDP dataset (dataset 24). Note that the reconstructed curvature closely matches the experimental data at least at the sensor locations. 3.6 A study of reconstruction error In this section we present our effort to quantify the error that might have crept in during the reconstruction method outlined in the previous sections. A systematic study of the various sources of error is first performed. This is followed by an attempt to quantify the error due to these sources of uncertainty Identifying sources of uncertainty during reconstruction Various sources of uncertainty are present during the response reconstruction method for Scenario I using Fourier decomposition. We present a study of the following sources of uncertainty:. Uncertainty from the presence of noise in experimental data. 2. Uncertainty from the use of a finite number of sine and cosine terms during reconstruction. 3. Uncertainty from the presence of both acceleration and strain measurements. 56

57 543 / 543 / , : : , , !"#$ &%('(") *& +'(* ".4.4.2!"#$% '&)()#*!+'!,()+!# Figure 3-7: Comparison of the CF experimental curvature time series (red), and the reconstructed CF curvature time series (blue) at select strain gage locations for a typical NDP dataset (dataset 246). Note that the reconstructed curvature closely matches the experimental data at least at the sensor locations Uncertainty from the presence of noise in experimental data In this section we will discuss how we can quantify the uncertainty in reconstruction arising from the presence of noise in the experimental data. Assume that the measured experimental data y e (z s, t) consists of the actual experimental data y(z s, t) with some noise n(z s, t), where z s corresponds to the location of the sensors as: y e (z s, t) = y(z s, t) + n(z s, t), (3.32) The above system of equations can be written in a matrix form as: Y e = Y + N, which in frequency domain can be written as Ŷ e = Ŷ + ˆN. (3.33) The noise terms n(z s, t) in each sensor signal are assumed to be independent, Gaussian white noise with a zero mean and a standard deviation σ n. Standard deviation is usually expressed as percentage of the signal strength. For simplicity, we non-dimensionalize standard deviation using the riser diameter (typical response amplitude) as σ n = Dσ n. The uncertainty arising 57

58 "!#$%"&('*)"+","-" )98:-")"A<; )98:-")>#; )98:->?<; )98:->)<; )98:-"-"=<; )98:-"-"+<; )98:-"-"-<; )98:-"B")<; )98:-"@"=<; )98:-"@"+<; )98:-"@"-<; )98:-"+"B<; )98:-"+")<; )98:-","?<; )98:-",",<; /# "!#% "2'*)"+","-" Figure 3-8: Depicts the displaced shape of the riser at several instances of time for NDP dataset 243. One can clearly observe the presence of disturbances which starts around z = and moves along the riser. from noise propagates to our estimation of ŵ (or w). If we write: Ŷ e = Φ ŵ e, and Ŷ = Φ ŵ, (3.34) where, ŵ e is the estimate using noisy data and ŵ is the actual one, then equation (3.33) can be rewritten as: Φ ŵ e = Φ ŵ + ˆN. (3.35) 58

59 Premultiplying the above equation (3.35) using the pseudo-inverse Φ + we obtain: ŵ e = ŵ + Φ + ˆN, or ŵe = ŵ + ê w, (3.36) where, ê w Φ + ˆN which in the time domain can be written as ew = Φ + N. e w is the error in estimating w, and can be quantified using the covariance matrix E w. The covariance matrix E w of the error in estimating w can then be written as: E w = E { e w e T w} = σ 2 n Φ + ( Φ +) T, (3.37) where, E {} is the expectation operator. Using E w, we can propagate the error in estimating w to the error in estimating the displacement y(z, t) using equation (3.2) which can be written in an expanded form as: y(z, t) = φ (z)w (t) + φ 2 (z)w 2 (t) φ i (z)w i (t) φ N (z)w N (t). (3.38) Note that w i (t) are not independent, and hence both the variance and covariance terms of E w are important. The variance of the displacement error due to noise in the experimental data at any location z along the riser σy N 2 (z) can then be obtained as: σ 2 y N(z) = P(z) E w P(z) T. (3.39) where, P(z) = [φ (z) φ 2 (z)... φ N (z)]. From equations (3.37) and (3.39) we can write σ 2 y N (z) as: σ 2 y N(z) = σ 2 n P(z) Φ + ( Φ +) T P(z) T, (3.4) Thus in terms of σ 2 n we can obtain the variance of the displacement error σy N 2 (z) due the presence of noise in the experimental data: σ 2 y N(z) = σ 2 n D 2 P(z) Φ + ( Φ +) T P(z) T. (3.4) Using the above formula for the standard deviation σ y N (z) we can obtain an error bound (y(z, t) ± σ y N (z)) on the reconstructed displacement. This method is applied to the reconstruction of experimental data from NDP tests. For a typical experimental dataset, the noise level can be obtained from the RMS of a segment of the signal when the riser is not being towed (zero velocity case). Figure 3-9 depicts one such study for NDP dataset 236, for which the σ n is estimated to be around.5%. Figure 3- depicts the RMS displacement y RMS (z) and the error bound (y RMS (z)±σ y N (z)) assuming two separate values of σ n =.5% and σ n = 5%, for two NDP datasets 236 and

60 %'&)(*+(-,. (/ ( * 4 /*)( ( 3 ( 4 3:<;=>?'@7ACBED F?HGJIK L $!#" Figure 3-9: The span averaged strain signal from NDP dataset 236 is subdivided into three segments. Segments within the solid red vertical lines correspond to zero flow velocity periods, and the segments between the dashed black vertical lines correspond to the period when the riser was towed. The σ n level can be obtained from the ratio of RMS of the zero velocity signal and RMS of the actual VIV signal. Clearly, when the assumed noise level in the experimental data is small, the variability during reconstruction is also small. As shown in Figure 3-, a larger noise level would result in a consequent increase in the level of uncertainty during reconstruction Uncertainty from the use of finite number of sine and cosine terms The reconstruction method uses a finite number of sine and cosine terms while assembling the Φ matrix. The use of a finite number of sine and cosine terms during reconstruction may introduce some error. We aim at quantifying this source of error during reconstruction. If the number of sine and cosine terms used in the Φ matrix (limited by number of sensors available) is large, then the error is close to zero. However, error arises due to the presence of a finite number of terms which may not able to represent the displaced shape of the riser completely. To quantify the error due to this uncertainty, we vary the number of sine and cosine terms and correspondingly obtain the resulting reconstructed displacements. If the reconstruction is sensitive to the number of sine and cosine terms used, then there is a large variability in each instance of the reconstructed riser response. This variability in the reconstructed response is captured in the variance σy H 2 (z) obtained by evaluating the various reconstructions for different number of sine and cosine terms. The standard deviation σ y H (z) is obtained as: σ y H (z) = S {y RMS (z), y RMS2 (z), y RMS3 (z),..., y RMSh (z),...}, (3.42) where, y RMSh (z) represents the RMS (taken in time) of the h th reconstruction y h (z, t), and S {} denotes the standard deviation operator. We apply the above method to obtain a reconstruction error bound (y RMS (z) ± σ y H (z)) 6

61 ; ; ;? ;?.9.8.7! #"$ #%'&(%!))*)(+,*-%*%.%*/-"$% :<; /&#=?>@AB!&"<& +,*-%C/ 3ED'F6G ! #"$ #%'&(%) #**+*(,-+.%+%)%+/."$% ;:=<>@? /&#BADCEF 9? #&"$&,-+.%G/ 3'HJI5K ! #"$ #%'&(%!))*)(+,*-%*%.%*/-"$% :<; /&#=?>@AB!&"<& +,*-%C/ 3ED'F6G ! #"$ #%'&(%) #**+*(,-+.%+%)%+/."$% ;:=<>@? /&#BADCEF 9? #&"$&,-+.%G/ 3'HJI5K Figure 3-: Variability in reconstruction arising due to the presence of noise in experimental measurements. RMS displacement and error bound assuming σ n = 5% (row I ) and σ n =.5% (row II ) for NDP dataset 236 (U max =.8ms ). RMS displacement and error bound assuming σ n = 5% (row III ) and σ n =.5% (row IV ) for NDP dataset 246 (U max =.8ms ). 6

62 on NDP data. The RMS displacement y RMS (z) and the error bound (y RMS (z) ± σ y H (z)) due to the use of a finite number of sine and cosine terms for various NDP datasets are produced in Figure 3-. We can see that with increase in flow velocity, the error bound increases in magnitude. Figure 3-2 illustrates this trend, where the mean and maximum values of the reconstruction error bounds are plotted for various linearly sheared flow profiles. This happens due to the fact that with increasing flow velocity, the peak response frequencies increases and consequently the excited wave number increases. The limited number of sine and cosine terms are not able to capture all the intricate details of such riser motions Uncertainty from the presence of both acceleration and strain measurements Depending on the length of the riser and the flow conditions, the relative magnitudes of the acceleration measurements and curvature measurements may vary. If the acceleration signals are several orders of magnitude smaller than the curvature signals, or vice-versa, then the data from these sensors may not be taken into account during reconstruction. In fact during reconstruction, both Φ and Ŷ matrices consist of rows corresponding to curvatures (Φ κ and Ŷκ) and rows corresponding to accelerations (Φ a and Ŷa). As a result, the Φ and Ŷ matrices can be written as: Φ [ Φκ Φ a ], and Ŷ [ Ŷκ Ŷ a ]. (3.43) The elements of Φ κ and Φ a are in fact the terms in front of the unknowns ŵ in the equation: [ Φκ Φ a ] ŵ = [ Ŷκ Ŷ a ]. (3.44) One measure of the relative importance of Φ κ and Φ a are their Frobenius norms 2 Φ κ F and Φ a F. Ideally the rows corresponding to curvature could be divided by the Frobenius norm Φ κ F, and the rows corresponding to acceleration could be divided by the Frobenius norm Φ a F. We introduce a parameter β which allows for changing the relative importance of Φ κ and Φ a respectively. Using β, we reassemble the system of equations (3.44) in the following form: ] ] [ Φκ βφ a ŵ = [ Ŷκ βŷa. (3.45) This section studies the uncertainty as a result of varying the relative magnitudes of the Φ κ and Φ a using the parameter β. The variability in the reconstructed response is captured by the variance σy β 2 (z) obtained by evaluating the various reconstructions for a range of values taken 2 Frobenius norm A F of a matrix A is defined as: A F = P P a 2 ij. i j 62

63 .7.6! #"$ #%'&)( #&"$& -H3%I> J'K7LNMOG ! #"$ #%'&)( *+,-&/.2&354%&6%7"8-9!,)3:;&;<,"=3%#6>?&#A@CBDEEF #&"$& -G3%H> I'J7KMLNF ! #"$ #%'&)( *+,-&/.2&354%&6%7"8-9!,)3:;&;<,"=3%#6>?&#A@CBDEFG #&"$& -H3%I> J'K7LNMOG ! #"$ #%'&)( *+,-&/.2&354%&6%7"8-9!,)3:;&;<,"=3%#6>?&#A@CBDEFG #&"$& -H3%I> J'K7LNMOG Figure 3-: Uncertainty arising due to the use of finite number of spatial harmonics during reconstruction. Row I : RMS displacement and error bound on noise for NDP dataset 236 (U max =.8ms ); row II : RMS displacement and error bound on noise for NDP dataset 24 (U max =.2ms ); row III : RMS displacement and error bound on noise for NDP dataset 243 (U max =.5ms ); row IV : RMS displacement and error bound on noise for NDP dataset 246 (U max =.8ms ). Note that as the flow velocity increases, the uncertainty in reconstruction increases due to the presence of higher spatial harmonics in the measured riser motions. 63

64 ! & %!#"$ GEF3<2-E7+3MN, -,.(4/EO+<P Q@, J3<4* 3/3,.G R.S:TVUXWZY4[ ]\_^2`ba S+cNUXWZY4[ ]\_^2`ba Figure 3-2: Uncertainty arising due to the use of finite number of spatial harmonics during reconstruction for various linearly sheared flow velocity cases. Note that as the flow velocity increases, the uncertainty in reconstruction increases due to the presence of higher spatial harmonics in the data. by β as: σ y β (z) = S { y RMS (z), y RMS2 (z), y RMS3 (z),..., y RMSg (z),... }, (3.46) where, y RMSg (z) represents the RMS (taken in time) of the g th reconstruction y g (z, t), and S {} represents the standard deviation operator. We apply our methodology to datasets from NDP experiments. The Frobenius norm of Φ κ F, and the Frobenius norm of Φ a F were found to be of similar magnitudes. Parameter β is varied from. to. In total 2 different reconstructions are performed for each dataset corresponding to 2 different values of β. The standard deviation σ y β (z) due to the variability in β is obtained, and the error bound on reconstruction is obtained as y RMS (z)±σ y β (z). Figure 3-3 depicts the RMS of displacements and the error bounds due to the variation in β for four separate datasets from NDP experiments. We observe that the variability in reconstruction increases with increasing flow velocity Total uncertainty We have developed methods to quantify the variability arising from three sources of uncertainty mentioned in Section 3.6. in terms of their variances σy N 2 (z), σ2 y H (z) and σ2 y β (z). However, we are interested in quantifying the total variance σy(z) 2 which takes into account each of the three sources of uncertainty. If we assume that the three sources of uncertainty are independent of each other, then we can obtain the total variance σy(z) 2 from the individual variances as: σy(z) 2 = σy N(z) 2 + σy H(z) 2 + σy β 2 (z). (3.47) We obtain the total variance for datasets from NDP experiments. Figure 3-4 depicts the es- 64

65 "!"#$!"%'&)(+*, -. /& $2!&.3$4!"-.!"%5&768!"9;:5&.=<>3;-7&.-? % % ""!"!"-? &.3;%A@>B &? C.!CDE;F;G HI.7.6 JCK"L"M>NPO;L"J QSR;TUVXW Y5Z\[^]`_ "!"#$!"%'&)(+*, -. /&2$3!&.4$5!"-.!"%6&879!":<;6&.>=?4<-8&.-@ % % ""!"!"-@ &.4<%BA?C &@ D.!DEF G<H<HI JDK"L"M?NPO<L"J QSR<TUVXW Y6Z\[^]`_ "!"#$!"%'&)(+*, -. /&2$3!&.4$5!"-.!"%6&879!":<;6&.>=?4<-8&.-@ % % ""!"!"-@ &.4<%BA?C &@ D.!DEF GH IJ KDL"M"N?OQP<M"K RTS<UVWYX Z6[]\_^a` "!"#$!"%'&)(+*, -. /&2$3!&.4$5!"-.!"%6&879!":<;6&.>=?4<-8&.-@ % % ""!"!"-@ &.4<%BA?C &@ D.!DEF GH IJ KDL"M"N?OQP<M"K RTS<UVWYX Z6[]\_^a` Figure 3-3: Uncertainty arising due to the use of both strain and acceleration signals during reconstruction for NDP datasets 236 (row I ), 24 (row II ), 243 (row III ) and 246 (row IV ). We can observe that with increasing flow velocity, the variability in reconstruction increases. 65

66 .7.6! #"$ #%'&)( #**+*),-+.% ! #"$ #%'&)( #**+*),-+.%. /&+2#&+*/34&#65879::;!&"<&,-+.%=3 >@?A; ! #"$ #%'&)( #**+*),-+.%. /&+2#&+*/34&#65879:;<!&"=&,-+.%>3?A@B< ! #"$ #%'&)( #**+*),-+.%. /&+2#&+*/34&#65879:;<!&"=&,-+.%>3?A@B< Figure 3-4: Depicts the RMS riser displacements and the bounds y RMS (z) ± σ y (z), which take into account three sources of uncertainty for NDP datasets 236 (row I ), 24 (row II ), 243 (row III ) and 246 (row IV ). We can observe that the total variability in reconstruction increases as the flow velocity increases. 66

67 timate of the RMS of the riser displacement y RMS (z) with the variability during reconstruction (y RMS (z)±σ y (z)) which takes into account all the three sources of uncertainty. This corresponds to an assumed noise level of.5%, β within the range [., ] and 8 different combinations of sine and cosine terms. We observe that as the flow velocity increases, the quality of reconstruction may decrease (variability of reconstruction increases) primarily due to an increase in the uncertainties arising from the use of a finite number of sine and cosine terms. Note that the total variance σy(z) 2 obtained during reconstruction is conservative in nature. This is in part due to our assumption that various sources of uncertainty are independent of each other. 3.7 Concluding remarks What we have achieved is a systematic and scalable approach to reconstruct the VIV response of a riser using the data from a limited number of sensors (strain gages, accelerometers) placed along the length of the riser. For number of sensors are large enough to capture the entire riser VIV motions, the problem is posed as a Fourier decomposition. This method requires no information on the riser response modes and is hence independent of riser VIV predictive capabilities. The criterion for full reconstruction is evaluated from both a matrix inversion perspective and from a signal processing perspective. Illustrative examples signify the importance of both the sine and cosine terms during reconstruction. We have also developed a systematic approach to quantify the various sources of uncertainty during the reconstruction. This method is accurate when the full reconstruction criterion is satisfied, and for most NDP datasets this allows the reconstruction of the harmonic part of the riser VIV response. Riser motions (CF and IL) hence reconstructed clearly depict the presence of traveling waves (with consistent direction in sheared profiles and irregular direction in uniform profiles). In addition, we observe clear evidence of the interaction of CF and IL motions during wave propagation. Finally, the reconstruction method enables us to perform several analyses mentioned later in the thesis. Specifically we can: Study the evolution and mechanism of traveling waves in riser response as mentioned in Section 4.4. Perform a modal decomposition of the riser VIV response to evaluate the peak response modes of the riser as mentioned in Section 4.3. Estimate the harmonic part of the external force from fluid to the riser as mentioned in Chapter 7. 67

68 68

69 Chapter 4 Riser VIV modal decomposition and traveling wave identification 4. Introduction As mentioned in great detail in Chapter, several aspects of vortex-induced vibration of flexible risers are not well understood. As such, there exist disparities between experimental observations and predictions based on empirical models of such risers. Understanding the reasons for such disparities requires innovative methods for observing the evolution of the phenomenon (data measured as strains, accelerations or displacements) over time. In this chapter we will develop the techniques critical to one of our research goals of comparing the appropriate experimental quantity with the appropriate theoretical quantity. With this underlying motivation, three aspects of riser VIV are studied in detail. In the first part of this chapter, we present a method to categorize a signal (or a part of it) as statistically stationary. An underlying assumption we use while making a theoretical estimate is that of a steady state response (a steady flow profile leading to a steady response). Though we may ask questions on the validity of the steady state assumption, it is a requirement for any empirical prediction program. One characteristic of a riser under a steady state response is that the measurements from such a riser would be statistically stationary. While performing a comparative study between a theoretical model predicting a steady state response and experimental measurements, obtaining a statistically stationary segment of the measurement becomes important. Secondly, for a statistically stationary segment of riser VIV measurement which is temporally and spatially dense, we can write the riser response using a Fourier expansion in terms of its response modes. Since VIV has a preferred frequency of excitation around the Strouhal frequency, we can in turn extract the riser peak response modes. Thus peak response modes can be thought 69

70 of as a useful description of the riser VIV response. Finally, we focus our attention on the presence of traveling waves in riser VIV response. Traveling waves may arise due to a forcing which travels along the riser, or due to the riser response at one location traveling to other locations. Identifying the presence of traveling waves in riser VIV response and characterizing them is important in developing appropriate models for predicting riser VIV. 4.2 Time evolution of VIV measurements using scalograms A plot of the measured signal against time (Figure 4-(top)) gives limited information on the evolution of a process; a plot of its finite-window spectrum (Figure 4-(middle)) can give much better information on the frequency content of the process, but presents difficulties when the frequency content of the signal changes with time. However, employing time evolution plots (scalograms) (Figure 4-(bottom)) offers far more insight into the evolution of the physical process. A scalogram is a contour plot of squared magnitude of a continuous wavelet transform [39, 4] which describes how the frequency content of a signal varies with time. As shown using examples in Figure 4-(bottom) the regions of high intensity in the contour plot correspond to the dominant frequencies which are excited over time. Please refer to Appendix D for a detailed approach on obtaining the scalograms. From a VIV perspective, a scalogram can be used to: Obtain a snapshot of the evolution of the underlying physical phenomenon. As illustrated in Figure 4-2(left), the evolution of the high intensity regions depict the evolution of the dominant frequency, presence of a multi-frequency (multi-modal) response and switching between the modes. Eliminate faulty data arising from instrumentation errors or processing errors as illustrated in Figure 4-2(right). Due to very complicated instrumentation of a riser during field experiments, a sensor could malfunction. The measured data from the sensor could depict a scalogram which is blank (no measurements), or a scalogram depicting a highly repeatable pattern (presence of buffer errors). Thus a scalogram also allows us to observe if a signal is artificially constructed by repeatedly copying and pasting rather than measured from a physical experiment. Isolate a region over which the process is statistically stationary as described in more detail in Section

71 Figure 4-: Three different representations of the CF strain signal measured at z = 8.6m for NDP dataset 2. Top: a plot of the time signal; middle: the spectrum of the signal; bottom: the scalogram of the signal. 7

