Reduced Velocity Ur Research Articles

Force Element Analysis in Vortex-Induced Vibrations of Side-by-Side Dual Cylinders: A Numerical Study

A numerical investigation was conducted in this study utilizing Force Element Analysis to explore the vortex-induced vibration (VIV) mechanism of side-by-side dual cylinders under the conditions of Reynolds number Re = 100, mass ratio m* = 10, and spacing ratios L/D ranging from 3 to 6. The hydrodynamic forces by force element formulas were incorporated into the vibration response calculations of elastically supported rigid cylinders using a User-Defined Function (UDF) and the fourth-order Runge–Kutta method. A comprehensive analysis was performed to elucidate the combined effects of the spacing ratio L/D and reduced velocity Ur on the vibration responses, quantifying the hydrodynamic forces involved in the mutual interaction during VIV for side-by-side dual cylinders. The influence mechanisms of inter-cylinder interaction and their effects on the resultant hydrodynamic phenomena were discussed. It was revealed that for side-by-side arranged dual cylinders outside the “lock-in region”, the lift and drag forces are predominantly supplied by the volume vorticity forces in conjunction with surface vortices (including frictional) forces. However, within the “lock-in region”, the surface acceleration lift forces provide greater force contributions, and the volume vorticity lift force contributes significantly to negative values. Notably, alterations to the spacing ratio do not change the proportion of force element components. The amplitudes of the cylinders’ mutual interaction forces are identical in magnitude but opposite in phase. Additionally, the “slapping” phenomenon near the “lock-in region” leads to “bounded” trajectories of cylinders.

Experimental study on the auto-parametric internal resonance of a frame structure excited by water vortex-induced vibration in the flume

A small external excitation can trigger a strong auto-parametric internal resonance response in underwater structure. Currently, research on auto-parametric internal resonance primarily relies on theoretical methods and experimental approaches involving the simulation of water flow forces using vibrators. To more realistically reflect the auto-parametric internal resonance of structures in water, this study investigates the phenomenon of auto-parametric internal resonance of a Γ-shaped frame structure induced by vortex-induced vibration in a laboratory flume. The results indicate that the auto-parametric internal resonance can be excited by water flow in the flume and the Γ-shaped frame structure is more prone to experience normal resonance than auto-parametric internal resonance. However, the vibration intensity caused by auto-parametric internal resonance is significantly greater than that of normal resonance. The range of h/L (the ratio of water depth to the length of the vertical beam) for auto-parametric internal resonance in both the clamped-pined joint system (CPJS) and the pined-pined joint system (PPJS) is 0.307–0.364, but the flow velocity range for auto-parametric internal resonance in CPJS is broader than that in PPJS. When h/L is 0.307, the reduced velocity Ur ranges for the auto-parametric internal resonance of CPJS and PPJS are 4.839–5.375 and 4.904–5.256, respectively. When h/L is 0.364, the Ur ranges for the auto-parametric internal resonance of CPJS and PPJS are 4.587–4.742 and 4.507–4.660, respectively.

Coupled response of flow-induced vibration and flow-induced rotation of a circular cylinder with a triangular fairing

In this paper, a numerical simulation investigation is carried out on the coupled response of the flow-induced vibration (FIV) and flow-induced rotation (FIR) of a circular cylinder attached with a triangular fairing at a low Reynolds number of Re = 100. The primary focus is on the impact of FIR on FIV. The vibration response, hydrodynamic coefficient, vortex shedding mode, and flow field characteristics are examined for the fairings within the vibrational reduced velocity Ur range of 3–16 with shape angle of α = 45°, 60°, and 90°. The results reveal that at low Ur, all the three considered fairings have a good suppression effect on the FIV. Nevertheless, the galloping response emerges as Ur increases when α = 45° and 60°. In contrast, the vibration response of 90° fairing presents a wider lock-in region. The rotatable 2-degree of freedom (2-DoF) fairing has a better performance in the reduction of response amplitude and hydrodynamic coefficients. The 2S (two single vortices) vortex shedding mode mainly occurs in the vortex-induced vibration (VIV) region, while 2S–8S (from two to eight vortices), 2P (two pairs of vortices), 2T (two triplets of vortices), and P + T (a pair of vortices and a triple of vortices) modes emerge in the galloping branch. Moreover, four modes of wake structures are identified according to the variation of recirculation region and the migration of boundary layer separation point. Finally, the reduced regions of drag, lift, and amplitude are highlighted compared to the bare cylinder.

