Fluctuating Lift Forces Research Articles

An Experimental Investigation of Forces on a Textured Cylinder at Subcritical Reynolds Numbers

Spanwise deviations from a perfect circular cross-section, affect the aerodynamics and hydrodynamics of cylindrical structures. In this study the ability of a novel wavy cylinder, the textured cylinder, to reduce applied drag and lift forces is investigated experimentally. Three different configurations of the proposed textured geometry, as well as a bare cylinder and a cylinder with helical strakes are 3D printed and tested in an open circuit wind tunnel. Tests are conducted at Reynolds numbers from 3.5×104 to 8.0×104. Results show that the time-averaged drag and fluctuating lift forces on the textured cylinder can be 20% and 15% lower than those of a bare cylinder, respectively. Amongst the tested cylinders, the straked cylinder exhibited the lowest fluctuating lift forces. However, the lowest time-averaged drag forces were measured on the textured cylinder, Tex30, through the range of Reynolds numbers tested. The Strouhal number of the Tex30 was found to be between those of the bare and straked cylinders.

The Effects of Aerodynamic Interference on the Aerodynamic Characteristics of a Twin-Box Girder

To investigate the aerodynamic characteristics of a twin-box girder in turbulent incoming flow, we carried out wind tunnel tests, including two aerodynamic interferences: leading body-height grid, and leading circular cylinder. In this study, the pressure distribution and the mean and fluctuating aerodynamic forces with the two interferences are compared with bare deck in detail to investigate the relationship between aerodynamic characteristics and the incoming flow characteristics (including Reynolds number and turbulence intensity). The experimental results reveal that, owing to the body-height flow characteristics around the deck interfered with by the body-height grid, the disturbed aerodynamic characteristics of the twin-box girder differ considerably from those of the bare twin-box girder. At the upstream girder, due to the vortex emerging from the body-height grid breaking the separation bubble, pressure plateaus in the upper and lower surface are eliminated. In addition, the turbulence generated by the body-height grid reduces the Reynolds number sensitivity of the twin-box girder. At a relatively high Reynolds number, the fluctuating forces are mainly dominated by turbulence intensity, and the time-averaged forces show almost no change under high turbulence intensity. At a low Reynolds number, the time-averaged forces change significantly with the turbulence intensity. Moreover, at a low Reynolds number, the wake of the leading cylinder effectively forces the boundary layer to transition to turbulence, which reduces the Reynolds number sensitivity of the mean aerodynamic forces and breaks the separation bubbles. Additionally, the fluctuating drag force and the fluctuating lift force are insensitive to the diameter and the spacing ratio.

Study on the fluctuating wind responses of constructing bridge towers with magnetorheological elastomer variable stiffness tuned mass damper

Constructing bridge towers are high-rise and flexible structures subjected to significant wind induced vibrations. Tuned Mass Dampers (TMDs) have been widely used to reduce dynamic responses of high-rise structures under fluctuating wind loadings. By resonance with main structure, TMD can effectively suppress structural vibrations. However, the natural frequencies of bridge tower usually decrease continuously during its construction progress. The frequency shift characteristic will cause the detune of TMD, leading to a great control efficiency loss. As a novel stiffness tunable device made of magnetorheological elastomer (MRE), MRE-TMD can adjust its natural frequency in real-time to track the main structure, avoid detuning and achieve better control performance. To study the wind induced vibration control performance of MRE-TMD, this paper explores the fluctuating wind responses of constructing bridge towers in along wind and cross wind directions. The fluctuating wind loads are generated by harmonic superposition method with the along wind fluctuating wind speed spectrum and the empirical power spectrum of fluctuating lift force. By comparing among the uncontrolled, TMD controlled and MRE-TMD controlled constructing bridge towers, the simulation results show that the MRE-TMD system can effectively maintain the tuning state, and significantly reduce the wind induced vibrations during whole construction process.

