不同展长比旗帜在均匀来流中摆动的实验研究

为研究展长比对旗帜摆动特性的影响,在低速风洞内开展了质量比M*为0.6,展长比H*为0.2～1.5的柔性旗帜在不同来流速度下的失稳实验.利用高速摄影机和图像处理技术分析了无量纲速度U*和展长比H*对旗帜运动学特征(频率和振幅)的影响;利用自行研制的天平测量了旗帜受到的阻力,首次给出了旗帜的动力学特征.结果发现,保持展长...     

Free-stream turbulence effects on vortex-induced vibration of two side-by-side elastic cylinders

The effect of free-stream turbulence on vortex-induced vibration of two side-by-side elastic cylinders in a cross-flow was investigated experimentally. A turbulence generation grid was used to generate turbulent incoming flow with turbulence intensity around 10%. Cylinder displacements in the transverse direction at cylinder mid-span were measured in the reduced velocity range 1.45<U_r0<12.08, corresponding to a range of Reynolds number (Re), based on the mean free-stream velocity and the diameter of the cylinder, between Re=5000–41 000. The focus of the study is on the regime of biased gap flow, where two cylinders with pitch ratio (T/D) varying from 1.17 to 1.90 are considered. Results show that the free-stream turbulence effect is to enhance the vortex-induced force, thus to restore the large-amplitude vibration associated with the lock-in resonance. However, the enhancement is significant at a different Strouhal number (St) for different pitch ratios. When the spacing between two cylinders is relatively small (1.17<T/D<1.50), the enhancement is significant at St≈0.1. When the spacing is increased, the Strouhal number at which the enhancement is significant shifts to St≈0.16. This enlarges the range of reduced velocity to be concerned. An energy analysis showed that free-stream turbulence feeds energy to the cylinder at multiple frequencies of vortex shedding. Therefore, the lock-in region is still of main concern when the approach flow is turbulent.     

Vortex shedding in cylinder flow of shear-thinning fluids: I. Identification and demarcation of flow regimes

An experimental study on the flow of non-Newtonian fluids around a cylinder was undertaken to identify and delimit the various shedding flow regimes as a function of adequate non-dimensional numbers. The measurements of vortex shedding frequency and formation length (l_f) were carried out by laser-Doppler anemometry in Newtonian fluids and in aqueous polymer solutions of CMC and tylose. These were shear thinning and elastic at weight concentrations ranging from 0.1 to 0.6%. The 10 and 20 mm diameter cylinders (D) used in the experiments had aspect ratios of 12 and 6 and blockage ratios of 5 and 10%, respectively. The Reynolds number (Re^*) was based on a characteristic shear rate of U_∞/(2D) and ranged from 50 to 9×10³ thus encompassing the laminar shedding, the transition and shear-layer transition regimes. Increasing fluid elasticity reduced the various critical Reynolds numbers (Re_etr^*, Re_lf^*, Re_bbp^*) and narrowed the extent of the transition regime. For the 0.6% tylose solution the transition regime was even suppressed. On the other end, pseudoplasticity was found to be indirectly responsible for the observed reduction in Re_otr^*: it increases the Strouhal number which in turn increases the vortex filaments, precursors of the transition regime. Elasticity was better quantified by the elasticity number Re′/We than by the Weissenberg number. This elasticity number involves the calculation of the viscosity at a high characteristic shear rate, typical of the boundary layer, rather than at the average value (U_∞/(2D)) used for the Reynolds number, Re^*.     