72 Figure 4-2: Left: the scalogram of CF response (strain signal) at a given location (z/d = 394) from the Gulf-stream test (B242954). From the scalogram we can observe a multi-frequency response, a sudden event happening around t = 5seconds, and the sporadic appearance and evolution of the third harmonic at reduced frequency fr.6. Right: the scalogram of CF response (acceleration signal) at a given location (z/d = 927) from the Lake Seneca test (47462). We can clearly observe the repeated patterns in the signal, which arise from either a sensor malfunction or a processing error Extracting a statistically stationary segment of VIV measurement As mentioned previously, present state of the art analysis of riser VIV assumes a steady state excitation and any benchmark study will require that the benchmarking signal to be statistically stationary. A statistically stationary signal has all its statistical moments independent of time. For a vibrating riser this implies that the fluid excitation is also statistically stationary. For a stationary signal, the energy content corresponding to each frequency in the signal remains constant. It is easier and meaningful to relax this criterion to categorize the VIV motions as second-order stationary (SOS), if the statistics such as the standard deviation and autocorrelation function (relevant to the frequency content) remain nearly constant, within a reasonable threshold. We identify segments over which the response is second-order stationary (SOS). Of the various methods employed in [4] to extract patches of stationarity of a signal, we prefer scalograms. A statistically stationary signal results in scalograms containing horizontal lines of constant intensity, implying that the energy distribution remains constant along those frequencies. In Figure 4-3 (right), the sharp horizontal lines imply that one of the frequencies is excited more than the others. The parallel lines of lesser intensity depict the other frequencies which are also excited. Figure 4-3 (left) depicts a non-stationary signal where the dominating frequency changes over time. However, we may identify segments of the time series over which the signal is considered to be stationary. For analyses mentioned in this thesis, we extracted statistically stationary segments of the response from a visual inspection of the scalograms. We illustrate the method to obtain a statistically stationary segment of the time series for two NDP datasets [5] in Section

73 Figure 4-3: Left: scalograms of select strain signals from NDP experiments [5] (for dataset 2) depicting a typical case of non-statistically stationary behavior; right: scalograms from another set of data (dataset:238) from NDP experiments showing a behavior typical of being statistically stationary Application of scalograms to NDP data For each dataset from the NDP experiment (refer to Section B.), we obtain the scalograms for each strain gage measurements. Two examples are illustrated as follows: Example I (NDP dataset 24): Figure 4-4 depicts the scalograms of the strain signal obtained at 6 representative locations along the riser for the NDP dataset 24. Based on the observations from the scalograms obtained at all the strain gage locations, we choose a segment from 2.5 seconds to 6.5 seconds as our choice of a statistically stationary segment of data. Example II (NDP dataset 24): In a similar fashion, Figure 4-5 depicts the scalograms of the strain signal obtained at 6 representative locations along the riser for the NDP dataset 24. Again we choose a statistically stationary region of the signal based on a visual observation of the scalograms obtained for all the strain signals. Our choice is from 25.5 seconds to 29 seconds. In addition to extracting a statistically stationary segment of data using scalograms, we can also observe the evolution of the dominant response frequencies, and the presence and evolution of the third harmonic (refer to Figure 4-3). Some of the key observations for the linearly sheared (triangular) velocity profiles in NDP data [4] are presented below: 73

74 Marintek Dataset: 24; ( t stationary = [ ]s ) Figure 4-4: Depicts the scalograms of strain signals obtained at 6 representative locations along the riser for NDP dataset 24, and our choice of the statistically stationary segment of data (t stationary = [2.5, 6.5]seconds) shown between the two red lines.. As expected, the magnitude of the scalograms progressively decreases at the locations where the flow velocities are smaller. At low velocity, the maximum intensity of the scalogram is two orders of magnitude lower than the maximum intensity observed at the higher velocity locations. 2. One can observe sharp horizontal lines corresponding to monochromatic excitation in some cases (e.g.: 236, 243); in other cases parallel horizontal lines indicate that several frequencies are excited (e.g.: 238). Such horizontal streaks of high intensity are separated with a spacing corresponding to the spacing between the natural frequencies. It is likely that the sharp horizontal streaks of high intensity represent lockin of the riser and the parallel streaks represents a response where multiple frequencies (modes) participate. These streaks of maximum intensity sometimes do shift from one possible frequency (mode) to another (e.g.: 236, 238). Sharp streaks are more prominent at locations close to the maximum velocity, while the shifting of frequencies from one mode to another occurs at locations between the maximum and minimum velocities (e.g.: 236). 3. In addition to the sharp streak observed at the dominant frequency corresponding to the Strouhal frequency, one can also observe significant energy in the third harmonic 74

75 Marintek Dataset: 24; ( t stationary = [ ]s ) Figure 4-5: Depicts the scalograms of strain signals obtained at 6 representative locations along the riser for NDP dataset 24, and our choice of the statistically stationary segment of data (t stationary = [25.5, 29]seconds) shown between the two red lines. component of the signal. A sharp single frequency response corresponding to the first harmonic (lockin) is often accompanied by a strengthening of the third harmonic. The presence of the third harmonic has the effect of stabilizing the lockin of the riser and vice versa (e.g.: 236, 243, 246). 4. Significant portions of the signals from NDP data were found to be statistically stationary. 4.3 Riser response modes from displacement measurements Consider a distributed system like a tensioned beam, which is excited by a force with a frequency Ω along the span. Depending on the choice of Ω, the response magnitude at a representative location along the structure would show several peaks similar to the one shown in Figure 4-6. When the forcing frequency Ω matches any one of the natural frequencies ω n of the structure, the response magnitude is amplified several times (the amplification factor depends inversely on the damping). This is the classic case of resonance. If the tensioned beam is placed in a uniform current, then a narrow banded response similar to that of the resonant vibration of a tensioned beam in air is observed. Such self-excited narrow 75

76 !#" $% & '()*+,$-!.+/$- )." 23 # /,!;:!.);/<=" >,/ :"??!;,"!.@, #465BADC E 7 23 #465BAGF;E 7 HJI KMLON;PQ R LOS.T;T.TVU QWLYX6Z[P9\^]_Q*` a b LOTVZ[TcX Modal Magnitude at ω=7.43hz Figure 4-6: Magnitude of response of a tensioned string at chosen locations excited by a forcing with a frequency Ω. φ(z) 8 banded (monochromatic) response is similar to that of a freely vibrating string or beam in air. 6 We can refer to the frequency of response as the peak response frequency and the narrow banded 4 response the peak response mode. Section z (m) Modal Phase Angle energy inputat each excitation cell is carried over to the rest of the cylinder and dissipated due (φ(z)) [π rad].8 a wide band.6for a non-uniform velocity profile. If the nominal natural frequencies (assuming.4.2 The more formal definition is mentioned in the following For current profiles which are not uniform, we may observe a similar narrow banded response (Figure 4-7 (left)) or a wide band response (Figure 4-7 (right)). For such velocity profiles, the vortex shedding happens in cells [54], and only a part of the flexible cylinder is in lockin. The to both hydrodynamic and structural damping. Additionally, the frequency of vortex shedding depends linearly on the local flow speed resulting in the excitation frequency being spread over an added mass coefficient of unity) are spaced apart from each other and if one of them lies in the excitation frequency band, then the response of the riser is primarily at a single frequency. But if the nominal natural frequency is densely packed together and several nominal natural frequencies lie within the excitation bandwidth, then the response of the riser will have multiple z (m) peaks as depicted in Figure 4-7 (right). The mechanics of a multi-frequency response is largely unknown since the parametric space is too large to explore experimentally or numerically, even at low Reynolds number [35] Definition of response modes and response frequencies By response modes we denote here the special (complex) spatial functions that describe the response of the riser at each specific frequency (response frequencies). The best way to view these response modes is to consider the spatially and temporally dense riser response (obtained typically after a response reconstruction mentioned in Chapter 3) y(z r, t j ). The bivariate signal 76

77 PSD of Axial Strain Signal between t = [48 z =6.45m PSD of Axial Strain Signal between t = [ z =6.45m 8 5 PSD (µ ε 2 ) 6 4 PSD (µ ε 2 ) Frequency (Hz) Frequency (Hz) PSD of Axial Strain Signal between t = [48 z =6.89m PSD of Axial Strain Signal between t = [ z =6.89m 8 5 PSD (µ ε 2 ) NT =24 samples NOverlap =892 samples NFFT =65536 samples PSD (µ ε 2 ) 5 NT =84 samples NOverlap =496 samples NFFT =32768 samples Frequency (Hz) Frequency (Hz) PSD of Axial Strain Signal between t = [48 z =2m PSD of Axial Strain Signal between t = [ z =2m 8 5 PSD (µ ε 2 ) 6 4 PSD (µ ε 2 ) Frequency (Hz) Frequency (Hz) Figure 4-7: Left: narrow-banded response observed from strain data measured at 3 locations along the riser for NDP dataset 234; right: multi-frequency response observed from strain data measured at 3 locations along the riser for NDP dataset 238. The vertical green lines represent the natural frequencies obtained using the empirical prediction program VIVA. can be written using its two dimensional Fourier transform, ψ(k m, ω l ) as: We can recast equation (4.) as: y(z r, t j ) = (2π) 2 M m= l= Λ ψ(k m, ω l )e iω lt j e ikmzr. (4.) y(z r, t j ) = Λ Y l (z r ) e iω lt j, (4.2) l= where Y l (z r ) are the l =, 2,...Λ complex response modes corresponding to the response frequencies ω l, and are given as: Y l (z r ) = (2π) 2 We may also use an alternative expression: y(z r, t j ) = M ψ(k m, ω l )e ikmzr. (4.3) m= M w m (t j ) e ikmzr, (4.4) m= 77

78 where w m (t j ) are the m =, 2,...M complex modal participation amplitudes, given as: w m (t j ) = (2π) 2 Λ ψ(k m, ω l ) e iω lt j. (4.5) l= Such an approach is essentially what we employ for response reconstruction mentioned in Chapter 3, where the displaced shape of the riser at any instance of time is represented using sine and cosine terms. In fact, the spatial dependence e ikmzr sine and cosine terms. is a convenient representation for the The reason for preferring the approach of equation (4.3) is simply that expressions (4.2) and (4.3) provide a more deeper insight in terms of the energy transfer along the riser than expressions (4.4) and (4.5). Peak response modes: For riser response depicting one single peak frequency or a few major peak frequencies along the entire riser, we can extract these frequencies of peak response (or peak response frequencies) which are denoted as ω n. Corresponding to these peak response frequencies, we may define the peak response modes Y n (z). Thus ω n is a subset of ω l and correspondingly Y n (z r ) is a subset of Y l (z r ). Thus the response of the riser may be approximated using the peak response modes as: Numerical method y(z r, t j ) N Y n (z r ) e iωnt j. (4.6) n= Based on the definition of response modes given in 4.3., we describe a practical method to extract these response modes from experimental data. Let y(z r, t j ) be a riser displacement signal which is band limited in both wavenumber k and frequency ω. Also assume that the signal is such that the sampling is dense in both time and space. We obtain the Fourier expansion of the time series at any given location z r as: { Λ } y(z r, t j ) = Re ŷ(z r, ω l ) e iω lt j, (4.7) l= where ŷ(z r, ω l ) is a complex quantity and represents the l th Fourier coefficient corresponding to a frequency ω l from the signal obtained at z r. The response modes denoted by Y l (z r ) can be extracted by obtaining the Fourier coefficients ŷ(z r, ω l ) at each point along the riser corresponding to the frequency ω l as: Y l (z r ) = ŷ(z r, ω l ). (4.8) Thus we may rewrite equation (4.7) in terms of the response modes Y l (z r ) and response frequen- 78

79 cies ω l as: { Λ } y(z r, t j ) = Re Y l (z r )e iω lt j. (4.9) l= The modal magnitude Y l (z r ) corresponding to the response mode Y l (z r ) is obtained as: Y l (z r ) = (Re {Y l (z r )}) 2 + (Im {Y l (z r )}) 2. (4.) The modal phase angle Y l (z r ) corresponding to the response mode Y l (z r ) is obtained as: [ ] Im Y l (z r ) = tan {Yl (z r )}. (4.) Re {Y l (z r )} For obtaining the peak response modes and peak response frequencies, we need to first obtain a span averaged spectrum. From a span averaged spectrum we find out the frequency of peak response. The Fourier coefficients corresponding to these peak frequencies obtained along the entire riser give the peak response modes. Using equations (4.) and (4.), we can subsequently obtain the modal magnitude Y n (z r ) and modal phase angle Y n (z r ) of the peak response modes. Notes on modal extraction The response modes of the risers are experimentally observed quantities and can be used to investigate the presence of traveling wave behavior in the riser. Since most of the riser response is captured by the peak response modes, they contain significant information on energy transfer along the riser. Peak response modes depicting a modal phase angle which varies almost linearly along the riser span, and modal magnitudes depicting the absence of nodes indicate the presence of traveling waves. Section 4.4 presents various methods that can be employed to identify the presence of traveling waves (disturbances which propagate along the riser) in riser VIV displacement measurements Application to NDP data and discussion Experimental data from NDP portray a vortex-induced response whose spectra show a sharp single peak frequency (single-modal response) or few peak frequencies (multi-modal response) depending on the flow velocity. Since the linearly sheared flow profiles are more challenging than a uniform flow profile, we obtain the peak response modes and the peak response frequencies for such cases. For this we first obtain the displacements using the response reconstruction procedure development in Chapter 3. In Figure 4-8 we depict all the linearly sheared cases which show a response with a single peak frequency. One can observe the span averaged spectra, the modal magnitudes, and the modal phase angles of the peak response modes. Clearly, these riser peak response modes are different from the free vibration modes of a tensioned string. Nodes are locations along a riser where the displacements are zero. 79

80 4S2T4 -. / :28 9=A2B C4D 9=E< 8 9F;4E< 4G2? 8 BH8 37@ I C ;24G BH2:432E@=A2;48 DC < I C ;42G-2P42<=8Q432:4G8 DC < *,+ 4S2S4 *,+ 4S2R4 *,+ 4S24 *,+ 4SV *,+ 4S24 *,+ 4R *,+ 4R2U4 *,+ 4R2T4 *,+ 4R2S4 *, ! "# %$ &' &( ) &( ( Figure 4-8: Modal decomposition of the riser response for several linearly sheared flow profile datasets (each row correspond to one dataset) from NDP experiments. Column I : depicts the maximum velocity Umax; column II : depicts the harmonic part of the span averaged displacement spectra for various NDP datasets; column III : depicts the peak response modal magnitudes for various NDP datasets; column IV : depicts the peak response modal phase angles for various NDP datasets

81 The following observations are made from the magnitudes of the peak response modes Y n (z). As the flow velocity increases, the number of troughs and crests increases. This is because the excitation frequency increases as U max increases leading to a corresponding increase in the wavenumber resulting in increased number of troughs and crests. The modal magnitude also depicts an absence of nodes. This is a clear indicator of a response involving traveling waves and will be discussed in a greater detail in Section 4.4. These propagating disturbances are excited close to the location of higher flow velocity, decay while traveling to the other end and result in standing waves close to the riser ends. Although there are no nodes in the modal magnitude, there are troughs which tend to deepen at the ends due to the reflection at the boundary. In addition these troughs are more pronounced at the boundary corresponding to the low flow velocity. The phase angle of the peak response modes Y n (z r ) is limited within a range of 2π due to the tan operator. Except for this artificial shift we observe several segments of the modal phase angle which are nearly linear with a non-zero slope. This linear variation in modal phase angle is due to the disturbances created at the high flow velocity region getting propagated to the low flow velocity region. Near the ends of the riser the modal phase angle becomes flat (zero slope), which is a further indication of standing waves. More detail on interpreting modal magnitudes and modal phase angles for traveling wave identification is described in Section Traveling waves in riser VIV response A traveling wave (progressive wave or propagating disturbance) is a mechanical disturbance created at some point on the riser (typically the excitation region) that subsequently travels along the riser (from one point to another). In fact, it is the inherent tendency of such a disturbance to travel and result in energy being transferred along the riser. Consider a sheared flow profile where one frequency is excited. It is expected qualitatively that energy is input from fluid to structure at a frequency close to the local Strouhal frequency over a part of the riser, and then carried away to be dissipated at another part of the riser, where the local Strouhal frequency is different and the fluid force resists the traveling wave, providing a damping force. A steady response is obtained under three circumstances mentioned below:. If the riser is long (in comparison with the characteristic wavelength), then the disturbance travels along the riser, but dies down before the disturbance reaches the other end. At steady state we reach a situation where the disturbances are developed continuously at the excitation region but travel along the riser and get dissipated. 2. For a relatively short riser the disturbance reaches the boundary before it dies down and gets reflected. The incident wave and the reflected wave superpose to form a standing wave. 8

82 3. For a riser in between, a mixture of the above two phenomena happens. Most risers (e.g.: NDP riser model) fall in this category and the associated steady response has both traveling wave and standing wave characteristics. Clearly the above classification depends on the length of the riser in comparison with the characteristic wavelength (related to the frequency of oscillation) and the fluid damping. The criterion for classifying a riser into each such category is given by Vandiver et al. [64] and is mentioned in terms of a single parameter nζ n which captures both the damping and the riser length. For the third circumstance, the riser response cannot be described in terms of standing waves alone as they do not allow energy to be carried along the riser. Identifying traveling waves and classifying their behavior (e.g.: wave speed, hydrodynamic properties like effective added mass) becomes important. These quantities are employed to understand the spatial variation of the excitation force, damping and frequency of peak response, and develop effective models for representing the fluid forces. This section deals with different methods for identifying the presence of traveling waves (energy propagation) from experimental data. These methods are then used to obtain key characteristics of traveling waves like their wave propagation speed and hydrodynamic properties Method : From magnitude of peak response modes Section 4.3. shows that the peak response modes of a riser undergoing VIV is a complex function. The absence of nodes in the magnitude of peak response modes is an indicator of waves traveling along the riser (energy propagation). To illustrate this we consider a theoretical model of a riser - a tensioned string (length L, linear mass density m and tension T ). Assume a harmonic forcing of the form f(z, t) = Re { F (z)e iωt}, where F (z) is the spatial variation of the force and Ω is the forcing frequency. For example we can consider a force with a semi-sinusoidal spatial variation and a frequency Ω = ω (the tenth natural frequency of the tensioned string) as shown in Figure 4-9(top). Corresponding to this force, the displacement is obtained using Green s function approach as: y(z, t) = Re { Y (z)e iωt}, (4.2) where, Y (z) is the displacement magnitude (for this example also the peak response mode) of the tensioned string under consideration. Y (z) is obtained from Green s function G(z; Ω, ξ) as: L Y (z) = F (ξ)g(z; Ω, ξ)dξ, (4.3) ξ= and, G(z; Ω, ξ) is the Green s function for a tensioned string with zero end conditions obtained 82

83 as: G(z; Ω, ξ) = kt [ ] sin k(ξ L) sin k(2ξ L) sin kz, < z < ξ, [ sin kξ sin k(2ξ L) ] sin k(z L), ξ < z < L. (4.4) Figure 4-9 depicts the magnitude and phase of the peak response modes for two specific values of damping ratios ζ % and ζ 35% for a riser with properties similar to the NDP riser model. As mentioned previously, at higher damping one expects the presence of traveling waves. For the instance of a larger damping ratio we can observe that the modal magntiude of the peak response mode clearly depicts the absence of nodes. For small damping ratios the modal magnitude closely resembles the free vibration modes of the tensioned string. This is because energy input at the excitation region depicted in green color is dissipated at regions away from it. For small damping, we observe nodes (sharp troughs) in the peak response modal magnitude. As a result, the energy at one section of the riser is not propagated to the other locations leading to a standing wave behavior Method 2: From phase angle of peak response modes The modal phase angle also gives significant information on the transfer of energy. For a purely standing wave, the modal phase angle of the peak response modes remain constant between two consecutive nodes, with sudden jumps occurring about the nodes. For riser response involving purely traveling waves, we observe a linearly varying phase with a non-zero slope along the entire riser. For a riser response having both traveling and standing wave characteristics, we observe a hybrid behavior where any deviation from the constant phase indicating the presence of traveling waves. This behavior of the modal phase angle is illustrated in the tensioned string example mentioned in Section 4.4. corresponding to the two cases of damping ratios. For the low damping ratio, the modal phase angle remains more or less constant between the consecutive nodes, while the modal phase angle varies linearly (with a non-zero slope) for high damping ratio revealing the existence of traveling waves Method 3: From nodal evolution curves We seek to study a riser response which has both standing and traveling wave characteristics. For this purpose let us consider a simple case of a riser exhibiting a monochromatic response (a bivariate signal) of the form: y(z, t) = a sin(kz ωt) + a 2 sin(kz)cos(ωt). (4.5) 83

84 a`! #"$%&'()#$%!$%* +($%'$,-&.+$,)#')#$%$/*!.&*!.#,$%**!.# :9<; CD5E7F=8AHGJILK MON PRQ PRQ SUT VBWBT.5 b _ M N ^ XZYJ[]\ Figure 4-9: Response of a tensioned string under two different damping condition (ζ % ( ) and ζ 35% ( )) acted upon by a force of the form f(z, t) = Re { F (z)e iωt}. Top: depicts the spatial variation F (z) of the applied force; middle: depicts the magnitude of the response; bottom: depicts the phase of the response. where, the first term represents a purely traveling wave and the second term represents a purely standing wave, with the magnitudes of a and a 2 determining their relative importance. For a purely standing wave response at a given frequency ω, there are locations along the riser where the displacements are zero known as nodes. On the other hand, for a propagating disturbance the nodes do not stay at a given location along the riser. One excellent way to observe the riser response is to look at the evolution of the nodes in 84

