柔性圆柱涡激振动流体力系数识别及其特性

涡激振动是诱发海洋立管、浮式平台系泊缆和海底悬跨管道等柔性圆柱结构疲劳损伤的重要因素.目前,海洋工程中用于柔性圆柱涡激振动预报的流体力系数主要来源刚性圆柱横流向受迫振动的实验数据,存在一定缺陷和误差.本文综合考虑横流向与顺流向振动耦合作用,建立了柔性圆柱涡激振动流体力模型,运用有限元法和最小二乘法确定升力系数、脉动阻力系数和附加质量系数.为了准确识别柔性圆柱涡激振动流体力系数,设计并开展了拖曳水池模型实验,实验用柔性圆柱模型的质量比为1.82,长径比为195.5.通过与刚性圆柱流体力系数对比,深入分析了柔性圆柱流体力系数的特性.结果表明:柔性圆柱在一阶模态控制区,流体力系数随约化速度变化趋势与刚性圆柱大致相似;二阶模态控制区,升力系数和脉动阻力系数显著增大;附加质量系数在响应频率较低时与振动位移的相关性增强;当响应频率较低时,振动位移较大区域为能量耗散区,当响应频率较高时,振动位移较大区域为能量输入区.     

Application of wake oscillators to two-dimensional vortex-induced vibrations of circular cylinders in oscillatory flows

A nonlinear time-domain simulation model for predicting two-dimensional vortex-induced vibration (VIV) of a flexibly mounted circular cylinder in planar and oscillatory flow is presented. This model is based on the utilization of van der Pol wake oscillators, being unconventional since wake oscillators have typically been applied to steady flow VIV predictions. The time-varying relative flow–cylinder velocities and accelerations are accounted for in deriving the coupled hydrodynamic lift, drag and inertia forces leading to the cylinder cross-flow and in-line oscillations. The system fluid–structure interaction equations explicitly contain the time-dependent and hybrid trigonometric terms. Depending on the Keulegan–Carpenter number (KC) incorporating the flow maximum velocity and excitation frequency, the model calibration is performed, entailing a set of empirical coefficients and expressions as a function of KC and mass ratio. Parametric investigations in cases of varying KC, reduced flow velocity, cylinder-to-flow frequency ratio and mass ratio are carried out, capturing some qualitative features of oscillatory flow VIV and exploring the effects of system parameters on response prediction characteristics. The model dependence of hydrodynamic coefficients on the Reynolds number is studied. Discrepancies and limitations versus advantages of the present model with different feasible solution scenarios are illuminated to inform the implementation of wake oscillators as a computationally efficient prediction model for VIV in oscillatory flows.     

Nonlinear modeling of combined galloping and vortex-induced vibration of square sections under flow

In this paper, we propose a model for the transverse oscillation of a square-section cylinder under flow. The fluctuating transverse force due to vortex shedding is represented using a coupled nonlinear wake oscillator, while the unsteady force for galloping caused by the varying incidence angle effects is modelled using the quasi-steady approach. First, we analytically investigate the lift behavior and phase angle variation of the square cylinder under forced vibrations. Comparison with experimental data is used to determine the form of the coupling terms and its values. The present model shows advantages in predicting the phase angle, and it successfully captures the change in sign of the phase. Second, the proposed model is directly applied in predicting free oscillation cases without any tuning. The dynamical behaviors predicted by this model are compared with published experiments under different Scruton numbers, and reasonable agreement can be found. The results indicate that the model can not only be applied in simulating the “pure galloping” and “pure VIV,” but also is able to capture the interactions of VIV and galloping, including combined and separate VIV-galloping motions.

     