Intrinsic features of flow-induced stability of a square cylinder

Vortex-induced vibrations and galloping of an elastically mounted square cylinder are investigated for cylinder mass ratio m* = 2–50, damping ratio ζ = 0–1.0, mass-damping ratio m*ζ = 0–50 and flow reduced velocity Ur = 1–80. We home in on the effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on the critical reduced velocity Urc marking the onset of galloping, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added mass ratio and $m_{ae}^\ast $ is the effective added mass ratio. Vibration responses, forces, vibration frequencies and added mass ratios are studied and discussed. The different branches of vortex-induced vibrations have different dependencies of $m_{ae}^\ast $ on Ur. The $m_{ae}^\ast $ in the initial branch is positive and drops rapidly with Ur, but that in the lower branch is negative and declines gently. In the galloping regime, $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping, declining slightly with increasing Ur. Our results and prediction equations show that when ζ = 0, Urc is independent of m* for m* ≥ 5, albeit slightly higher for m* = 3. The latter is ascribed to mode competition. When ζ > 0, Urc linearly increases with increasing ζ. Detailed analysis substantiates that m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ does not serve as the unique criterion to predict the galloping occurrence. Here, we propose a new combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta$ in the relationship between galloping onsets and structural properties, which successfully scales all data of Urc at different m* and ζ values.

Energy Harnessing Performance of Oscillating Foil Submerged in the Wake of a Fixed Cylinder

The energy harnessing from flow-induced vibrations (FIV) by an oscillating foil placed tandemly behind a circular cylinder (which serves as a vortex generator) is investigated. The foil is submerged in the wake produced by the fixed cylinder and could oscillate in the direction perpendicular to the incoming flow with single-degree freedom. The spacing ratio ranges from 1.0 to 5.0. The oncoming fluid velocity is U = 1–10 m/s, corresponding to the reduced velocity Ur = 3.81–38.08 and the Reynolds number Re = 9.58 × 103–9.58 × 104. Four harnessing damping ratios (ζharness = 0.0054–0.0216) are used to simulate the energy conversion conditions. The main conclusions are: (1) The optimal oscillation pattern related to the highest harnessed energy emerges as the spacing ratio close to 1.0. (2) The airflow energy converted by the foil is positively correlated with the harnessing damping ratio because the amplitude responses are similar at various harnessing damping ratios. A high velocity yields the highest harnessed power. (3) The harnessing efficiency of the foil could reach 48.89%, which is much more than that of an isolated flapping foil.

Flow-induced vibrations of ten tandem cylinders at low Reynolds number

Flow-induced vibrations of ten cylinders in tandem arrangement are numerically investigated by using the immersed boundary method with a low Reynolds number (Re = 100). Seven spacing ratios L/D (where L = center–center spacing between tandem cylinders and D = diameter of the cylinder) are selected from 1.1 to 4.0, and the reduced velocity Ur ranges from 2.0 to 13.5 with an increment of 0.5. Small (L/D < 2.0) and large (L/D > 2.0) spacing ranges are identified, both including two types of responses: wake-induced vibration (WIV; Ur = 2.0–9.0∼10.5 for a small L/D and Ur = 2.0–6.0∼6.5 for a large L/D) and wake-induced galloping (WIG; Ur > 9.0∼10.5 for a small L/D and Ur > 6.0∼6.5 for a large L/D). The largest vibration amplitude of each cylinder is obtained in the WIG region for the small L/D condition. The presence of downstream cylinders suppresses the vortex shedding of upstream cylinders and thus postpones the vibration of upstream cylinders at a small Ur, whereas the downstream cylinder enhances the vibration at a larger Ur due to the wake interference. For a small L/D, three flow regimes with the extended-body, reattachment, and co-shedding patterns are successively presented as Ur increases. For a large L/D, four types of flow regimes, namely, EB-2S (the extended-body with “2S” pattern), RT-2S (the reattachment with “2S” pattern), TR-2S (two-row vortex street with “2S” pattern), and CS-VS (co-shedding with variation shedding), are classified. Two new vortex shedding patterns, “2G (two counter-rotating vortices shed from each side per vibration cycle)” and “2C (two co-rotating vortices shed from each side per vibration cycle),” have been identified. In the WIV region, there is only one dominant vibration frequency for upstream cylinders (C1–C7), while a sub-harmonic frequency emerges and dominates C8–C10 when L/D is large. The fluctuating lift force spectra show a broad-band frequency distribution due to the irregular positions of the vortex generation and merging, and the dominant frequency in the WIG region decreases consecutively from C1 to C10.