Vortex shedding from two finite circular cylinders in a staggered configuration

Wind tunnel experiments were conducted to measure the vortex shedding frequencies for two circular cylinders of finite height arranged in a staggered configuration. The cylinders were mounted normal to a ground plane and were partially immersed in a flat-plate turbulent boundary layer. The Reynolds number based on the cylinder diameter was Re_D=2.4×10⁴, the cylinder aspect ratio was AR=9, the boundary layer thickness relative to the cylinder height was δ/H=0.4, the centre-to-centre pitch ratio was varied from P/D=1.125 to 5, and the incidence angle was incremented in small steps from α=0° to 90°. The Strouhal numbers were obtained behind the upstream and downstream cylinders using hot-wire anemometry. From the behaviour of the Strouhal number data obtained at the mid-height position, the staggered configuration could be broadly classified by the pitch ratio as closely spaced (P/D<1.5), moderately spaced (1.5?P/D?3), or widely spaced (P/D>3). The closely spaced staggered finite cylinders were characterized by the same Strouhal number measured behind both cylinders, an indication of single bluff-body behaviour. Moderately spaced staggered finite cylinders were characterized by two Strouhal numbers at most incidence angles. Widely spaced staggered cylinders were characterized by a single Strouhal number for both cylinders, indicative of synchronized vortex shedding from both cylinders at all incidence angles. For selected staggered configurations representative of closely spaced, moderately spaced, or widely spaced behaviour, Strouhal number measurements were also made along the vertical lengths of the cylinders, from the ground plane to the free end. The power spectra showed that for certain cylinder arrangements, because of the influences of the cylinder–wall junction and free-end flow fields, the Strouhal numbers and flow patterns change along the cylinder.     

Self-sustained oscillations between two tandem cylinders at Reynolds number 1,000

This study focuses on the self-sustained oscillatory flow characteristics between two tandem circular cylinders of equal diameter placed in a uniform inflow. The Reynolds number (Re _D ), based on the cylinder diameter, was around 1,000 and all experiments were performed in a recirculating water channel. The streamwise distance between two tandem cylinders ranged within 1.5 ≤ X _c/D ≤ 7.0. Here X _c denotes the center-to-center distance between two tandem cylinders. For all experiments studied herein, quantitative velocity measurements were performed using hot-film anemometer and the LDV system. The laser sheet technique was employed for qualitative flow visualization. The wavelet transform was applied to elucidate the temporal variation and phase difference between two spectral components of the velocity signals detected in the flow field. The remarkable finding was that when two tandem circular cylinders were spaced at a distance within 4.5 ≤ X _c/D ≤ 5.5, two symmetrical unstable shear layers with a certain wavelength were observed to impinge onto the downstream cylinder. The responding frequency (f _u ), measured between these two cylinders, was much higher than the natural shedding frequency behind a single isolated cylinder at the same Re _D . This responding frequency decreased as the distance X _c/D increased. Not until X _c/D ≥ 6.0, did it recover to the natural shedding frequency behind a single isolated cylinder. Between two tandem cylinders, the Strouhal numbers (St _c = f _u X _c/U_c) maintained a nearly constant value of 3, indicating the self-sustained oscillating flow characteristics with a wavelength X _c/3. Here U _c is the convection speed of the unstable shear layers between two tandem cylinders. At Re _D = 1,000, the self-sustained oscillating characteristics between two tandem circular cylinders were proven to exhibit a sustained flow pattern, not just a sporadic phenomenon.     

Large eddy simulations of flow interference between two unequal sized square cylinders

A uniform flow past two unequal sized square cylinders arranged in a side-by-side pattern and at a Reynolds number of 50,000 has been investigated using large eddy simulation (LES) technique. The modelling of sub-grid scales of turbulence is done using the Smagorinsky model. The effect of the transverse gap ratio (T/D) on the flow characteristics has been studied. Numerical simulations are carried out for five different transverse gap ratios (T/D), namely 1.120, 1.250, 1.375, 1.750 and 2.500. Results in terms of the aerodynamic forces, Strouhal number, mean base pressure coefficient, streamlines, vorticity, surface pressure distribution, normal and shear stresses are presented. A shift in the stagnation point for the small square cylinder from the centre of its front face towards its gap side is seen at smaller T/D ratios. The presence of a jet-like flow seen in the gap side is more pronounced at T/D = 1.12. A biased gap side flow towards the near wake of the small square cylinder is seen at smaller T/D ratios. No interference effect is seen at T/D = 2.5. The flow behaviour is similar to that of the isolated square cylinder at this gap ratio.     