Multi-modes approach to modelling of vortex-induced vibration

In this work the fluid–structure interactions are considered by investigating a straight but slender pipe interacting with uniform water flow. Two configurations are studied, namely vertically and horizontally positioned pipes, which are modelled as an Euler–Bernoulli beam with flexural stiffness. Both pretension and length-wise mass distribution are considered. The structure is assumed to be moving only in the direction normal to flow (cross-flow motion) hence its in-line motion is neglected. The external fluid force acting on the structure is the result of the action of sectional vortex-induced drag and lift forces. Only mean drag force is considered, with time varying lift force modelled using a non-linear oscillator equation of the Van der Pol type. The obtained coupled system of non-linear partial differential equations is simplified employing Galerkin-type discretisation. The resulting ordinary differential equations are solved numerically providing multi-mode approximations of cross-flow displacement and non-dimensional lift coefficient. The comparison between the responses of vertical and horizontal structures shows that, as expected, due to a balancing between pretension and weight, in general a higher amplitude of vibration is observed for the vertical configuration than in the same location along the pipe for the horizontal configuration in the lower part of the structure. However, lower amplitudes are obtained in the upper part of the pipe. The horizontal configuration solutions are identical in symmetrical locations along the pipe due to constant pretension. The influence of the wake equation coefficients and the fluid force coefficients on the response amplitudes has been also considered together with the length of the pipe and pretension level, and the appropriate response curves are included. Finally, for the higher mode approximations it has been shown that the vibrations level at lower frequencies is predicted reasonably well by retaining only a small subset of modes.     

Optimal lift force coefficient databases from riser experiments

Significant past effort has gone into understanding the complicated flow–structure interaction problem of vortex-induced vibration (VIV) of long flexible cylindrical structures (e.g., risers, mooring lines, tendons, conductors) in the ocean environment. However, major challenges persist with regard to riser VIV modeling and response prediction. The existing prediction schemes are based on a number of hypotheses, experimental facts and data like strip theory, energy balance, correlation length and, most importantly, the use of lift force coefficient databases. Recent advances in observing the VIV motions on experimental risers with high confidence shows that some of these assumptions may not be valid. One important source of the discrepancies between theoretical estimates and experimental observations arise from the use of experimentally obtained lift coefficient databases. These databases were obtained under the laboratory conditions of limited Reynolds number, and under the assumption that the cross-flow motions are not influenced by restraining the in-line motions. In this paper we develop a method to improve the modeling capability of riser VIV by extracting empirical lift coefficient databases from field riser VIV measurements. The existing laboratory-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 objective is to minimize the error between the prediction using a theoretical model and the experimental data. Application of the method to data from the Norwegian Deepwater Programme experiments shows that the new optimal databases significantly reduce the error in estimating the riser VIV cross-flow response.     

Shear flow induced vibrations of long slender cylinders with a wake oscillator model

A time domain model is presented to study the vibrations of long slender cylinders placed in shear flow. Long slender cylinders such as risers and tension legs are widely used in the field of ocean engineering. They are subjected to vortex-induced vibrations(VIV) when placed within a transverse incident flow. A three dimensional model coupled with wake oscillators is formulated to describe the response of the slender cylinder in cross-flow and in-line directions. The wake oscillators are distributed along the cylinder and the vortex-shedding frequency is derived from the local current velocity. A non-linear fiuid force model is accounted for the coupled effect between cross-flow and in-line vibrations. The comparisons with the published experimental data show that the dynamic features of VIV of long slender cylinder placed in shear flow can be obtained by the proposed model,such as the spanwise average displacement,vibration frequency,dominant mode and the combination of standing and traveling waves. The simulation in a uniform flow is also conducted and the result is compared with the case of nonuniform flow. It is concluded that the flow shear characteristic has significantly changed the cylinder vibration behavior.     

An experimental investigation of one- and two-degree of freedom VIV of cylinders

In the paper,an experiment investigation was conducted for one-and two-degree of freedom vortex-induced vibration(VIV) of a horizontally-oriented cylinder with diameter of 11 cm and length of 120 cm.In the experiment,the spring constants in the cross-flow and in-line flow directions were regulated to change the natural vibration frequency of the model system.It was found that,in the one-degree of freedom VIV experiment,a "double peak" phenomenon was observed in its amplitude within the range of the reduced velocities tested,moreover,a "2T" wake appeared in the vicinity of the second peak.In the two-degree of freedom VIV experiment,the trajectory of cylinder exhibited a reverse "C" shape,i.e.,a "new moon" shape.Through analysis of these data,it appears that,besides the non-dimensional in-line and cross-flow natural vibration frequency ratios,the absolute value of the natural vibration frequency of cylinder is also one of the important parameters affecting its VIV behavior.     

Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities

A general low-order fluid–structure interaction model capable of evaluating the multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures is presented. Cross-flow motions due to unsteady lift forces of inclined sagged cables and tensioned beams in uniform currents are investigated. In contrast to a linear equation governing the transverse motion of straight beams or cables typically considered in the literature, coupled horizontal/vertical (axial/transverse) displacements and geometric nonlinearities of curved cable (straight beam) are accounted for. A distributed nonlinear wake oscillator is considered in the approximation of space–time varying hydrodynamics. This semi-empirical fluid force model in general depends on the mass-damping parameter and has further been modified to capture both the effects of varying initial curvatures of the inclined cylinder and the Reynolds number. Numerical simulations are performed in the case of varying flow velocities and parametric results highlight several meaningful aspects of vortex-induced vibrations of long flexible cylinders. These comprise multi-mode lock-in, sharing, switching and interaction features in the space and time domains, the estimated maximum modal and total amplitudes, the resonant nonlinear modes of flexible cylinders and their space–time modifications, and the influence of fluid/structure parameters. A shortcoming of single-mode or linear structural model is underlined. Some quantitative and qualitative comparisons of numerical/experimental results are discussed to demonstrate the validity and required improvement of the proposed modelling and analysis predictions.     

On the validity of the independence principle applied to the vortex-induced vibrations of a flexible cylinder inclined at 60°

The vortex-induced vibrations (VIV) of a flexible cylinder inclined at 60° are investigated by means of direct numerical simulation, at a Reynolds number equal to 500, based on the cylinder diameter and inflow velocity. The cylinder has a circular cross-section and a length to diameter aspect ratio equal to 50; it is modeled as a tension-dominated structure which is free to oscillate in the in-line and cross-flow directions. The behavior of the coupled fluid–structure system is examined for two values of the tension. Particular attention is paid to the validity of the independence principle (IP) which states that the inclined and normal-incidence body cases are comparable if the inflow velocity normal component is used to scale the physical quantities.The flexible cylinder exhibits regular VIV for both values of the tension. In the high-tension configuration, where the in-line bending of the structure remains small, the IP is shown to be valid for the prediction of the cylinder responses and the fluid forces. In contrast, in the lower-tension configuration, the behavior of the fluid–structure system deviates from the IP. It is shown that this deviation is connected to the larger in-line bending of the structure which leads to considerably different profiles of the flow velocity locally perpendicular to the body in the inclined and normal cylinder cases. Since the system behavior appears to be mainly driven by this component of the flow, the profile modification induced by the larger in-line bending results in distinct responses: multi-frequency vibrations are observed in the inclined cylinder case whereas mono-frequency oscillations of larger amplitudes develop at normal incidence.     

大柔性圆柱体两自由度涡激振动试验研究

基于模型试验研究了柔性圆柱体两自由度涡激振动问题, 研究了顺流向涡激振动和横向涡激振动的频率与振幅关系, 提出了考虑流固耦合的两自由度涡激振动非线性分析模型. 研究表明, 在不同的流速(雷诺数)范围, 柔性圆柱体顺流向涡激振动与横向涡激振动的频率比和幅值比是不同的; 在非锁定区, 圆柱体的顺流向振动频率与横向振动频率相同, 在锁定区, 圆柱体的顺流向振动频率是横向振动频率的两倍; 在非锁定区, 顺流向振幅与横向振幅比约为1, 而在锁定区, 顺流向振幅与横向振幅比约为1/3~2/3.      

Multi-frequency vortex-induced vibrations of a long tensioned beam in linear and exponential shear flows

The multi-frequency vortex-induced vibrations of a cylindrical tensioned beam of aspect ratio 200, free to move in the in-line and cross-flow directions within first a linearly and then an exponentially sheared current are investigated by means of direct numerical simulation, at a Reynolds number equal to 330. The shape of the inflow profile impacts the spectral content of the mixed standing-traveling wave structural responses: narrowband vibrations are excited within the lock-in area, which is limited to a single region lying in the high flow velocity zone, for the linear shear case; in contrast, the lock-in condition occurs at several spanwise locations in the exponential shear case, resulting in broadband responses, containing a wide range of excited frequencies and spatial wavenumbers. The broadband in-line and cross-flow vibrations occurring for the exponential shear current have a phase difference that lies within a specific range along the entire span; this differs from the phase drift noted for narrowband responses in linear shear flow. Lower vibration amplitudes, time-averaged and fluctuating in-line force coefficients are observed for the exponential shear current. The cross-flow force coefficient has comparable magnitude for both inflow profiles along the span, except in zones where the broadband vibrations are under the lock-in condition but not the narrowband ones. As in the narrowband case, the fluid forces associated with the broadband responses are dominated by high frequencies related to high-wavenumber vibration components. Considerable variability of the effective added mass coefficients along the span is noted in both cases.     

