Vortex-induced Vibration Lock-in Research Articles

Deep learning for predicting frequency lock-in of a freely vibrating sphere

In this paper, we present a deep learning-based reduced-order model (DL-ROM) for the stability prediction of unsteady three-dimensional fluid–structure interaction systems. The proposed DL-ROM has the format of a nonlinear state-space model and employs a recurrent neural network with long short-term memory (LSTM) cells. We consider a canonical fluid–structure system of an elastically mounted sphere coupled with the incompressible fluid flow in a state-space format. Specifically, we develop a nonlinear data-driven coupling for predicting the unsteady forces and the vortex-induced vibration (VIV) lock-in of the freely vibrating sphere in a transverse direction. We design an input–output relationship as a temporal sequence of force and displacement datasets for a low-dimensional approximation of the fluid–structure system. Based on the prior knowledge of the VIV lock-in process, the input function contains a range of frequencies and amplitudes, which significantly affects the system realization ability and enables an efficient DL-ROM without the need for a massive training dataset for the low-dimensional modeling. Once trained, the network provides a nonlinear mapping of input–output dynamics that can predict the coupled fluid–structure dynamics for a longer horizon via the feedback process. By integrating the LSTM network with the eigensystem realization algorithm (ERA), we construct a data-driven state-space model for the reduced-order stability analysis. We investigate the underlying mechanism and stability characteristics of VIV of a freely vibrating sphere via an eigenvalue selection process. Using the reduced-order eigenvalue analysis, we characterize the vibrating sphere-wake lock-in phenomenon and demonstrate that the lock-in responds at preferred vibration frequencies. To understand the frequency lock-in mechanism, we study the eigenvalue trajectories for a range of the reduced oscillation frequencies and the mass ratios. Consistent with the full-order simulations, the frequency lock-in branches are accurately captured by the combined LSTM-ERA procedure. The proposed DL-ROM aligns with the development of physics-based digital twin of engineering systems involving moving boundaries and fluid–structure interactions.

Large eddy simulations and modal decomposition analysis of flow past a cylinder subject to flow-induced vibration

Large eddy simulations (LES) are carried out to investigate the flow around a vibrating cylinder in the subcritical Reynolds number regime at Re = 3900. Three reduced velocities, Ur = 3, 5, and 7, are chosen to investigate the wake structures in different branches of a vortex-induced vibration (VIV) lock-in. The instantaneous vortical structures are identified to show different coherent flow structures in the wake behind the vibrating cylinder for various branches of VIV lock-in. The combined effects of the frequency and amplitude of the oscillation on the flow pattern in the wake region, the hydrodynamic quantities of the cylinder, and the spanwise length scale of the energetic wake flow structures are discussed in detail. It is found that the typical spanwise lengths of the flow structures are 0.22D at Ur = 5 and 0.3D at Ur=[3,7] in the near-wake region and level out at 0.5D further downstream. Furthermore, multiscale proper orthogonal decomposition (mPOD) is used to analyze the dominant flow features in the wake region. With the increasing Ur, the total kinetic energy contribution of superharmonic modes increases and the contribution of subharmonic modes decreases. The dominant flow characteristics associated with the vortex shedding and their super harmonics, and the low-frequency modulation of the wake flow can be captured by the mPOD modes.

Secondary lock-in of vortex-induced vibration and energy transfer characteristics of a vibrating cylinder subject to cross buoyancy

The hydrodynamic and thermal characteristics of a freely vibrating circular cylinder subject to cross buoyancy are numerically investigated at low Reynolds numbers. The structural responses and onset of vortex-induced vibration (VIV) are documented over a range of parameter space, 2.0≤ reduced velocity (Ur) ≤10.0, 0.7≤ Prandtl number (Pr) ≤10.0 and 0.5≤ Richardson number (Ri) ≤2.0. The fluid and structural coefficients are chosen as Reynolds number (Re)=100, mass ratio (m*)=10.0, and damping ratio (ζ)=0.01. A phenomenon of secondary VIV lock-in is found in the cases of Ri = 2.0 (the cross buoyancy effect becomes influential), Pr≲2.0 and Ur≳7.0. An extended VIV lock-in region is formed over a wide range of reduced velocity values together with a tremendous kinetic energy transfer between fluid and structure. This finding is significant for the research of hydropower harvesting. On the other hand, the influence of structural dynamics on heat convection over the surface of a heated circular cylinder is recorded and discussed as well. The significance and mutual influence between Prandtl and Richardson numbers on hydrodynamics, structural dynamics, and heat convection are discussed in detail. The temperature contours are found concentrating around the cylinder's surface in the cases of high Prandtl numbers, which are also associated with higher mean Nusselt number (Nu¯) values. The influence on heat convection over a cylinder's surface is quantified via the computation of Nu¯ and its fluctuation for different circumstances. The energy transfer coefficient is employed to quantify the kinetic energy transfer between the fluid and a heated structure in mixed convective flow. The phase angle difference between the transverse displacement and lift force is used to support the discussions of energy transfer in fluid.