85 the displaced shape of the riser. These nodal evolution curves are the locus of points in z-t plane through which the nodes evolve (the zero-crossing points in time and space), and allows us to gather valuable insights into the presence of traveling waves and the wave propagation mechanism. Figure 4- depicts the evolution of the nodes for 8 different combinations of a and a 2. For a purely standing wave (a =, a 2 = ; Figure 4-(g)), there are points along the riser where the response is zero at every instance of time. In addition, at specific instances of time at every period ( 2π ω ) the displaced shape of the entire riser is zero. This results in a mutually perpendicular net like pattern. For a purely traveling wave (a =, a 2 = ), the location of zero displacements (nodes) do not remain at a single point but moves along the span of the riser. We observe a pattern as depicted in Figure 4-(h) corresponding to parallel straight lines inclined at a constant angle to the time axis. The slope of these lines gives us the speed of wave propagation. For a riser depicting both standing and traveling wave behavior, we expect staggered equidistant curves as depicted by Figures 4-(a) to 4-(f ). Again the average slope of these curves give the average speed of wave propagation. In Section we obtain the nodal evolution curves for a variety of NDP datasets corresponding to linearly sheared and uniform flow profiles. These nodal evolution curves from NDP datasets will be used to understand important characteristics of wave propagation. 85

86 LHK(O(^ CQNH[_JR:À [N(?abUc)dBegf)hiVkj!lPm)nopc)qSd>rtsufvhxwyj!zTm)nopc)qSdh){) :mcs}fvh)~8 R:A?)À q VkX( ƒ( : Sj( ( :ˆM~gsV HŠ( ƒ: j( ( (m! 3 2 " # $! % # & $ 3 2 ' # $() *,+ # & $ &(),9 # *:) / ' / '; Figure 4-: Figure depicting the nodal evolution plots for 8 different cases of riser displacements of the form given by equation (4.5). Figure 4-(g) depicts a purely standing wave. Figure 4-(h) depicts a purely traveling wave. Figures 4-(a) to 4-(f ) illustrates responses indicating varying presence of standing and traveling waves. 86

87 4.4.4 Traveling waves in NDP experimental data Traveling wave identification methods presented previously are applied to data from NDP experiments. Figure 4-8 depicts the magnitude and phase angle of the peak response modes. Clearly the modal magnitude (Figure 4-8(column III )) depicts the absence of nodes indicating the presence of traveling waves. The deepening of the troughs in modal magnitude near the boundary is again an indicator of the increased importance of standing waves near the boundary. The modal phase angle (Figure 4-8(column IV )) also indicates the presence of traveling waves as observed from the near linear variation of the modal phase angle. NDP datasets clearly depict the presence of traveling waves which can be observed from the nodal evolution curves as depicted in Figure 4- (for linearly sheared flow profiles) and Figure 4-2 (for uniform flow profiles). The important conclusions from our study using nodal evolution curves are produced below. For linearly sheared flow profiles: Linearly sheared flows clearly depict the presence of disturbances (pulses) in CF displacements which travel along the length of the riser (observed as lines inclined to the z and the t axis as in Figure 4-) with a consistent direction and velocity. As the flow velocity increases, the presence of traveling waves become more apparent due to the increasing number of pulses generated per second (frequency), and the increasing number of nodes (wavenumber) within the riser span (compare Figure 4-(top left) and 4- (top right)). However, the wave propagation speed in the CF direction (obtained from the slope of the curves in Figure 4-) remains more or less constant over a wide range of flow velocities. This is illustrated in Figure 4-3, where the wave propagation speed in CF direction for various maximum flow speeds are plotted. The in-line (IL) motions (refer to Figures 4-(bottom right) and 4-(bottom left)) are much more irregular depicting the presence of both standing waves (lines parallel to z axis) and traveling waves (lines inclined to z axis) over a part of the riser. The wave propagation speed observed for CF & IL (over the traveling part) are found to be nearly the same as shown in Figure 4-3. For uniform flow profiles: CF displacements for uniform flows depict the presence of standing waves, or disturbances which travel over a part of the riser length, with no directional consistency. The standing waves are not purely sinusoidal modes as expected by prediction programs. In fact this could be a case where the given standing wave may be combination of several purely sinusoidal modes with their corresponding modal frequency converging to a single value. The nodal evolution curves for uniform profiles (Figure 4-2) shows a completely irregular pattern with slopes changing both direction and magnitude. Also an estimation of the propagation speed as shown in Figure 4-4 depicts a larger variation in the wave propagation speed for uniform cases than the nearly steady speed observed for linearly sheared cases. 87

88 LNMO&PRQ S4TUWV4XEPYQ S4PRZW[ES4\ ]&TP_^` a b cedfehgjirkmlnceop_qsrtvu KMLN&OQP R4STVU4WDOXP R4OQYVZDR4[ \&SO^]_ ` _ adbdcfehgqikjml n`^oqprts! #"%$& %')(*,+ -./243! #"%$& %')(*,+ -./ A BCD$E ABC$D %')(EGFH-.I/J ;:=<?> !:<;>= Figure 4-: Contour plots depicting the evolution of the nodes in riser VIV for two linearly sheared flow profiles; Top left: CF displacements for dataset 236 (Umax =.8ms ); bottom left: IL displacements for dataset 236 (Umax =.8ms ); top right: CF displacements for dataset 242 (Umax =.4ms ); bottom right: IL displacements for dataset 242 (Umax =.4ms ). 88

89 KML&NOP Q4R S&Q4P TDUV WX Y Z YD[]\_^M`badceYDfgihkjdlnm KML&NOP Q4R S&Q4P TDUV WXDYZX [D\^]`_MaZbdceY fgihkjdlnm! #"%$& %')(*,+ -./243! #"%$& %')(*,+ -./ ABC$D ABC$D %')(EGFH-.I/J !:<;>= !:<;>= Figure 4-2: Contour plots depicting the evolution of the nodes in riser VIV for two uniform flow profiles; Top left: CF displacements for dataset 26 (Umax =.8ms ); bottom left: IL displacements for dataset 26 (Umax =.8ms ); top right: CF displacements for dataset 22 (Umax =.4ms ); bottom right: IL displacements for dataset 22 (Umax =.4ms ). 89

90 Figure 4-3: Observed CF wave propagation speed for different linearly sheared velocity profiles Figure 4-4: Observed CF wave propagation speed for different uniform velocity profiles Traveling waves in third harmonic of riser VIV response We have presented conclusive evidence of traveling waves in riser VIV response. Another question which seeks an answer is the presence/absence of traveling waves in the third harmonic component of the riser response. Observing the third harmonic component of riser response is complicated due to the requirement of a relatively large number of sensors (3 times the number required for the harmonic part) during reconstruction (refer Section 3.4.). As a result, we focus on the low velocity cases of NDP data, where this criterion is met. Specifically we would expect the presence of traveling waves for a linearly sheared profile. Figure 4-5 depict the nodal evolution curves obtained for two such NDP datasets (23 and 234). As mentioned previously, the nodal evolution curves of the harmonic component clearly depicts the presence of waves which travel with a consistent direction. The nodal evolution curves of the third harmonic component on the other hand depict the presence of traveling waves (sloping curves), and also depict a standing wave behavior. In addition, the third harmonic component of response also depicts no consistent direction of travel. 9

91 KLNMO PRQTSVU! "#!$&%'"# )(+*-,/. 23 LMONP QSRUTWV! "#!$&%'"# )(+*-,/ : "#!$&%'"# )(+*-;/ : "#!$&%'"# )(+*-;/ <+=?>A@BDC@EFGIHCJ <>=+?A@CBED@FGHJID/K Figure 4-5: Left: nodal evolution curves for the harmonic and third harmonic components of riser CF displacements for the NDP dataset 23; right: nodal evolution curves for the harmonic and third harmonic component for the NDP dataset 234. The third harmonic component of the response clearly depict the presence of traveling waves, but not necessarily the same consistency which the harmonic component portray A discussion on traveling waves in riser VIV response Vortex-induced vibrations are generated as a result of flow instability. They are not free vibrations, but forced vibrations generated as a result of vortex shedding. These vortex-induced force vibrations are self-limited in nature. The vibrations of an elastically mounted rigid cylinder are self-limited by reaching a lift force in phase with velocity which is very close to zero (or C lv ). The vibrations of a riser, on the other hand, are more complicated. The self-limited response can be attributed either to the fact that the lift force in phase with velocity reaches zero (C lv ), or that the lift force in phase with velocity is positive over a part of the riser (C lv > ) where energy is input from the fluid to the riser, and the energy is transported away to points where the energy is dissipated to the fluid (C lv < ). The traveling waves were found to travel at nearly uniform speed over the riser length for various linearly sheared profiles, although this may not be true in other velocity profiles. 9

92 4.5 Concluding remarks In this chapter, we developed methods to study the evolution of vortex-induced response in marine risers. For this, we first used scalograms to identify patches of experimental measurements which are statistically stationary. Next, methods to evaluate the peak response modes of a riser from experimental data were presented. This was followed by three methods to identify the presence of traveling waves. These methods were then applied to NDP data and some important characteristics of these traveling waves were obtained. Even when the response of the riser is at a single frequency, the peak response modes (spatial patterns) do not resemble the free vibration modes. This is primarily because of the variability in the effective added mass and hydrodynamic damping due to wake capture effects during vortex shedding. These peak response modes are complex functions and for several linearly sheared flow profiles depict traveling waves indicated by the presence of linear phase variation (non-zero slope) and the absence of nodes in the modal magnitude. Linearly sheared flow profiles are almost always characterized by the presence of waves in CF motion which travel with nearly a consistent velocity and direction over the entire riser. Uniform flow profiles, are on the other hand, characterized by the absence of traveling waves or the presence of waves which travel only over a short span of the riser. In addition, for uniform profiles there is no consistent direction of travel, and increased variability in the wave propagation speed. The conclusive evidence of traveling waves and their dominant role in riser VIV response especially under sheared flows present a paradigm shift needed in riser VIV prediction. The programs developed to predict riser VIV should capture both traveling wave and standing wave phenomena. 92

93 Chapter 5 Optimal lift force coefficient databases from riser experiments 5. Introduction Codes for predicting the riser vortex-induced vibration (VIV) often subdivides this complicated fluid-structure interaction problem into two simply stated problems: a hydrodynamics problem which quantifies the action of the fluid on the structure, and secondly a structural dynamics problem, which predicts the response of the structure given excitation from the fluid. The structural dynamics problem is more straight forward and can be linearized since the riser motions are of the order of the diameter [5]. The problem is made rich due to the challenging hydrodynamics, where the fluid dynamics behind the wake results in a net force from the fluid to the riser (at excitation region), or force from the riser to the fluid (damping region), or even forces which travel along the riser due to the formation of vortex streets sheared off from the riser [78, 25, 24, 6]. This makes the hydrodynamics problem nonlinear (excitation dependent on response), requiring one to go back and forth between these subproblems. The hydrodynamics problem involves predicting the harmonic part of the vortex-induced forces f fluid (z, t) on a section of the flexible cylinder corresponding to a flow velocity profile U(z) and fluid density ρ f. This boils down to estimating the two components of the fluid force; the excitation force in phase with velocity, and the excitation force in phase with acceleration [4, 5, 59]. Assuming a quasi-uniform approach, the excitation force in phase with velocity and the excitation force in phase with acceleration can be written in terms of the empirical rigid cylinder lift coefficient in phase with velocity C lv and the added mass coefficient C m respectively [5, 24]. Both C lv and C m are functions of non-dimensional frequency of oscillation (V r = 2πD and amplitude ratio (A = A D ) as shown in Figure 5-. Thus accurately obtaining the fluid forcing for a velocity profile U(z) requires accurate C lv (V r, A ) and C m (V r, A ) lift force coefficient databases. 93 ωu ),

94 5.. A summary of related research Standard codes for predicting VIV response (e.g.: VIVA [6, 59], SHEAR7 [66, 6], VI- VANA [27]) uses a strip-theory based (quasi-uniform) approach which employs this C m and C lv databases to estimate the excitation force from fluid to cylinder [57]. These C lv and C m databases were obtained by Gopalkrishnan, Smogeli et al. [4, 53] from simplified experiments at laboratory (MIT towing tank) where a cylinder is forced to oscillate at prescribed trajectories at select Reynold s numbers for specific frequencies [8, 4, 65] Figure 5-: The extensive C m and C lv database obtained by Gopalkrishnan [4]. The databases were obtained by Gopalkrishnan (referred as the nominal databases) under certain conditions: ) databases are available for limited Reynold s number regimes; 2) cylinders undergo simplified motion tests at a single frequency; 3) cylinder motions are restricted to crossflow (CF) direction instead of both cross-flow and in-line (IL) motions. It has been observed from various field experiments (Smogeli et al. [53]) that the C m and C lv databases for increased Reynold s number will preserve all the major features (peaks and valleys) in the database, but these features are found to move and scale (a warping transformation). In a similar manner, allowing IL motions have the effects of adding another peak in the C lv database (Marcollo and Hinwood [36]). However, these databases obtained by Gopalkrishnan are extensive, and it is impossible to obtain similar databases for a variety of experiments conducted in the field. Due to the above said limitations, the response predicted by the empirical programs (VIVA, SHEAR7, VIVANA) differ significantly from observations from the field experiments. To overcome these limitations, there is a need to develop a method which can extract information from more realistic experiments (e.g.: NDP experiment, Lake Seneca test, Gulf stream test [5, 7, 7]) and build on the presently available extensive databases of C m (V r, A ) and C lv (V r, A ). The method we develop should be such that we should be able to understand and preserve the key features of the C m (V r, A ) and C lv (V r, A ) databases important to accurately predicting the 94

95 VIV response of the riser. 5.2 Problem statement and solution overview We will assume that the the model used for predicting the response is accurate, while the ambiguity in the prediction and measurements arise from the uncertainty in the empirical databases alone. We aim at systematically modifying the nominal lift coefficient databases of C m (V r, A ) and C lv (V r, A ), to take into account the effects of ) higher Reynold s number regimes; 2) effect of allowing IL motions on the CF motions. The modifications of the databases are to be performed so that predictions from a semi-empirical program like VIVA using the modified databases match the experimentally observed results Working hypothesis The working hypothesis for correcting the databases is based on the observations of various researchers mentioned in Section 5.. [53, 36, 73]. Based on these observations we assume that we can obtain modified databases which will account for the discrepancy between the measurements and the predictions. The modifications we propose are: Correction for high Reynold s number warping of the nominal databases Correction for allowing in-line motions adding another peak in the C lv database Solution overview Appropriately parameterizing the relevant features of the databases using a chosen set of parameters p will allow these databases to be flexible (allowing for warping and second peak). The flexible databases hence created can absorb information from the experimental data. The theoretical predictions from these modified C m and C lv databases will be compared with the experimentally observed riser response. As a first step we will specifically identify (define) the experimental measurements which will be compared with the theoretical estimates. This will be followed by understanding the key features of the C m and C lv databases which determine the theoretical estimates. Next, we will pose the problem as an optimization problem, where the C m and C lv databases are parameterized using a chosen set of parameters p. The optimization problem requires an optimization index or error metric which will be defined. The choice of the appropriate solution technique will be discussed, which will be followed by a method to obtain the universal C lv and C m databases which will minimize the error over a family of NDP datasets. 95

96 5.3 Experimental measurements and theoretical estimates For a simple single degree of freedom (SDOF) system like an elastically mounted rigid cylinder, the input is the flow velocity U and the observables are the response amplitude A and the Optimal force coefficient database correction response frequency ω. Refer Table 5.. In a similar fashion, a distributed system like a riser has as input a steady velocity profile U(z) in the in-line direction, while the observables are a set of A comparison of SDOF & MDOF systems & Problem statement excited peak response frequencies ω n and the corresponding set of peak response modes Y n (z).! Typical input and output of a VIV problem So while formulating a comparison of the riser CF response both ω n and Y n (z) are important. Input Output SDOF System (elastically mounted rigid cylinder) U ω A Distributed System (riser) U (z) ω n Y n (z) Table 5.: A comparison of the typical input and observables of a VIV problem.! Prediction of riser VIV requires empirically obtained force coefficients of C m and C lv! Due to various assumptions, the theoretical estimates differ from observations! 5.3. Modifying Experimental force coefficient measurements databases in a systematic way such that: A detailed description of the theory and methodology for obtaining the peak response frequencies ωtheoretical n exp and peak estimates response modes match Y n exp (z) from Experiments experimental data is provided in Section 4.3. Here, we provide a only a brief description of the procedure, as illustrated by the ω n-th & Y n-th (z) ω n-exp & Y n-exp (z) flow chart given in Figure 5-2. Given experimental data from sensors located along the riser, we identify a statistically stationary segment of experimental data using scalograms. A response reconstruction is then performed to obtain the displacement time series at any point along the riser y(z, t). Next, we compute the Fourier transform of the time signals at every location along the span to obtain the Fourier coefficients ŷ(z, ω). The set of peak response frequencies ω n exp are obtained from the peaks of a span averaged spectrum of the response. 25 Finally, the corresponding peak response modes Y n exp (z) are obtained by extracting the Fourier coefficients corresponding to each of the peak frequencies along the entire riser. Choose statistically stationary segment of experimental data y( z, t) s Response reconstruction to obtain displacement at any location y( z, t) Compute Fourier transform ) y( z, ω) = t F { y( z, )} Pick peak response frequencies ω n Obtain peak response modes ) Y z) = y( z, ω ) n( n Figure 5-2: A flow chart describing our method to obtain the peak response frequencies and the peak response modes from experimental data. 96

97 5.3.2 Theoretical estimates Detailed derivation for obtaining the theoretical estimates of the peak response frequencies ω n th and the peak response modes Y n th (z) can be found in Chapter 2. In short, given the velocity profile U(z), and the lift coefficient databases of C m and C lv the theoretical estimates of peak response frequencies ω n th and peak response modes Y n th (z) are obtained by solving the set of two integral equations. These integral equations were obtained by grouping the real and imaginary parts of the weak form of the nonlinear eigenvalue problem mentioned in equation (2.7): L z= T dy dz 2 + EI d 2 Y dz 2 L z= 2 dz = ω 2 bω Y 2 dz = L z= L z= [m + C m ρ f πd 2 4 ] Y 2 dz (5.) ρ f U 2 D C lv Y dz (5.2) 2 The empirical prediction program VIVA [59] can solve the system of equations (5.) and (5.2), and we use it to evaluate the theoretical estimates Y n th (z) and ω n th. 5.4 Formulation as an optimization problem The systematic modification of the C lv and C m (to allow for warping and second peak) proposed in the earlier section can be performed by first parameterizing the relevant features of databases using a chosen set of parameters p. Thus the C lv and C m databases are now functions of the parameters p. Allowing these parameters to be modified results in the databases to be ( ) ( flexible C Y (z) lv D, U(z) ωd, p and C Y (z) m D, U(z) ωd )., p This flexible database hence created can absorb the information from experiments. More details on parameterizing the databases are given in Section We aim at obtaining a systematic method for varying this set of parameters p such that the prediction using this modified databases (functions of the parameters) will match the experiments. This problem can be viewed as a standard inversion problem of parameter estimation where a nonlinear forward model given in equation (5.3 and 5.4 ) is to be fitted on to experimental data of the form Y n exp (z; U(z)) and ω n exp (U(z)). The optimal choice of parameters also require an appropriate error metric J( p) and an appropriate solution technique which will minimize J( p) during the solution process. The solution to such a procedure is a new optimal set of parameters p opt which will minimize the error metric. 97

98 5.4. Parameterization of lift coefficient databases The proposed warping transformation and adding the second peak requires one to first represent the C lv and C m databases using a chosen set of parameters p. The parameterization should be such that the features of the databases relevant to the prediction of riser VIV are made flexible using the least number of parameters p. Numerous parameterizations of the C lv and C m databases were considered. These include representing the C lv and C m databases in an analytical form, representing C lv and C m as Bezier surfaces with the control points as the parameters, representing C lv and C m using a set of B-spline basis functions, representing C lv and C m as a meshed surface with each data point as a parameter and several combinations of these. All the methods suffers from some or several limitations, the most common being the required number of parameters and the required amount of flexibility. The eventual choice of parameterization is described below. C lv database parameterization Based on the observations from a careful study of the SDOF system, the most important feature of the C lv database was found to be the C lv = contour (all the free vibration rigid cylinder results align along this curve). Hence any parameterization we consider should be such Parameterization of C lv Database that, this contour can be made flexible using the chosen parameterization of C lv. Chosen Gopalkrishnan C lv Database Parameters Parameters for Correcting for Inline Motion p5 p8 p p2 p3 p4 p6= max(clv) p7 p9= max(clv_add) Figure 5-3: Parameterization of the C lv database. Thus the C lv database is parameterized using 9 parameters which reflect these important features. The specific definition of each of the parameters are produced in Table 5.4. and are also illustrated in Figure 5-3. The first 6 parameters (p to p 6 ) can describe the warping transformation of the nominal C lv database, and allows for corrections due to Reynold s number effects. The parameters p 7, p 8 and p 9 allows us to specify the location of a second peak (depicted 98