Strouhal numbers,forces and flow structures around two tandem cylinders of different diameters

This paper presents a detailed investigation of Strouhal numbers, forces and flow structures in the wake of two tandem cylinders of different diameters. While the downstream cylinder diameter, D, was fixed at 25 mm, the upstream cylinder diameter, d, was varied from 0.24D to D. The spacing between the cylinders was 5.5d, at which vortices were shed from both cylinders. Two distinct vortex frequencies were detected behind the downstream cylinder for the first time for two tandem cylinders of the same diameter. The two vortex frequencies remained for d/D=1.0–0.4. One was the same as detected in the gap of the cylinders, and the other was of relatively low frequency and was ascribed to vortex shedding from the downstream cylinder. While the former, if normalized, declined progressively from 0.196 to 0.173, the latter increased from 0.12 to 0.203 with decreasing d/D from 1 to 0.24. The flow structure around the two cylinders is examined in the context of the observed Strouhal numbers. The time-averaged drag on the downstream cylinder also climbed with decreasing d/D, though the fluctuating forces dropped because vortices impinging upon the downstream cylinder decreased in scale with decreasing d/D.     

An immersed interface method for flow past circular cylinder in the vicinity of a plane moving wall

Two‐dimensional flows past a stationary circular cylinder near a plane boundary are numerically simulated using an immersed interface method with second‐order accuracy. Instead of a fixed wall, a moving wall with no‐slip boundary is considered to avoid the complex involvement of the boundary layer and to focus only on the shear‐free wall proximity effects for investigating the force dynamics and flow fields. To analyze the convergence and accuracy of our implementation, numerical studies have been first performed on a simple test problem of rotational flow, where the second order of convergence is confirmed through numerical experiments and an optimal range of relative grid‐match ratio of Lagrangian to Eulerian grid sizes has been recommended. By comparing the force quantities and the Strouhal number, the accuracy of this method has been demonstrated on the flow past a stationary isolated cylinder. The cylinder is then put in proximity to the wall to investigate the shear‐free wall proximity effects in the low Reynolds number regime (20≤Re≤200). The gap ratio, e/D, where e denotes the gap between the cylinder and the moving wall and D denotes the diameter of the cylinder, is taken from 0.10 to 2.00 to determine the critical gap ratio, (e/D)_critical, for the alternate vortex shedding, where the fluid forces, flow fields and the streamwise velocity profiles are studied. One of the key findings is that the (e/D)_critical for the alternate vortex shedding decreases as the Reynolds number increases. We also find that, in this low Reynolds number regime, the mean drag coefficient increases and peaks at e/D = 0.5 with the increase of e/D and keeps decreasing gently from e/D = 0.5 to e/D = 2.0, while the mean lift coefficient decreases monotonically with the increase of e/D. New correlations are then proposed for computing force coefficients as a function of Re and e/D for a cylinder in the vicinity of a moving plane wall. Copyright © 2015 John Wiley & Sons, Ltd.     

Cross-flow past a pair of nearly in-line cylinders with the upstream cylinder subjected to a transverse harmonic oscillation

An experimental investigation is presented for the cross-flow past a pair of staggered circular cylinders, with the upstream cylinder subject to forced harmonic oscillation transverse to the flow direction. Experiments were conducted in a water tunnel with Reynolds numbers, based on upstream velocity, U, and cylinder diameter, D, in the range 1440⩽Re⩽1680. The longitudinal separation between cylinder centres is L/D=2.0, with a transverse separation (for the mean position of the upstream cylinder) of T/D=0.17; the magnitude of the harmonic oscillation is 0.44D peak-to-peak and the nondimensional frequency range of the excitation is 0.05⩽f_eD/U⩽0.44. Flow visualization of the wake-formation region and hot-film measurements of the wake spectra are used to investigate the wake-formation process. An earlier study showed that stationary cylinders in this nearly in-line configuration straddle two very different flow regimes, the so-called shear-layer reattachment (SLR) and induced separation (IS) regimes. The present study, demonstrates that oscillation of the upstream cylinder causes considerable modification of the flow patterns around the cylinders. In particular, the wake experiences strong periodicities at the frequency of the oscillating cylinder; in addition to the usual fundamental lock-in, both sub- and superharmonic resonances are obtained. It is also observed that, although the flow exhibits regions of SLR and IS for excitation frequencies below the fundamental lock-in, for frequencies above the lock-in range the flow no longer resembles either of these flow regimes and vortices are formed in the gap between the cylinders.     

Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows

Steady incident flow past a circular cylinder for sub- to supercritical Reynolds number has been simulated as an unsteady Reynolds-averaged Navier–Stokes (RANS) equation problem using nonlinear eddy-viscosity modelling assuming two-dimensional flow. The model of Craft et al. (Int. J. Heat Fluid Flow 17 (1996) 108), with adjustment of the coefficients of the ‘cubic’ terms, predicts the drag crisis at a Reynolds number of about 2×10⁵ due to the onset of turbulence upstream of separation and associated changes in Strouhal number and separation positions. Slightly above this value, at critical Reynolds numbers, drag is overestimated because attached separation bubbles are not simulated. These do not occur at supercritical Reynolds numbers and drag coefficient, Strouhal number and separation positions are in approximate agreement with experimental measurements (which show considerable scatter). Fluctuating lift predictions are similar to sectional values measured experimentally for subcritical Reynolds numbers but corresponding measurements have not been made at supercritical Reynolds numbers. For oscillatory ambient flow, in-line forces, as defined by drag and inertia coefficients, have been compared with the experimental values of Sarpkaya (J. Fluid Mech. 165 (1986) 61) for values of the frequency parameter, β=D²/νT, equal to 1035 and 11240 and Keulegan–Carpenter numbers, KC=U₀T/D, between 0.2 and 15 (D is cylinder diameter, ν is kinematic viscosity, T is oscillation period, and U₀ is the amplitude of oscillating velocity). Variations with KC are qualitatively reproduced and magnitudes show best agreement when there is separation with a large-scale wake, for which the turbulence model is intended. Lift coefficients, frequency and transverse vortex shedding patterns for β=1035 are consistent with available experimental information for β≈250−500. For β=11240, it is predicted that separation is delayed due to more prominent turbulence effects, reducing drag and lift coefficients and causing the wake to be more in line with the flow direction than transverse to it. While these oscillatory flows are highly complex, attached separation bubbles are unlikely and the flows probably two dimensional.     

In the intermixing region behind circular cylinders with stepwise change of the diameter

The flow within the intermixing region behind circular cylinders with stepwise change of the diameter of diameter ratio d/D of 0.5 has been examined. Based on the statistical analysis and conditional sampling of the velocity fluctuations and of flow visualization, the vortex wakes associated with the big and small cylinders have been established. Both wakes are found under the dominant primary mode, which corresponds to the vortex shedding Strouhal number of two dimensional cylinder, and the less dominant secondary mode. The Strouhal number of the secondary mode of the big vortex wake is higher than that of the primary mode and the opposite is found for the small vortex wake. Both vortex wakes and their modes are found convecting downstream and into region behind the other cylinder. Both wakes are observed to be different from that of two dimensional cylinder.List of symbols D, d diameter of big and small cylinder - f frequency - R ₁₂ (f) cross-power spectral function - R ₁₁, R ₂₂ auto-power functions - Re _D, Re_d Reynolds numbers U ₀ D/v, U ₀ d/v - t time relative to triggering instant - U ₀ freestream mean velocity - U, V, W streamwise, lateral and spanwise mean velocity, respectively - u, v, w streamwise, lateral and spanwise velocity fluctuations, respectively - U _f phase velocity - U _T convection velocity - u _R, v_r recovered u and v velocity fluctuations - uv Reynolds stress - x, y, z streamwise, lateral, and spanwise coordinates, respectively - separation - ₁₂ ² (f) coherence function - _R recovered coherent vorticity fluctuation - phase - ₁₂ (f) phase spectral function     