Vortex-induced vibration of bridge decks: Describing function-based model

A describing function (DF)-based model is introduced for the simulation of vortex-induced vibration (VIV) of bridge decks. Similar to the linear frequency response function, the DF is the complex ratio of the first-order component of the nonlinear output to the harmonic input. The DF can be either identified using the forced vibration technique or based on the VIV nonlinear response time history. An iterative procedure is accordingly developed to predict the VIV response with the DF-based model, and an equivalent-damping-ratio-based simplified method is further proposed to efficiently obtain the limit cycle oscillation (LCO) amplitude of VIV. It is demonstrated that the conventional van der Pol-type model is equivalent to a special case of the DF-based model for VIV. Three case studies involving various cross-sections are utilized to validate the simulation accuracy and efficiency of the proposed DF-based model for typical features at VIV lock-in such as LCO and hysteresis phenomena. The vertical VIVs can be well captured by the DF-based model, while its capability of simulating the torsional VIVs requires further improvement. Furthermore, the predictive capability of DF-based model for vertical VIVs of bridge decks within a wide range of mass-damping conditions is highlighted.

An experimental investigation of flow-induced vibration of high-side-ratio rectangular cylinders

This study reports an experimental investigation on the flow-induced vibration (FIV) of elastically mounted rectangular cylinders with high-side-ratio in free-stream flow. The side ratio (σ), defined as the ratio of the cross-flow side width (h) to the streamwise side width (b) of the cylinder, namely σ=h∕b, was varied from 2.0 to 5.0. The fluid–structure system was modelled using a low-friction air-bearing system in conjunction with a free-surface water channel facility. The structural vibration was characterised over the reduced velocity range of 2⩽U∗=U∕(fnwh)⩽16, where U is the free stream velocity and fnw is the natural frequency of the system in quiescent water. The corresponding Reynolds number varied in the range of 940⩽Re=Uh∕ν⩽8200, where ν is the fluid kinematic viscosity. The mass ratio, defined as the ratio of the oscillating mass to the displaced fluid mass, varied from 6.56 to 12.18, depending on the cylinder models. It was found that the vibration response was dominated by vortex-induced vibration (VIV) response for U∗≲8.4 for all the σ cases tested. In the VIV lock-in regime, the local peak amplitude response was found to increase with σ (i.e. from A∕h≃1.05 for σ=2.0 to A∕h≃1.69 for σ=5.0). Interestingly, beyond the VIV lock-in regime, while a galloping response was observed for the cases of σ⩽4.0, where the vibration amplitude increased linearly with U∗, the σ=5.0 case exhibited a bounded galloping regime with the amplitude increasing up to A∕h≃1.44 at U∗=10.8, prior to an abrupt drop to A∕h≈0.35 of a desynchronisation regime for higher U∗ values. The results suggest that this unexpected collapse of galloping response could be due to exceeding the relative incidence angle threshold for possible galloping.

Dynamics and stability of gap-flow interference in a vibrating side-by-side arrangement of two circular cylinders

In this work, the coupled dynamics of the gap flow and the vortex-induced vibration (VIV) of a side-by-side (SBS) arrangement of two circular cylinders is numerically investigated at Reynolds numbers $100\leqslant Re\leqslant 500$ . The influence of VIV is incorporated by allowing one of the cylinders to vibrate freely in the transverse direction, which is termed as a vibrating side-by-side (VSBS) arrangement. A comparative three-dimensional study is performed between the stationary side-by-side (SSBS) and the VSBS arrangements to examine the characteristics of the complex coupling between the VIV and the gap flow. The results are also contrasted against the isolated configurations without any proximity and gap-flow interference. Of particular interest is to establish a relationship between the VIV, the gap flow and the near-wake instability behind bluff bodies. We find that the kinematics of the VIV regulates the streamwise vorticity concentration, which accompanies a recovery of the two-dimensional hydrodynamic response at the peak lock-in. Moreover, the near-wake instability may develop around an indeterminant two-dimensional streamline saddle point along the interfaces of a pair of imbalanced counter-signed vorticity clusters. The interaction between the imbalanced vorticity clusters and the gap-flow momentum are closely interlinked with the prominence of streamwise vortical structures. In both SSBS and VSBS arrangements, the flip-flopping frequency is significantly low for the three-dimensional flow, except at the VIV lock-in for the VSBS arrangement. While an early onset of VIV lock-in is observed for the vibrating configuration, a quasi-stable deflected gap-flow regime with stably deflected gap flow is found at the peak lock-in. The increase of the gap-flow proximity interference promotes the energy transfer and stabilizes the VIV lock-in. Finally, we employ the dynamic mode decomposition procedure to characterize the space–time evolution of the vortex wake system behind the cylinders.

