Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.     

URANS vs. experiments of flow induced motions of multiple circular cylinders with passive turbulence control

Two-dimensional Unsteady Reynolds-Average Navier–Stokes equations with the Spalart–Allmaras turbulence model are used to simulate the flow induced motions of multiple circular cylinders with passive turbulence control (PTC) in steady uniform flow. Four configurations with 1, 2, 3, and 4 cylinders in tandem are simulated and studied at a series of Reynolds numbers in the range of 30 000<Re<120 000. Simulation results are verified by experimental data measured in the Marine Renewable Energy Laboratory. Good agreement was observed between the values of vorticity, amplitude ratio, and frequency ratio predicted by numerical simulations and experimental measurements. The amplitude and frequency response show the initial and upper branches in vortex induced vibration (VIV), transition from VIV to galloping, and galloping branch for all PTC-cylinders. The maximum amplitude of 2.9 diameters for the first cylinder is achieved at Re=104 356 in the numerical results. Compared with the first cylinder, the VIV initial branch starts at higher Re for the downstream cylinders due to the presence of the upstream cylinder(s). 2P and 2P+2S vortex patterns are observed at Re=62 049 and Re=90 254 for the single PTC-cylinder. Furthermore, the shed vortices of the downstream cylinders are strongly disrupted and modified by the vortices shed from the upstream one in the cases of multiple PTC-cylinders.     

On shear flow single mode lock-in with both cross-flow and in-line lock-in mechanisms

A long flexible cylinder exposed to ocean currents is known to undergo vortex-induced vibration (VIV). In a spatially sheared flow the response of a riser to VIV can vary from single mode lock-in to multimodal. A new experimental facility was designed and built to investigate the above-mentioned areas. The facility consisted of a long flexible cylinder in either a uniform or a simplified vertically sheared flow. The instrumentation consisted of direct local fluid force measurement at two locations on the cylinder as well as accelerometers spaced along the cylinder axis. The simplified shear flow was a 2-slab flow, with each slab having uniform velocity. Test conditions included forcing the cylinder simultaneously at resonance in both regions to investigate modal competition issues and multimodal response patterns. Resonant VIV excitation of two different modes simultaneously, was conducted which revealed single mode lock-in of the higher frequency through an unexpected mechanism. The higher frequency mode's damping region underwent in-line excitation at four times the predicted shedding frequency that provided a power-in effect to support the dominant mode's cross-flow response.     

两端铰接的细长柔性圆柱体涡激振动响应特性数值研究

目前细长柔性圆柱体涡激振动响应的研究方法主要包括实验方法、计算流体动力学方法以及半经验模型方法. 鉴于实验方法研究成本较高、计算流体动力学方法计算时间较长,本文基于尾流振子模型对线性剪切来流下两端铰接的细长柔性圆柱体涡激振动响应特性进行了半经验模型方法研究. 先建立了柔性圆柱体结构振子以及尾流振子之间的耦合模型,紧接着基于二阶精度中心差分格式对耦合模型先离散后迭代进行求解. 对不同剪切参数下柔性圆柱体涡激振动响应的振动波长、振动频率、振动位移以及响应频率随时间的变化特性等参数进行了分析. 分析结果表明:圆柱体的涡激振动响应由驻波和行波混合组成. 当无量纲弯曲刚度较小时,在圆柱体两端附近,驻波占主导;而在圆柱体中间段附近,行波占主导. 当无量纲弯曲刚度较大时,在圆柱体整个长度区间上均为驻波占主导. 随着剪切参数的增大,振动位移以及振动波长均逐渐减小,而振动频率和频率带宽均逐渐增大.      

