Wake Oscillator Model Research Articles

Numerical and experimental investigation of nonlinear vortex induced vibration energy converters

In this paper, effects of bi-stable stiffness and hardening stiffness on the performance of a Vortex induced vibration (VIV) energy converter are theoretically and experimentally studied through a wake oscillator model and a computer-based force-feedback testing platform. Based on the simulation and experimental results, it is found that the bi-stable stiffness is potential to allow the system to operate at low velocity water flows, while the hardening stiffness will extend the operating range at high velocity flows. Subsequently, in order to take the advantages of both types of stiffness, a combined nonlinear stiffness is proposed and verified experimentally to demonstrate its capability in improving the overall operating range of the VIV energy converter.

VIM simulation on a cylindrical floating structure

Vortex-induced motion (VIM) on a cylindrical floating structure can be easily found in ocean strong current. Since the VIM on the structure with long-term low-frequency motion causes fatigue damage of the structure’s mooring lines and risers, precise VIM assessment is needed for the safe evaluation of them. In the standard of the International Organization for Standardization ISO19901-7, ‘Specific requirements on stationkeeping systems for floating offshore structures and mobile offshore units’, for instance, a concrete method of assessing VIM displacement is not represented in the standard document, though the requirement on the VIM demands for an assessment on the basis of a proper way. In this paper, a VIM time-domain simulation method on a floating structure with circular cylinder form, that is, for example a Spar, an MPSO (mono-column type floating production storage and off-loading) and so on, is shown using the wake oscillator model. Transverse and in-line VIM are treated for the floater’s motion. The assessment quality of the simulation on the VIM amplitudes of floaters in current is confirmed by model test data.

Design and experiment of controlled bistable vortex induced vibration energy harvesting systems operating in chaotic regions

Abstract Vortex induced vibration based energy harvesting systems have gained interests in these recent years due to its potential as a low water current energy source. However, the effectiveness of the system is limited only at a certain water current due to the resonance principle that governs the concept. In order to extend the working range, a bistable spring to support the structure is introduced on the system. The improvement on the performance is essentially dependent on the bistable gap as one of the main parameters of the nonlinear spring. A sufficiently large bistable gap will result in a significant performance improvement. Unfortunately, a large bistable gap might also increase a chance of chaotic responses, which in turn will result in diminutive harvested power. To mitigate the problem, an appropriate control structure is required to stabilize the chaotic vibrations of a VIV energy converter with the bistable supporting structure. Based on the nature of the double-well potential energy in a bistable spring, the ideal control structure will attempt to drive the responses to inter-well periodic vibrations in order to maximize the harvested power. In this paper, the OGY control algorithm is designed and implemented to the system. The control strategy is selected since it requires only a small perturbation in a structural parameter to execute the control effort, thus, minimum power is needed to drive the control input. Facilitated by a wake oscillator model, the bistable VIV system is modelled as a 4-dimensional autonomous continuous-time dynamical system. To implement the controller strategy, the system is discretized at a period estimated from the subspace hyperplane intersecting to the chaotic trajectory, whereas the fixed points that correspond to the desired periodic orbits are estimated by the recurrence method. Simultaneously, the Jacobian and sensitivity matrices are estimated by the least square regression method. Based on the defined fixed point and the linearized model, the control gain matrix is calculated using the pole placement technique. The results show that the OGY controller is capable of stabilizing the chaotic responses by driving them to the desired inter-well period-one periodic vibrations and it is also shown that the harvested power is successfully improved. For validation purpose, a real-time experiment was carried out on a computer-based forced-feedback testing platform to validate the applicability of the controller in real-time applications. The experimental results confirm the feasibility of the controller to stabilize the responses.

A vortex-induced vibration model for the fatigue analysis of a marine drilling riser

ABSTRACTThe objective is to propose a quick and accurate vortex-induced vibration (VIV) model to predict the fatigue life of marine drilling risers. The nonlinear wake oscillator model was applied to represent the lift force, which is described by a fourth-order partial differential equation set. Next, we adopted the generalised integral transform technique (GITT) to transform it into second-order ordinary differential equation system, which was solved by the Runge-Kutta method. Validation of the numerical model was implemented and a good agreement was achieved. Next, an S-N curve was adopted to conduct fatigue prediction. The results demonstrate that under the conditions set forth in this paper, an increase in the current and outer diameter could reduce fatigue life, except when the outer diameter reaches a certain value, at which point the tendency is reversed. Moreover, when the top tension coefficient reaches a certain value, the fatigue life tendency is also reversed.