Influence of upstream staggered cylinder on vortex-induced vibration responses of side-by-side cylinders in turbulent flow

A direct-forcing immersed boundary method with large-eddy simulation was used to simulate the phenomenon of the vortex-induced vibration (VIV) of multiple cylinders in a flow field. The present study analyzed the influence of an upstream stationary cylinder on the vibration behavior of two side-by-side cylinders downstream in a staggered position. The latter two side-by-side cylinders were allowed to vibrate in the cross-flow direction. By using different center-to-center distances between cylinders, damping ratios, mass ratios, Reynolds numbers, and diameters of the upstream stationary cylinder, the VIV response and energy conversion efficiency of the vibrating cylinders were studied. The results showed that the amplitude and efficiency of the vibrating cylinders are significantly enhanced at reduced velocity UR*≥6.0 when compared with a single vibrating cylinder. The maximum values of amplitude and efficiency can be shifted and enhanced, respectively, by adjusting the mass ratio and damping ratio. Reducing the diameter of the stationary upstream cylinder can effectively improve efficiency, especially in the lock-in region.

Control of flow-induced vibration of a circular cylinder using a splitter plate

A circular cylinder attached by a rigid splitter plate of different lengths was tested to examine its effects on the control of flow-induced vibration. Tests were carried out in a closed-loop water channel. A cylinder of diameter D = 20 mm and a mass ratio m* ≈ 50 was installed to oscillate in the transverse direction. A wide range of splitter length was considered, i.e., L/D = 0–3.5, at a range of reduced velocity Ur = 1–25 and the Reynolds number Re = 800–11 000. Numerical simulations were also conducted to reveal the flow structures associated with the vibration modes observed in the experiment. It is found that, as L/D increases from 0 to 0.25, the peak value of cylinder oscillation amplitude increases and appears at higher reduced velocities. When the splitter length continues to rise, galloping-type oscillations occur at L/D = 0.5 and 0.75. The transition stage has been found at L/D = 1.0. Oscillation is then significantly suppressed when the splitter length is larger than L/D = 1.5.

Aspect ratio influence on the vortex induced vibrations of a pivoted finite height cylinder at low Reynolds number

The effect of the aspect ratio on the vortex induced vibrations (VIV) of a pivoted finite length circular cylinder is investigated. A fixed value of the Reynolds number Re = 100 with four values of the aspect ratio AR=2, 3, 5, 7 is considered. Different values of the reduced velocity ur* in the range 2≤ur*≤11 were used for each AR value with a fixed value of the reduced mass mr*=5. Results on the oscillatory response of the cylinder, hydrodynamic forces, and wake structures are reported. In order to compare the VIV of the different length cylinders, the displacement of the center of mass (which coincides on each case) was analyzed. It is found that the maximum oscillation amplitudes, the extent of the synchronization region, and the wake structures are influenced by the aspect ratio. Also, a steady symmetrical flow is obtained for the small AR=2, 3 cases with relatively low values of ur*, which is found to be unstable when increasing ur*.