Vortex-Induced Vibrations of a square prism free to oscillate in the inline direction

We study the response of a prism placed in flow and free to oscillate in the direction of flow (the inline direction) for different possible angles of attack. The tests were conducted in a water tunnel for a range of reduced velocities of 2.4 <U* < 5.8 and Reynolds numbers of 757 <Re < 1900, and over angles of attack of 0° ≤α≤ 45°, where α= 0° corresponded to the case where one of the square’s sides was perpendicular to the incoming flow. We observed Vortex-Induced Vibrations in the inline direction for angles of attack of α= 0°, 2.5°, 5°, 7.5°, and we showed that lock-in has occurred in the cases where oscillations are observed, by showing that the shedding frequency and the oscillation frequency were equal in this range. Throughout the lock-in range, the fluctuating force and the displacement remained in phase as well. The width of the lock-in range and the maximum amplitude of oscillations decreased with increasing angles of attack, and no oscillations were observed for α > 7.5°. The wake structures were found to be predominantly symmetric for α= 0°, and transitioned toward asymmetry with increasing α. By calculating the ratio of the fluctuating lift force to the fluctuating drag force, we quantified how the asymmetry in fluid forces increased with increasing both the angle of attack and the reduced velocity. We also showed how for larger angles of attack where oscillations were not observed, giving an initial disturbance to the prism resulted in formation of vortices for only few cycles and the prism’s oscillations decayed and were damped by the fluid forces.

Nonlinear vibration analysis of vortex-induced vibrations in overhead power lines with nonlinear vibration absorbers

Vortex-induced vibrations are one of the major factors in fatigue failure of power transmission lines and can be mitigated using vibration absorbers in the form of Stockbridge dampers. Since power transmission lines play an important role in modern infrastructure, a thorough understanding of the nonlinear dynamical interactions between conductors, dampers, and wind forces is crucial. Although different nonlinear models exist for conductor vibration with attached dampers or under wind force, no work combines all these nonlinearities in a single model and examines the dynamics of the conductor along with dampers. In an attempt to fill this gap, this work combines the nonlinearities from the mid-plane stretching of the conductor, equivalent cubic stiffness of the Stockbridge damper, and fluctuating lift force modeled as a Van der Pol oscillator in a single model to investigate the nonlinear vortex-induced vibrations. In this work, the conductor is modeled as a simply supported beam and the Stockbridge damper as a mass–spring–damper–mass system with a combination of cubic and linear stiffness. The governing equations of motion are solved analytically using the method of multiple scales for the case of primary resonance between the fluctuating lift-force and conductor. Analytical findings are further validated by comparing against the numerical integration of a reduced-order system, and the results show an excellent match. The analysis is extended by conducting a parametric study to investigate the effect of different system parameters on the frequency response curves. These findings are promising and further provide a direction to design an optimal vibration absorber.

Numerical studies on flow-induced motions of a semi-submersible with three circular columns

Abstract The semi-submersible with three circular columns is an original concept of efficient multifunctional platform, which can be used for marginal oil, gas field, and floater of wind turbines (FOWT). However, under certain flow conditions, especially in uniform current with specific velocities, the eddies will alternatively form and drop behind columns, resulting in the fluctuating lift force and drag force. Consequently, the semi-submersible will subject to the flow-induced motions (FIM). Based on the detached eddy simulation (DES) method, the numerical studies were carried out to understand the FIM characteristics of the three-column semi-submersible at two different parameters, i.e., current incidences (0°, 30°, and 60°-incidences) and reduced velocities ( 4 ≤ U r ≤ 14 ). The results indicate that the lock-in range of 6 ≤ U r ≤ 10 for the transverse motions is presented, and the largest transverse non-dimensional nominal amplitude is observed at 60°-incidence, with a value of A y / D = 0.481 . The largest yaw amplitude A y a w is around 3.0° at 0°-incidence in the range of 8 ≤ U r ≤ 12 . The motion magnitude is basically the same as that of a four-column semi-submersible. However, smaller responses are presented compared to those of the three-column systems revealing the mitigation effect of the pontoon on FIM.