99 as a bi-normal distribution) in the database and allows compensating for the motions in the inline direction. This modification of the nominal C lv database to a new C Y (z) ( ) lv D, U(z) ωd, p is obtained after making the appropriate transformations in the V r A domain. p intersection of C lv = contour and the V r axis corresponding to the lowest V r. p 2 intersection of C lv = contour and the V r axis corresponding to the second lowest V r. p 3 intersection of C lv = contour and the V r axis corresponding to the third lowest V r. p 4 intersection of C lv = contour and the V r axis corresponding to the fourth lowest V r. p 5 maximum A value taken by the C lv = contour. p 6 maximum value of the C lv database. p 7 V r location of the second peak added to compensate for in-line motion. p 8 A location of the second peak added to compensate for in-line motion. maximum value of the second peak added to compensate for in-line motion. p 9 Table 5.2: Definition of the parameters p to p 9 corresponding to the C lv database. Please refer to Figure 5-3 for a depiction of these parameters. The nominal values of the C lv parameters (represented by p nom ) are given in Table 5.3. Thus using the nominal values of C lv parameters ( p nom ), will result in the nominal lift coefficient database of C lvn. Thus corresponding to a modified parameter-set p, we obtain the modified ( database of C Y (z) lv D, U(z) ωd )., p Figure 5-5(left) shows an example where the nominal C lvn database and the modified C lv database obtained when only one of the parameters p 3 is modified from its nominal value of p 3 nom = 5.4 to a modified value of p 3 = 7.2. p nom p 2 nom p 3 nom p 4 nom p 5 nom p 6 nom p 7 nom p 8 nom p 9 nom Table 5.3: Nominal values of the parameters used for representing C lv database. C m database parameterization The major features important to predicting the VIV response in the C m database are the sudden transition of C m corresponding to the Strouhal frequency, the peak C m value and the asymptotic value attained by the C m for high flow velocities. Thus the C m database is parameterized using 3 parameters (p to p 2 ) and their definitions can be found in Table The parameters are also illustrated in Figure 5-4. p The mean value of V r corresponding to the C m = contour. p The averaged constant value which C m takes for large values of V r. p 2 The maximum value of the C m database. Table 5.4: Definition of the parameters p to p 2 corresponding to the C m database. The nominal values of the C m parameters (represented by p nom ) are given in Table 5.5. Similar to the C lvn, using the nominal values of parameters ( p nom ), will result in the nominal 99

100 Parameterization of C m Database Chosen Gopalkrishnan C m Database Parameters p2= max(cm) p p = avg(cm) Figure 5-4: Parameterization of the C m database. p nom p nom p 2 nom Table 5.5: Nominal values of the parameters used for representing C m database. C mn database. Again, variations of the parameters from p nom to another arbitrary value p will ( result in a modified database C Y (z) m D, U(z) ωd )., p Once again, this modification of the nominal ( ) C mn database to the new C Y (z) m D, U(z) ωd, p database is obtained after making the appropriate transformations in the V r A domain. Figure 5-5(right) depicts an example where the nominal database C mn and the modified database obtained when only one of the parameters p is modified from its nominal value of p nom = 6.2 to a modified value of p = Nonlinear forward model ( ) ( ) This parametrized C Y (z) lv D, U(z) ωd, p and C Y (z) m D, U(z) ωd, p are used to obtain the theoretical estimate of the peak response mode Y n th (z, p) and the peak response frequency ω n th ( p) given the velocity profile U(z) and a set of parameters p using the two equations as: L z= T dy dz 2 + EI d 2 Y dz 2 L z= 2 dz = ω 2 bω Y 2 dz = L z= L z= [ ( Y (z) m + C m D, U(z) ωd, p ( Y (z) C lv D, U(z) ωd, p ) ρf πd 2 4 ] Y 2 dz, (5.3) ) ρf U 2 D Y dz. (5.4) 2

101 Figure 5-5: Left top: nominal C lv database corresponding to the nominal value of parameter p 3 ; right top: nominal C m database corresponding to the nominal value of parameter p ; left bottom: example of a modified C lv database by varying the parameter p 3 from its nominal value 5.4 to 7.3; right bottom: example of a modified C lv database by varying the parameter p from its nominal value 6.2 to 8.2. An iterative scheme VIVA [57, 6, 59] is employed to obtain these peak response modes Y n th (z, p) and the peak response frequencies ω n th ( p). The above system of equations to predict ω n th ( p) and Y n th (z, p) is called the forward model. The forward model and the parameterization needs to be appropriately chosen so that the theoretical model is able to represent the experimental measurements Error metric (Optimization index) We have to take into account, the error in both the peak response frequency ω n and the peak response mode Y n (z). To reflect this error in frequency and span varying amplitude we choose the following optimization index or error metric as: J( p) = ϖ RMS { Y n th (z, p) Y n exp (z) } + ω n th ( p) ω n exp. (5.5) where, ϖ is a factor which allows one to weigh the relative importance of reducing the error in peak response frequency or the error in peak response modes. An alternate choice of the error metric was explored and takes into account the normalized modal magnitudes as: { J( p) = RMS Y n th (z, p) max z { Y n th (z, p) } } Y n exp (z) max z { Y n exp (z) } + ϖ 2 ω n th ( p) ω n exp. (5.6)

102 where, ϖ 2 is a factor which allows one to weigh the relative importance of the peak response frequency and peak response modes Choice of solution technique and computational framework PC- PC-2 CPU Intel Pentium 4 (2 3GHz each) Intel Xeon (4 2.4GHz each) RAM GB 3GB OS MS Windows XP MS Windows Vista Table 5.6: Two desktop PCs with the above specifications were employed to run the optimization codes. The solution technique should be chosen taking into account the conflicting demands of accuracy and speed. Table 5.6 illustrates the computational resources available to us. A single evaluation of the optimization index J( p) requires close to 3 seconds of computational time. Thus in 24 hours approximately 29 evaluations of the error metric is possible. The large number of parameters in p (2 in number) renders grid search or random search methods computationally impossible. Our choice of error metric J( p) is highly nonlinear and involves several heuristics making it discontinuous. As a result, gradient based or quasi Newton methods will not work. A variant of simulated annealing method which is a directed random search method was chosen due to its simplicity and speed [37]. This algorithm is depicted as a flow chart in Figure 5-6. The input to a simulated annealing algorithm is a range for each parameter within which the solution is expected to lie, and the output is the parameter-set p opt which minimizes the error metric over the parameter range. Overall computational framework Matlab programming environment was employed for the overall scheme due to the ease of programming, in built libraries and visualization routines. The first major part of the overall scheme was to develop the simulated annealing code. The simulated annealing code requires several evaluation of the error metric J( p) corresponding to various parameter-sets p as mentioned in Figure 5-6. ( ) Another set of Matlab code was developed to obtain the modified C Y (z) m D, U(z) ωd, p and ( ) C Y (z) lv D, U(z) ωd, p databases corresponding to a given a parameter-set p. The steps involved in evaluating one instance of the optimization index is given as a flow chart depicted in Figure 5-7. Each evaluation of the optimization index involves invoking VIVA for which the modified 2

103 Input: Parameter range ( PR ) Rate of cooling ( ) r cool Obtain: r random p within PR r J = J ( p ) Update: i = i +, T = T i i att =, r i- cool i, succ = Evaluate: PRi = r r r pi = pi + r J = J p i ( ) i cool PR i [ PR PR ] ( PR )[ rand. 5] i = r cool i i Ji Ji rand < exp Ti Ji Yes No No p r soln Yes If: J i < J tol Yes i i att succ No < N & < N att succ i i r p att succ soln = i att = i r = p succ + + i i r p att succ soln = i att = i r = p succ soln + Figure 5-6: Flow chart describing the simulated annealing algorithm to obtain the optimal parameter-set p opt. Experimental measurements Theoretical estimates New parameter set p r C lvn Nominal databases * * r ( Vr, A ), Clm ( Vr, A ), pnom N Experimental data ω Peak response frequency n exp Peak response modes Y exp (z) n Modification method C Modified databases * r * r ( V, A, p) & C ( V, A, p) lv r Forward model (using VIVA) r Peak response frequency ωn th( p) r Peak response modes Y ( z, p) m r n th Optimization index J(p) r Figure 5-7: Flow chart illustrating a single evaluation of the optimization index. ( ) ( ) C Y (z) m D, U(z) ωd, p and C Y (z) lv D, U(z) ωd, p databases are first written on to a file in the format required by VIVA. VIVA is then called from within the Matlab code. The output from VIVA (ω n th ( p) and Y n th (z, p)) is then read by the Matlab code and the corresponding error metric J( p) is evaluated. VIVA together with the Matlab code provide a framework for systematically varying the modified lift coefficient databases as required by the simulated annealing 3

104 +! ' +! ' * ), - * ), -,, ;< B, B+ *, ' * - : ' ;< B, B+ *, * ' - : & & methodology. The developed algorithm was thoroughly tested using benchmark datasets to verify its speed and accuracy. The same dataset used for benchmarking the response reconstruction (mentioned in Section 3.5.) was used to benchmark the optimal data assimilation algorithm also. The benchmark dataset was constructed using one of the VIVA modes. This VIVA mode was generated using a modified lift coefficient database with the values of parameters known a priori. x 3 * ) ( ' & %$ $ $# "!,! ', * ) ( ',! ', *, + - & ; A@? = > ;< + ; D C = > & ; ; D C = > ;< E ; A@? = > ; ;<. + *, + - & ; A@? = > ;< + ; D C = > & ; ; D C = > ;< E ; A@? = > ; ;< / / Figure 5-8: Left: a comparison of the experimental and nominal theoretical prediction; right: a comparison of the experimental and optimal theoretical prediction. As expected, using the nominal databases for prediction results in significant difference between the experimental and theoretical peak response frequencies and peak response modes as depicted in Figure 5-8(left). This is a condition very similar to the one encountered in real life. The optimal C lvopt and C mopt databases are obtained using the simulated annealing methodology and correspond to a parameter-set p opt. The results are reproduced in Figure 5-8 where the left hand half provides the comparison between the experimental data and the prediction using the nominal databases. The right hand half of Figure 5-8 provides the comparison between the experimental data and the prediction using the optimal databases. We note that the prediction using the optimal databases (corresponding to p opt ) closely match the experimental data which verifies the accuracy of our algorithm. The small differences are attributed to our choice of error metric (and weighing factor ϖ) and the complicated intermediate data processing. The shaded region represents the error in the normalized modal magnitude. The output from such a code is an optimal parameter-set p opt for each of the NDP datasets we may consider. In addition, we can also gather parameter-sets where the error metric is slightly bigger than J( p opt ). The method we developed is now ready to be applied to NDP experimental data. We 4

105 apply the methodology specifically to NDP experimental data involving linearly sheared velocity profiles depicting a single peak response frequency. Figure 5-9 depicts a typical comparison between the experimental data and the prediction using the nominal databases (left side) and optimal databases (right side) for NDP dataset 243. We can repeat similar procedure for all the NDP datasets under consideration and consequently obtain the optimal databases for each such dataset. INITIAL COMPARISON (datacase : 243) Span averaged displacement spectrum FINAL COMPARISON Span averaged displacement spectrum Frequency [Hz] Frequency[Hz] Modal magnitude Modal magnitude z [m] z [m] * * C ( V, A ) C ( V, A ) lv r m r ϖ Figure 5-9: Application of the optimization method to NDP dataset 243. Left: comparison of the experimental and nominal theoretical prediction; right: comparison of the optimal and experimental prediction. Contour plots below depict the corresponding C lv and C m databases. 5.5 Universal C lv and C m databases As mentioned in the previous section, each NDP dataset yields an optimal parameter-set p opt (or optimal databases). Having one different optimal database for each NDP dataset (corresponding to each velocity profile) is inconvenient. In addition, the optimal parameter-set for one NDP dataset may not be the optimal parameter-set for the other NDP datasets. This could be due to the minor but non negligible data processing errors, limitations of empirical models, or due to the internal working of VIVA. What is required is a C lv and C m database which is 5

106 universal in the sense that it minimizes error (in frequency and modal magnitude) for all the NDP datasets under consideration. We call such databases the universal database and are denoted as C lvuniv and C muniv. The following sections will describe how we obtain the universal databases Obtaining candidate databases (candidate parameter-sets) First we consider a few selected datasets from the NDP experiments. For each such dataset, we run the simulated annealing (optimization) algorithm several times. Again for each such run, in addition to the optimal parameter-set we also collect the parameter-sets where the error metric J( p) is below a certain percentage of the optimal error metric J( p opt ) as: J( p cand ) [ + γ] J( p opt ), (5.7) where, γ (we use γ 8%) is a factor which controls the number of candidate parameter-sets isolated from each optimization run. Thus for each NDP dataset we have several candidates from various runs of the simulated annealing algorithm. This procedure is repeated now for each of the chosen NDP datasets. The several runs of the simulated annealing algorithm on each NDP dataset increases the probability of finding the best solution. In practice nearly 8 to runs are performed on the same dataset. A specific candidate parameter-set is henceforth denoted by p candj, and around 2 such candidate parameter-sets were isolated Obtaining the best candidate parameter-set Corresponding to each candidate parameter-set p candj, we have the associated candidate ( ) ( databases C Y (z) lv D, U(z) ωd, p cand j and C Y (z) m D, U(z) ωd, p cand j ). Using each of the candidate databases, we evaluate the optimization index for each NDP dataset dataset i and we denote it by J dataset i ( p candj ). Figure 5- illustrates the result from the use of a typical candidate parameter-set to evaluate the error in frequency and the error in modal magnitude for all the considered NDP datasets. Similar figures can be obtained for each candidate parameter-set p candj. To obtain the candidate which minimizes the error for all the datasets we establish the universal error metric: J univ ( p candj ) = J dataset ( p candj ) + J dataset 2 ( p candj ) J dataset i ( p candj ) (5.8) The candidate database which minimizes the universal error metric J univ ( p candj ) is chosen as the universal database (or universal parameter-set p univ ). That is: p univ = argmin { Juniv ( p candj ) }. (5.9) p candj 6

107 &! ( * & # 6 uv w\xzy{@ v {@}N{@y~/w\~/}N yxzˆ@{ Kv {@}N{@y~/w\~/}N _ ƒ _@t _%ƒ _ b Q@ƒ b %ƒ t t Q ƒ t@p _ ƒzq@w _%ƒ _ _ _%ƒ _ _ _%ƒ _ _ _ ƒ _@_ _ ƒ _@_ _ ƒ _@_ "!$#% & "! #% ')( * ( +,*-(. #%! +/( )2435! #%! B qed F GIHKJ/LNMPORQTSVl@l XYGZHKL\[N]^Ò _asvp@b XcEdfe^ghJ/L\MfOi_jSV_ _@s Xkc dpe^ghln[\]oi_jsv_ _@b 6 798;: <%= 798;7>@? 6 798;> =%A B qeq F GIHKJ/LNMPORQTSnW/b XYGZHKL\[N]^Ò _asvs@s XcEdfe^ghJ/L\MfOi_jSV_ _%W Xkc dpe^ghln[\]oi_jsv_ _@s B qec F GIHKJ/LNMPORQTSVp%W XYGZHKL\[N]^OoQTSVp@p XcEdfe^ghJ/L\MfOi_jSV_ Q/p Xkc dpe^ghln[\]oi_jsv_ _@l B qeb F GIH J/LNM Or_aSnW/U XYGZH L\[N] Ò _asvt%xcedfe^g J/LNM Or_aSV_%Q Q@X%cEdfe^g L\[N] Or_aSV_@_ l B qef F GIHKJ/LNMPORQTSVb@p XYGZHKL\[N]^OoQTSVb@l XcEdfe^ghJ/L\MfOi_jSV_ Q@X%cEdfe^ghL\[N]Or_aSV_@_ s B CEm F GIHKJ/LNMPORQTSnQ/b XYGZHKL\[N]^OoQTSV_@b XcEdfe^ghJ/L\MfOi_jSV_ Q/p Xkc dpe^ghln[\]oi_jsv_ _%W B CED F GIHKJ/LNMPORQTSVU%W XYGZHKL\[N]^Ò _asvb%q XcEdfe^ghJ/L\MfOi_jSV_ _@U Xkc dpe^ghln[\]oi_jsv_ _@l Figure 5-: The predictions ( of Y n (z) ) and ω n using ( nominal databases ) (red dashed) and candidate databases C Y (z) lv D, U(z) ωd, p cand j and C Y (z) m D, U(z) ωd, p cand j (green) are compared with the experimentally observed ones (blue) for various datasets from the NDP experiments. Obtaining the parameter-set p univ corresponding to the universal lift coefficient databases is illustrated in the flow chart given in Figure 5-. Figure 5-2(left) depicts the J dataset i ( p candj ) evaluated for each NDP dataset corresponding to all the candidate parameter-sets p candj that were isolated. Consequently we obtain the universal error metric J univ ( p candj ) from equation (5.8) for each of the identified candidate parameter-sets as shown in Figure 5-2(right). Care should be maintained to eliminate certain physically meaningless databases (parameter-sets) from our collection of parameter-sets. Finally, the universal databases C lvuniv and C muniv which minimizes the J univ ( p candj ) is obtained. 7

108 % % ' $ & : < = : > < = : < < % % ' $ & = ; : < = : : < 3 2 /. = = : < =?;: = >;: = <;: Candidate no. p r cand Candidate no. 2 p r cand 2 Candidate no. N p r cand N r J univ ( p cand ) r J univ ( p cand 2 ) r ( ) J univ p cand N r p univ Universal database r = arg min J ( p r pcand j { ) } univ cand j Figure 5-: Flow chart illustrating the process of obtaining the universal parameter-set p univ corresponding to the universal C lvuniv and C muniv databases. # "! DA ; =CB A!# +, ") ' ( : : D B > +, ) +, *) - +, *) ' ( Figure 5-2: Left: a comparison of the optimization index J dataset i ( p candj ) for all the candidates isolated; right: depicts the universal error metric J univ ( p candj ) for all the candidates isolated. Note that the use of universal databases result in a 44% reduction in J univ ( p candj ). 8

109 5.5.3 A discussion on the universal C m database The universal added-mass coefficient database C muniv is depicted in Figure 5-4. One important observation is that the characteristic jump in the C m database has a shift in the reduced frequency (/V r ). This shift is found to be.295, and remains stable for several runs of the optimization algorithm. As a result, the prediction of the oscillation frequency of the flexible cylinder is expected to be slightly smaller than the predictions using the nominal databases (refer Figure 5-7(left)). This observation is in close agreement with the high mode number experiments conducted by Vandiver et al. [72] where a reduced frequency in the range of.52 to.82 is observed. In addition, as seen from Figure 5-3, this apparent reduction in reduced frequency can also be understood from the dependence of Strouhal number on Reynold s number. The Gopalkrishnan experiments were performed for Reynold s number corresponding to, where as the present study is based on NDP experiments were performed for a maximum Reynold s number variation between 6 to 46. For Reynold s number within the range 6 to 46, the corresponding Strouhal frequency is lower than the Strouhal frequency at Reynold s number equal to. Thus the observations from the universal databases are in agreement with the presently available experimental observations. Other modifications includes an increase in the maximum value taken by C m and the width of the peak in the C m database as shown in Figure 5-4. C m is found to produce a significant impact on both the peak response frequency ω n and also the peak response modal magnitude Y n (z). The importance of C m on ω n can easily be deduced from equation (5.3). The impact of C m on the modal magnitude arises due to the dispersion relation connecting the response frequency ω n and the wavenumber k n. The dispersion relation contains the C m term in addition to linear density m and stiffness terms T and EI. Figure 5-3: Strouhal number variation as a function of the Reynold s number, also depicts the Reynold s number corresponding to the Gopalkrishnan database (Reynold s number = ) and the present study (maximum value of Reynold s number in the range 6 to 46). For the range of Reynold s number in the present study, the Strouhal number is expected to be lower than that for Reynold s number around. Figure is adapted from [2, 3, 48, 58]. 9

110 Figure 5-4: Depicts the universal Cmuniv database obtained from the optimization code in conjunction with VIVA. A total of nine datasets from NDP experiments (for linearly sheared flow profiles) were used. The maximum value of Reynold s number lies between 6 to 46.

111 Figure 5-5: Depicts the universal database obtained from the optimization code in conjunction with VIVA. A total of nine Clvuniv datasets from NDP experiments (for linearly sheared flow profiles) were used. The maximum value of Reynold s number lies between 6 to 46.