Study on pure IL VIV of a free spanning pipeline under general boundary conditions

Pipeline spans may occur due to natural seabed irregularities or local scour of bed sediment. The pure in-line (IL) vortex-induced vibrations (VIV) analysis of the free spans is an important subject for design of pipeline in uneven seabed. The main objective of this paper is to analyze the characteristics of pure IL VIV of a free spanning pipeline under general boundary conditions. An IL wake oscillator model which can describe the coupling of pipeline structure and fluctuating drag is introduced and employed. The coupled partial differential equations of structure and wake are transformed into a set of ordinary differential equations using two-mode Galerkin method. Some case studies are presented and thoroughly discussed in order to investigate the effects of internal fluid, axial force and boundary conditions on the pure IL VIV.

Vortex-induced vibrations of a freely vibrating cylinder near a plane boundary: Experimental investigation and theoretical modelling

This work reports on experiments that were performed with a freely vibrating cylinder exposed to currents and placed near a plane boundary parallel to the cylinder axis. It is observed that the proximity of the boundary affects the vertical response of the cylinder in two ways: (i) for gaps between 0.75 and 2 diameters (D), the amplitude of oscillation is reduced; (ii) for gaps smaller than 0.75D, the cylinder impacts the boundary, resulting in an increase of amplitudes and frequencies of oscillations as the flow is accelerated. The in-line force acting on the cylinder is also examined, and the dependency of its harmonic components on the flow velocity and distance to the boundary is evaluated. Besides the typical amplification of the mean component inside the lock-in region, it is also observed that as the cylinder is placed closer to the boundary, the harmonic component with the frequency of the vertical oscillations increases, while the component with twice that frequency decreases in similar amount.Based on the experimental observations, an existing wake-oscillator model for vortex-induced vibrations is enhanced in order to account for the effect of the boundary. The proposed model introduces an effective damper that is activated when the cylinder reaches a certain distance from the boundary, and a damper/spring set representing the rigidity of the boundary and the dissipation of energy due to impact.

Interference of Vortex-Induced Vibration and Galloping: Experiments and Mathematical Modelling

This paper deals with the phenomenon of interference between vortex-induced vibration and transverse galloping, which occurs in the case of low-damped rectangular cylinders with a side ratio comprised between about 0.75 and 3. The results of a wide experimental campaign carried out in the wind tunnel are discussed, focusing on the key role played by the mass-damping parameter (Scruton number) of the oscillating body. It was found that a high value of the Scruton number is required to decouple completely the ranges of excitation due to vortex-induced vibration and galloping, and for the quasi-steady theory to predict correctly the galloping critical wind speed. Moreover, the problem is analytically addressed through a nonlinear wake-oscillator model, which is able to reproduce several features of the complicated phenomenon, provided that a crucial parameter is correctly calibrated.

Experimental chaotic quantification in bistable vortex induced vibration systems

The study of energy harvesting by means of vortex induced vibration systems has been initiated a few years ago and it is considered to be potential as a low water current energy source. The energy harvester is realized by exposing an elastically supported blunt structure under water flow. However, it is realized that the system will only perform at a limited operating range (water flow) that is attributed to the resonance phenomenon that occurs only at a frequency that corresponds to the fluid flow. An introduction of nonlinear elements seems to be a prominent solution to overcome the problem. Among many nonlinear elements, a bistable spring is known to be able to improve the harvested power by a vortex induced vibrations (VIV) based energy converter at the low velocity water flows. However, it is also observed that chaotic vibrations will occur at different operating ranges that will erratically diminish the harvested power and cause a difficulty in controlling the system that is due to the unpredictability in motions of the VIV structure. In order to design a bistable VIV energy converter with improved harvested power and minimum negative effect of chaotic vibrations, the bifurcation map of the system for varying governing parameters is highly on demand.In this study, chaotic vibrations of a VIV energy converter enhanced by a bistable stiffness element are quantified in a wide range of the governing parameters, i.e. damping and bistable gap. Chaotic vibrations of the bistable VIV energy converter are simulated by utilization of a wake oscillator model and quantified based on the calculation of the Lyapunov exponent. Ultimately, a series of experiments of the system in a water tunnel, facilitated by a computer-based force-feedback testing platform, is carried out to validate the existence of chaotic responses. The main challenge in dealing with experimental data is in distinguishing chaotic response from noise-contaminated periodic responses as noise will smear out the regularity of periodic responses. For this purpose, a surrogate data test is used in order to check the hypotheses for the presence of chaotic behavior. The analyses from the experimental results support the hypothesis from simulation that chaotic response is likely occur on the real system.

