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.     

Flow-induced vibrations of a side-by-side arrangement of two flexible circular cylinders

Laboratory experiments with a side-by-side arrangement of two vertical, high aspect ratio (length over diameter) and low mass ratio (mass over mass of displaced fluid) cylinders, pin-jointed at the ends and vibrating at low mode number, were carried out in a free-surface water channel. The dynamic response of the models under two different wake interference situations is presented here. Initially, one of the cylinders was fixed and the other was completely free to move. In a second battery of experiments both cylinders were free to vibrate. A very large parameter space was covered by varying the free-stream flow speeds, the natural frequencies of the system and the separation between the models, allowing the identification of vortex-induced vibrations (VIV) and wake-coupled VIV (WCVIV). Amplitudes, frequencies and phase synchronisation between the models are presented.     

Vortex Induced Vibrations of a Pair of Cylinders at Reynolds Number 1000

Vortex induced vibrations of two equal-sized cylinders in tandem and staggered arrangement placed in uniform incompressible flow is studied. A stabilized finite element formulation is utilized to solve the governing equations. The Reynolds number for these 2D simulations is 1000. The cylinders are separated by 5.5 times the cylinder diameter in the streamwise direction. For the staggered arrangement, the cross-flow spacing between the two cylinders is 0.7 times the cylinder diameter. In this arrangement, the downstream cylinder lies in the wake of the upstream one and therefore experiences an unsteady inflow. The wake looses its temporal periodicity, beyond a few diameters downstream of the front cylinder. The upstream cylinder responds as an isolated single cylinder while the downstream one undergoes disorganized motion. Soft-lock-in is observed in almost all the cases.     

A simplified method for time domain simulation of cross-flow vortex-induced vibrations

A new method for time domain simulation of cross-flow vortex-induced vibrations of slender circular cylindrical structures is developed. A model for the synchronization between the lift force and structure motion is derived from already established data for the cross-flow excitation coefficient. The proposed model is tested by numerical simulations, and the results are compared to experimental observations. When a sinusoidal cross-flow motion is given as input to the algorithm, the generated force time series are generally in good agreement with experimental measurements of cross-flow force in phase with cylinder velocity and acceleration. The model is also utilized in combination with time integration of the equation of motion to simulate the cross-flow vibration of a rigid cylinder. The resulting amplitude and frequency of motion as functions of reduced velocity are compared to published experimental results. In combination with the finite element method, the model is used to simulate cross-flow vibrations of a flexible cylinder in shear flow. Comparison with experiments shows that the model is capable of reproducing important quantities such as frequency, mode and amplitude, although some discrepancies are seen. This must be expected due to the complexity of the problem and the simple form of the present method.     

Vortex-induced vibrations of two mechanically coupled circular cylinders with asymmetrical stiffness in side-by-side arrangements

Vortex-induced vibrations of two mechanically coupled circular cylinders with asymmetrical stiffness in side-by-side arrangements are numerically investigated in a uniform flow at a low Reynolds number of 100. The oscillation system is restricted to the cross-flow direction, giving rise to a coupled two-degree-of-freedom response. Attention is placed on the two cylinders with a center-to-center gap ratio of 4 and a mass ratio of 10. The flow dynamics are described by the two-dimensional incompressible Navier–Stokes equations and resolved by the Characteristic-Based-Split finite element method. The stiffness of the first spring that connects the lower cylinder to the wall is chosen such that the vortex-induced vibration of the associated single cylinder with the same stiffness undergoes a pre-synchronization (state A), synchronization (state B) and post-synchronization (state C), respectively. In each state, the stiffness of the second spring connecting the lower and upper cylinders is varied to cover both synchronization and de-synchronization regimes. Numerical results show that the mechanically coupled system locks on the first-mode natural frequency in state A, while on the second-mode natural frequency in states B and C. In such a lock-in regime, the amplitude ratios of the two oscillating and coupled cylinders collapse well onto the corresponding first or second free-vibration mode. The overall coupling mechanism is further explained in terms of the hydrodynamic coefficients, frequency characteristics, wake patterns and effective added mass, quantifying the associated fluid-structure interactions against those governing a single-degree-of-freedom, single-cylinder system.     