112 5.5.4 A discussion on the universal C lv database The universal lift coefficient in phase with velocity database C lvuniv is depicted in Figure 5-5. As mentioned earlier, the C lv = contour is an important feature of the C lv database. The two resonant lobes (primary and secondary excitation regions) observed in the nominal C lvn C lvuniv database is seen to merge together to create one large region of C lv > in the universal database. In addition, the area of the resonant peaks seem to have spread out over a larger region of the V r axis. This may be explained as follows: the VIV of flexible cylinders are fundamentally different from the VIV of elastically mounted rigid cylinders, where some parts of the riser may encounter oscillations forced from other locations. This phenomenon could result in a C lv database which may be much less sensitive to the sharp valley separating the two resonant peaks. Thus, it is not necessary that the span varying amplitude ratio of riser VIV lie close to the C lv = contour line and may result in some averaging resulting from vibrations forced from other locations along the riser..2!"#$ #&%('#)*+, -/ /. 23: #B"$#&%(CDEF G #, HH <; 2 4 Figure 5-6: VIV oscillation amplitude of an elastically mounted freely oscillating cylinder in uniform flow. Left: experiments conducted by Smogeli [53] for various values of reduced velocity V r ; right: experiments conducted by Vikestad [27] for various values of nominal reduced velocity V rn. Note that the amplitude ratio A takes values as high as.5. The maximum A value (A.24) taken by the C lv = contour is much higher than the value taken by the nominal C lv database of A.8. Similar observations may be seen from independent experiments performed by Vikestad [27] (refer Figure 5-6(right)) using a freely oscillating cylinder shows a significantly larger response with A reaching values as high as.5. The results for freely vibrating cylinders by Smogeli [53] (refer Figure 5-6(left)) once again depicts a similar trend of high A values. Another important feature of the C lv database is the slope of C lv (V r, A ) surface in the 2

113 A direction corresponding to the resonant frequency. This slope corresponds to the linearized hydrodynamic damping (referred to as b h ) in the empirical model. The universal C lv database has a slope (b h.32) which is lower than the slope in the Gopalkrishnan database of (b h.7). Thus the transition from C lv > to C lv < near the resonant frequency is found to be much less drastic than the observation by Gopalkrishnan <; A B=;>! #" $&%('*), / ,9;: <;C =;>? B=;> Figure 5-7: A snapshot of the results from the use of the universal database with VIVA (for linearly sheared flow profiles). The figure compares experimental with the theoretical prediction using nominal (+ sign) and universal ( sign) lift coefficient databases for various linearly sheared flow profile datasets from NDP experiments. The use of universal databases improves our estimate of both the peak response frequency and the peak response modal magnitude. The agreement between the experiments and the predictions using the universal databases result in 44% reduction in error A discussion on the optimization method and practical issues The method presented above addresses several concerns in the empirical databases, but these improvements are by no means complete. This section presents some of the valuable insights gained during the exercise to significantly reduce the error in the present prediction methods by improving the empirical lift coefficient databases. We have assumed that the empirical models for VIV prediction to be accurate, and that the uncertainty in predictions stems from the empirical databases of C lv and C m alone. However, several researchers [77, 9, 9] have observed that allowing IL motions have a significant impact on the CF motions. The ultimate solution seems to be a coupled analysis taking into account both CF and IL motions. Since such a model has not yet been evolved, our method presents the best alternative available to this date. As observed from various tests performed (refer 3

114 Figure 5-7), the universal databases C lvuniv and C muniv provides significant improvements in the prediction capability than the nominal databases C lvn and C mn. Ideally, we would like to obtain a universal parameter-set by minimizing J univ over the range of parameters. However in practice evaluating the J univ is prohibitive. Even if we consider only separate datasets from the NDP experiments, each evaluation of J univ require runs of VIVA and consequently times the time required to evaluate J. This would make the present algorithm extremely time consuming and hence impractical. Instead we perform a two step strategy of collecting a reduced set of candidate parameters and then choosing the best among them. This two step strategy is also advantageous in the sense that it allows parallelization by letting us employ several PCs to independently obtain the candidate parameter-sets. With improving computational abilities, it is expected that our optimization framework can be used in a wide variety of context including more complicated parameterization of the databases, improved empirical models, several datasets from a wide variety of environments, and faster and more accurate optimization algorithms. It is also expected that the framework developed in this thesis will also act as a touchstone for testing new VIV modeling techniques. Another point of interest is our choice of parameters to represent the C lv and C m databases. In fact, neither this parameterization nor our choice of parameters is unique. Of the various parameterizations considered, the parameterization used was found to best accommodate the conflicting requirements of minimum number of parameters and accurate representation of C lv and C m. Our choice of parameterization is based on accurately representing the features of C lv and C m databases which determine the free vibration solutions of an elastically mounted rigid cylinder. However, the VIV of flexible cylinders (risers) are fundamentally different from the VIV of elastically mounted rigid cylinders. It is not necessary that the span varying amplitude ratio of vortex-induced vibration of the riser lie close to the C lv = contour line. As a result the parameterization we have chosen may not be adequate in representing all the features of the riser VIV response. All of our analyses are based on linearly sheared cases of NDP data where the harmonic part of vortex-induced response shows only one peak frequency. Introduction of additional frequencies into the riser motions typically results in diffusion of energy between the participating frequencies and may result in cancellations [5]. In that sense, the use of universal databases may result in a more conservative estimate. Evolving a database extraction scheme for multi-frequency response (beating motion) is much more complicated. 5.6 Concluding remarks We have developed a systematic method to update the lift coefficient databases using experimental data from risers. The lift coefficient databases are represented in a flexible form, with 4

115 the information from extensive experiments previously conducted as the backbone. The relevant features of the databases are then made flexible using a set of carefully chosen parameters. An optimization scheme was then developed to obtain the best set of parameters such that the theoretical estimates closely follow the experimental observations. This scheme when applied to a family of datasets from NDP experiment allows obtaining the universal C lv and C m database which minimizes the error over each of the datasets. The use of the universal databases within VIVA results in significant improvements in predicting the riser VIV cross-flow response than the use of the nominal databases. This method will allow us to absorb information from more realistic field experiments, and is expected to bring in the effects of Reynold s number, and the effects of in-line motion in VIV prediction. The method was developed in such a way that no extensive change in the prediction program (e.g.: VIVA) user interface is required. Another application of the method is to produce calibrated in-situ fatigue monitoring in marine risers. The live VIV data measured by loggers placed on a riser is used to build a customized empirical lift coefficient database, which is subsequently used to produce accurate riser VIV fatigue predictions. This application is described in detail in Chapter 6. 5

116 6

117 Chapter 6 Response reconstruction from few sensor data 6. Introduction It is a common practice to monitor the health of marine risers using few sensors (loggers) placed along its length for extended periods of time. The measurements from these sensors may be used directly to predict the fatigue life of the riser at the sensor locations. However, we do not have information on how any arbitrary point along the riser may behave. In fact, the location of the stress hot-spots could be far away from the location of the sensors and depend on the profile of the current. Thus our estimates of fatigue life (at sensor locations) may be far off from the reality at the locations of these fatigue hot-spots. In order to obtain a good estimate of the riser fatigue life, we require a capability to obtain the response of the riser at each point along its length. The method to reconstruct the riser vortex-induced vibration (VIV) response when the available number of sensors are greater than the full reconstruction criterion is presented in Chapter 3. In this chapter we consider the problem of obtaining the riser VIV response, when the number of sensors making measurements are relatively low. For the scenario of few sensors, the sensors are not able to pickup all the information related to riser VIV. This missing information is filled by using our prior understanding of the riser VIV motions (riser peak response modes and peak response frequencies) for which we require a predictive capability. Since the predictive capabilities are limited by the accuracy of the lift coefficient databases, we need to solve an associated problem of extracting lift force coefficients using data from few sensors. This chapter presents a method to perform response reconstruction for the case of few sensors aided by the predictions based on VIVA. A local lift force coefficient database correction methodology is also developed to improve the predictive capability of VIVA, thereby allowing 7

118 improvements in the quality of the response reconstruction. 6.. Problem setting We are given a riser taut between z = and z = L (refer Figure 6-). The riser is instrumented with N s sensors placed along its length at locations z s. We may assume for simplicity that these sensors typically measure accelerations signals y tt (z s, t). The number of sensors N s available is much lesser than the criterion for full reconstruction mentioned in Section In addition, we are also provided with the following information on the riser and flow properties: Riser properties: The riser has a diameter D, length L, mass per unit length m, structural damping constant b, bending stiffness EI, and effective tension T. All these quantities may vary along the span of the riser. Flow properties: The riser encounters a steady current U(z) along the in-line (IL) direction. The fluid around it has a density ρ f, and we have a good estimate of the empirical lift force coefficient databases of C lv and C m. Figure 6-: Figure depicting the typical arrangement of N s sensors at locations z s on a riser. We are required to use the data measured by the sensors y tt (z s, t), the flow and the riser properties, and obtain the displacements y(z, t) at any arbitrary location z along the riser. 6.2 Response reconstruction from few sensor data Since we do not have enough sensors to satisfy the criterion for full reconstruction, the measurements from these sensors do not contain all the information pertinent to riser VIV. In such a scenario, we complete the missing information on riser motions by providing the additional information based on our prior understanding of riser VIV. To do this we need to pose the problem in a slightly different way from the case where we satisfy the full reconstruction criterion. 8

119 6.2. Solution overview We pose the problem from a structural dynamics perspective where we express the displaced shape of the riser at any instance of time as a linear combination of the peak response modes Y n (z) oscillating at the peak response frequencies ω n as: { Nm } y(z, t) = Re Y n (z) e iωnt. (6.) n= The above statement is identical to the statement of the modal decomposition presented in Section 4.3. It is important to note that these Y n (z) and ω n are not the classical free vibration modes and the free vibration frequencies. The additional information we intend to provide for reconstruction are these peak response modes Y n (z) and the peak response frequencies ω n of the riser motions. These are a result of an equilibria arising from a power balance where the fluid excites some part of the riser and the riser dissipates the energy at other parts. The N m number of Y n (z) and ω n can be obtained using prediction programs like VIVA [59] after solving the nonlinear eigenvalue problem mentioned in Section 2.3. Since a physical process is never purely harmonic, we can generalize the equation (6.) to write the displaced shape of the riser as: { Nm } y(z, t) = Re w n (t) Y n (z). (6.2) n= where, w n (t) are the narrow banded modal participation factors with spectra depicting one dominant peak frequency centered around ω n (ideally w n (t) e iωnt ). Taking the Fourier transforms of both sides of equation (6.2), we obtain: ŷ(z, ω) = F {y(z, t)} = N m n= ŵ n (ω) Y n (z). (6.3) The ŵ n (ω) and ŷ(z, ω) are respectively the Fourier transforms of w n (t) and y(z, t) taken along time. Corresponding to the measurements at the sensor locations z s we can write: ŷ(z s, ω) = N m n= ŵ n (ω) Y n (z s ). (6.4) Thus corresponding to the N s sensors, we have N s equations and N m unknowns corresponding to each ŵ n. Since N s is few in comparison to N m, writing the equation (6.4) in a matrix form and solving it directly is not possible due to a condition where there are more unknowns than the number of equations. We reformulate the equation (6.4) in a different form by employing the band-separated nature 9

120 of the riser VIV. What we do is to subdivide the spectrum of the original signal ŷ(z s, ω) into N m separate spectra ŷ n filt (z s, ω) by bandpass filtering them around each of the peak response frequencies ω n as depicted in Figure 6-2. Typically this bandpass filtering is performed in the frequency domain, by choosing those frequency components of the spectrum which lies within the frequency bands. As a result the following equation holds: ŷ n filt (z s, ω) = ŵ n (ω) Y n (z s ). (6.5) Now for each n, we have N s equations and only one unknown ŵ n, resulting in scenarios where the number of equations are greater than the number of unknowns. Thus, for each n we solve for ŵ n (ω), and take the inverse Fourier transform to obtain w n (t). ŵ n (ω) = Y + n (z s ) ŷ n filt (z s, ω). (6.6) Instead of posing it as one matrix inversion problem with more unknowns than equations, we pose the problem as N m inversion problems each of which is well behaved. Once we obtain w n (t), we can reconstruct the riser displacements as: { Nm } y(z, t) = Re w n (t) Y n (z). (6.7) n= Figure 6-2: The spectrum of signal ŷ(z s, ω) from a sensor is subdivided by bandpass filtering them around each of the peak response frequencies ω n to obtain N m separate spectra ŷ n filt (z s, ω). 2

121 6.2.2 Overall algorithm The algorithm is described in Figure 6-3. A statistically stationary segment of the data Y(z s, t) is first extracted using the method mentioned in Section 4.2. Next, the Fourier transform of Y(z s, t) is taken to obtain Ŷ(z s, ω). Corresponding to the input flow profile U(z), a prediction program like VIVA is used to obtain the peak response modes Y n (z) and peak response frequencies ω n. Next, the spectrum of each of the sensor data Ŷ(z s, ω) is subdivided into N m separate spectra to obtain Ŷn filt(z s, ω). These bands are each centered around the predictions of ω n. Response reconstruction from few sensor data Statistically stationary segment of data Y(, t) z s Compute Fourier transform ) Y( z, ω) = Y( z, t s F{ )} s Peak response modes ( Y n (z) ) Peak response frequencies ( ) from VIVA ω n Corrected databases of C lv and C m Filtering sensor data in a band around ω Y ) n filt ( z, ω) s n for each VIVA mode n Optimal reconstruction in frequency domain ) Ψ ˆ n ( zs ) wn ( ω) = Yn filt ( zs, ω) ) + wˆ ( ω) = Ψ ( z ) Y ( z, ω) n n s n filt s Compute inverse Fourier transform: IF { ˆ ( )} w ( t) = ω n w n Obtain displacements at any location N m y( z, t) = Re w n= n ( t) Yn ( z) Figure 6-3: Overview of the algorithm for reconstructing riser response using measurements from few sensors. For each n, we need to evaluate the modal matrix Ψ n (z s ) and the data matrix Ŷn filt(z s, ω). Ψ n (z s ) is identically equal to the peak response mode Y n evaluated at the sensor locations z s. The data matrix Ŷn filt(z s, ω) is obtained as mentioned in the previous paragraph. A pseudo-inverse of Ψ n (z s ) is calculated for each n, and the corresponding ŵ n (ω) is evaluated. An inverse Fourier transform of ŵ n (ω) is taken to obtain w n (t). Once we obtain each w n (t), the displacements are obtained using equation (6.7) Illustrative example To validate the algorithm, we apply the developed methodology to a benchmark data set constructed synthetically. The benchmark data corresponds to a riser with properties similar to that of the NDP riser described in Appendix B.. The riser encounters a triangular flow profile U(z) = U max ( z L) corresponding to a maximum flow velocity Umax =.5ms. VIVA 2

122 *) ( ' & produced peak response modes Y n (z) and peak response frequencies ω n obtained corresponding to the above riser and flow properties, and are employed to produce the displacements as: { Nm } y(z, t) = Re c n Y n (z)e iωnt. (6.8) n= Accelerations are extracted for a variety of cases involving different number of sensors N s, located at the corresponding z s. The resultant acceleration signals are around 6 seconds in time, and as shown in Figure 6-4(left) has 4 primary peaks depicting a multi-frequency response. In addition to the acceleration data, we are also provided with the flow profile U(z) from which we can obtain Y n (z) and ω n using VIVA..8 # # "!.5 - #! + # "!, %$ Figure 6-4: Left: the span averaged spectrum of the benchmark data; right: the RMS of the displacements measured along the riser. In this benchmark example, VIVA produced 22 peak response frequencies and peak response modes. Figure 6-3 illustrates how we subdivide the spectrum of a signal into several spectra each corresponding to the given ω n. The RMS of the original signal and the reconstructed signals for various N s cases are presented in Figure 6-6 (N s = 2), Figure 6-7 (N s = 3) and Figure 6-8 (N s = 4). Figure 6-9 illustrates a typical comparison of the original and the reconstructed signals at the sensor locations for N s = 3 and N s = 4. 22

123 Figure 6-5: Left: the span averaged spectra of the benchmark data. The vertical lines represents the VIVA predicted ω n. Right: the spectra of sensor signals ŷ(z s, ω) are subdivided by bandpass filtering them around each ω n to obtain N m separate spectra ŷ n filt (z s, ω). 23

124 & $%!#".5.5 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-6: Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 2 are illustrated. & $%!#".5.5 )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-7: Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 3 are illustrated. 24

125 - -.. & $%!#".5.5 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-8: Results comparing the RMS of the original and the reconstructed signals obtained using the proposed reconstruction algorithm for benchmark data. Various cases of sensor locations (marked by the sign) for number of sensors N s = 4 are illustrated..! /% " )! $!2!" #$%'2%$&3 " (! $&) "* (3)#$%')54!"7698;: <>=@?A>=CBB>=ED /! &% " )! $!23!" #$&%'3%$54 " (! $5) "* (46)#$&%')87 2!":9<;>=?A@CBDA@EDFA@EGGA@IH , !" #$&%' " (! $&) "* (! $ -,, !" #$&%' " (! $&) "* (! $ Figure 6-9: Left: depicts the original and reconstructed signals over 5 seconds for a case when N s = 3; right: depicts the original and reconstructed signals over 5 seconds for a case when N s = 4. 25

126 Results from illustrative examples As seen from Figures 6-6, 6-7, and 6-8, the reconstruction is quite accurate even for number of sensors as low as 2. The comparison of the original and the reconstructed signals at the sensors locations (Figure 6-9) also depict excellent agreement. It was found that the reconstruction depends on ) the sensor locations (z s ), and 2) the number of sensors (N s ). A more uniform distribution of sensors yields a more accurate reconstruction. Another observation is the fact that accurate reconstruction depends heavily on the accuracy with which we can obtain Y n (z) and ω n. Our predictive capabilities could be compromised due to the minor errors in measuring the flow profile or the empirical lift coefficient databases. Thus the illustrative examples clearly show that accurate reconstruction is possible even for very small N s, provided we are able to accurately evaluate Y n (z) and ω n. Sensitivity to flow profile As mentioned previously, the peak response modes Y n (z) and peak response frequencies ω n are not the classical free vibration modes and frequencies and hence depend critically on the flow profile U(z). Thus a reconstruction employing Y n (z) and ω n also depends on U(z). If there is an inaccuracy in obtaining the flow profile, the accuracy of reconstruction may be affected. The sensitivity of the flow profile on the reconstruction is studied using the benchmark example. The benchmark data remains the same but we use a slightly perturbed flow profile while performing the reconstruction. The perturbed flow profile is again a linearly sheared profile but given by: ( U p (z) = [U max + U max ] z ). (6.9) L where U max is the error assumed to be introduced while measuring the flow profile. The Y n (z) and ω n corresponding to various such perturbed flow profiles U p (z) are obtained, and used to reconstruct the data corresponding to a flow profile U(z) = U max ( z L). Figure 6- depicts the sensitivity of the reconstruction for various number of sensors N s placed uniformly along the riser. We measure the reconstruction error (average error and maximum error) corresponding to different values of error Umax U max in the flow profile. We can clearly observe that the reconstruction errors become zero when the error in the flow profile becomes zero. For small errors in the flow profile, the reconstruction error is small but grows considerably as the error in the flow profile grows. The R R two non-dimensional reconstruction error metrics are ) average error = D LT yorig(z, t) y recon(z, t) dt dz, and 2) maximum error = max yorig(z, t) yrecon(z, t), where D T is the time interval over which we perform the reconstruction, y orig(z, t) is the original riser response and y recon(z, t) is the reconstructed riser response. 26

127 6> = C \ [V 5= < B \ [V ] = < D G F y(z,t) y rec(z,t) dzdt =?> < 8:9:; 576 XZY[ [ U:VW 9;:;< 7 AB >? 9;:;< 687 JIH 8:9:; 576 KJI XZY[ [ U;VW \W Y[Z\ \ V:WX TLD 8:9:; = > max y(z,t) yrec(z,t) y(z,t) y rec(z,t) dzdt D TLD max y(z,t) yrec(z,t) D log(error) # $ % & $ ' $ (&% ))*+ &,-.'/ $ 324* $5 "! "! Reconstruction! error using a perturbed velocity profile #L $ % & $Ḧ $ (&% ))*+ &,+.H/ $ 32H* $4 )MONPRQ Reconstruction error using a perturbed velocity profile Percentage change in U #K $ % & $ 4 $ (&% ))*- &,-.4/ $ L24* $5 )MONPRQ max [%] "! " # $ % # & # '%$ (()* % +, -&. /# 23) #4/ ST ]/) ^ _' Reconstruction G H error using a perturbed! velocity profile "! Percentage change in U "L # $ % # 3 # '%$ ((), % +, -3. /# M3) #4/ (NPOQSR max [%] ST ]/) ^ _' Reconstruction error using a perturbed velocity profile Percentage change in U max [%] Reconstruction error! using a perturbed velocity profile (N s =8) TU ^.( _ `& avg err max err Percentage change in U max [%] Reconstruction error using a perturbed velocity profile (N s = ) Percentage change in U max [%] avg err max err log(error) Percentage change in U max [%] Figure 6-: Depicts the average and maximum reconstruction error as a function of the error in the flow profile Umax U max used to obtain Y n (z) and ω n for various N s cases. We can observe that the accuracy of reconstructed response is highly sensitive to the error in the flow profile used during reconstruction. 27

128 6.2.4 A discussion on response reconstruction using few sensor data The method critically relies on accurate prediction of peak response frequencies and peak response modes of the riser. At present, our predictive capabilities are handicapped by the approximations involved while obtaining the empirical lift force coefficient databases. It is proposed that for effective use of this method, we are required to use a database correction algorithm to correct these databases. The predictions of peak response modes require the flow velocity U(z) which is required as an input to the problem. It has been found that the estimates of the peak response modes are sensitive to the variation in flow velocity and hence affects the reconstruction itself. The reconstruction method for few sensor case differs from the reconstruction method for large number of sensors in many ways:. The present method is applicable even when the number of sensors are very few (as low as 2 or 3), unlike the case of reconstruction method for large number of sensors where the full reconstruction criterion needs to be satisfied. 2. The proposed method requires us to accurately obtain the flow profile U(z), fluid properties, and the riser properties. This is in contrast to the reconstruction method for large number of sensor case which is independent of the predictive capabilities and hence the riser properties or the flow properties. 3. To obtain the peak response modes and frequencies, we are required to assume a steady state response. However in reality this assumption is very difficult to be satisfied. However, we may be able to find stationary segments of the data over which this assumption is valid. 4. Instead of posing the reconstruction problem as one pseudo-inverse problem, we subdivide the problem into several well behaved problems by band pass filtering the signals around each peak response frequency ω n. 5. We employ the use of peak response modes and frequencies obtained by solving a nonlinear eigenvalue problem. These modes and frequencies are very different from the classical free vibration modes used in the previously considered methods [2, 6, 29]. 6. Since our predictive capabilities are limited to the riser CF motions, reconstruction method in its present form is applicable to riser CF motions only. However, if a model for predicting the IL peak response modes and frequencies are evolved, then the reconstruction method is expected to work for IL motions also. 28