Some new characteristics of the confined flow over circular cylinders at low reynolds numbers

This study summarises some new characteristics of the fluid flow over a confined circular cylinder at low Reynolds numbers. Results from both two- and three-dimensional direct numerical simulations are presented at blockage ratio between 0.1 and 0.9 and Reynolds number between 120 and 500. Floquet stability analysis of selected cases will also be presented. From the two-dimensional simulations, it is found that the fluctuating lift forces decreases with blockage ratio and becomes zero (where the flow is steady) at blockage ratio of approximately 0.7–0.8. Upon further increasing the blockage ratio to 0.9, the simulations show a dramatic increase in the fluctuating lift forces, nearly an order of magnitude greater than previously reported for an unconfined cylinder flow. It is also found that for blockage ratio of 0.5, there is a long term two-dimensional instability that becomes more prominent with increasing Reynolds number. This instability has a time scale of approximately 105 time units (D/Umax) at Reynolds number of 500. In addition, the transition between two- and three-dimensional flow at blockage ratios up to 0.5 is investigated. It is shown that the transition Reynolds number decreases with increasing blockage ratio. At high blockage ratio of 0.5, as we increase the Reynolds number, the transition to three-dimensional flow is shown to go from unsteady two-dimensional to steady three-dimensional before transitioning to unsteady three-dimensional flow.

Experimental investigation of the angle of attack dependence of the flow past a cactus-shaped cylinder with four ribs

The aerodynamic properties of a two-dimensional cactus-shaped cylinder with four ribs are studied experimentally. The cross section of the cylinder corresponds to a modified square cylinder with rounded corners and concave sides inspired by tall succulents with four ribs. The mean aerodynamic coefficients, fluctuating lift coefficient, Strouhal number, and mean surface pressure distribution are measured as a function of the angular orientation for Reynolds numbers ranging from 50,000 to 150,000. Hot-wire measurements are conducted at two locations in the wake for several angles of attack to provide insight into the vortex shedding frequencies. The results show that the studied shape exhibits a strong angle of attack dependence while no Reynolds number dependence is found within the tested range. As for the square cylinder, a critical angle of attack of approximately 13° is observed at which drag and fluctuating lift forces are minimised while mean lift and Strouhal number reach their highest values. Reattachment of the shear layer at and above the critical angle of attack is confirmed using PIV. Overall, succulents with four ribs retain some of the aerodynamic benefits that have been observed for cactus-shaped cylinders with many ribs, albeit over a limited range of angles of attack.

Tonal-Noise Assessment of Quadrotor-Type UAV Using Source-Mode Expansions

The present work deals with the modeling of the aerodynamic sound generated by the propellers of small-size drones, taking into account the effects of horizontal forward flight with negative pitch and of installation on supporting struts. Analytical aeroacoustic formulations are used, dedicated to the loading noise. The fluctuating lift forces on the blades are expanded as circular distributions of acoustic dipoles, the radiated field of which is calculated by using the free-space Green’s function. This provides descriptions of the sound field, valid in the entire space. The stationary mean-flow distortions responsible for the lift fluctuations and at the origin of the sound are estimated from existing numerical flow simulations and from ad hoc models. Installation and forward-flight effects are found to generate much more sound than the steady loading on the blades associated with thrust. Therefore, the models are believed reliable fast-running tools that could be used for preliminary low-noise design through repeated parametric calculations, or for noise-impact estimates corresponding to prescribed urban traffic.