FLOW-INDUCED VIBRATION OF A CIRCULAR CYLINDER AT LIMITING STRUCTURAL PARAMETERS

Transverse oscillation of a dynamically supported circular cylinder in a flow at Re=100 has been numerically simulated using a high-resolution viscous-vortex method, for a range of dynamical parameters. At the limiting case with zero values of mass, damping and elastic force, the cylinder oscillates sinusoidally at amplitudeA /D=0·47 and frequency fD/U_∞=0·156. For zero damping, the effects of mass and elasticity are combined into a new, “effective” dynamic parameter, which is different from the classic “reduced velocity”. Over a range of this parameter, the response exhibits oscillations at amplitudes up to 0·6 and frequencies between 0·15 and 0·2. From this response function, the classic response in terms of reduced velocity can be obtained for fixed values of the cylinder/fluid ratio m^*. It displays “lock-in” at very high values of m^*.     

Effect of blockage on free vibration of a circular cylinder at low Re

The effect of the blockage on vortex‐induced vibrations of a circular cylinder of low non‐dimensional mass (m^*=10) in the laminar flow regime is investigated in detail. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. The transverse response of the cylinder is found to be hysteretic at both ends of synchronization/lock‐in region for 5% blockage. However, for the 1% blockage hysteresis occurs only at the higher Re end of synchronization/lock‐in region. Computations are carried out at other blockages to understand its effect on the hysteretic behavior. The hysteresis loop at the lower Re end of the synchronization decreases with decrease in blockage and is completely eliminated for blockage of 2.5% and less. On the other hand, hysteresis persists for all values of blockage at the higher Re end of synchronization/lock‐in. Although the peak transverse oscillation amplitude is found to be same for all blockage (～0.6D), the peak value of the aerodynamic coefficients vary significantly with blockage. The r.m.s. values show lesser variation with blockage. The effect of streamwise extent of computational domain on hysteretic behavior is also studied. The phase between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and is independent of blockage. Copyright © 2008 John Wiley & Sons, Ltd.     

A numerical investigation of the flow over a pair of identical square cylinders in a tandem arrangement

This paper describes a numerical study of the two‐dimensional and three‐dimensional unsteady flow over two square cylinders arranged in an in‐line configuration for Reynolds numbers from 40 to 1000 and a gap spacing of 4D, where D is the cross‐sectional dimension of the cylinders. The effect of the cylinder spacing, in the range G = 0.3D to 12D, was also studied for selected Reynolds numbers, that is, Re = 130, 150 and 500. An incompressible finite volume code with a collocated grid arrangement was employed to carry out the flow simulations. Instantaneous and time‐averaged and spanwise‐averaged vorticity, pressure, and streamlines are computed and compared for different Reynolds numbers and gap spacings. The time averaged global quantities such as the Strouhal number, the mean and the RMS values of the drag force, the base suction pressure, the lift force and the pressure coefficient are also calculated and compared with the results of a single cylinder. Three major regimes are distinguished according to the normalized gap spacing between cylinders, that is, the single slender‐body regime (G < 0.5), the reattach regime (G < 4) and co‐shedding or binary vortex regime (G ≥4). Hysteresis with different vortex patterns is observed in a certain range of the gap spacings and also for the onset of the vortex shedding. Copyright © 2011 John Wiley & Sons, Ltd.     

Aerodynamic characteristics of a square cylinder with a rod in a staggered arrangement

The aerodynamic characteristics of a square cylinder with an upstream rod in a staggered arrangement were examined. The pressure measurement was conducted in a wind tunnel at a Reynolds number of Re_D=82,000 (based on the width of the square cylinder) and the flow visualization was carried out in a water tunnel with the hydrogen bubble technique at Re_D=5,200. When the rod and the square cylinder were in tandem, the reduction of drag was mainly caused by the increase of the rear suction pressure. When the staggered angle was introduced, the shield and disturbance effect of the rod on the square cylinder diminished, which results in the increase of the cylinder drag. The side force induced by the staggered angle is small (the maximum value is 20% of the drag of the isolate square cylinder). There were six different flow modes with various staggered angles and spacing ratios, and the corresponding flow patterns are presented in present paper.     