Combined action of transverse oscillations and uniform cross-flow on vortex formation and pattern of a circular cylinder

This paper attempts to study the roles of lateral cylinder oscillations and a uniform cross-flow in the vortex formation and wake modes of an oscillating circular cylinder. A circular cylinder is given lateral oscillations of varying amplitudes (between 0.28 and 1.42 cylinder-diameters) in a slow uniform flow stream (Reynolds number=284) to produce the 2S, 2P and P+S wake modes. Detailed flow information is obtained with time-resolved particle-image velocimetry and the phase-locked averaging techniques. In the 2S and 2P mode, the flow speeds relative to the cylinder movement are less than the uniform flow velocity and it is found that initial formation of a vortex is caused by shear-layer separation of the uniform flow on the cylinder. Subsequent development of the shear-layer vortices is affected by the lateral cylinder movement. At small cylinder oscillation amplitudes, vortices are shed in synchronization with the cylinder movement, resulting in the 2S mode. The 2P mode occurs at larger cylinder oscillation amplitudes at which each shear-layer vortex is found to undergo intense stretching and eventual bifurcation into two separate vortices. The P+S mode occurs when the cylinder moving speeds are, for most of the time, higher than the speed of the uniform flow. These situations are found at fast and large-amplitude cylinder oscillations in which the flow relative to the cylinder movement takes over the uniform flow in governing the initial vortex formation. The formation stages of vortices from the cylinder are found to bear close resemblance to those of a vortex street pattern of a cylinder oscillating in an otherwise quiescent fluid at Keulegan–Carpenter numbers around 16. Vortices in the inclined vortex street pattern so formed are then convected downstream by the uniform flow as the vortex pairs in the 2P mode.     

MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m^*ζ), or a high mass-damping (highm^*ζ ). For our low m^*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm^*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m^*ζ), whereas the regime of synchronization (measured by the range of velocity U^*) is dependent primarily on the mass ratio, m^*ζ. At low (m^*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m^*=1–20), when plotted against (m^*+C_A) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m^*+C_A)ζ 0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m^*+C_A) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (f_N), to form a normalized velocity (U^*/f^*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm^* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m^*.     

单窄条控制件对横向振荡柱体尾流 2P模式 旋涡脱落的改变

横向强迫振荡柱体尾流控制是柱体涡激振动控制的基础,在海洋、土木等工程中具有重要意义. 横向强迫振荡柱体尾流中存在一种锁频旋涡脱落模式,即在一个振荡周期内柱体上、下侧各脱落旋转方向相反的一对涡,称为2P模式. 本文将相对宽度b/D=0.32的窄条控制件置于横向强迫振荡柱体下游,对振幅比A/D=1.25, 无量纲振频f_e D/V_∞=0.22,雷诺数Re=1 200的2P模式旋涡脱落进行干扰,并通过改变控制件位置,研究旋涡的变化规律. 采用二维大涡模拟和实验验证方法进行研究,在控制件位置范围0.8≤X/D≤3.2, 0.4≤Y/D≤3.2内,得到了2P, 2S, P+S和另外6种新发现的旋涡脱落模式,并对各模式旋涡的形成过程作了详细描述. 在控制件位置平面上给出了各旋涡模式的存在区域,画出了旋涡脱落强度的等值线图,并发现在一个相当大的区域内,旋涡脱落强 度可减小一半以上,尾流变窄. 发现柱体大幅振荡引起的横向剪切流在旋涡生成中起关键作用. 探讨了控制件对横向剪切流的影响,分析了控制件在每种旋涡模式形成中的作用机制.      

A circular cylinder undergoing large-amplitude transverse oscillations in a slow uniform cross flow

This study explores the vortex patterns formed by a circular cylinder undergoing lateral cylinder oscillations with large amplitudes and in the presence of a slow uniform cross flow. It is an extension of our previous study (Lam et al., 2010b) in which formation of the 2S, 2P and P+S vortex modes were discussed from the viewpoint of interaction of a uniform cross-flow with the vortex street patterns of a cylinder oscillating in an otherwise quiescent fluid at Keulegan–Carpenter numbers up to KC=8.9. The present paper reports three additional experimental sets in which the amplitudes of cylinder oscillations have even larger values, at A/D>2.5, and lie beyond the vortex mode map usually quoted from Williamson and Roshko (1988). It is found that the slow uniform cross-flow at λ/D≈3 and Reynolds number based on cross-flow velocity at 232 acts to convect the corresponding vortex patterns in the absence of cross-flow downstream across the line of cylinder oscillation. Vortex–vortex interaction and vortex–cylinder interaction are observed to affect the subsequent development of vortices. The P+S vortex mode is found to occur up to KC=16. At KC between 16 and 24, a new vortex mode is observed in which only one vortex pair can be convected downstream every cylinder oscillation cycle. Another new vortex mode with two vortex pairs and two stationary vortices are found at KC>24.     