Interference effect of an upstream larger cylinder on the lock-in vibration of a flexibly mounted circular cylinder

Experiments have been carried out to investigate the flow-induced vibration response of a flexibly mounted circular cylinder located in the vicinity of a larger cylinder and subjected to cross-flow. The interfering larger cylinder was placed upstream and had a diameter twice that of the vibrating cylinder. Complex interaction was observed between the flow over the two cylinders. The vibration responses of the flexible cylinder were classified into different regimes according to the relative positions of the two cylinders. In the-side-by-side arrangement and the tandem or near-tandem arrangement, flow-induced vibrations of the flexible cylinder were greatly suppressed. In the staggered arrangement which covered a large portion of the relative cylinder positions being investigated, vibrations of the smaller cylinder were greatly amplified. The vibration response curves were also largely modified with a broadening of the lock-in resonance range. A shift of the peak reduced velocity for maximum vibration response was also found. Flow visualizations and wake velocity measurements suggested that the modifications of the vibration responses were related to the presence or absence of constant or intermittent flow through the gap region between the two cylinders. The proposed mechanisms of flow interactions and the resulting vibration response characteristics could explain previous observations on flow-induced vibrations of two equal-sized circular cylinders reported in the literature.     

Flow around three rectangular cylinders arranged in connected and separated Y-shape

Characteristics of cross flow around three rectangular cylinders with two aspect ratios of breadth to width arranged in connected and separated Y-shape at various angles of incident flow were studied by means of force measurement in a wind tunnel. Flow visualizations with smoke-wire technique for typical cases were also given. Different types of flow patterns were formed for individual models at different angles of incident flow. From the results of fluctuating velocity measurement in the wake, features of vibration were determined. It shows that as the wind blows along the lines of one limb or rectangular cylinder of the model, oscillation is weak, whereas when the wind blows along the bisector lines of two limbs or cylinders, strong vibration is observed. It is associated with the regular vortex shedding.The project supported by the National Natural Science Foundation of China (10172008)The English text was polished by Keren Wang     

Flow-induced vibrations of two circular cylinders in tandem with shear flow at low Reynolds number

The flow-induced vibrations of two elastically mounted circular cylinders subjected to the planar shear flow in tandem arrangement are studied numerically at Re=160. A four-step semi-implicit Characteristic-based split (4-SICBS) finite element method is developed under the framework of the fractional step method to cope with the vortex-induced vibration (VIV) problem. For the computational code verification, two benchmark problems are examined in the laminar region: flow-induced vibration of an elastically mounted cylinder having two degrees of freedom and past two stationary ones in tandem arrangement. Regarding the two-cylinder VIVs in shear flow, the computation is conducted with the cylinder reduced mass M_r=2.5π and the structural damping ratio ξ=0.0. The effects of some key parameters, such as shear rate (k=0.0, 0.05, 0.1), reduced velocity (U_r=3.0–18.0) and spacing ratio (L_x/D=2.5, 3.5, 4.5, 8.0), are demonstrated. It is observed that the shear rate and reduced velocity play an important role in the VIVs of both cylinders at various center-to-center distances. Additionally, in comparison with the single cylinder case, a further study indicated that the gap flow has a significant impact on such a dynamic system, leading it to be more complex. The results show that, the performances of ‘dual-resonant’ are discovered in the shear flow. A valley is formed in transverse oscillation amplitude of DC for each spacing ratio when U_r is about 6.0. For the X–Y trajectories of the circular cylinders, figure-eight, figure-O and oval shape are obtained. Finally, the interactions between cylinders are revealed, together with the wake-induced vibration (WIV) mechanism underlying the oscillation characteristics of both cylinders exposed to shear flow. Besides, the “T+P” wake pattern is discovered herein.     

A linear stability approach to vortex-induced vibrations and waves

The motion induced by vortex shedding on slender flexible structures subjected to cross-flow is considered here. This phenomenon of vortex-induced vibration (VIV) is analysed by considering the linear stability of a coupled system that includes the structure dynamics and the wake dynamics. The latter is modelled by a continuum of wake oscillators, distributed along the span of the structure. In the case of uniform flows over a straight tensioned cable, VIV are found to arise as an instability related to the merging of two waves. In the case of a cable of finite length, the selection of modes that experience lock-in with the wake is found using the same stability argument. In non-uniform flows, several unstable wave systems are identified, and competition between them is discussed. Comparison is then made with existing experimental and computational data of VIV of slender structures under uniform and non-uniform flows. Phenomena previously identified in these systems, such as mode switching when the flow velocity is varied, time sharing of the response between two frequencies, or the coexistence of several regions of VIV with different dynamics in the same structure, are discussed with the help of the proposed model.     