129 6.3 Local lift coefficient database correction As mentioned in the previous section, for accurate reconstruction it is important to accurately predict the peak response modes Y n (z) and the peak response frequencies ω n. One source of error in the prediction of Y n (z) and ω n is the various assumptions made while obtaining the empirical lift force coefficients of C lv and C m. These assumptions are described in detail in Chapter 5. Ideally we would prefer to use the universal C lv and C m databases while generating the predictions for C lv and C m. Since we have the measured data from the few sensors, we could also think of a method to use the data from these sensors to correct the lift coefficient databases in a local sense. However, the C lv and C m extraction algorithm described in Chapter 5 requires the experimental modes Y n exp (z) at every point along the riser which in turn requires the reconstructed displacements. Since we do not have the reconstructed displacements a priori, we require a method to correct the C lv and C m databases using data from the few sensors which are available. These corrected databases are then used to improve the accuracy of predicting Y n (z) and ω n respectively Formulation as an optimization problem The local lift force coefficient correction method we propose is very similar to the method mentioned in Chapter 5, except for the fact that we are performing a correction based only the local values of the peak response modes and not the peak response modes over the entire riser. We formulate this problem once again as a parameter estimation problem and the overall algorithm is illustrated in Figure 6-. Experimental measurements The acceleration signals y ttexp (z s, t) at the sensor locations are used to obtain the displacement signals y exp (z s, t). The Fourier transform of the signals are taken with respect to time. We identify the peak response frequency ω n exp first, and then corresponding to the ω n exp, we extract the peak response modes at the sensor location Y n exp (z s ). Thus the experimental data consists of ω n exp and Y n exp (z s ). Theoretical estimates Given the flow profile U(z), and the lift coefficient databases of C m and C lv the theoretical estimates of peak response frequencies ω n th and peak response modes Y n th (z) are obtained by solving the set of two integral equations (5.) and (5.2). For obtaining the theoretical estimates Y n th (z) and ω n th we once again use the empirical prediction program VIVA [59]. This is 29

130 Few sensor reconstruction (Marintek dataset 243) Local force coefficient database correction methodology Given Data y tt exp ( z, t) s Prediction r y ( z, ω ; p) th n yexp ( zs, t) yˆ exp ( z s, ω) ω n exp Experimental Data ω Peak response frequency n exp Peak response modes z ) Y n exp ( s Forward Model r Peak response frequency ωn th ( p) r Peak response modes Y ( z, p) n th s Error metric (optimization index) { Yn th ( z s ) Yn exp ( z s ) } + ϖ RMS { Yn th ( z s ) Yn exp ( z s ) } + ω n th ω exp J ( p) = ϖ RMS 2 n Solution Figure 6-: technique: Overview of the method simulated used for annealing local lift coefficient database correction. followed by extracting the peak response modes Y n th (z s ) at the sensor locations z s. Error metric (optimization index) As discussed with great detail in Chapter 5, we have to minimize the error in both peak response frequency ω n, and peak response mode Y n (z s ). However it was found that for the reconstruction involving few sensors, both the modal magnitude and the modal phase angle of Y n (z s ) are important. Based on this we choose the optimization index or error metric in the following form as: J( p) = ϖ RMS { Y n th (z s, p) Y n exp (z s ) } + ϖ 2 RMS { Y n th (z s, p) Y n exp (z s ) } + ω n th ( p) ω n exp. (6.) where, ϖ, ϖ 2 are factors which allows us to weigh the relative importance of minimizing error in the modal magnitude, modal phase angle and peak response frequency. It is to be noted that this weighing factor is quite different from the values used when we have a large number of sensors allowing reconstruction first. The notation Y n exp (z s ) = Y n exp (z s+ ) Y n exp (z s ) represents the difference in modal phase angle between two consecutive sensors. 3

131 Solution technique The local lift coefficient correction method is very similar to the lift coefficient correction method for large number of sensors mentioned in Chapter 5 except for the way the error metric is evaluated. Hence our choice of solution technique was easy and we used the simulated annealing method which is described in Chapter Application to NDP data Typical spectral data from NDP depict a narrow banded response with a single peak response frequency or a response involving few peak response frequencies as depicted in Figure 4-7. We aim at applying our reconstruction method to such datasets from NDP experiments. Specifically we consider NDP dataset 243, corresponding to a linearly sheared flow profile with a maximum velocity U max =.5ms. First of all to demonstrate the applicability of the method, we use a library of modes and frequencies obtained previously (refer Figure 4-8). As we can observe from Figures 6-2, 6-3 and 6-4, the reconstruction is accurate for various configurations of N s and z s. The underlying message is simple: this method can reconstruct complicated motions of risers provided we have a capability to accurately predict Y n (z) and ω n Application employing the VIVA generated modes and frequencies Next, we use VIVA to generate the Y n (z) and ω n corresponding to the flow profile U(z) =.5 ( L) z. The nominal Clv and C m databases are used by VIVA to obtain the Y n (z) and ω n. Figures 6-5 and 6-6 depict the reconstructed response for the various configurations of N s and z s. As seen from the figures, it is very clear that the reconstruction is indeed not accurate with large over estimations and under estimations. This is due to the assumptions in the nominal C lv and C m databases used for predicting Y n (z) and ω n. We employ the local lift coefficient correction algorithm using data from 8 accelerometers. The optimization index we used contains one term to account for the error in the modal phase angle. The result from such a correction method is illustrated in Figure 6-7. Figure 6-7(left) shows the comparison of Y n (z) and ω n using the nominal databases, while Figure 6-7(right) shows the comparison using the optimal databases. The Y n (z) and ω n obtained using the optimal databases are used for reconstruction. The results can be observed from Figures 6-8, 6-9 and 6-2. We can observe that the reconstruction depicts reasonably good agreement for various cases of N s and z s. We apply this reconstruction method for various number of sensors and sensor placements. Using the data from 8 accelerometers available in the NDP data, we obtain the reconstruction 3

132 for all the combinations of number of sensors N s and the sensor locations z s. The error between the reconstructed signal using few sensors and the original signal (using all the available sensors using the method described in Chapter 3) is obtained. The Figure 6-2 illustrates how the average error vary as a function of number of sensors N s. For the use of a small number of sensors, there is a large variability in the reconstruction error. In addition, the maximum value of the average error is as high as 35%. As the number of sensors increase, the maximum value of the average reconstruction error and the variability in the reconstruction error both decreases. Physically it means that the confidence in our reconstruction increases with the use of increasing number of sensors. In addition, it was also observed that a more uniform distribution of sensors placed along the riser provides more accurate reconstruction. & $%!#".5.5 )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-2: Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s = 2 for various cases of sensor locations. A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign. 32

133 & $%!#".5.5 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-3: Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s = 3 for various cases of sensor locations. A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign. & $%!#".5.5 )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-4: Results comparing the RMS of the original and reconstructed signals for NDP dataset 243 when the number of sensors N s = 4 for various cases of sensor locations. A library of modes and frequencies obtained previously were used during the reconstruction. The locations of the sensors are marked by the sign. 33

134 & $%!#".5.5 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-5: Results comparing the RMS of the original and reconstructed signals (using nominal lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 2 for various cases of sensor locations. The locations of the sensors are marked by the sign. & $%!#".5.5 )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-6: Results comparing the RMS of the original and reconstructed signals (using nominal lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 4 for various cases of sensor locations. The locations of the sensors are marked by the sign. 34

135 INITIAL COMPARISON (datacase:243) Span averaged FFT of acceleration FINAL COMPARISON Span averaged FFT of acceleration FFT [ms 2 ] Frequency [Hz] Frequency [Hz] Modal magnitude Modal magnitude Yn [m].3.2 experimental nominal VIVA correct experimental optimal VIVA correct z[m] z[m] C lv(v r,a ) C m(v r,a ) Figure.4 6-7: Application of the local lift coefficient extraction method to NDP dataset Left:.2 comparison of the experimental and nominal theoretical prediction; right: comparison of.8 the optimal and experimental prediction..5 A & $%!#" V r )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3* V r.5 Chosen weight : α = 2; Time taken :27hrs and 54min Nominal parameters: (J nominal =4.9) Optimal parameters: (J optimal =2.8) )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3* & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-8: Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 2 for various cases of sensor locations. The locations of the sensors are marked by the sign. 35

136 & $%!#".5.5 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-9: Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 3 for various cases of sensor locations. The locations of the sensors are marked by the sign. & $%!#".5.5 )+*-,G.?-234*-5>*-7H8+:I;6=34.?@-AB,GB 5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 )+*-,/.-234*-56*-798+:<;>=?34.@-Ä B,/B5DC4E-7CB2E2B A*-5?4C2FÄ C3*-5 & $%!#".5.5 & $%!#".5.5 & $%!#".5.5 ' ( Figure 6-2: Results comparing the RMS of the original and reconstructed signals (using corrected lift coefficient databases) for NDP dataset 243 when the number of sensors N s = 4 for various cases of sensor locations. The locations of the sensors are marked by the sign. 36

137 !#" $% & " $ '& ( '&$)'&.45.4 *$+-,.$ " $% / Figure 6-2: The average error during reconstruction for all possible combinations using data from the 8 accelerometers for NDP dataset Concluding remarks We have developed a method to reconstruct the response of the entire riser when we have data from few sensors. We employed a modal decomposition approach in terms of the riser peak response modes and the peak response frequencies, and utilized the property of band-separated nature of the riser VIV response. In Chapter 3, we posed the problem as a Fourier decomposition view point without any information on the riser properties or the flow properties. On the other hand, the method described in this chapter requires both the riser and flow properties in addition to the need for a prediction program to get the peak response modes and peak response frequencies. Since, we have a theoretical prediction model for the vortex-induced motions in the cross-flow direction only, the reconstruction method in its present form is applicable only for reconstructing the cross-flow motions of the riser. It was shown that the reconstruction accuracy critically depends on our estimate of the peak response modes and peak response frequencies. Since the peak response modes and peak response frequencies depend on the flow profile, it is shown that reconstruction depends on the accuracy of obtaining the flow profile. To improve the accuracy of reconstruction we develop a method to locally correct the lift coefficient databases. The reconstruction method is applied to experimental data from NDP experiments which clearly depicts the importance of using the corrected databases while reconstruction. 37

138 38

139 Chapter 7 On origin and resonance of third harmonic of VIV on risers 7. Introduction One characteristic of the vortex-induced vibration (VIV) of marine risers which is not fully understood is the presence and behavior of the higher harmonic components in the vortexinduced force [77, 9]. The presence of the higher harmonic components of the vortex-induced force may result in a substantial increase in the number of stress cycles endured by the riser. Recent findings [, 7] have shown that not considering these higher harmonics has the effect of significantly overestimating the fatigue life of marine risers. Several features of the higher harmonics remain unexplained even today. This chapter will focus on identifying two key features of the higher harmonic components of the vortex-induced force based on realistic experimental data measured on a riser. These features will provide insights into the origin of the higher harmonics, and the absence of resonance in the third harmonic component of riser VIV. Using these features we can develop a conceptual model for the vortex-induced force on a riser for predicting its fatigue life. 7.. A summary of related research Researchers have observed the presence of higher harmonic components in the acceleration and strain spectra in the VIV related field experiments as early as 98s [62, 45, 63, 64, 44, 5]. Since displacement spectra of the response rarely portrayed these higher harmonics, its significance was assumed to be limited. However, recent findings by Vandiver et al. [7], Dahl et al. [], and Jhingran and Vandiver [2] showed that fatigue life estimates based on the first and the third harmonics were up to two order of magnitudes smaller than the fatigue life considering the harmonic part alone. Thus, it is important to understand the features of the third harmonic 39

140 for accurately modeling the vortex-induced force for predicting the riser fatigue life. Williamson and Jauvtis [77, 9] were successful in reproducing the third harmonics under laboratory conditions on a flexibly mounted rigid cylinder (of mass ratios less than 6) which was free to oscillate in both cross-flow (CF) and in-line (IL) directions. They observed significant third harmonic component in fluid force and cylinder response, and attributed it to a distinct 2T periodic vortex wake mode, where a triplet of vortices are shed in each half cycle. The associated CF motions Ay D were large and the third harmonic component was found to be excited over a select regime of Ay D, Ax D (in-line motions) and V r (reduced velocity). They note that in spite of the fact that the forces were non-sinusoidal, the resulting displacements observed on an SDOF system were sinusoidal. Such arguments are not necessarily valid for a distributed system like a riser having several natural frequencies. Dahl et al. [9, ] extended the study further using a flexibly mounted rigid cylinder and a flexible cylinder with adjustable natural frequency in CF and IL direction. They were able to reproduce several features of the higher harmonics. Systematic conditions on V rn under which the third harmonic is predominant were evaluated (V rn [5, 8.5]). They also associated the third harmonic to the cylinder motion with respect to the the wake, using particle image velocimetry (PIV). However, the cylinder used in the experiment was short and was bending dominated unlike the long tension dominated systems found in real world scenario. Vandiver et al. [7, 68, 69, 7] performed three experiments using long flexible cylinders measuring axial strains due to bending or CF accelerations at closely located positions. For such long cylinders, they again confirmed the presence of a preferred V rn regime for significant third harmonic component. However any information aiding the modeling of the third harmonic component of forces was not within their scope of study. Jhingran and Vandiver [2] followed a statistical approach for fatigue life prediction. They observed that the earlier approach using a Rayleigh distribution assumed a narrow banded process. They also noted that the introduction of higher harmonics made the above narrowbanded assumption invalid. As a solution they proposed a modified version of Dirlik distribution to reduce error in fatigue life estimates. This approach in its present form is empirical and only provide limited insights into the origin and prediction of the magnitude and the location of the higher harmonics Topics of present study Though the studies mentioned in the previous section presented several interesting features of the higher harmonic components in the vortex-induced force, several pieces of the puzzle are still missing. Two such previously unanswered questions are:. Is the third harmonic due to a separate process or is it due to the same underlying phe- 4

141 nomenon that results in the harmonic part? Is the presence of the third harmonic due to the nonlinearity arising in the fluid dynamics or from the structural dynamics? Concrete answers to this question remains elusive even today. 2. Why do distributed systems show only a single peak in its displacement spectra? This happens in spite of the fact that the third harmonic component of forcing has a magnitude comparable to harmonic component, and the system have several natural frequencies which could potentially be excited. This question hints towards a hypothesis that the third harmonic is non-resonant. A proof of this argument is important in describing the spatial variation of the third harmonic component of the vortex-induced force. 7.2 Origin of higher harmonics in vortex-induced fluid force As mentioned before, it is critical to understand the origin of the high harmonic components in the experimental data. That is, we are interested in knowing whether the same underlying phenomenon which causes the harmonic part of response causes the higher harmonics as well. One way to answer this question is to look into the phase between the harmonic and the third harmonic component. If the higher harmonics are phase-locked to the dominant one, the underlying phenomenon is the same. A good way to study this is by using Lissajous figures (or phase plots) Lissajous figures A phase plot or a Lissajous curve is a curve plotted between two signals. For the case involving two monochromatic signals x(t) = A x cos(ω x t + ψ x ) and y(t) = A y cos(ω y t + ψ y ), where ω x t + ψ x and ω y t + ψ y represent the phases, and A x and A y represent the constant amplitudes of the two signals respectively. Depending on the ratio of ωx ω y we can classify the curves that are obtained (refer to Figure 7-). For integer ωx ω y, the curves close and the closed curve remains stable (a, b, c, d, e and f in Figure 7-) [74]. Such a condition is referred to as a phase-locked behavior. Distinct lobes in Lissajous plots (d, e and f in Figure 7-) are obtained when ψ x ψ y is a constant and not equal to nπ Application to VIV higher harmonics We first filter the CF acceleration or strain signal at a band around the dominant frequency f peak, the third harmonic of dominant frequency f 3peak and the fifth harmonic of dominant frequency f 5peak to obtain three signals CF fpeak (t), CF f3peak (t) and CF f5peak (t) respectively using If ωx ω y is not rational the curves never close and the point of crossing of the lobes keeps shifting as time passes. 4

142 % $ # "! "! Figure 7-: Different Lissajous figures obtained for different values of frequency ratio ( ωx ω y ) and (ψ x ψ y ). A x = A y is assumed to be constant. a band-pass filter. The filtering is done in the frequency domain and induces no phase change which may contaminate our analysis. If CF fpeak (t) and CF f3peak (t) are phase-locked and are remarkably periodic, then this would mean that the third harmonics observed are caused by the same phenomenon causing the harmonic part. That is, for each period of the dominant frequency, we expect the occurrence of another minor event three times. Similar phase plots can be obtained between CF fpeak (t) and CF f5peak (t). This phenomenon observed in the measured data is an indicator of some nonlinearity in the system we are studying. The nonlinearity could be due to structural nonlinearity or hydrodynamic nonlinearity Inference from NDP data To demonstrate the concept of phase locked behavior we apply the method to several NDP datasets. Phase plots from two datasets are depicted in Figure 7-2 (for dataset: 234), and Figure 7-3 (for dataset: 243). The phase plots clearly depict periodic closed lobes. This points 42

143 D8 6 B&: 78D>: <lk B&7 8 I =?A@ B& 8@ 78C&: ED B&D ;: <>=?A@ B& 8@ 78C&: ED D J K L M NPO RQSL L^T N?VaY RQSK [^T [ X Y RQSK `^TWV ` Y RQSK?V T L _ Y 6 4 )* ( )* ( 2 ' $&% " #! )* ( 5 $&% " # 5 " # 5 $&% RQ]V X^T [ _ Y 6 ' /, -. 2 # 5 43,2 # 5! 5! 5 " RQ]V L^T X [ Y ' " RQS[UT XPV\Y RQSMUTWV XZY b B& 8@ 78C&: 89d: C& 7 O> Y egf hg 5 Figure 7-2: Sample phase-plots from NDP experiment (dataset: 234) depicting the phase-locked behavior of CFfpeak (t) with CFf3peak (t), and CFfpeak (t) with CFf5peak (t). The figure illustrates the remarkable periodicity between the harmonic and the higher harmonic components of the response. 43

144 !. #! 3 #! ;: <>=?A@ B& 8@ 78C&: ED D J K L M NPO D8 6 B&: 78D>: <lk B&7 8 I=?A@ B& 8@ 78C&: ED B&D 4 RQSM M^TN?VaY RQSK [^T[ X Y RQSK `^TWV` Y RQSK?V TM _ Y )* ( ' $&% # " RQ]VX^T[ _ Y RQ]VM^TẌ [ Y )* ( ' $&% " # /,- " )* ( ' $&% " # 4,2 " RQS[UTXPV\Y RQSLUTWV XZY b B& 8@ 78C&: 89d: C& 7 O> Yegf hg Figure 7-3: Sample phase-plots from NDP experiment (dataset: 243) depicting the phase-locked behavior of CF fpeak (t) with CF f3peak (t), and CF fpeak (t) with CF f5peak (t). The figure illustrates the remarkable periodicity between the harmonic and the higher harmonic components of the response. 44

145 toward an underlying phenomenon where the third and the fifth harmonics (CFf3peak (t) and CFf5peak (t)) are phase locked to the harmonic part CFfpeak (t). As mentioned in the previous section, such a phase locked behavior is observed when the underlying cause of the higher harmonics and the harmonic part are the same. Such an event usually happens when there is an underlying nonlinearity. VIV results from a nonlinear dynamic equilibria arising from a structure interacting with the surrounding fluid. The result is a resonant matching between fluid excitation and smallamplitude (hence linearizable) structural response. Since the structural motions have small amplitude, the nonlinear behavior can entirely be attributed to forcing from the fluid on the structure. Experiments by Dahl et al. [9] and Williamson et al. [77, 9] clearly show that there is a 2T vortex shedding mode where three vortices are shed at every half cycle as shown in Figure 7-4. This leads us to the belief that the observed nonlinearity may be due to this vortex triplet and is primarily hydrodynamic in nature. Figure 7-4: 3D iso-contours of instantaneous vorticity obtained from DNS simulations by Dahl et al. []. We can observe the presence of spatial persistence of vortex triplets being shed during every half cycle. 45