Flow interference between a stationary cylinder and an elastically mounted cylinder arranged in proximity

The flow interference between two circular cylinders, one stationary and the other free to oscillate in the transverse direction, are studied numerically at Re=150. The incompressible Navier–Stokes equation in two space dimensions, an assumption that is expected to be valid at the considered Re, is solved by the characteristic-based-split (CBS) finite element method using the T4/C3 MINI triangular element. The center-to-center spacing between the two cylinders is fixed at 4D, where D is the cylinder diameter. The angle of incident flow, α, with respect to the line through the two cylinder centers, varies within the range from α=0° to 90°. For the elastically mounted cylinder, the reduced mass considered is M_r=2.0; the structural damping coefficient is assigned to be zero, which encourages high amplitude oscillations. For each α, the computations are conducted for a wide range of reduced velocities, U_r. The flow interference is examined by scrutinizing (i) the frequency characteristics of the vortex shedding and oscillation; (ii) the dynamic response of the oscillating cylinder, including the amplitude of displacement, the drag and lift force characteristics and the phase relationship between the lift and the displacement series; and (iii) the flow response in terms of the instantaneous vorticity field. It was found that the flow interference type is significantly affected by the angle of the incident flow. As the cylinder is oscillated outside of the region of the wake behind the stationary cylinder (α≥30°), it behaves similarly to its isolated counterpart. In contrast, if the cylinder is partially or entirely submerged within the upstream wake (α<30°), then both the flow and body responses are substantially modified due to the vigorous interaction between the upstream wake and the oscillating cylinder; the response therefore belongs to the wake-induced regime. The U_r range associated with the higher amplitude response is significantly shifted toward a higher U_r. The maximum vibration amplitude builds up to a significantly higher level, even increasing the U_r far beyond the resonance regime. In general, the wake flow associated with the wake-induced vibration (WIV) regime appears to be more unperiodic than does that corresponding to the vortex-induced oscillation regime. It is also revealed that both the vortex-cylinder and the shear layer-cylinder interaction mechanisms are responsible for the characteristics of the responses of oscillating cylinder. The larger momentum required for the higher oscillation amplitude is obtained from the duration of the energy transfer from the fluid to the cylinder, which is ascribed to the phase lag between the lift force and the cylinder displacement.     

不同交通流状态下桥梁气动导数的风洞试验研究

为了考虑实际运营车辆对桥梁气动导数的影响,根据车辆密度模拟了三种交通流状态,基于强迫振动装置,分别对每个交通流和无车状态下的桥梁气动导数进行风洞试验研究,讨论了不同攻角下不同车流的车辆对桥梁气动导数的影响,探究了车辆对气动导数影响的百分比以及气动导数变化量的变化规律。研究结果表明:不同攻角下不同车流的车辆均对直接导数A*2、H*4和交叉导数A*4、H*2影响显著,A*2、A*3变化量随着折减风速有一定的变化规律。虽然不同攻角下不同车流的车辆对气动导数的影响程度及影响规律不同,并且车流的繁忙程度对大多数气动导数的影响规律不明显,但是车辆对桥梁气动导数的影响不容忽略。     