Vortex-induced vibration of two elastically coupled cylinders in side-by-side arrangement

Vortex-induced vibration (VIV) of two elastically coupled circular cylinders in side-by-side arrangement is investigated numerically. The Reynolds-averaged Navier–Stokes equations are solved by the finite element method for simulating the flow and the equation of motion is solved for calculating the vibration. The mass ratio (the ratio of the mass of the cylinder to the displaced fluid mass) is 2 and the Reynolds number is 5000 in the simulations. Simulations are carried out for one symmetric configuration (referred to be Case A) and one asymmetric configuration (referred to be Case B). In both Case A and Case B, the primary response frequencies of the two cylinders are found to be the same both inside and outside the lock-in regimes. Five response regimes are found in both cases and they are the first-mode lock-in regime, the second-mode lock-in regime, the sum-frequency lock-in regime and two transition regimes. When the vibration is transiting from the first- to the second-mode lock-in regimes, the vibration of each cylinder contains both first- and the second-mode natural frequencies, and the vibrations are usually irregular. In the transition regime between the second-mode lock-in and the sum-frequency lock-in regimes, the response frequencies of both cylinders increases with an increase in the reduced velocity until they are close to the sum of the two natural frequencies. In both cases, the lower boundary reduced velocity of the total lock-in regime (the sum of the five lock-in regimes) is about 3 and the upper boundary reduced velocity is about 11 times the first-to-second-mode natural frequency ratio.     

Vortex shedding modeling using diffusive van der Pol oscillatorsModélisation du détachement tourbillonnaire avec des oscillateurs de van der Pol interagissant par diffusion

A simple model for the near wake dynamics of slender bluff bodies in cross-flow is analyzed. It is based on a continuous distribution of van der Pol oscillators arranged along the spanwise extent of the structure and interacting by diffusion. Diffusive interaction is shown to be able to model cellular vortex shedding in shear flow, the cell size being estimated analytically with respect to the model parameters. Moreover, diffusive interaction succeeds in describing qualitatively the global suppression of vortex shedding from a sinuous structure in uniform flow. To cite this article: M.L. Facchinetti et al., C. R. Mecanique 330 (2002) 451–456.     

On the spatio-temporal structure of cylinder wakes

The wake of a short aspect ratio cylinder placed in a uniform flow is experimentally investigated. After having characterized the temporal behavior of the Bénard–Von Kàrmàn vortex shedding by the use of a classical hot-wire anemometer, an ultrasound anemometry technique is applied to study the spatial critical behavior of the envelope of the transverse velocity of the wake. It is shown that this envelope which represents the spatial form of the global mode of the wake, follows universal scaling laws which are in agreement with a second order phase transition. In a second set of experiments, the behavior of the longitudinal velocity fluctuations is also investigated. It is discovered that there is a special point several diameters behind the cylinder, which plays a role of a wave maker. Finally, for very small aspect ratio cylinders, symmetric vortex shedding is reported and interpreted using a system of coupled oscillators. Received: 15 May 1997/Accepted: 20 January 1998     

结构振动对湍流近尾迹的影响

研究了圆柱绕流中流体与结构的相互作用,侧重结构振动对湍流尾迹的影响,用激光测振仪测量圆柱在升力方向的位移;用热线和LDA（二维）测量湍流的近尾迹,通过变化自由流的速度和圆柱体直径（特征尺寸）来变化雷诺数,用两个振动特性不同的（一个相对刚性,一个相对弹）圆柱来产生尾迹,研究固体结构振动对湍流近尾迹的平均速度场和湍流场的影响,结果表明,结构自由振动对湍流近尾迹场影响明显,该影响随雷诺数的变化不明显。     

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 characteristics of two-degree-of-freedom vortex induced vibration of two low-mass circular cylinders in proximity at low Reynolds number