Hydrodynamic and Geometric Stiffening Effects on the Out-of-Plane Waves of Submerged Cables

This study focuses on the relative importance of two sources of nonlinearities affecting submerged cable response. The first of these is the added fluid damping offered by the surrounding medium while the second is the geometric stiffening offered by the cable through finite extensions of its centerline. The contribution of each nonlinear effect, taken separately and in tandem, is evaluated herein through the study of structural waves that form in the (out-of-plane) direction normal to the cable equilibrium plane.Numerical solutions are pursued herein using a finite difference algorithm which is brought to bear on two nonlinear cable/fluid models including: (1)~a nonlinear submerged cable model in which hydrodynamic drag is the sole nonlinear mechanism (referred to herein as the 'nonlinear drag model'); and (2)~a nonlinear submerged cable model in which hydrodynamic drag and geometric stiffening are both active nonlinear mechanisms (the 'nonlinear elastic-drag model'). Numerical solutions for propagating cable waves are developed for the case of a long suspension subjected to a concentrated harmonic excitation source. Conclusions are subsequently drawn regarding the spatial decay of the resulting out-of-plane waves and the dynamic cable tension induced by these waves. The effect of these two nonlinear mechanisms is further explored through the analysis of two additional, linear models: (3)~a simple linear taut string model without drag (the 'simple model'); and (4)~a linear taut string model with linear drag (the 'linear drag model'). The results of all models are critically compared and the range of validity of the linear/cable fluid models are assessed.     

Effect of internal flow on vortex-induced vibration of risers

Although there are many studies dedicated to the problem of vortex-induced vibration (VIV) of marine risers, VIV experiments with internally flowing fluid have not been carried out before. In order to investigate this area, the present experiment with an internally flowing fluid and external current was designed. The riser was towed in the water flume with varying internal flow speeds. The tests in still water and in a current were conducted successfully. Various measurements were obtained including the frequency responses and the time-domain tracing of in-line and cross-flow responses. The experimental results exhibit several valuable features. First, with an increase in internal flow speed, the response amplitude increases while the vibration frequency decreases. Secondly, internally flowing fluid lessens the correlation of the vibration between different sections. In addition, by plotting both in-line strain and cross-flow strain simultaneously, a figure-of-eight for bending strain is also observed, and the trajectories in different cycles are more concordant with the increase of internal flow speed.     

An experimental investigation on vortex induced vibration of a flexible inclined cable under a shear flow

In the present study, an experimental investigation was performed to characterize the vortex induced vibration (VIV) of a flexible cable in an oncoming shear flow. The VIV tests were conducted in a wind tunnel with a flexible cable model. It was found that, under different oncoming velocity profiles, the cable model behaved in single-mode and multi-mode VIVs. The displacement amplitudes of the single mode VIVs were found to be larger than those of multi-mode VIVs, and the cross-flow (CF) response was larger than that of in-line (IL) direction for either the single mode or multi-mode VIVs. For a single mode vibration, the largest CF response occurs in the 1st mode VIV, and the motion trajectory of the 1st mode VIV was found to be an inclined figure of eight shape, while other single mode VIVs behaved in ellipse or straight line trajectories. For multi-mode VIVs, no stable vibration trajectories were found to exist since the vibration frequency bands covered two or more vibration modes. The vortex-shedding frequencies in the wake behind the inclined cable were also characterized in the present study. The shedding frequencies of the wake vortices were found to coincide well with the vibration modes: for a single mode VIV, they were close to the dominant vibration mode; for a multi-mode VIV, they could also cover the appearing vibration modes.     

Three-phase coupled modelling research on rain–wind induced vibration of stay cable based on lubrication theory

Cables of cable-stayed bridges may vibrate with large amplitude under wind and rain, which is known as rain–wind induced vibration (RWIV). According to the pervious researches, the formations and oscillations of rivulets on stay cable surface play important roles in RWIV. In this paper, four different 2D models are presented based on lubrication theory, and the best way of simulating RWIV through lubrication theory is confirmed by the comparisons of rivulet motions and cable vibration responses between these four models and pervious researches. On this basis, the relations among rivulet motions, cable aerodynamic forces and vibration responses are investigated to reveal the mechanism of RWIV. Numerical simulation results show that when RWIV occurs, the periodic oscillations of rivulets around cable lead to the periodic fluctuations of cable lift and drag, whose frequencies are almost equal and close to cable natural frequency. Under the periodically fluctuant lift and drag, cable vibrates with large amplitude in across-wind and along-wind directions, which may further enhance the circumferential oscillations of rivulets conversely. These confirm the conclusion that the resonance between rivulets and cable oscillation may be one of the main reasons for RWIV.     