ENERGY DISTRIBUTION IN MODES IN THE WAKE OF A FINITE-LENGTH CYLINDER BEFORE AND AFTER TRANSITION

The flow in the wake of a finite-length cylinder has been studied experimentally both before and after the transition to turbulence. This instability occurs at Reynolds numbers around 180–190. One end of the cylinder was fixed to the bottom of the test-section of a wind tunnel, whilst the other terminated in the open flow (free end). For these boundary conditions four main frequency modes within the wake can be identified. These are a centre-cell mode at the Strouhal frequency, end-cell modes with a frequency below the Strouhal frequency, a mode exhibiting the difference frequency between the centre-cell and end-cell modes, and a low-frequency mode (appearing only after transition to turbulence). In this work the energy content of these four modes has been determined throughout the wake, both before and after transition to turbulence. For three of the modes, the energy content is the same before and after transition, whereas the low-frequency mode exhibits energy two to four orders of magnitude greater after transition than before. Hence it is clear that the additional turbulence energy appearing in the wake after transition is located predominantly in this low-frequency mode. The appearance of this low-frequency mode is characterized by the simultaneous appearance of a peak in the power spectra of the velocity fluctuations centred about zero frequency (but with finite width). Consequently, the appearance of this zero frequency peak can be taken as the signature of the onset of turbulence. By considering the downstream growth rates of this low-frequency mode, evidence is presented which suggests that transition to turbulence may occur as a result of wake transition in the downstream central plane of the cylinder.     

Three-dimensional numerical simulation of vortex-induced vibration of an elastically mounted rigid circular cylinder in steady current

Vortex-induced vibration (VIV) of an elastically mounted rigid circular cylinder in steady current is investigated by solving the three-dimensional Navier–Stokes equations. The cylinder is allowed to vibrate only in the cross-flow direction. The aim of this study is to investigate the variation of the vortex shedding flow in the axial direction of the cylinder and to study the transition of the flow from two-dimensional (2D) to three-dimensional (3D) for VIV of a cylinder. Simulations are carried out for a constant mass ratio of 2, the Reynolds numbers ranging from 150 to 1000 and the reduced velocities ranging from 2 to 12. The three-dimensionality of the flow is found to be the strongest in the upper branch of the VIV response and weakest in the initial branch. The 2S and 2P vortex shedding modes are found to coexist along the cylinder span in the upper branch, leading to strong variations of the lift coefficient in the axial direction of the cylinder. The difference between the flow transition from 2D to 3D in the VIV lock-in regime and that in the wake of a stationary cylinder is identified. The transition mode B found in the wake of a stationary cylinder is also found in the wake of a vibrating cylinder. The critical Reynolds number for flow transition from 2D to 3D of a cylinder undergoing cross-flow VIV at a reduced velocity of 6 is found to be greater than that for a stationary cylinder. For a constant reduced velocity of 6, the wake flow changes from 2D to 3D as the Reynolds number is increased from 250 to 300. Some 2D numerical simulations are performed and it is found that the 2D Navier–Stokes (NS) equations are not able to predict the VIV in the turbulent flow regime, while the 2D Reynolds-averaged Navier–Stokes (RANS) equations improve the results.     