Optimal energy harvesting from vortex-induced vibrations of cables.

Vortex-induced vibrations (VIV) of flexible cables are an example of flow-induced vibrations that can act as energy harvesting systems by converting energy associated with the spontaneous cable motion into electricity. This work investigates the optimal positioning of the harvesting devices along the cable, using numerical simulations with a wake oscillator model to describe the unsteady flow forcing. Using classical gradient-based optimization, the optimal harvesting strategy is determined for the generic configuration of a flexible cable fixed at both ends, including the effect of flow forces and gravity on the cable's geometry. The optimal strategy is found to consist systematically in a concentration of the harvesting devices at one of the cable's ends, relying on deformation waves along the cable to carry the energy towards this harvesting site. Furthermore, we show that the performance of systems based on VIV of flexible cables is significantly more robust to flow velocity variations, in comparison with a rigid cylinder device. This results from two passive control mechanisms inherent to the cable geometry: (i) the adaptability to the flow velocity of the fundamental frequencies of cables through the flow-induced tension and (ii) the selection of successive vibration modes by the flow velocity for cables with gravity-induced tension.

2DOF CFD calibrated wake oscillator model to investigate vortex-induced vibrations

In this study a new two degrees-of-freedom wake oscillator model is proposed to describe vortex-induced vibrations of elastically supported cylinders capable of moving in cross-flow and in-line directions. The total hydrodynamic force acting on the cylinder is obtained here as a sum of lift and drag forces, which are defined as being proportional to the square of the magnitude of the relative flow velocity around the cylinder. The two van der Pol type oscillators are then used to model fluctuating drag and lift coefficients. As the relative velocity around the cylinder depends both on the fluid flow velocity and the velocity of the cylinder, the equations of motions of the cylinder in cross-flow and in-line directions become coupled through the fluid forces. It is shown that such approximation of the fluid forces allows to obtain the well known low dimensional models in the limit case, and the model proposed by Facchinetti et al. [1] to describe the cross-flow vibrations is used as an example. Existing experimental data and Computational Fluid Dynamics (CFD) results are used to calibrate the proposed model and to verify the obtained predictions of complex fluid-structure interactions for different mass ratios. The "super upper" branch phenomenon, exclusive for a two degrees-of-freedom motion at low mass ratios, has been observed. The influence of the empirical parameters of the wake oscillators and fluid forces coefficients on the dynamic responses is also discussed.

A novel Wake Oscillator Model for simulation of cross-flow vortex induced vibrations of a circular cylinder close to a plane boundary

This paper presents a novel Wake Oscillator Model for predicting Vortex-Induced Vibration (VIV) of a circular cylinder close to a plane boundary. The innovation lies in its extension of the classic Van der Pol equation by introducing an additional nonlinear term and two empirical coefficients to account explicitly for the effects of the gap ratio and mass ratio. The model parameters are calibrated against available experimental data and a number of validation tests are then performed. Despite the intrinsic limitations of the Wake Oscillator Model approach, the simulation results demonstrate that the predicted variations of vibration amplitudes with the reduced velocity are broadly consistent with the experimental data and the model is also capable of capturing the different vortex shedding modes. With further refinements, the proposed modelling approach is expected to provide an efficient engineering means to predict both the overall variation trend and maximum value of vibration amplitude of a circular cylinder close to a plane boundary over the full range of reduced velocities.

VIV fatigue reliability analysis of marine risers with uncertainties in the wake oscillator model

Uncertainties are rife in the fatigue life prediction of marine risers subjected to vortex-induced vibration (VIV). Industry deals with this issue by imposing large factors of safety that may not be properly justified, resulting in over-conservative riser designs in general. One important source of uncertainty arises from the VIV prediction models. This paper focusses on identifying the uncertainties of a wake oscillator model which approximates the fluctuating hydrodynamic force coupled with the riser equation of motion for nonlinear fluid–structure interaction analysis. This van der Pol-type oscillator relies on two wake coefficients which are described deterministically by empirical equations obtained via curve-fitting. However, the underlying data exhibit wide scatter; thus, it is proposed to model the two key coefficients as random variables. Based on experimental data, the joint probability density function of the variables is approximated. A new fast reliability approach is proposed for the VIV fatigue reliability analysis, while Monte Carlo simulations are performed for comparisons. Case studies of a vertical riser in a uniform flow show that the proposed method compares favorably with Monte Carlo in terms of predicting the failure probability as well as safety factors conforming to prescribed reliability levels. Moreover, this study reveals that the randomness of wake coefficients leads to large variability in the riser fatigue damage. The correlation between the coefficients should be properly incorporated as it affects the fatigue reliability of risers experiencing VIV.