Hydrodynamic Forces of a Semi-Submerged Cylinder in an Oscillatory Flow

This work experimentally investigated the performance of hydrodynamic forces on a semi-submerged cylinder under an oscillatory flow. To generate the equivalent oscillatory flow, the semi-submerged cylinder is forced to oscillate in several combinations of different periods and amplitudes. The mean downward lift force was observed to be significant and the fluctuating lift forces show dominant frequency is twice that of oscillatory flow and amplitude that is the same as the mean lift force. Based on this main hydrodynamic feature, a novel empirical prediction formula for the lift forces on semi-submerged cylinder under oscillatory flow is proposed where the lift forces expression is proportional to the square of oscillatory flow velocity. This novel empirical formula directly assigns the fluctuating lift force with frequency twice of oscillatory flow and the amplitude that is the same as the mean lift force. This assignment of empirical lift force formula reduces parameters required to determine a dynamic lift force but is demonstrated to well predict the fluctuating lift force. The lift coefficient can reach 1.5, which is larger than the typical value 1.2 of the drag coefficient for a fully submerged cylinder with infinite depth. Moreover, relationships among hydrodynamic coefficients, Keulegan-Carpenter (KC) number, Stokes number and Froude number are studied. With the increase of KC number, the Froude number has a more significant influence on the distribution of hydrodynamic coefficients. As Froude number is increasing, the drag coefficient shows a nonlinear decay (KC &lt; 20) but a linear increase (KC &gt; 20), while the added mass coefficients show a nonlinear (KC &lt; 20) and a linear (KC &gt; 20) increase trend. The present work can provide useful references for design of the relevant marine structures and serve as the useful guideline for future research.

Numerical study of wake and aerodynamic forces on a twin-box bridge deck with different gap ratios

Two-dimensional Delayed Detached Eddy Simulation (DDES) was carried out to investigate the uniform flow over a twin-box bridge deck (TBBD) with various gap ratios of L/C=5.1%, 12.8%, 25.6%, 38.5%, 73.3% and 108.2% (L: the gap-width between two girders, C: the chord length of a single girder) at Reynolds number, Re=4x104. The aerodynamic coefficients of the prototype deck with gap ratio of 73.3% obtained from the present simulation were compared with the previous experimental and numerical data for different attack angles to validate the present numerical method. Particular attention is devoted to the fluctuating pressure distribution and forces, shear layer reattachment position, wake velocity and flow pattern in order to understand the effects of gap ratio on dynamic flow interaction with the twin-box bridge deck. The flow structure is sensitive to the gap, thus a change in L/C thus leads to single-side shedding regime at L/C&#880425;.6%, and co-shedding regime at L/C&#880435;.8% distinguished by drastic changes in flow structure and vortex shedding. The gap-ratio-dependent Strouhal number gradually increases from 0.12 to 0.27, though the domain frequencies of vortices shedding from two girders are identical. The mean and fluctuating pressure distributions is significantly influenced by the flow pattern, and thus the fluctuating lift force on two girders increases or decreases with increasing of L/C in the single-side shedding and co-shedding regime, respectively. In addition, the flow mechanisms for the variation in aerodynamic performance with respect to gap ratios are discussed in detail.

Direct Numerical Simulations of Horizontally Oblique Flows Past Three-Dimensional Circular Cylinder Near a Plane Boundary

Abstract Understanding hydrodynamics of a free-spanning pipeline subjected to omni-directional flows is important to engineering design. In this study, horizontally oblique flows past a three-dimensional circular cylinder in the vicinity of a plane boundary are numerically investigated using direct numerical simulations. Parametric studies are carried out at the normal Reynolds number of 500, a fixed gap-to-diameter ratio of 0.8 and five flow inclination angles (α) ranging from 0 deg to 60 deg with an increment of 15 deg. Two distinct vortex-shedding modes are observed: parallel (α ≤ 15 deg) and oblique (α ≥ 30 deg) vortex shedding. The wake evolution is further divided into two or three stages depending on α. The occurrence of the oblique vortex shedding is accompanied by the base pressure gradient along the cylinder span and the resultant axial flows near the cylinder base. The total hydrodynamic drag and lift force coefficients decrease from being the parallel mode to the oblique mode, owing to the intensified three-dimensionality of wake flows and the phase differences in the spanwise vortex shedding. The independence principle (IP) is found to be valid in predicting hydrodynamic forces and wake patterns when α ≤ 15 deg. This IP might produce unacceptable errors when α &amp;gt; 15 deg. In comparison with the mean drag force, the fluctuating lift force is more sensitive to the inclination angle. The IP validity range is substantially smaller than that in the case of flow past a wall-free cylinder. Such finding would be practically useful for vortex-induced vibration prediction.