Modelling of coupled cross-flow and in-line vortex-induced vibrations of flexible cylindrical structures. Part II: on the importance of in-line coupling

Nonlinear Dynamics - To illustrate the influence of the in-line coupling on the prediction of vortex-induced vibration (VIV), the simulation results of the coupled cross-flow and in-line VIVs of...     

Modelling of coupled cross-flow and in-line vortex-induced vibrations of flexible cylindrical structures. Part I: model description and validation

Nonlinear Dynamics - This paper is first of the two papers dealing with the nonlinear modelling and investigation of coupled cross-flow and in-line vortex-induced vibrations (VIVs) of flexible...     

Two-degree-of-freedom vortex-induced vibrations of a circular cylinder at Re=3900

The vortex-induced vibrations of an elastically mounted circular cylinder are investigated on the basis of direct numerical simulations. The body is free to move in the in-line and cross-flow directions. The natural frequencies of the oscillator are the same in both directions. The Reynolds number, based on the free stream velocity and cylinder diameter, is set to 3900 and kept constant in all simulations. The behavior of the coupled flow-structure system is analyzed over a wide range of the reduced velocity (inverse of the natural frequency) encompassing the lock-in range, i.e. where body motion and flow unsteadiness are synchronized. The statistics of the structural responses and forces are in agreement with prior experimental results. Large-amplitude vibrations develop in both directions. The in-line and cross-flow oscillations are close to harmonic; they exhibit a frequency ratio of 2 and a variable phase difference across the lock-in range. Distinct trends are noted in the force-displacement phasing mechanisms in the two directions: a phase difference jump associated with a sign change of the effective added mass and a vibration frequency crossing the natural frequency is observed in the cross-flow direction, while no phase difference jump occurs in the in-line direction. Higher harmonic components arise in the force spectra; their contributions become predominant when the cylinder oscillates close to the natural frequency. The force higher harmonics are found to impact the transfer of energy between the flow and the moving body, in particular, by causing the emergence of new harmonics in the energy transfer spectrum.     

Parametric study of a two degree-of-freedom cylinder subject to vortex-induced vibrations

This numerical work is an attempt to build accurate and continuous response surfaces of two degree-of-freedom vortex-induced vibrations (VIV) of flexibly mounted cylinders for a wide range of transverse and in-line natural frequencies. We consider both the structure and the flow to be two-dimensional. The structure has a low mass damping, with the transverse and in-line mass ratios as well as the transverse and in-line damping coefficients being equal. The goal is to capture the sensitivity of the response to the change in the natural frequencies of the structure. The system is studied for a wide range of transverse natural frequency within the synchronization region. The extent of variation of the in-line natural frequency is chosen to be larger than the one of the transverse natural frequency in order to favor multi-modal responses. No preferred frequencies are emphasized within the intervals of study. The numerical technique uses a multi-element stochastic collocation method coupled to a spectral element based deterministic solver.     

Dynamic analysis of suspended cables carrying moving oscillators

An efficient procedure for analyzing in-plane vibrations of flat-sag suspended cables carrying an array of moving oscillators with arbitrarily varying velocities is presented. The cable is modelled as a mono-dimensional elastic continuum, fully accounting for geometrical nonlinearities. By eliminating the horizontal displacement component through a standard condensation procedure, the nonlinear integro-differential equation governing vertical cable vibrations is derived. Due to the dynamic interaction at the contact points with the moving oscillators, such equation is coupled to the set of ordinary differential equations ruling the response of the travelling sub-systems. An improved series representation of vertical cable displacement is proposed, which allows to overcome the inability of the traditional Galerkin method to reproduce the kinks and abrupt changes of cable configuration at the interface with the moving sub-systems. Following the philosophy of the well-known “mode-acceleration” method, the convergence of the series expansion of cable response in terms of appropriate basis functions is improved through the introduction of the so-called “quasi-static” solution. Numerical results demonstrate that, despite the basis functions are continuous, the improved series enables to capture with very few terms the abrupt changes of cable profile at the contact points between the cable and the moving oscillators.