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

Results are presented for the numerical simulation of vortex-induced vibrations (VIVs) of a cylinder at low Reynolds numbers (Re). A stabilized space–time finite-element formulation is utilized to solve the incompressible flow equations in primitive variables. The cylinder, of low nondimensional mass (m^*=10), is free to vibrate in, both, the transverse and in-line directions. To investigate the effect of Re and reduced natural frequency, F_n, two sets of computations are carried out. In the first set of computations the Reynolds number is fixed (=100) and the reduced velocity (U^*=1/F_n) is varied. Hysteresis, in the response of the cylinder, is observed at the low- as well as high-end of the range of reduced velocity for synchronization/lock-in. In the second set of computations, the effect of Reynolds number (50Re500) is investigated for a fixed reduced velocity (U^*=4.92). The effect of the Reynolds number is found to be very significant for VIVs. While the vortex-shedding mode at low Re is 2S (two single vortices shed per cycle), at Re300 and larger, the P+S mode of vortex shedding (a single vortex and one pair of counter-rotating vortices are released in each cycle of shedding) is observed. This is the first time that the P+S mode has been observed for a cylinder undergoing free vibrations. This change of vortex-shedding mode is hysteretic in nature and results in a very large increase in the amplitude of in-line oscillations. Since the flow ceases to remain two-dimensional beyond Re200, it remains to be seen whether the P+S mode of shedding can actually be observed in reality for free vibrations.     

Flow-induced oscillation of two circular cylinders in tandem arrangement at low Re

Results are presented for flow-induced vibrations of a pair of equal-sized circular cylinders of low nondimensional mass (m^*=10) in a tandem arrangement. The cylinders are free to oscillate both in streamwise and transverse directions. The Reynolds number, based on the free-stream speed and the diameter of the cylinders, D is 100 and the centre-to-centre distance between the cylinders is 5.5D. The computations are carried out for reduced velocities in the range 2≤U^*≤15. The structural damping is set to zero for enabling maximum amplitudes of oscillation. A stabilized finite element method is utilized to carry out the computations in two dimensions. Even though the response of the upstream cylinder is found to be qualitatively similar to that of an isolated cylinder, the presence of a downstream cylinder is found to have significant effect on the behaviour of the upstream cylinder. The downstream cylinder undergoes very large amplitude of oscillations in both transverse and streamwise directions. The maximum amplitude of transverse response of the downstream cylinder is quite similar to that of a single cylinder at higher Re beyond the laminar regime. Lock-in and hysteresis are observed for both upstream and downstream cylinders. The downstream cylinder undergoes large amplitude oscillations even beyond the lock-in state. The phase between transverse oscillations and lift force suffers a 180 jump for both the cylinders almost in the middle of the synchronization regime. The phase between the transverse response of the two cylinders is also studied. Complex flow patterns are observed in the wake of the freely vibrating cylinders. Based on the phase difference and the flow patterns, the entire flow range is divided into five sub-regions.     

Finite element study of vortex‐induced cross‐flow and in‐line oscillations of a circular cylinder at low Reynolds numbers

Vortex‐induced vibrations of a circular cylinder placed in a uniform flow at Reynolds number 325 are investigated using a stabilized space–time finite element formulation. The Navier–Stokes equations for incompressible fluid flow are solved for a two‐dimensional case along with the equations of motion of the cylinder that is mounted on lightly damped spring supports. The cylinder is allowed to vibrate, both in the in‐line and in the cross‐flow directions. Results of the computations are presented for various values of the structural frequency of the oscillator, including those that are sub and superharmonics of the vortex‐shedding frequency for a stationary cylinder. In most of the cases, the trajectory of the cylinder corresponds to a Lissajou figure of 8. Lock‐in is observed for a range of values of the structural frequency. Over a certain range of structural frequency (F_s), the vortex‐shedding frequency of the oscillating cylinder does not match F_s exactly; there is a slight detuning. This phenomenon is referred to as soft‐lock‐in. Computations show that this detuning disappears when the mass of the cylinder is significantly larger than the mass of the surrounding fluid it displaces. A self‐limiting nature of the oscillator with respect to cross‐flow vibration amplitude is observed. It is believed that the detuning of the vortex‐shedding frequency from the structural frequency is a mechanism of the oscillator to self‐limit its vibration amplitude. The dependence of the unsteady solution on the spatial resolution of the finite element mesh is also investigated. Copyright © 1999 John Wiley & Sons, Ltd.