Numerical study on suppression of vortex-induced vibration of circular cylinder by helical wires

The mechanism of suppression for the vortex-induced vibration (VIV) of circular cylinder by the helical wires is investigated using LES turbulence model. Numerical models for the free-vibrating circular cylinders with and without the helical wires are proposed and validated by the experiments. Simulations are carried out for a large mass ratio of 248 and a small damping ratio of 0.00257 with the Reynolds numbers ranging from 16000 to 24500 and the reduced velocities ranging from 4.5 to 6.8. It is noticed that the helical wires of diameters ratio d/D ​= ​0.1 effectively suppress the amplitude of VIV by nearly 80% and avoid the “lock in”. The steady and unsteady aerodynamic forces of circular cylinders with and without the helical wires as well as the flow patterns are also examined to clarify the mechanism of VIV suppression. It is found that the fluctuating lift forces and their spanwise correlation for the wired cylinder are significantly reduced comparing with those for the bare cylinder, due to the enhancement of three-dimensional disturbance to the wake caused by helical wires. The aerodynamic damping for the oscillated bare cylinder is negative, while that for the wired cylinder is positive with a vibration amplitude A/D ​= ​0.1 ​at the resonance velocity of bare cylinder.

Measurements of fluctuating lift forces on rectangular cylinders in turbulent flow

The fluctuating lift on a rectangular cylinder induced by turbulent flow is investigated by force measurement experiments. A square uniform grid is installed downstream of the entrance of the working section of the wind tunnel to generate approximately homogeneous and isotropic turbulence. The rectangular cylinder models of two different cross sections (width to depth ratio B/D = 5, 10) are tested in the experiments. Each rectangular cross section has two aspect ratios (length to width ratio θ = 1, 7) and a fixed width (B = 0.15 m). The measurements include the spectra of fluctuating lift force and vertical turbulent velocity component. From these measurements, the experimental values of generalized transfer functions of these rectangular cylinder models are obtained, which are compared with the calculated values based on three aerodynamic force theories. The one-wavenumber transfer functions and the spanwise influence terms of these rectangular cylinders are used in the calculations, which were determined by the pressure measurement experiments in the previous work. Comparisons of the measured and calculated values of generalized transfer functions demonstrate that the three-dimensional theory is satisfactory and confirm that the one-wavenumber transfer functions and spanwise influence terms determined by the pressure measurements are valid. The results also show that both the two-dimensional theory and the strip theory would overestimate the generalized transfer function, but for the rectangular cylinder with the larger aspect ratio, the strip theory is very close to the three-dimensional theory. The one-wavenumber transfer functions of these rectangular cylinders are therefore determined based on the strip theory. This indicates that the approximate approach to determination of the one-wavenumber transfer function, which was proposed for an airfoil in the previous work, could also be applicable to bluff bodies.

A wake oscillator model with nonlinear coupling for the vortex-induced vibration of a rigid cylinder constrained to vibrate in the cross-flow direction