Flow-induced vibration of a cantilevered cylinder in the wake of another

This work presents an experimental study of the flow-induced vibration of a cantilevered circular cylinder in the wake of another cylinder, fix-supported at both ends. The diameter and spacing ratios, d/D and L/d are 0.24 – 1.0 and 1.0 – 5.5, respectively, where d and D are the diameters of the upstream and downstream cylinders, respectively, and L is the separation from the upstream cylinder center to the front stagnation point of the downstream cylinder. Measurements are performed over the reduced velocity Ur=3.8 – 68.3 (based on D and freestream velocity), with the downstream cylinder vibration, vortex shedding frequency, surface pressure distribution and flow structures simultaneously captured. The downstream cylinder accompanied by a smaller d/D and/or L/d is more susceptible to galloping vibration and hysteresis in the galloping response. The mechanism behind initiation and sustenance of galloping is unveiled. It has been found that the effective mass and damping play a key role in initiating and sustaining the vibrations. The variation in the flow structure is discussed in detail in both streamwise and spanwise directions, along with the fluid–structure interactions.

Effects of spacing ratio on vortex-induced vibration of twin tandem diamond cylinders in a steady flow

Vortex-induced vibration of twin tandem square cylinders at an inclined angle of 45° to the fluid, i.e., twin diamond cylinders of mass ratio m* = 3, is numerically investigated at Reynolds number Re = 100 and reduced velocity Ur = 3–18. This paper focuses on the effects of cylinders' spacing ratio L* (=L/B, where L is cylinders' center-to-center spacing and B is the characteristic length) ranging from 2 to 6 on the oscillation responses of two-degree-of-freedom cylinders. The results indicate that the wake structure experiences two gap flow patterns, the reattachment and co-shedding regimes, and eight different wake modes. At a small spacing (L* = 2–3), the reattachment regime occurs for the lower or higher Ur with the approximate range of 3 and 16–18. Meanwhile, the reattachment regime mainly occurs for other ranges of Ur at L* = 2–6. The more significant oscillation of each spacing appears in the cross-flow direction, and the maximum cross-flow amplitude of the upstream cylinder is smaller than that of the downstream cylinder. Additionally, although significant cross-flow oscillations occur at small spacings (L* = 2–3) with the Ur ≈ 5–9 and 12–14, the intrinsic mechanisms are entirely different. For the cross-flow oscillation characteristics of larger spacings (L* = 4–6), they are virtually similar.

Triggering of galloping in structures at low Reynolds numbers

The fundamental mechanisms that are important for the triggering of galloping in a flow-induced vibration (FIV) system consisting of the flow past an elastically-mounted body (D-section, isosceles-triangular, rectangular) is investigated in this paper using three key-enabling methodologies: namely, high-fidelity full-order model/computational fluid dynamics simulations, modal analysis based on the determination of an data-driven model using the eigensystem realization algorithm, and use of the Den Hartog stability (galloping) criterion for the assessment of the aerodynamically unstable behavior of the system obtained from classical quasi-steady theory. The synthesis of the results from the application of these three key-enabling technologies is used to study the effect of the Reynolds number Re, the mass ratio m∗, the cross-sectional geometry and the angle of attack of the incident wind direction relative to the orientation of the elastically-mounted body on the suppression or initiation of galloping. In the application of data-driven modal analysis to a FIV system with coupled modes, the importance of identifying correctly which of the coupled modes correspond to the structure mode (SM) possibly taking into account mode switching is stressed, and the failure to do so is shown to lead to a significant underestimation of the value of the reduced velocity Ur associated with the onset of galloping. The range of values of Ur where the SM is positive is related to the flutter lock-in regime for smaller values of Ur (typically for Ur≤7) and to the galloping regime for the larger values of Ur (typically for Ur≥10). The application of data-driven modal analysis to a FIV system shows that increasing Re and decreasing m∗ has two effects: namely, enhancement of the coupling between the structure and wake modes and increases of the range of Ur values where the SM exhibits a positive growth rate. Data-driven modal analysis and the Den Hartog stability criterion is applied to explain the recently observed phenomenon of the collapse of galloping for a rectangular cylinder when the side ratio is decreased below a critical threshold value (between about 0.1 and 0.2). It is shown that galloping appears and disappears when the flat face of a body in directed into (windward of) or away from (leeward of) the incident wind direction. An investigation of the effects of the cross-sectional geometry of the body on the triggering of soft- or hard-galloping is undertaken.