Mode competition in streamwise-only vortex induced vibrations

Time-resolved Particle-Image Velocimetry (PIV) has been used to study mode competition and transient behaviour in the wake of a cylinder experiencing Vortex-Induced Vibrations (VIV) in the streamwise direction. The cylinder response regime contained two branches, occurring above and below the onset of synchronisation between the wake and the cylinder motion (lock-in). During the first branch, the wake exhibited both the S-I mode (in which two vortices are shed simultaneously per vibration cycle) and the alternate A-II mode (similar to the well known von Kármán vortex street). An extended PIV data set acquired in this region revealed mode switching between the S-I and A-II modes. A criterion based on Proper-Orthogonal Decomposition was developed to identify which mode was dominant as a function of time. The A-II mode was found to be dominant for over 90% of the instantaneous fields examined, while the S-I mode appeared to be more unstable.Symmetrically shed vortices were found to rearrange downstream into an alternate structure in which the wake was no longer synchronised to the cylinder motion. The dominant frequency of transverse velocity fluctuations was measured throughout the wake in order to study the effects of this breakdown in more detail. For the majority of the wake, the fluctuations occurred at the Strouhal frequency, while in a region in the near wake the fluctuations occurred at the frequency of the cylinder motion. It is thought that during the first response branch vortices are formed at the cylinder response frequency, but tend to quickly rearrange downstream into an alternate structure which is no longer synchronised to the cylinder motion. As a result, the fluctuating drag will be synchronised to the structural motion, and is capable of providing positive energy transfer in the apparent absence of lock-in. Finally, the spatial dependence of the frequency of velocity fluctuations throughout the wake is used to explain some of the conflicting results in the literature regarding streamwise VIV, and the implications for the general study of VIV are discussed.     

A consideration on pre- and post-instability of an axisymmetric elastic beam subjected to axial leakage flow

The dynamical behavior of an axisymmetric elastic beam subjected to axial leakage flow is investigated numerically and experimentally. The coupled equations of motion for a fluid and a beam structure are derived using the Navier–Stokes equation for an axial leakage flow-path and the Euler–Bernoulli beam theory. Performing complex eigenvalue analysis, the variation of the dynamic behavior during pre- and post-instability is investigated with respect to increasing axial leakage flow velocity. Also, an experiment was performed to determine the critical velocity of the unstable dynamic behavior of an axisymmetric elastic beam confined in a concentric cylinder subjected to axial leakage flow through a small annulus, and to measure the variation of the dynamic behavior on pre- and post-instability when the unstable phenomenon with the lower predominant frequency is shifted to the higher one. The relationships between the axial flow velocities and the unstable phenomena are clarified for the transition from the lower mode to the higher mode by comparing the theoretical calculations with experimental observations. Especially, the generation of traveling waves and the energy balance for the distortion of vibration response in the axial direction are discussed and considered at the transition region of the complex coupled vibration response of an axisymmetric elastic beam subjected to an axial leakage flow. Numerical and experimental results are found to be in quite good agreement.     

Insights on vortex-induced,traveling waves on long risers

This paper is based on portions of a keynote presentation, which explored the boundaries of what is understood about the vibration behavior of long cylinders excited by vortex shedding. The source of data is a recent field experiment on a long flexible cylinder, densely instrumented with fiber optic strain gauges. The paper emphasis is on previously unknown or unexplained phenomena and in some cases offers provocative insights as opposed to conclusive proofs. Three particular topics are covered: (i) the occurrence of peak strain and fatigue damage rates at unexpected locations, (ii) the dominance of traveling wave rather than standing wave response, and (iii) the appearance of stable cylinder trajectories, such as figure eights and crescents in pure traveling wave regions. Potential explanations are offered and areas for promising additional research are proposed.     

Numerical study on the suppression of the vortex-induced vibration of an elastically mounted cylinder by a traveling wave wall

In the present paper, the commercial CFD code “Fluent” was employed to perform 2-D simulations of an entire process that included the flow around a fixed circular cylinder, the oscillating cylinder (vortex-induced vibration, VIV) and the oscillating cylinder subjected to shape control by a traveling wave wall (TWW) method. The study mainly focused on using the TWW control method to suppress the VIV of an elastically supported circular cylinder with two degrees of freedom at a low Reynolds number of 200. The cross flow (CF) and the inline flow (IL) displacements, the centroid motion trajectories and the lift and drag forces of the cylinder that changed with the frequency ratios were analyzed in detail. The results indicate that a series of small-scale vortices will be formed in the troughs of the traveling wave located on the rear part of the circular cylinder; these vortices can effectively control the flow separation from the cylinder surface, eliminate the oscillating wake and suppress the VIV of the cylinder. A TWW starting at the initial time or at some time halfway through the time interval can significantly suppress the CF and IL vibrations of the cylinder and can remarkably decrease the fluctuations of the lift coefficients and the average values of the drag coefficients; however, it will simultaneously dramatically increase the fluctuations of the drag coefficients.     