Model reduction and mechanism for the vortex-induced vibrations of bluff bodies

We present an effective reduced-order model (ROM) technique to couple an incompressible flow with a transversely vibrating bluff body in a state-space format. The ROM of the unsteady wake flow is based on the Navier–Stokes equations and is constructed by means of an eigensystem realization algorithm (ERA). We investigate the underlying mechanism of vortex-induced vibration (VIV) of a circular cylinder at low Reynolds number via linear stability analysis. To understand the frequency lock-in mechanism and self-sustained VIV phenomenon, a systematic analysis is performed by examining the eigenvalue trajectories of the ERA-based ROM for a range of reduced oscillation frequency $(F_{s})$, while maintaining fixed values of the Reynolds number ($Re$) and mass ratio ($m^{\ast }$). The effects of the Reynolds number $Re$, the mass ratio $m^{\ast }$ and the rounding of a square cylinder are examined to generalize the proposed ERA-based ROM for the VIV lock-in analysis. The considered cylinder configurations are a basic square with sharp corners, a circle and three intermediate rounded squares, which are created by varying a single rounding parameter. The results show that the two frequency lock-in regimes, the so-called resonance and flutter, only exist when certain conditions are satisfied, and the regimes have a strong dependence on the shape of the bluff body, the Reynolds number and the mass ratio. In addition, the frequency lock-in during VIV of a square cylinder is found to be dominated by the resonance regime, without any coupled-mode flutter at low Reynolds number. To further discern the influence of geometry on the VIV lock-in mechanism, we consider the smooth curve geometry of an ellipse and two sharp corner geometries of forward triangle and diamond-shaped bluff bodies. While the ellipse and diamond geometries exhibit the flutter and mixed resonance–flutter regimes, the forward triangle undergoes only the flutter-induced lock-in for $30\leqslant Re\leqslant 100$ at $m^{\ast }=10$. In the case of the forward triangle configuration, the ERA-based ROM accurately predicts the low-frequency galloping instability. We observe a kink in the amplitude response associated with 1:3 synchronization, whereby the forward triangular body oscillates at a single dominant frequency but the lift force has a frequency component at three times the body oscillation frequency. Finally, we present a stability phase diagram to summarize the VIV lock-in regimes of the five smooth-curve- and sharp-corner-based bluff bodies. These findings attempt to generalize our understanding of the VIV lock-in mechanism for bluff bodies at low Reynolds number. The proposed ERA-based ROM is found to be accurate, efficient and easy to use for the linear stability analysis of VIV, and it can have a profound impact on the development of control strategies for nonlinear vortex shedding and VIV.

Experimental study of the effects of surface roughness on the vortex-induced vibration response of a flexible cylinder

The effects of surface roughness on the vortex-induced vibration (VIV) of a flexible cylinder have been studied experimentally. The drag, lift, tension, vortex-shedding frequency, vibration frequency, and displacement response of cylinders with different roughnesses are compared. The results indicate that a pure VIV lock-in phenomenon occurs in the in-line direction earlier than in the cross-flow direction, and thus the VIV response in the in-line direction is larger than in the cross-flow direction in the low-reduced-velocity range. The tension in the cylinder has two peak frequencies. One peak is the dominant frequency, which coincides with the dominant frequency of the drag and is induced by the VIV in the in-line direction. The other peak is at half of the dominant frequency, at the dominant frequency of the lift; it is caused by the VIV in the cross-flow direction. Rough cylinders have a smaller displacement response, a narrower lock-in region, and a higher vortex-shedding frequency than a smooth cylinder.

Flow-induced vibrations of square and rectangular cylinders at low Reynolds number

One-degree-of-freedom (1-dof) and two-degrees-of-freedom (2-dof) flow-induced vibrations (FIVs) of square and rectangular cylinders at a mass ratio of 10 and a low Reynolds number of 200 are studied numerically by solving the two-dimensional incompressible Navier–Stokes equations. The aim of the study is to identify the effects of the aspect ratio α, defined to be the ratio of the cylinder dimension in the cross-flow direction to that in the inline direction, on the vortex-induced vibration (VIV) and galloping responses. Simulations are conducted for aspect ratios of 0.3, 0.5, 0.7, 1 and 1.25 and reduced velocities ranging from 1 to 30. Distinct VIV lock-in and galloping regimes are found for all the aspect ratios except α = 0.3, for which only VIV lock-in is found. The VIV lock-in regime and the galloping regime are separated by a reduced velocity range, where the response amplitude is very small and the response frequency is a linear function of the reduced velocity. It is found that the maximum amplitude in the VIV lock-in regime decreases with increasing aspect ratio. Galloping does not start until the reduced velocity exceeds a critical value. The critical reduced velocity for galloping increases with increasing aspect ratio. For α = 0.5, galloping starts at Vr = 7 and 6 for 1-dof and 2-dof vibrations, respectively. The critical reduced velocity for galloping is increased to 17 at α = 1.25 for both 1-dof and 2-dof vibrations. Because the response amplitude in the inline direction is much smaller than that in the cross-flow direction, the response amplitude and frequency in 2-dof vibration are very similar to their counterparts in 1-dof vibration. However, the response amplitude in 2-dof galloping is greater than that in 1-dof galloping. A 2T vortex shedding mode is observed in the VIV lock-in regime for α = 0.3 and 0.5.

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

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