Fatigue Analysis of Deep-Water Risers under Vortex-Induced Vibration Considering Parametric Excitations

ABSTRACT Zhang, J., and Tang, Y., 2015. Fatigue analysis of deep-water risers under vortex-induced vibration considering parametric excitations. This paper presents vortex-induced vibration (VIV) and fatigue analysis of deep-water risers in shear flow considering parametric excitations due to the heave of floating platform. As a first step, the governing equation of transverse motion of the riser is established; the dynamic response of the riser is calculated numerically by the finite difference method based on Van der Pol wake-oscillator model. Then the fatigue damage of the riser is obtained by applying Miner cumulative damage theory and S-N curve. The results show that the VIV response of the riser, especially in the flow with low speed, is much larger due to the parametric excitation. The bending stress and the fatigue damage at the bottom of the riser are significantly increased under parametric excitations.

Time-delay feedback controller for amplitude reduction in vortex-induced vibrations

We investigate the possibility of using a time-delay feedback control to suppress the vortex-induced vibrations of an elastically mounted circular cylinder. An appropriate reduced-order wake-oscillator model is employed to determine the cylinder’s displacement and lift fluctuating coefficient in a coupled manner. A parametric study is performed to investigate the effects of the time-delay feedback control on the cross-flow oscillations of the circular cylinder. To study the effects of this controller on the coupled frequency and damping of the aeroelastic system, a linear analysis is performed. It is demonstrated that the presence of time-delay feedback control can result in an increase or decrease in the coupled damping of the aeroelastic system, varying from negative to positive values periodically. Then, the effects of this time-delay feedback controller on the nonlinear responses of the circular cylinder are determined. The results show that a good choice of time-delayed controller parameters can be efficiently implemented to significantly decrease or suppress the vortex-induced vibrations amplitudes in the lock-in or synchronization region and hence greatly reduce the potential oscillation hazard in engineering applications.

Wake oscillator model for assessment of vortex-induced vibration of flexible structures under wind action

The wake oscillator model is considered to be the most proper semi-empirical model for qualitatively or even quantitively assessing the vortex-induced vibration of a structure in the design stage. However, the complexities in parameter determination limit its usage in real engineering fields. Moreover, this kind of model is traditionally used to describe the experimental phenomena of two-dimensional cross-section models, the gaps between the results of two-dimensional cross-section model tests and the prototype responses of the three-dimensional real structures should be filled up. In this paper, a detailed study of the wake oscillator model is firstly presented, which demonstrates the influences of model parameters on the VIV features described by the wake oscillator model. Then, a new method for conveniently identifying the model parameters in the wake oscillator model is proposed. Besides, a new procedure, based on the wake oscillator model, for interpreting the wind tunnel test results into real structures is also proposed. The results are validated by wind tunnel tests and compared with field measurements.

Modeling of fluid–structure interaction for simulating vortex-induced vibration of flexible riser: finite difference method combined with wake oscillator model

This paper proposes a numerical simulation method for the dynamic motion of a flexible riser pipe undergoing vortex-induced vibration (VIV). The method is based on a finite difference scheme for solving nonlinear structural dynamics of the pipe and wake oscillator model for quantifying vortex-induced forces acting on the pipe, the combination of which can offer a very efficient and stable computation. To investigate the accuracy of the method, we performed simulations of the VIV of riser pipes under uniform flow and sheared flow conditions; and then compared obtained results with experiments of preceding works. We consequently confirmed that the present method can simulate a couple of important aspects of the VIV of the pipes: frequency, mode shape, and amplitude of displacement of cross-flow displacement.