Hydrokinetic energy harvesting from flow-induced motion of oscillators with different combined sections

Flow-Induced Motion (FIM) of elastically-supported oscillators with "circle-polygon-attachments"sections are experimentally investigated in a water channel to examine the effects of combined sections on hydrokinetic energy harnessing. The incoming flow velocity considered is U = 0.55 – 1.35 m/s, corresponding to reduced velocity Ur = 5.3 –13. A controllable magnetic damping system is applied to change the total damping ratio (ζtotal) of the flow-induced motion energy conversion system (FIMECS) by varying excitation voltages VB (VB = 0 – 108 V, ζtotal = 0.037 – 0.398). Particularly, to calculate fluid force with experimental results, a torque sensor is introduced into FIMECS. The amplitude, frequency, fluid force, active power and efficiency are analyzed. Results suggest that an oscillator with symmetric sharp attachments and no vortex reattachment is conducive to galloping with self-excitation under large damping, and the circle-T-attachments oscillator is optimum. Its best branch for energy conversion is the galloping branch, the peak of active power is Pharn is 19.28 W and the peak of efficiency is ηharn is 26.22% (VB = 99 V, Ur = 11.28). Under the same condition, the circular-T-attachments oscillator has a better energy conversion capacity than the existing ones (triangular prism).

The space effect on WIV interference between a fixed and oscillating diamond cylinder at a low Reynolds number of 100

To clarify the space effect on wake-induced vibration (WIV) of twin tandem diamond cylinders, the fixed upstream cylinder and the downstream cylinder freely oscillating in both in-line and transverse directions have been numerically investigated. The nondimensional center-to-center spacing ratio L* is varied in the range of 2–6, the Reynolds number Re is 100, the reduced velocity Ur is 3–20, and the mass ratio m* is 3. According to the wake modes and dynamic responses, the mechanism for the wake excitation of twin diamond cylinders with different spacing is further discussed. At a smaller spacing (L* = 2 and 3), the reattachment and co-shedding regimes appear with the increasing Ur, but when it comes to a larger spacing (L* = 4–6), only the co-shedding one occurs. The downstream cylinder shares a similar lock-in region at each spacing, while the corresponding flow patterns are not the same. The vortex shedding mode of the downstream cylinder within the lock-in region is 2S at L* = 2, while it is 2P at L* = 3–6. Additionally, the downstream diamond cylinder displays significant in-line and transverse oscillations at each spacing ratio, with the largest amplitude appearing at L* = 3.

Vortex induced vibration of four cylinders configurations at critical spacing in 0° and 45° flow incidence angle

This work presents a numerical study on vortex-induced vibration of four circular cylinders with one and two degrees of freedom subjected to uniform free-stream velocity U at α=0°and α=45°angle of incidence, Reynolds number of Re=UD/ν=150, center-to-center cylinders distance of L=3.5D and a mass ratio of ρc/ρ=2.5and24/π. The structural damping is set to zero for reduced velocity UR=U/(fnD) from 1 to 14. The high-precision code Incompact3d coupled to a moving immersed boundary method is applied to solve the governing equations. The oscillation amplitudes (Ax, Ay), transverse force frequency ratio f/fn, wake Strouhal number St, vorticity fields, and force coefficients statistics are investigated. A wake-stiffness is inferred by f/fn=2 of the lateral cylinders in the 45°rotated configuration. For the one degree of freedom case, when α changes from 0°to 45°, the overall peaks of the force coefficients statistics and oscillation amplitudes increase. In contrast, for the two degrees of freedom case (2dof), only the |C¯L| and CL′ overall peaks increase about 10%, and the Ax peak decreases around 37%. Besides, when L/D is reduced from 5 to 3.5 in 2dof, the overall maximum Ay is slightly affected. However, for α=0°, the |C¯L| , CD′ and Ax peaks respectively are 106%, 284% and 177% higher at L/D=3.5. For α=45°, the CD′ and Ax peaks increase 42% and 33%, respectively.