146 7.3 Non-resonant third harmonic in vortex-induced force For a system like a flexibly mounted rigid cylinder, Jauvtis and Williamson [9] present arguments using the transfer function between the force and the displacement. They argue that the single resonant peak in the transfer function is the reason why the higher harmonics are not excited. Distributed systems on the other hand have several natural frequencies which could be potentially excited by the third harmonic component of vortex-induced force. However experimental data clearly show that even when the third harmonic component of force has similar magnitude as the first, the third harmonic component of displacement (spectra at various locations) is very small compared to the harmonic component. This hints towards a non-resonant third harmonic. We present experimental and theoretical evidence to support the above hypothesis. Based on the evidence from theoretical and experimental study, we argue that the spatial variation of the third harmonic component of the force closely follows the spatial variation of the harmonic component of the force. For identifying resonance in the harmonic and third harmonic, we separately compare the net inertia force and the stiffness force (restoring force). Resonance is indicated by a situation where the inertia force and the stiffness force terms are nearly identical for both the harmonic and the third harmonic part Validation based on experimental data from NDP Experimental datasets from NDP presents an ideal platform for identifying the presence/absence of resonance in riser VIV. Specifically, we consider datasets in which the riser model encounters a linearly sheared velocity profile, and the harmonic part of response depicts a single peak frequency. Both curvature ( 2 y obtained from strain) and accelerations ( 2 y ) were recorded at z 2 t 2 8 locations along the span of the riser in the NDP experiments. Figure 7-5 depicts the span averaged spectrum (averaged over the 8 locations) for all the sheared velocity profile cases in which the harmonic part of response depicts single peak frequency. Measuring curvature and acceleration allows us to obtain the local fluid forcing f fluid (z, t) at 8 locations along the riser as mentioned below. Obtaining vortex-induced forces from experimental data If we know the displacement y(z, t) in the cross-flow direction, its first two temporal derivatives ( y t and 2 y ), and the spatial derivatives ( 2 y and 4 y ), the external force acting from the t 2 z 2 z 4 fluid to the riser in the cross-flow direction f fluid (z, t) can be obtained as: f fluid (z, t) = m 2 y t 2 + b y t T 2 y z 2 + EI 4 y z 4, (7.) 46

147 , /? * +/ + VX[Y 2 CD$E E&F EG HI D J ELKMON PIJ EQ C$DE ERF EG H%I D J ELKMSN E TU E J QUK 4 2 VXXY 5 4 VXWY VXVY VX]Y VXYY VW\Y VW[Y > <.= 9.:; )4 /,.-.2 /,.- +* ')( B> A 8 57@ * /,.- +* ' ( VWZY VWXY !"# 2 3 $ $! %"& Figure 7-5: Depicts span averaged spectra for various NDP datasets corresponding to different linearly sheared flow profiles. Left: the maximum velocity U max ; middle: the span averaged spectra of CF strain; right: the span averaged spectra of CF acceleration. where, m is the mass per unit length in air, T is the tension, b is the structural damping constant and EI is the bending stiffness and are all assumed to be constant along the span of the riser. 47

148 If the structural damping is negligible and the riser is tension-dominated (the riser behaves like a tensioned string rather than a beam under bending), then the fluid force f fluid (z, t) can be approximated as: f fluid (z, t) m 2 y t 2 T 2 y z 2. (7.2) The expression (7.2) for fluid force f fluid (z, t) is much simpler than (7.) and requires only the estimation of the curvature 2 y z 2 and the acceleration 2 y t 2 signals at the locations of interest. Vortex-induced force is dominated by tension and inertia terms The stiffness of the riser model used in NDP experiment is tension dominated, and to illustrate this we need to compare the tension term T 2 y z 2 and the bending term EI 4 y z 4. For this we assume that the riser is oscillating in its n th sinusoidal mode Y n (z) = sin ( nπ L z) and compare the ratio β n of the tension term to the beam bending term as: β n = T 2 y z 2 EI 4 y z 4 T EI L 2 n 2 π 2 = T L 2 n 2 EIπ 2. (7.3) For NDP data, the mode numbers n = 4 to n = 5 are found to be excited by the first harmonic, and mode numbers n = 2 to n = 5 to be excited by the third harmonic. The β n corresponding to each of the mode numbers are obtained as: β 4 = 23 for smallest participating mode in first harmonic β 5 = 87.4 for highest participating mode in first harmonic. β 5 = 7.86 for highest participating mode in third harmonic β =.97 (for reference) tension dominated. Thus even if the 5 th sinusoidal mode is excited, the tension force is nearly 8 times the beam bending force. Similarly, the damping force is much smaller due to the very small value of the structural damping constant b. Thus the external force from the fluid to the riser consists mainly of the two components m 2 y t 2 (7.2). (or my tt ) and T 2 y z 2 (or T y zz ) and can be obtained using equation Obtaining stiffness and inertia terms To check for resonance we need to compare the net inertia force and the net stiffness force. The total inertia force consists of the added mass term (m a y tt ) in addition to the term due to the mass (my tt ). As mentioned previously, the stiffness force is dominated by the tension force 48

149 ( T y zz ). Thus we can write: f inertia (z, t) = (m + m a ) y tt, (7.4) f stiffness (z, t) T y zz. (7.5) To separately look for the harmonic and third harmonic components we obtain the fluid forces in the frequency domain as: ˆf inertia (z, ω) = F{f inertia (z, t)} = F{(m + m a ) y tt } = [m + m a ] ŷ tt, (7.6) ˆf stiffness (z, ω) = F{f stiffness (z, t)} = F{ T y zz } = T ŷ zz. (7.7) The harmonic part and the third harmonic part are evaluated by extracting the approximate frequency components ˆf inertia (z, ω ), ˆf inertia (z, ω 3 ), ˆf stiffness (z, ω ) and ˆf stiffness (z, ω 3 ) respectively. The inertia force however requires an added mass m a. A mean added mass coefficient C m is used to compute the added mass m a as m a = C m πd 2 ρ f 4. For a monochromatic riser motion, the C m (ω ) for first harmonic can be computed in two ways:. By evaluating the external fluid force f fluid (z, t) first and then finding the part in phase with acceleration and then taking the mean value. Refer Appendix E for more detail. 2. By evaluating the mean added mass coefficient C m (ω ) which satisfies the weak form of the governing equation obeyed by the harmonic part of forcing [57]: L z= T dŷ 2 dz dz ω 2 L z= [m + C m (ω ) ρ f πd 2 4 ] ŷ 2 dz =. (7.8) For the third harmonic we use a value of C m (ω 3 ) =, corresponding to the added mass coefficient when the cylinder is excited at three times the Strouhal frequency. Thus we are now in a position to estimate the harmonic part and the third harmonic part of the inertia force ( ˆf inertia (z, ω ), ˆfinertia (z, ω 3 )), and the stiffness force ( ˆf stiffness (z, ω ), ˆf stiffness (z, ω 3 )) at 8 locations along the riser where we measure both strains and accelerations. The RMS of ˆf inertia (z) and ˆf stiffness (z) taken along the span of the riser (over 8 locations) is an indicator of their magnitudes relative to each other. The ratio of the inertia force to the stiffness force (RMS{ ˆf inertia }/RMS{ ˆf stiffness }) is calculated for both the harmonic and the third harmonic component. This ratio is an indicator whether the inertia force balances the stiffness force, or if the residual excitation force ((m + m a )y tt T y zz ) is dominated by either the inertia force or by stiffness force. By definition, during resonance the inertia force matches the stiffness force, and hence this ratio is a key indicator of resonance. 49

150 Results from NDP data Force calculations using equation (7.4) is performed on all the sheared velocity profile cases in which the harmonic part of response depicts single peak frequency. This is followed by evaluating the mean added mass coefficients of the harmonic component C m (ω ) using the two methods described previously. These mean added mass coefficients obtained for each dataset corresponding to the maximum velocity U max is depicted in Figure 7-6(bottom) (*) ! "#%$'& + Figure 7-6: Top: peak response frequency obtained for various NDP datasets corresponding 3 to the maximum velocity U max. Bottom: the C m (ω ) for the harmonic component soln obtained 2.5 exp for different datasets corresponding to the maximum velocity U max ; ( ) corresponds to estimate Jason 2 obtained by evaluating the force in phase with acceleration; (+) corresponds to estimate obtained.5 from equation (*) (7.8). Note the remarkable agreement between the C m (ω ) estimated using the two methods..5 Figure 7-7 depicts the RMS of each of the forcing terms RMS{ ˆf inertia }, RMS{ ˆf stiffness }, RMS{ ˆf residual } and the ratio RMS{ ˆf inertia,.-/ }/RMS{ ˆf stiffness } for both the harmonic and the 8:9<; third harmonic components for different NDP datasets as a function of the maximum velocity U max. We can observe that for the harmonic component, the inertia force balances the stiffness force for all the flow cases (U max ) considered. This clearly indicates resonance in the harmonic part. However for the third harmonic component, as the flow velocity increases the ratio of inertia 5

151 $ $ ) ) /. - J I NPO " ) ) A D D? = < > C B ; 7 : 9 > 8 7 /. 6 - and stiffness force shows a clear departure from unity (refer to Figure 7-7(bottom right)). The inertia term is larger than stiffness term by up to a factor of.5 especially for datasets where there is significant component of the forcing in the third harmonic. This is a strong evidence pointing to a non-resonant third harmonic observed from riser data. "! #! " 32!" "! #! "! ), %'&( *+& ), % &( *+& !" #! " ! $ 5 ) # %'&(!, *+& R R NM SHG L3R KE Q F M J I NPO HGF O H3MN L KE J H3I I G E3F ), % &( *+& Figure 7-7: Comparison of harmonic (+) and third harmonic ( ) components of the stiffness and inertia forces for different flow velocities. Top left: RMS of the harmonic and third harmonic components of stiffness force; top right: RMS of the harmonic and third harmonic components of net inertia force; bottom left: the RMS of the harmonic and third harmonic components of residual excitation force (inertia force minus stiffness force); bottom right: the ratio of RMS of the harmonic and third harmonic components of inertial force by the RMS of stiffness force. Note that for the harmonic component, the inertia forces nearly matches the stiffness force. However for the third harmonic component the inertia force dominate over the stiffness force Validation based on theoretical models Absence of resonance from the viewpoint of the theoretical model of a simple system like a flexibly mounted rigid cylinder was indicated by Jauvtis and Williamson [9]. They claim that the transfer function between force and displacement has only one peak frequency which is effective in removing significant displacements corresponding to the third harmonic component of the fluid force. From the transfer function of an SDOF system (Figure 7-8) we may also understand that at resonance the inertia force balances the stiffness force. Away from the 5

152 resonance we can either have a stiffness dominated system or an inertia dominated system.!"#!$%&!$(')+*-,."/"# :<;=8 9>>:<?@8 >BADCE FHGG ;JF 4L!"%#E.5. ;JFPI 6FHGG 'QR""S)NO 4LM!#"%#E 6FHGGI ;JF KL!M)NO 4LM!#"%#E Figure 7-8: Magnitude of the transfer function of a single degree of freedom system. At resonant frequency the inertia force balances the stiffness force, at high frequencies we observe an inertia dominated system and at low frequencies we have a stiffness dominated system. In the following section we will explain the absence of resonance from the viewpoint of a more complicated distributed system like a tensioned string. This theoretical study supports the experimental observation mentioned in Section Using theoretical models, we will show that this apparent absence of resonance is possible due to the incompatible spatial variation of the third harmonic of forcing. To study this effect we will simulate the response of the riser modeled as a tensioned string using Green s function. Theoretical model of a distributed system (tensioned string) We examine the displacement magnitude corresponding to different forcing cases of the form f(z, t) = Re { F (z)e iωt}, where Ω is the forcing temporal frequency and F (z) is the spatial variation of the forcing. For obtaining the displacement magnitude of a tensioned string taut between z = and z = L we use a Green s function approach. For a given forcing frequency Ω we obtain the Green s function G(z; Ω, ξ) as: G(z; Ω, ξ) = kt [ ] sin k(ξ L) sin k(2ξ L) sin kz, < z < ξ, [ sin kξ sin k(2ξ L) ] sin k(z L), ξ < z < L. where, k mω 2 T ( iζ) and ζ is the damping ratio which is assumed to be small. Then, corresponding to a forcing of the form f(z, t) = Re { F (z)e iωt}, the magnitude of displacement 52 (7.9)

153 Y (z) is obtained from the Green s function G(z; Ω, ξ) as: L Y (z) = F (ξ)g(z; Ω, ξ)dξ. (7.) ξ= The displacement of the string y(z, t) can then be obtained as y(z, t) = Re { Y (z)e iωt}. An arbitrary forcing f(z, t) along a riser can be decomposed using a 2D Fourier transform in temporal forcing frequency Ω and spatial forcing frequency K. The structure under consideration (tensioned string) however, has its own intrinsic temporal natural frequencies ω n and spatial natural frequencies k n which related to each other through the dispersion relationship. The condition for resonance in such a system happens when the temporal forcing frequency Ω matches the temporal natural frequency ω n and the spatial forcing frequency K matches the spatial natural frequency k n. This is only possible when Ω and K are compatible obeying the dispersion relationship. Any other combination of Ω and K will not result in resonance. This is illustrated for a tensioned string model with properties similar to that of the NDP riser (m =.52Kg/m, L = 38m, ζ =.5, T = 5N). Specifically, we will understand three cases of force of the form f(z, t) = Re { sin(kz)e iωt}, where: Case I : Ω = ω n, K = nπ L ; (7.) Case II : Ω = ω 3n, K = nπ L ; (7.2) Case III : Ω = ω 3n, K = 3nπ L ; (7.3) Case I and Case III corresponds to a force where the spatial and the temporal forcing frequencies are compatible, while Case II corresponds to a force where the spatial and temporal forcing frequencies are incompatible. Results from the simulation for each of these three forcing cases for n = 5 are depicted in Figure 7-9. Because of the incompatible spatial and temporal forcing frequencies, we observe that the displacement magnitude corresponding to the Case II is much smaller than the displacement magnitude corresponding to the Case I or Case III. The cases where Ω and K are compatible results in large displacements due to resonance. An incompatible Ω, K combination predicts a non-resonant motion of the string model. The analysis using theoretical model hints toward a harmonic component of the fluid force, with a spatial variation which is compatible similar to the Case I (Figure 7-9(top)). The third harmonic force is then expected to follow the Case II (Figure 7-9(middle)). Such a fluid force will produce the displacements similar to the ones observed from NDP data, and validate our hypothesis that the spatial variation of the third harmonic is dictated by the spatial variation of the harmonic part. 53

154 !"$# -./#!2"3#$45#$6 768 :6;=< E7 768 :6;=< >?@BFAC E7 E768 G:H;I< >?@BFADC %'&)(+* %'&)(+* Figure 7-9: Displacement magnitudes of a tensioned string (similar properties as the NDP riser) for three cases of forcing of the form f(z, t) = Re { sin(kz)e iωt}. Note that Case I (top), and the Case III (bottom) will result in increased displacement magnitude (resonance) much higher than the displacement magnitude in Case II (middle) ,,, 54

155 Again as observed from Figure 7-9, even though the harmonic and third harmonic forces are bound, the displacements are not. The dominant wavelength λ 3 of the third harmonic of displacement is comparable to rd 3 of the dominant wavelength λ of the harmonic part. The curvatures due to the third harmonic of displacements is then amplified 9 times in comparison with the harmonic part. This is the reason why we observe peaks of nearly the same magnitudes in the harmonic and third harmonic components of the curvature spectra. The resemblance of the dominant wavelength of the third harmonic component of the force to that of the harmonic component can also be thought of from a wake creation argument. The wake behind the riser is created due to the displacement of the fluid by the riser resulting in a force which depends on the wake itself Comparison with laboratory observations Free vibration experiments by Dahl et al. [] shows that the third harmonic forcing is dominant over a certain reduced velocity V r range. Response observed from NDP data shows that V r varies linearly along riser due to the linearly varying flow velocity and a single dominant response frequency. However, NDP third harmonic forcing does not show any pattern along the riser as predicted by Dahl et al.. This happens because Dahl et al. conducts free vibration tests but NDP riser consists of riser sections involving forced vibration. In NDP experiments, close to the excitation region, the third harmonic forcing is a function of V r but in regions away from the excitation, the riser behaves like a forced cylinder (with different prescribed orbits). This results in a leakage of motion from the excitation region to other regions of the riser which need not obey the pattern shown by Dahl et al.. This leakage is the reason no pattern is observed in the third harmonic component of the force as a function of V r. Dahl et al. [8] have performed a series of 2D forced oscillation experiments, where a rigid cylinder is made to follow a particular trajectory at specific frequencies in CF and IL directions. The resultant lift forces were measured and a database of the lift coefficients were extracted. Two lift coefficients are obtained corresponding to both the harmonic component of the lift force C L and the third harmonic component of the lift force C L3. Each of these lift coefficients are functions of four parameters which describe the trajectory and the frequency. The four parameters are the reduced velocity V r, the non-dimensional CF and IL amplitudes Y/D and X/D and the phase between the CF and IL motions φ xy. Reconstruction method described in Chapter 3 allows us to obtain each of the four parameters (V r, Y/D, X/D, φ xy ) along the length of the riser. Using the C L (V r, Y/D, X/D, φ xy ) and C L3 (V r, Y/D, X/D, φ xy ) databases we evaluate each of these coefficients along the riser, and then multiply it by the normalizing factor (.5ρ f DU(z) 2 ) to obtain the forces F (z) and F 3 (z) along the riser. The following Figure 7- shows the F (z) and F 3 (z) obtained for various cases of NDP 55

156 QP D O MF ONT D M J MF ONT F M KD R QP O F KD R QP O L J L R S R S P Q P Q R S R S a c a c c a ACB >@? FMEN CED YQ X UWV CED ACB >@? DKCL ACB YQ X UWV CED DKCL ACB WO V SUT ACB WO V FMEN CED SUT ACB YQ X UWV MF CED T FMEN CED L ;HGIKJ Z[ \^] `ba caedf`_g!c"$#&%(')g*!-,/ a d? !"$#&%(')+*,-/ :"$;<6.45.4=> ;HGIKJ.25 7 "$;<6 => Z[ \^]_ `a`bdc ekcgfhb`ie!a"$#&%j')+*!-,/. k2d4 6 c f? !"$#&%(')+*,-/ FEG.IH 798:"$;<6 => "$;<6 => XY Z\[] ^_^`ba ciaedf`^gc _!#"%$h&(*) +,-. /ib2j FEG.IH !#"%$'&(*) +,-. / XY Z\[] ^_^`ba ciaedf`^gc _!#"%$h&(*) +,-. /ib !#9:4 ;< = 6!#9:4 ;< ;HGIKJ Z[ \^]_ `a`bdc ekcgfhb`ie!a"$#&%j')+*,-/. k2d4 6 ;HGIKJ a d a d c f YQ X UWV MF CED T Z[ \^]_ `a`bdc ekcgfhb`ie!a"$#&%j')+*,-/. k2d4l6 c f Figure 7-: Depicts the harmonic component F (z) and third harmonic component F 3 (z) of the fluid force obtained using lift coefficient database from Dahl et al., for various cases from NDP experimental data. We can observe that the dominant wavelength of F 3 (z) closely follows the dominant wavelength of F (z). 56

157 experiments. We can observe that though the amplitude of F (z) and F 3 (z) are very different, the dominant wavelength associated with F (z) and that of F 3 (z) closely follows each other. This once again confirms our hypothesis that the dominant wavelength of the third harmonic component of the lift force closely follows the dominant wavelength of the harmonic component of the lift force. 7.4 A conceptual model for the third harmonic force Section 7.2 shows how the third harmonic in VIV is caused by the same underlying phenomenon that gives rise to the harmonic part. This is attributed to the shedding of multiple (triplets in case of 2T mode) vortices in each half cycle as depicted in Figure 7-4. The second feature we have observed is the absence of resonance in the third harmonic component of VIV specifically for higher flow velocity cases in NDP data. The reason for nonresonance is a spatial variation of the third harmonic force which follows the harmonic part. That is, the dominant wavelength of the spatial variation of the third harmonic force is comparable with the dominant wavelength of the spatial variation of the harmonic component. Thus the third harmonic force spatial variation can be thought of as on accompanying structure on the main wake features. However, it is interesting to note that the riser motions are such that the dominant wavelength of the third harmonic component of riser displacement, are one third the dominant wavelength of the harmonic component of the riser displacement. Based on the above arguments a conceptual model of the third harmonic component of the fluid force can be evolved. Figure 7- depicts such a model where the harmonic part is depicted in Figure 7-(b). The corresponding spatial variation of the third harmonic force resembles the first as shown in the various alternatives depicted in Figure 7-(c). Obtaining the detailed spatial variation and the relative magnitude could depend on the Reynold s number, the local reduced velocity, the correlation-length during vortex-shedding and the riser properties. 7.5 Concluding remarks VIV occurs as a result of the nonlinear dynamic equilibria arising from an interaction between the riser and the surrounding fluid. Appearance of higher harmonics are observed when motions in the IL direction are not restrained as is generally the case in offshore applications. Even under these conditions, the response of the riser is primarily in the CF direction. However, it is critical to consider the IL motion as it drastically alters the wake, resulting in significant changes in the transverse lift force and response itself. Use of phase plots on NDP riser data reveals that the higher harmonics are caused by the same phenomenon which caused the harmonic component of the response. This is attributed to 57

158 A 5? < D 3 A 5? < D D 3 D 3!#"%$&('*)+&,$ A@?> :> 9=< 9:; )-.!## /243 KL M*"N$#O$"N $#P $&(&,$ TL '*$ #! &U$ VRWYX[Z#"N$#O$"N $P $&(&,$ B CEDF 24 B/GIH+J B CEQR 2S B/GIH+J 3 B CEQR 2S B/GIH+J B CEQF 2S B/GIH+J Figure 7-: (a) Typical span averaged spectrum of CF fluid force. (b) Typical spatial variation of the harmonic part of the CF fluid force. (c) Proposed concepts for the spatial variation of the third harmonic of the CF fluid force. 58