In this paper, a new wake oscillator model with nonlinear coupling is proposed for the modelling of vortex-induced vibration. The purpose is to develop a model that is capable of reproducing both free and forced vibration experiments. To achieve this goal, an existing van der Pol wake oscillator model is first reviewed. The limitations of the model are discussed and the influence of different drag force models on the dynamic characteristics of the fluctuating lift force that matches the forced vibration experiments are studied. Based on this model, nonlinear coupling terms are introduced to improve its predictive capabilities. The tuning of this improved model to the forced vibration shows a good agreement with experiments in terms of the added damping. However, the model failed to capture the negative added mass at high reduced velocities. As a result, the new model underpredicts both the range and frequency of lock-in in free vibration tests. To eliminate this discrepancy, the model is further enhanced by introducing frequency dependent nonlinear couplings, which are achieved in the time domain by means of convolution integrals. A single set of frequency dependent, complex-valued functions – which are the Laplace transforms of corresponding convolution kernels – that reproduce the forced vibration experiments is determined over a limited range of frequencies. However, no analytical extension of these functions to the infinite frequency domain was found such that the causality principle and the energy conservation would be satisfied. The latter is a major challenge for all existing wake oscillator models that aim at reproducing the forced vibration experiments.

Experimental investigation on the effects of inter-layer gaps on wind loads of cantilevered stadium roofs

To meet the ventilation requirement, an engineer needs to design the cantilevered roof of a stadium as a layered one with gaps between the layers. In order to further understand the wind resistance performance of such structures, a wind tunnel test was carried out on a 1:200 scaled stadium model considering two roof inter-layer cases: opened gaps and blocked gaps. The laws of lift force coefficients and net wind pressure coefficients were compared for these two gap conditions. The study results indicate that the roof experiences larger mean lift force on both top and bottom surfaces when the inter-layer gaps are blocked. However, the roof gaps can reduce the fluctuating lift force on top roof surface and the net fluctuating lift force, weakening the vibration of the flexible large-span roof. The negative pressures (suctions) in the inner and middle roof areas on the leeward side decrease for the blocked gap case. The positive pressure in the outer roof area generally increases along the windward direction, while the negative pressure decreases in other wind directions, when the roof inter-layer gaps are blocked. In the highest and transition roof areas, the fluctuating pressures increase for most wind directions, while in the lowest roof area, they decrease when the roof inter-layer gaps are blocked. The test and analysis results provide a valuable basis for the wind design of this kind of roof structures.

Nonlinear vortex-induced vibration dynamics of a flexible pipe conveying two-phase flow

In this article, a vortex-induced vibration prediction model of a flexible riser conveying two-phase flow, including geometric and hydrodynamic nonlinearity, is established. A van der Pol wake oscillator is utilized to characterize the fluctuating lift forces. The finite element method is chosen to solve the coupled nonlinear fluid–structure interaction equations. The natural frequencies of the flexible riser are calculated to validate the method through comparisons with results from the literature. The modal analyses show that geometric nonlinearity has a significant effect on the natural frequency, and the critical internal velocity is reduced than those in linear analyses. The impacts of the gas volume fraction as functions of cross-flow velocity on the synchronization region, the displacement amplitudes, and the maximum stresses and frequency spectra have been investigated. The results show that an increase in the gas fraction results in the linear increase in natural frequencies and a wider synchronization region, and an increase in liquid flow rate led to the slight decrease in displacement amplitude and maximum stress within a small flow range.

On the flow around two circular cylinders in tandem arrangement at high Reynolds numbers