Dynamics and wake structure of a near-wall flexible cylinder

Interactions between the structural dynamics and wake of a vibrating flexible cylinder close to a wall are investigated using three-dimensional direct numerical simulation at reduced velocity Ur = 5-25, gap-to-diameter ratio G/D = 0.8, aspect ratio L/D = 50, and Reynolds number Re = 500. The 1st to 4th cross-flow (CF) vibration modes and the 1st to 9th in-line (IL) vibration modes are excited, and four types of modal groups for the ratio of IL to CF modes are identified, including groups ‘n:n’, ‘2n-1:n’, ‘2n:n’, and ‘2n+1:n’, where n refers to the excited mode in the CF direction. With the increase of Ur, the vibration responses are first dominated by one specific mode and then co-dominated by multi-modes with comparable intensities. Within the same modal group, the maximum CF and IL amplitudes and mean drag/lift coefficients decrease with Ur under smaller Ur conditions, whereas they increase with Ur under larger Ur conditions. The cylinder oscillation exhibits a standing-wave behavior and a mixture of standing and traveling behavior at the lower and higher Ur cases, respectively. Compared to the isolated case, the reduction of the IL vibration frequency by half and the enhanced IL amplitude are caused by the vortex shedding suppression from the lower side of the cylinder and by the corresponding IL lock-in state along the span. The vortex sheddings at the antinode and node of the CF displacement are in ‘2S’ and weak ‘2S ’ modes, with the energy input and output from the wake, respectively; whereas, for the isolated case, they are in ‘2P’ and ‘2S’ modes, respectively.

Mode transformation and interaction in vortex-induced vibration of laminar flow past a circular cylinder

An investigation of the mode transformation and interaction underlying the behavior of vortex-induced vibration (VIV) of a flow past a circular cylinder elastically mounted on a linear spring is conducted using a high-fidelity full-order model (FOM) based on computational fluid dynamics (CFD), a reduced-order model (ROM), and a dynamic mode decomposition (DMD) of the velocity. A reduced-order model for the fluid dynamics is obtained using the eigensystem realization algorithm (ERA), which is subsequently coupled to a linear structural equation to provide a state space model for the coupled VIV system, in order to provide a simplified computationally inexpensive mathematical representation of the system. This methodology is used to study the dynamics of laminar flows past an elastically mounted circular cylinder with Reynolds number Re ranging from 20 to 180, inclusive. The results of the simulations conducted using FOM/CFD and ROM/ERA, in conjunction with the power spectral analysis and DMD, are used to identify the characteristic natural frequencies and the growth/decay of various modes (including the complex interactions between the myriad wake modes and the structural mode) of the VIV system as a function of the Reynolds number and the reduced natural frequency Fs (or, equivalently, the reduced velocity Ur). A detailed analysis of the distribution of the eigenvalues of the transfer (or, system) matrix of the reduced VIV system shows that the frequency range of the lock-in can be partitioned into resonance and flutter lock-in regimes. The resonance lock-in (lower branch of the VIV response) dominates the fluid-structure interaction. Furthermore, it is shown that when the structural natural frequency is close to one of the eigenfrequencies associated with the wake modes, resonance lock-in (rather than flutter lock-in) will be the primary mechanism governing the VIV response even though the real part of the eigenvalues associated with structural mode is positive. With increasing Reynolds number, the instability of each wake mode is enhanced resulting in a transformation of the wake modes interacting with the structural mode. It is suggested herein that the weakened interaction between the wake modes and the structural mode at Re = 180 (associated with the greater separation between the root loci of the modes) results in the premature termination of the resonance lock-in at Fs=0.155 with increasing Ur. The DMD and power spectral analysis of the time series of the transverse displacement and lift coefficient are used to support the results obtained from ROM/ERA and, more specifically, to provide a clear demonstration of the balanced interaction between the wake modes and the structural mode. This result is used to explain the beating phenomenon, which occurs in the initial branch and the significant lag time that arises between the initial branch and the occurrence of a fully developed response in the lower branch.