159 inherent nonlinearities during VIV when there is a resonant matching between fluid excitation and small-amplitude (hence linearizable) structural response. Since the structural motions have small amplitude, the nonlinear behavior can entirely be attributed to hydrodynamic forcing from the fluid on the structure. This is confirmed by experiments by Dahl et al. [9] and Williamson et al. [77, 9] who observe a distinct 2T vortex shedding mode where three vortices are shed during every half cycle. NDP experiments allows estimating the fluid force acting on the riser. A comparison of inertia force and the stiffness force along the riser allows us to detect the presence/absence of resonance. It is found that the harmonic component is in resonance over a range of triangular velocity profiles. However, for large velocity cases, the third harmonic component is non-resonant. With the help of theoretical models we show that resonance of a distributed system like a tensioned string is possible if i) both temporal forcing frequency Ω matches a temporal natural frequency ω n, and ii) the spatial forcing frequency K matches the spatial natural frequency k n. The spatial natural frequency and the temporal natural frequency are related using the dispersion relationship and hence the condition for resonance require Ω and K to be compatible obeying the dispersion relation. This supports the experimental evidence and validates our hypothesis that the spatial variation of the harmonic component of force obeys condition for resonance, but spatial variation of third harmonic does not obey condition for resonance. This led us to believe that the spatial variation of the third harmonic is dictated by the spatial variation of the harmonic part. 59

160 6

161 Chapter 8 Conclusions 8. Summary and principal contributions from each chapter The work presented in this thesis provides a systematic approach to estimate and analyze the vortex-induced motions and forces on a marine riser, and develop suitable methods to improve riser VIV modeling and response prediction. This thesis may be subdivided into three major themes. In the first part of the thesis, a systematic framework is developed, which allows reconstruction of the riser motion from a limited number of sensors placed along its length. In the second part of the thesis, a method is developed to improve the modeling of riser VIV by extracting empirical lift coefficient databases from field riser VIV measurements. In the final part of the thesis, data from a comprehensive experiment is utilized to show some fundamental characteristics of the higher harmonics in VIV of marine risers. The following sections present a detailed chapter wise list of principal contributions from this thesis. 8.. Riser VIV response reconstruction method In Chapter 3 of this thesis, we develop algorithms to reconstruct the response of a riser from experimental data consisting of a combination of strain and acceleration measurements. The reconstructed response obtained using this method is extensively used in other chapters. The specific contributions of this chapter are as follows: Systematic evaluation of full reconstruction criterion: The criterion to determine when a full reconstruction is feasible is systematically evaluated. This criterion gives the minimum number of sensors which is required for this reconstruction. This full reconstruction criterion is also interpreted from both a signal processing perspective as well as a matrix inversion perspective. Development of a systematic and scalable algorithm for riser VIV response reconstruction: A systematic and scalable method to reconstruct the response of riser is devel- 6

162 oped when we are provided with measurements from a limited number of sensors (strain gages, accelerometers) placed along the length of the riser. This method is applicable when the number of sensors are such that the full reconstruction criterion is satisfied. The problem is posed as a spatial Fourier decomposition which requires no VIV prediction capabilities. This approach is found to be more accurate than the previous approaches and independent of predicting the riser classical free vibration modes or response modes a priori. Response reconstruction error analysis: Three sources of error during response reconstruction was identified. This was followed by a study to quantify the error from each source during the reconstruction. Illustrative examples signify the importance of both the sine and cosine terms when a mixture of accelerations and strains are measured. The above method is applied to data from NDP experiments and the reconstructed riser response is extensively used in other chapters of the thesis Riser modal decomposition and traveling wave identification Chapter 4 presents the theoretical basis and practical methods to obtain the riser response modes from experimental data. The riser peak response modes are shown to contain most of the information on the experimental data itself. The unique contributions from this chapter are: Extract a statistically stationary segment of VIV measurement: A method to extract a statistically stationary region of a signal using scalograms (time-frequency representation) are presented. This method has several added benefits in the form of identifying faulty data, and obtaining an overall picture of how the underlying physical phenomenon evolves in time. Extract riser VIV peak response modes: A simple practical method to extract the riser peak response modes from experimental data is developed. This method is applied to NDP experimental data and a library of peak response modes are obtained. The significance of the magnitude and the phase angle of the peak response modes are also presented. Traveling wave identification methods: Several methods for identifying traveling waves in experimental data are listed and applied to high mode VIV tests data from NDP. These include methods using ) nodal evolution plots, 2) phase of peak response modes, and 3) magnitude of peak response modes. Using these methods conclusive evidence of traveling waves in riser response were presented. Also for the first time, we were able to demonstrate the presence of traveling waves in the third harmonic component of the riser response. The conclusive evidence of traveling waves and their dominant role in riser VIV response especially under sheared flows present a needed paradigm shift in riser VIV prediction methods. 62

163 8..3 Optimal lift coefficient databases from riser experiments In Chapter 5 we develop a method to improve the VIV prediction based on measurements obtained from risers in the field. Specific contributions from this chapter are listed below: Zero-contour method solving elastically mounted rigid cylinder problem: One outcome of thesis was a robust method to evaluate the solution of an elastically mounted rigid cylinder VIV response amplitude and frequency. This method is more robust than the present methods (Newton s method) and can even capture the phenomenon of multiple solutions (due to hysteresis during VIV). It is hoped that this method will be of use in the prediction programs for risers which employs a quasi-uniform approach. Flexible representation of lift force coefficients: Based on the physics of the VIV problem, we develop a method to represent the C lv and C m databases in a flexible format using just 2 parameters. The databases which are flexibly represented is able to capture most important aspects of the VIV prediction problem. Optimal lift coefficient databases from experimental data: A method to extract VIV lift force coefficient databases from experimental data conducted in the field was formulated as an optimization problem (parameter estimation problem) for the first time. The method is designed to account for the high Reynold s number effects and the effect of allowing in-line motion into VIV prediction. Another advantage of the developed method is that it require minimum change in the present VIV prediction programs like VIVA. Universal lift coefficient databases: An universal database of lift force coefficients, called the universal C lv and C m databases are obtained using the data from a family of linearly sheared flow experiments. It is shown that the use of the universal C lv and C m databases reduces the overall error between prediction and experimental measurements by 44 percent for linearly sheared flow profiles Response reconstruction using data from few sensors Chapter 6 of the thesis considers the case of response reconstruction when the number of sensors is much fewer than that required by the full reconstruction criterion. We understand that under this condition, the measurements does not contain all information about the VIV response of the riser. Based on this understanding, the following methodologies are developed in this chapter: A method to reconstruct response using data from few sensors: A method to reconstruct the response of the riser using data measured by a few sensors (number of sensors could be as low as 2 or 3) is developed. The solution to the problem was posed using a modal decomposition approach where the displaced shape of the riser at any instance of time is expanded as a superposition of the riser peak response modes oscillating at the peak response frequencies. It 63

164 was shown that the response reconstruction accuracy critically depends on accurately predicting the riser peak response modes and the peak response frequencies. Lift force coefficient database extraction from few sensors: The reconstruction accuracy depend on accurately predicting the riser peak response modes and the peak response frequencies. The present prediction schemes suffer from the insufficiencies in the empirical lift coefficient databases of C lv and C m. We developed a method to improve the prediction by extracting the lift coefficient database information using measurements from few sensors. It is shown that the use of the improved estimates of the peak response modes and frequencies yield improved accuracy during reconstruction Insights on the third harmonic component in riser VIV This chapter extends the present knowledge on the third harmonic component of riser VIV. Several new findings based on the data from NDP experiments were presented. Origin of higher harmonics of vortex-induced forcing: Using Lissajous figures, we show that the high harmonics are phase-locked with the first harmonics. This result nicely corroborates the findings by hydrodynamicists who observe multiple vortices being shed in every cycle when high harmonics are present. Multi-frequency C m and C lv from experiments: Lift force coefficients were extracted from experimental data involving risers. The added mass coefficient was used to illustrate the resonance in the harmonic part of the vortex-induced response. Non-resonant third harmonic in vortex-induced force: NDP experiments allows estimating the fluid force acting on the riser. A comparison of inertia force and the stiffness force along the riser allows one to detect the presence/absence of resonance. It is found that the harmonic component is in resonance over a range of triangular velocity profiles. However, for large flow velocity cases, the third harmonic component is non-resonant. This is the first time the presence of resonance is evaluated systematically for experimental data involving a riser. Insight into dominant wavelength of third harmonic component of the vortexinduced force: With the help of theoretical models we show that resonance of a distributed system like a tensioned string is possible if both temporal forcing frequency Ω matches a temporal natural frequency ω n, and the spatial forcing frequency K matches the spatial natural frequency k n. The spatial natural frequency and the temporal natural frequency are related using the dispersion relationship and hence the condition for resonance require Ω and K to be compatible obeying the dispersion relation. This supports the experimental evidence and validates our hypothesis that the spatial variation of the harmonic component of force obeys condition for resonance, but spatial variation of third harmonic does not obey condition for resonance. This led us to believe that the spatial variation of the third harmonic is dictated by the spatial 64

165 variation of the harmonic part. A conceptual model for the third harmonic component of the vortex-induced force: Based on the observation on the spatial variation of the third harmonic component of the vortex-induced force, a conceptual model for predicting it was evolved. It is expected that this conceptual model will help us in developing a detailed mathematical model for the third harmonic of the vortex-induced force. 8.2 Recommendations for future research During the progress of the thesis several problems previously considered difficult have been solved. However, it has also provoked several questions which at this point remains unanswered and is not within the purview of the thesis. Some of these are presented below: Benchmarking and benchmark datasets: One significant drawback which the VIV community faces is the absence of benchmarking data or benchmark databases, and the rules for creating such a database. Development and identification of accurate prediction programs for predicting VIV depends on the mutually accepted benchmark datasets with which we compare the prediction. This database should also contain experiments at high Reynold s number on long tensioned beams with sufficiently dense measurements. A standardized, commonly developed, well understood and publicly available database is essential for consolidating the VIV work by various researchers. The author have been an active participant and developer of the collaborative effort in creating a Vortex-Induced Vibration Data Repository (VIVDR) initiative [], which in its present form already contains several such experimental datasets and databases. Transient data and statistical modeling of VIV: For large part of our analyses we utilized a statistically stationary segment of the measured data. These stationary segments of the data were small parts of the entire datasets. A study into the larger segments of the nonstationary (transient) data may yield more insights on the evolution of the riser VIV. Real world experience also indicate a stochastic/chaotic behavior of systems undergoing VIV. Methods based on modeling VIV as a stochastic process may be the solution to phenomenon like multifrequency response, variation in hydrodynamic and structural properties. Filling VIV databases: The force coefficient databases can be extracted from experimental data for a variety of structural and hydrodynamic parameters like very high Reynold s number, riser configurations with strakes and fairings, roughness, traveling v.s. standing waves, and multi v.s. single frequency response. Multi-frequency VIV: Multi-frequency riser response is something which is less understood and which was beyond the scope of the current effort. The issue here is the evident but not well understood problem of leakage of energy from one frequency to the other. We need methods to establish these frequencies of lockin, competition between the modes and span of 65

166 the riser over which these frequencies are dominant. Another closely related issue is the extraction of lift force coefficient databases during a multi-frequency VIV. In a way similar to the work mentioned in Chapter 5, we can then extract the lift force coefficient databases during multi-frequency (beating) VIV. The other questions which require significant thought are more philosophical in nature like the one between reality and experiments. How correct is the measured experimental data? Can we compare one model to other models? Can we compare the experimental measurements to the theoretical predictions? We should also be aware of the fact that by definition all models are necessarily approximate (wrong). It is just that some models are better than the rest and closely follow the reality [55]. 66

167 Appendix A Nomenclature A. Coordinate system definition Figure A-: Definition of coordinate system used throughout the thesis. 67

168 A.2 Definition of various coefficients Consider a single degree of freedom system (SDOF) like an elastically mounted rigid cylinder of length L and diameter D placed in a constant flow with a flow velocity U and fluid density ρ f. The force from fluid to the structure can be written in two parts. One part in phase with acceleration and another one in phase with velocity as depicted in equation (A.). F fluid (t) = F a (t) + F v (t). (A.) For a harmonic loading at frequency f (ω = 2πf), one also expects a harmonic displacement y(t) = A cos(ωt) such that the forcing lags behind the displacement by a phase angle ψ as: F fluid (t) = F cos(ωt ψ), (A.2) = F cos(ψ) cos(ωt) + F sin(ψ) sin(ωt), (A.3) [ ( ρf πd 2 )] [ ( L ρf U 2 )] DL = C la cos(ωt) + C lv sin(ωt), (A.4) 4 2 = [ C m ( ρf πd 2 L 4 The phase angle (ψ) is obtained as: ψ = tan [ ) ] [ ( ( Aω 2 ρf U 2 DL ) cos(ωt) + C lv 2 C lv 2π 3 C m A fd 2 D U The lift coefficient in phase with acceleration (C la ) is defined as: ] )] sin(ωt). (A.5). (A.6) C la = F cos(ψ) ). (A.7) ( ρf U 2 DL 2 The lift coefficient in phase with velocity (C lv ) is defined as: The added mass coefficient (C m ) is defined as: C lv = F sin(ψ) ). (A.8) ( ρf U 2 DL 2 C m = ( ρf πd 2 L 4 F cos(ψ) ) ( Aω 2 ). (A.9) 68

169 If we assume that the SDOF system has a spring constant k, damping constant b and mass m, then we can define the following quantities. The mass ratio (m ) is defined as: m = m ρ f ( πd 2 L 4 ). (A.) The structural damping ratio (ζ) is obtained from the structural damping constant b as: b = 2ζ(m πd 2 L + C m )ρ f f. (A.) 4 The non-dimensional frequency (f ) is obtained from frequency (f) as: f = fd U. (A.2) The reduced velocity (V r ) is obtained from flow velocity U as: V r = U fd = f. (A.3) One can define the nominal natural frequency (f n ) of the cylinder in water as: f n = [ k 2π πd m + ρ 2 L f 4 ]. (A.4) Further, the nominal natural frequency (f n ) can be non-dimensionalized to obtain non-dimensional nominal natural frequency (f n) as: f n = f nd U. (A.5) 69

170 7

171 Appendix B Available riser VIV experimental datasets Throughout the thesis several experimental datasets were used to observe the phenomenon of VIV, and extract pertinent information from the data collected. Further, these datasets were used to illustrate and substantiate the theory and methodologies which have been developed in this thesis. The following VIV experimental datasets will be used throughout the thesis:. Norwegian Deepwater Programme (NDP) riser high mode VIV datasets 2. Lake Seneca experiment datasets 3. First Gulf-stream experiment datasets We describe the setup, the properties of the riser model, the instrumentation, and the peculiarities of each of the experiments. 7

172 B. Norwegian deepwater programme high mode VIV datasets This set of data is obtained from experiments, sponsored by Norwegian Deepwater Programme (NDP) and conducted at Marintek s ocean basin at Trondheim (width: 5m, length: 8m and depth: m) in December 23 [6, 5]. High mode VIV model tests with high length to diameter ratio (L/D) riser were the focus of the tests. A 38m long riser model made of fiber glass pipe (properties described in Table B.) was horizontally towed at different speeds, simulating uniform and triangular shear currents as depicted in Figure B.. The riser was instrumented with strain gages and accelerometers to record both in-line (IL) and cross-flow (CF) bending strains and accelerations along the riser. The maximum number of strain gages in CF direction was 24 and in the IL direction was 4 (refer [5] for the sensor locations). There were 8 biaxial accelerometers recording accelerations in CF and IL directions. Outer Diameter (D).27 m Thickness of Riser (t).3 m Length of Riser (L) 38 m Effective Tension (T ) 4 to 6 N Section Modulus (EI) 37.2 Nm 2 Table B.: Important properties of riser model used in NDP experiment as reproduced from Trim et al. [6, 5]. The riser model was taut horizontally from a crane using two inclined pendulums on each ends as shown in Figure B.. The pathological riser end motions and frame motions were stabilized using clump weights suspended from the pendulums at each end of the riser using a spring mechanism. One of the pendulums was connected to the crane using a gondola which allows for producing prescribed motions of that riser end. For a detailed description of the experimental setup please refer to the NDP technical report [5]. Two types of flow profiles (uniform and triangularly sheared) were simulated by varying the relative velocities at the ends of the riser. To achieve a uniform flow profile, both ends of the riser were towed simultaneously at the same velocity. For a triangular sheared flow profile, one end of the riser was kept at rest while the other side (Gondola side) was made to follow a predefined circular arc. The velocity profiles are obtained by interpolating the velocities between the two ends of the riser. Experiments were conducted for various instances of such flow profiles where the maximum flow velocity varies from U max =.3ms to U max = 2.2ms with steps of.ms. 72

173 Figure B-: Setup of NDP experiment as reproduced from Trim et al. [6, 5]. 73

174 B.2 Lake Seneca experiment datasets This field experiment was sponsored by DeepStar and conducted by the research group of Prof. Kim Vandiver and is described in great detail in references [69, 7]. Two experiments were conducted at the Naval Underwater Warfare Center (NUWC) Lake Seneca Test facility. The preliminary mechanical test was held in November 23, and the final testing was completed in July 24. The experiment setup is described in detail in references [69, 7]. As depicted Figure B-2: Setup of the Lake Seneca experiment as reproduced from Vandiver et al. [7] in Figure B.2 a fiberglass composite pipe with physical properties described in Table B.2 was deployed on a boat. The pipe was tensioned using a railroad wheel suspended at its bottom. The boat was then towed to achieve the necessary flow velocity. Outer Diameter (D).333 m Length of Riser (L) m Effective Tension (T ) 3225 N Section Modulus (EI) 429 Nm 2 Linear Mass Density in Water (m).35 kg/m Nominal Added Mass Density (m an ).76 kg/m Table B.2: Important properties of pipe used in Lake Seneca experiment as reproduced from Vandiver et al. [7]. The pipe was instrumented with 25 tri-axial accelerometers, a top-end tilt meter, a load cell, and two current meters (top and bottom). The pipe was made in several segments each of one foot in length, joined together to complete the entire specimen. This was done to facilitate the placement of accelerometers and cables inside. Flexible epoxy was used to fill the pipe and hold the instrumentation in place within the pipe. The current meters at the top and bottom end of the pipe were designed to measure the current velocity at top and bottom respectively. Each accelerometer measurement was sampled at 6Hz using an analog to digital converter and 74

175 micro-processors located locally at each accelerometer unit. For a detailed description of the experiment please refer [7]. The design of the pipe (its physical dimensions), loading (tension) and flow conditions were done so as to excite the 25 th mode in the cross-flow. The boat was towed with velocities ranging from.3ms to.ms. Test Runs Current Velocity (m/s) S Bare Riser Cases S S S S S % Straked Cases S S S Table B.3: An overview of the various test runs conducted during the Lake Seneca experiment as reproduced from [69, 7]. Acceleration data was acquired at 25 locations along the span of the riser. However, only 7 channels or less contained valid data depending on experiments. The acceleration along each of the CF and IL directions were extracted and was made available to the Deepstar project by Prof. Vandiver s group. In addition, of a maximum of two current meters only one was working and hence a uniform current profile is assumed. 75

176 B.3 First Gulf-stream experiment datasets The Gulf-stream is the fastest ocean current in the world with peak velocities reaching 2ms. This makes the Gulf Stream ideal to perform field experiments which best simulate the VIV in typical offshore structures. This set of data is obtained from experiments performed at offshore Miami in the Gulf Stream on October 29, 24 by Prof. Kim Vandiver s research group. The details of this experiment is given in detail in [7, 72, 7]. Figure B-3: Left: first Gulf Stream experimental setup as reproduced from Vandiver et al. [72]; right: cross-sectional view of the riser depicting placement of the optical strain gage fibers. There are four sets of fiber couples (Q, Q2, Q3, Q4), each containing 7 strain locations. This experiment (depicted in Figure B-3) was similar to the one conducted at Lake Seneca B.2, with a fiberglass pipe tensioned using a railroad wheel at the bottom, and towed using a boat. The exceptions were, strains were measured instead of accelerations, and that the velocity profile was not uniform. For a detailed description of the experiment setup please refer [7, 72]. The riser used in the experiments was instrumented with strain-measuring fiber optic lines. If we consider the cross-section of the riser, we can see two adjacent fibers at each quadrant, for a total of eight fiber optic lines; each fiber with 35 strain gratings. For each pair of adjacent fibers, the strain gratings are at alternate locations along the span of the riser. Hence, if you combine the signals from both fibers we should get a better resolution along the span. Thus, effectively there are four sets of strain data along the span of the riser, denoted as Q, Q2, Q3 and Q4, and arranged along the four quadrants of the riser as shown in the Figure B-3(right). It should however be noted that for the Miami data that the optical fibers were wound in a helical fashion along the span of the cable with a helix-angle of 6. Various sheared velocity profiles were obtained by towing the pipe at different directions along the Gulf Stream current. The sheared velocity profiles (both magnitude and direction) were recorded using an Acoustic Doppler Current Profiler (ADCP). Further two mechanical current meters were placed at the top and bottom to validate the data from ADCP. A tilt meter 76