Two-degree-of-freedom vortex induced vibrations (VIVs) of two identical spring-supported circular cylinders in proximity with the mass ratio of 2 and zero damping at Re of 100 are numerically studied. Totally 20 arrangements of cylinders are investigated combining four stagger angles and five normalized center-to-center spacings. Results show that the in-line vibration amplitude is comparable to the transverse one for most arrangements and usually accompanies irregular cylinder trajectories. Extremely slender figure-8 cylinder trajectories usually seen in single-cylinder VIVs exist only for the tandem arrangements. Arranging the trailing cylinder to vibrate near the wake boundary of the leading cylinder enhances the possibility of irregular trajectories and impacts of both cylinders. Impact between cylinders must occur in cases with irregular cylinder trajectories; however, irregular cylinder trajectories could be found in impact-free cases. The stagger angle significantly changes the attribute of the transverse vibration frequency, toward either the single-cylinder VIV frequency or natural structure frequency in still fluid. The major transverse vibration frequency and the natural structure frequency in still fluid are decoupled for all the side-by-side arrangements and some far spaced tandem arrangements and highly correlated for non-tandem and non-side-by-side arrangements. The time-averaged impact frequency increases with decreasing normalized center-to-center spacing for most combinations of stagger angle and reduced velocity. Apart from the side-by-side arrangements, high-frequency impacts occur when the trailing cylinder is initially located in or near the wake zone of the leading cylinder. The mechanism of trailing cylinder chopping the gap-flow vortices plays an important role in determining the near-wake vortex structures for all non-side-by-side arrangements.     

Vortex and wake-induced vibrations of a tandem arrangement of two flexible circular cylinders with near wake interference

Results showing the dynamic response of a tandem arrangement of two vertical high aspect ratio (length over diameter) and low mass ratio (mass over mass of displaced fluid) flexible cylinders vibrating at low mode number are presented in this paper. Two circular cylinder models were aligned with the flow, so the downstream or trailing cylinder was immersed in the wake of the leading one. Centre-to-centre distances from 2 to 4 diameters were studied. The models were very similar in design, with external diameters of 16 mm and a total length of 1.5 m. Reynolds numbers up to 12 000 were achieved with reduced velocities, based on the fundamental natural frequency of the downstream cylinder in still water, up to 16. The trailing model had a mass ratio of 1.8 with a combined mass-damping parameter of 0.049, whilst the corresponding figures for the leading cylinder were 1.45 and 0.043, respectively. The dynamic response of the trailing model has been analysed by studying cross-flow and in-line amplitudes, dominant frequencies and modal amplitudes. The dynamic response of the leading one is analysed by means of its cross-flow amplitudes and dominant frequencies and it is also related to the motion of the trailing cylinder by studying the synchronisation between their instantaneous cross-flow motions. Planar digital particle image velocimetry (DPIV) was used to visualise the wake. Different response regimes have been identified based on the type of oscillations exhibited by the cylinders: vortex-induced (VIV), wake-induced (WIV) or combinations of both.     

Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.     

Scale-adaptive simulation of flow past wavy cylinders at a subcritical Reynolds number

The Xu & Yan scale-adaptive simulation (XYSAS) model is employed to simulate the flows past wavy cylinders at Reynolds number 8 × 10 3.This approach yields results in good agreement with experimental measurements.The mean flow field and near wake vortex structure are replicated and compared with that of a corresponding circular cylinder.The effects of wavelength ratios λ/D m from 3 to 7,together with the amplitude ratios a /D m of 0.091 and 0.25,are fully investigated.Owing to the wavy configuration,a maximum reduction of Strouhal number and root-meansquare (r.m.s) fluctuating lift coefficients are up to 50% and 92%,respectively,which means the vortex induced vibration (VIV) could be effectively alleviated at certain larger values of λ/D m and a /D m.Also,the drag coefficients can be reduced by 30%.It is found that the flow field presents contrary patterns with the increase of λ/D m.The free shear layer becomes much more stable and rolls up into mature vortex only further downstream when λ/D m falls in the range of 5-7.The amplitude ratio a /D m greatly changes the separation line,and subsequently influences the wake structures.     

Development of models correlating vibration excitation forces to dynamic characteristics of two-phase flow in a tube bundle

Recent experiments revealed significant quasi-periodic forces in both the drag and lift directions in a rotated triangular tube bundle subjected to two-phase cross-flow. The quasi-periodic drag forces were found to be related to the momentum flux fluctuations in the main flow path between the cylinders. The quasi-periodic lift forces, on the other hand, are mostly correlated to the oscillation in the wake of the cylinders. In this paper, we develop semi-analytical models for correlating vibration excitation forces to dynamic characteristics of two-phase flow in a rotated triangular tube bundle for a better understanding of the nature of vibration excitation forces. The relationships between the lift or drag forces and the dynamic characteristics of two-phase flow are established through fluid mechanics momentum equations. A model has been developed to correlate the void fraction fluctuation in the main flow path and the dynamic drag forces. A second model has been developed for correlating the oscillation in the wake of the cylinders and the dynamic lift forces. Although still preliminary, each model can predict the corresponding forces relatively well.