Flow-induced vibration of a cantilevered cylinder in oscillatory flow at high KC

Flow-induced vibrations of a cantilevered circular cylinder are measured in sinusoidal, oscillatory, water flows with amplitude of reduced velocity in the range 1.9≤Ur≤4.4 and Keulegan–Carpenter number in the range 120≤KC≤900 respectively. Flow velocities are measured using laser Doppler anemometry, and forces and moments are measured using a 6-axis load cell; the two-degree-of-freedom (2-DOF) cylinder motions are determined from the measured moments. The dominant type of vibration occurring within the flow half-period is shown to depend mainly on Ur, with predominantly in-line vibration occurring for Ur⪅2.7, figure-8 vibration occurring for 2.7⪅Ur⪅4, and transverse vibration occurring for Ur⪆4. In-line vibration frequency, fx, is close to, or slightly higher than the cylinder’s natural frequency in still-water, while transverse vibration frequency, fy, is generally close to the vortex shedding frequency given by Strouhal number St=0.2. Some unsteadiness is seen in the transverse vibration frequency in that accelerating flow fy is consistently higher than decelerating flow fy for the same instantaneous reduced velocity ur. The most notable unsteady effect is seen in the in-line vibration amplitude, Ax, which is much higher during flow deceleration than during flow acceleration; maximum Ax occurs at decelerating ur≈2 for all three vibration types. Transverse vibration amplitude, Ay, increases with increasing ur and shows only slight asymmetry between accelerating and decelerating flow. Experiments with the cylinder placed within a large array of similar cylinders with a spacing between cylinders of six cylinder diameters, show that cylinder vibrations within the array are more variable than those of the isolated cylinder, but exhibit similar average vibration amplitudes and frequencies as the isolated cylinder. An empirical model for unsteady in-line vibration based on theoretical considerations and the experimental data is presented. Model-predicted and measured in-line vibration amplitudes through the flow half-period show good agreement for in-line, figure-8 and transverse vibrations.

Numerical study on vortex-induced vibration of circular cylinder with two-degree-of-freedom and geometrical nonlinear system

The vortex-induced vibration (VIV) of circular cylinder in uniform flow with two-degree-of-freedom (2DOF) and geometrical nonlinear system is investigated numerically using the combined sliding and layering dynamic meshes. The characteristics of vibration amplitude, fluid force coefficients, branching behaviour, vibration trajectory, energy transfer and flow pattern of circular cylinder in VIV are studied with two-dimensional shear stress transfer (SST) k-ω turbulence model and three-dimensional large eddy simulation (LES). It is found that in the simulation with LES model the predicted vibration amplitude and fluid force coefficients show good agreement with the experimental data, while in the simulation with SST k-ω model these parameters are significantly underestimated. The two triplets (2T) vortex shedding mode happens in the super upper branch of circular cylinder with 2DOF linear system and leads the large-magnitude positive energy transferring from fluid to cylinder, which causes the maximum vibration amplitude. For the circular cylinder with 2DOF nonlinear system, the vibration simulated by LES model shows a galloping-like phenomenon that the amplitude keeps increasing with the reduced velocity Ur. The crescent-shaped trajectory and the 2T vortex shedding mode happen upon the critical Ur, which is similar to that in the super upper branch of circular cylinder with 2DOF linear system. This phenomenon is caused by the amplitude-dependent stiffness of nonlinear system and the modified reduced velocity U∗ reveals the universality of vibration amplitude for the circular cylinders with linear and nonlinear systems.

Energy harvesting from passive oscillation of inverted foil

A numerical study is carried out to investigate the energy harvesting from an inverted foil undergoing flow-induced pitching oscillation for reduced velocity Ur = 1–45 and damping ratio ζ = 0–0.295. The benchmark results with undamped foil (ζ = 0) indicate that the foil does not oscillate for Ur ≤ 27 but does oscillate with increasing amplitude for 27 < Ur < 34 and with constant amplitude for Ur ≥ 34. Lissajous diagrams of moment coefficient against the foil displacement are linked to the energy harvesting, showing how Ur and ζ affect the oscillating amplitude, reduced frequency, wake structures, and power exchange between the foil and the flow. The energy harvesting efficiency η up to 15.06% is achieved at Ur = 37 and ζ = 0.130 with a reduced frequency f* = 0.151 that is used by the cruising aquatic animals. The foil oscillation with negative power enhances the growth of vortices while that with positive power weakens the growth.