Numerical Modeling of the Dynamical Interaction Between Slug Flow and Vortex Induced Vibration in Horizontal Submarine Pipelines

Fatigue life of submarines pipelines is known to depend on many factors, e.g., materials, spanning length, vortex induced vibration (VIV), pipe soil-interaction, etc. In the case of VIV, the external flow due to marine currents can generate the well known alternate vortex shedding regime. In this regime, a time-varying pressure distribution over the surface of the pipeline imposes forces upon the pipeline itself, thus producing vibration with a defined frequency. On the other hand, in recent years some authors have shown that slug flow regime may produce a cyclic damage that could reduce in a significant way the fatigue life of submarine pipelines, thus constituting a governing mechanism in their design. In slug flow regime, slugs traveling in the pipe act as moving gravity loads for the pipeline structure, producing a dynamic response, especially important for the free spans. If both frequencies of the before mentioned effects are closer and, in addition, are in the same range of the natural frequency of the pipeline span, resonance effects can be expected to be reinforced and drastic changes in the dynamics of the pipeline could appear. In this work, a first study of the interaction between slug flow induced vibration in horizontal pipelines and cross-flow response due to vortex shedding is presented. The fluid model was based on the classical wake oscillator model. A numerical model based on the finite difference method was implemented for the structure. Two particular extreme cases were modeled to analyze the pipeline dynamics for “small-size” and “large-size” slugs, for a range of marine current velocities. For the case involving small-size slugs, it was observed a near 10% increment in the vibration amplitude (compared to a reference value), while in the case of “large-slugs” the VIV was overweighed by the slug induced vibration (SIV) phenomenon.

On the efficiency of energy harvesting using vortex-induced vibrations of cables

Many technologies based on fluid–structure interaction mechanisms are being developed to harvest energy from geophysical flows. The velocity of such flows is low, and so is their energy density. Large systems are therefore required to extract a significant amount of energy. The question of the efficiency of energy harvesting using vortex-induced vibrations (VIV) of cables is addressed in this paper, through two reference configurations: (i) a long tensioned cable with periodically-distributed harvesters and (ii) a hanging cable with a single harvester at its upper extremity. After validation against either direct numerical simulations or experiments, an appropriate reduced-order wake-oscillator model is used to perform parametric studies of the impact of the harvesting parameters on the efficiency. For both configurations, an optimal set of parameters is identified and it is shown that the maximum efficiency is close to the value reached with an elastically-mounted rigid cylinder. The variability of the efficiency is studied in light of the fundamental properties of each configuration, i.e. body flexibility and gravity-induced spatial variation of the tension. In the periodically-distributed harvester configuration, it is found that the standing-wave nature of the vibration and structural mode selection plays a central role in energy extraction. In contrast, the efficiency of the hanging cable is essentially driven by the occurrence of traveling wave vibrations.

Using vortex strength wake oscillator in modelling of vortex induced vibrations in two degrees of freedom

A new wake oscillator model is established to predict the structural response characteristics of vortex induced vibration (VIV) in two degrees of freedom. Based on the two-dimensional potential flow approach, the streamwise and transverse fluctuating fluid forces acting on structure are simplified and quantified. The work–energy balanced between the fluid and the structure leads to establish the coupled dynamic model by introducing a displacement variable related to the strength of nascent vortex. Analysis and prediction of the amplitudes, frequencies and phase angles of x–y motions of a rigid 2-D circular cylinder, along with a comparison to the existing experimental results, show that the numerical solutions of the present model can qualitatively and quantitatively capture the important features of VIV. Moreover, the x–y trajectory displays a crescent shape at cross-flow approaching peak amplitude. The reduced-order model also used in predicting VIV response of a 3-D top tensioned riser undergoing a stepped current is presented. The simulation results highlight the combination of standing wave and traveling wave occurring in structure vibration. The trajectories exhibit various patterns of figure-of-eight orbital motions and phase angles between the in-line and cross-flow motion change more rapidly in the standing wave region.

Vortex-Induced Dynamic Response of Submerged Floating Tunnel Tether Based on Wake Oscillator Model

To study the vibration response of submerged floating tunnel tether under the combined action of vortex-induced vibration and parametric vibration, a non-linear vibration equation based on wake oscillator model is set up taking the geometric nonlinearity of tether into account, in which effect of tube on tether is simplified as axial excitation. An approximate numerical solution of planning submerged floating tunnel tether is obtained by applying Galerkin method and Runge-kutta method. The variation degree of mid-span displacement response and axial force of tether is analyzed when the vortex-induced resonance and parametric resonance occur. The results show that, when vortex-induced resonance and parametric resonance occur simultaneously, the mid-span displacement and axial force of tether increase dramatically; the tether sag effect results in the asymmetry of tether mid-span vibration amplitude.