The flows around two tandem circular cylinders in the subcritical and supercritical flow regimes are studied through three-dimensional numerical simulations. Different spacing ratios L/D (where L is the center-to-center distance between the two cylinders with D being the diameter of the cylinders) from 2.0 to 5.0 is considered. The instantaneous flow structures, pressure distributions and hydrodynamic forces on two tandem cylinders are analyzed at subcritical (Re=2.2×104) and supercritical (Re=3.0×106) Reynolds numbers. The present numerical solutions demonstrated that at the two Re, for L/D<3.5, the shear layer from the upstream cylinder reattaches on the surface of the downstream cylinder and vortex street is only formed behind the downstream cylinder, while the shear layer from the upstream cylinder rolls up alternately for L/D≥3.5, and vortex shedding occurs from both upstream and downstream cylinders. Compared with the flow characteristics at Re=2.2×104, for Re=3.0×106, the flow separation positions on the upstream cylinder move backward and the width of the wake behind the upstream cylinder becomes narrower, which leads to forward movement of the reattachment position. At the supercritical Reynolds number, the mean drag and fluctuating lift coefficients of the upstream cylinder are nearly independent of L/D while the sharp increase of force coefficients of the downstream cylinder occurs for L/D=3.5. Comparing to the case at the subcritical Reynolds number, higher vortex-shedding frequencies from the upstream and downstream cylinders are identified. It is also revealed that the slope of the best fit line for φ (the phase lag of the fluctuating lift force between two cylinders) and L/D increases at the supercritical Reynolds number. In addition, for L/D=2.5, the mean drag coefficients of the upstream cylinder decrease sharply for Re>1.0×105, while those of the downstream cylinder rise slightly with the growing Re. Moreover, the fluctuating lift coefficients of the upstream cylinder are almost independent of Re, but those of downstream cylinder drop as Re increases.

Wake structures and acoustic resonance excitation of a single finned cylinder in cross-flow

This paper presents an experimental and numerical investigation to clarify the effect of circular fins on the wake structures, the dynamic loading, as well as the flow–sound interaction mechanism downstream of a single finned cylinder in cross-flow. It is revealed that adding circular fins to the cylinder enhances the flow coherence along the cylinder’s span, strengthens the intensity of the vortex shedding process downstream of the cylinder, reduces the vortex formation length, and increases the dynamic lift force acting on the cylinder. Thus, adding circular fins to the cylinder makes the flow more susceptible to acoustic resonance excitation, compared to an equivalent bare cylinder. Three-dimensional numerical simulations of the flow field around the cylinder show that the flow is entrained between the fins which results in the generation of an accelerated jet-like flow in the developing shear layer region. The jet-like flow emanating from the fins leads to a closer and stronger shear layer roll-up, and hence a shorter vortex formation region. This, in turn, increases the fluctuating lift force on the cylinder, as the strong oscillating shear layers occur closer to base of the cylinder. Additionally, with the reduction of the vortex formation length, the base pressure coefficient decreases and therefore the drag force on the cylinder increases. These intrinsic flow features explain the global effect of circular fins on the vortex shedding process and there by on the excitation mechanism of acoustic resonance.

Vibrations of a square cylinder submerged in a wake

A numerical investigation is conducted on the flow around and vibration response of an elastic square cylinder (side width $D$) in the wake of a stationary cylinder at Reynolds numbers of $Re=100$ and 200 based on $D$ and the free-stream velocity. The downstream cylinder, referred to as the wake cylinder, is allowed to vibrate in the transverse direction only. The reduced velocity $U_{r}$ is varied from 1 to 30. Cylinder centre-to-centre spacing ratios of $L^{\ast }(=L/D)=2$ and 6 are considered. Simulations are also conducted for a single isolated cylinder, and the results are compared with those for the wake cylinder. The focus is given to vibration response, frequency response, fluctuating lift force, phase relationship between the lift and displacement, work done and the flow structure modification during the cylinder vibration. The results reveal that the dependence of the Strouhal number $St$ on $U_{r}$ can distinguish different branches more appropriately than that of the vibration amplitude on $U_{r}$. The vibration response of the single cylinder at $Re=100$ is characterized by the initial, lower and desynchronization branches. On the other hand, that at $Re=200$ undergoes initial, lower and galloping branches. The galloping involves the characteristics of both the initial and the lower branches or the initial and the desynchronization branches depending on $U_{r}$. For the wake cylinder, the gap flow has a significant impact on the vibration response, leading to (i) the absence of galloping at either $Re$ and $L^{\ast }$, (ii) the presence of an upper branch at $Re=200$, $L^{\ast }=6$ and (iii) an initial branch of different characteristics at $Re=100$, $L^{\ast }=6$. The different facets are discussed in terms of wake structures, work done and phase lag between lift and displacement.