Distorted wave response of ultrasonic annular stator incorporating non-uniform geometry

The traveling wave ultrasonic stator is normally fabricated with teeth. The tooth geometry improves the driving speed, but it creates natural frequency splitting and mode contamination, especially a distorted traveling wave. A dynamic model of a stepped-plate periodic stator is developed to examine the distortion. The stator is treated as an annular supported by a thin mid plate, and the support stiffness is formulated by using equivalent energy principle. The effects of the tooth and mid plate on the natural frequency and vibration mode are examined by using the perturbation method. The rules governing the frequency splitting, frequency perturbation as well as mode contamination are also identified. The traveling wave response and elliptical trace on stator surface are obtained by using the mode superposition method and they are proved to be distorted due to the tooth geometry. The response at the repeated doublets becomes coupled forward and backward traveling waves, but that at the split doublets becomes coupled forward traveling, standing and backward traveling waves. The results indicate that the tooth mass instead of the stiffness decreases the vibration amplitude and driving speed of the dominant wave, but their effects are different at the repeated and split doublets. Inspection of the model implies that the distortion can be suppressed by using a suitable combination of the wavenumber, tooth count, tooth height and occupying fraction. Numerical calculations are carried out to demonstrate the tooth geometry effect on the transient waveform, driving speed and elliptical trace. The optimization of the tooth geometry that can help achieve a purer traveling wave is discussed.     

Reconstructing the vortex-induced-vibration response of flexible cylinders using limited localized measurement points

A series of experiments was conducted in a re-circulating water tunnel with a uniform flow profile, in which a flexible tension-dominated cylinder was held fixed at both ends and placed perpendicular to the incoming flow direction. The circular cylinder had an aspect ratio of L/D=67 and a low mass ratio of m*=0.43. Dynamic response of the system was studied in the reduced velocity range of U*=2.9–14.5 and Reynolds numbers of Re=315–1580. The oscillations of the cylinder in the inline (IL) and crossflow (CF) directions were captured using two synchronized high speed cameras. Continuous response of the cylinder was reconstructed from limited number of measurement points based on modal expansion theorem modified using Modal Assurance Criterion (MAC). The MAC enhanced the response reconstruction by adjusting the contributions from each structural mode. A distinct advantage of implementing the MAC in this method compared with the previously used methods was reconstructing the VIV response accurately when the experimental measurement points were clustered in a small region along the length of the cylinder. Mono- and multi-frequency excitation responses as well as transition from a low mode number to a higher one were observed. Flow forces acting on the cylinder were calculated and a consistent relation between the regions where the cylinder was being excited by the flow and the counterclockwise figure-eight trajectories of oscillations was observed.     

周向导波在空心圆柱体中传播的数值模拟研究

在石油、化工、食品和城市供水等行业,由于腐蚀等原因而引起泄漏事故,造成巨大的经济损失和资源浪费.因此,能够预先检测到管道缺陷避免事故发生显得十分重要.基于弹性动力学理论,利用Matlab编程实现周向导波在空心圆柱体中传播的数值模拟.利用特征函数展开法计算出周向导波的频散曲线及0.6MHz和1MHz各个模态的时域波形图并据此分析激励方式同周向导波传播的关系,频率为1MHz或0.6MHz时,1和2模态在空心圆柱壳中传播的周向导波中处于主导地位:各模态的径向或周向位移幅度会受到激励入射角的影响而变化并且周向导波的位移幅度随着激励入射角的增加而增加:得到特定频率下,周向导波各模态径向、周向位移的变化关系.本文结果为进一步利用实验方法研究周向导波在空心圆柱体中传播特性和缺陷检测奠定了基础.     

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional(2D) second mode with oblique modes are studied numerically in a Mach 6.0 flat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation(PSE) method, which is able to simulate interaction of two modes with different frequencies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.