Transverse vortex-induced vibration of a circular cylinder on a viscoelastic support at low Reynolds number

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Abstract

The effect of a viscoelastic-type structural support on vortex-induced vibration (VIV) of a circular cylinder has been studied computationally for a fixed mass ratio (m=2.546) and Reynolds number (Re=150). Unlike the classical case of VIV where the structural support consists of a spring and damper in parallel, this study considers two springs and one damper, where the two springs are in parallel and the damper is in series with one of the springs. This spring/damper arrangement is similar to the Standard Linear Solid (SLS) model used for modelling viscoelastic behaviour. The viscoelastic support (SLS type) is governed by the following two parameters: (a) the ratio of spring constants (R), and (b) the damping ratio (ζ). The focus of the present study is to examine and understand the varied response of the cylinder to VIV as these parameters are varied. For small ζ and R, the cylinder response shows characteristics similar to the classical case, where the amplitude response is composed of an upper- and the lower-type branch. The presence of upper-type branch at low Re is evident through the peak lift force, frequency and phase response of the cylinder. As the damping ratio is increased, the vibration amplitude decreases and hence the upper-type branch disappears. There exists a critical value of ζ=1 beyond which the amplitude again increases asymptotically. The non-monotonic variation of amplitude response with ζ is presented in the form of the ”Griffin plot”. The amplitude, force, frequency and phase-difference response of cylinder were found to be mirror symmetric in log(ζ) about ζ=1. In addition, the effect of R at the critical value of damping, ζ=1, was studied. This show that the amplitude decreases with an increase of R, with suppression of the response branches for high R values. The results suggest that a careful tuning of the damping may be effectively employed both to enhance power output for energy extraction applications or to suppress flow-induced vibration.

Introduction

Vortex-induced vibration of a cylinder, and in particular the case where a rigid circular cylinder is elastically mounted and constrained to oscillate transversely to a free stream, has been well-studied and reported on, as can be seen from comprehensive reviews of Sarpkaya, 1979, Bearman, 1984, Parkinson, 1989, Sarpkaya, 2004, Williamson and Govardhan, 2004, Williamson and Govardhan, 2008, Bearman, 2011 and Wu et al. (2012). Vortex-induced vibration (VIV) occurs when vortex shedding exerts an oscillatory or quasi-oscillatory force on a structure causing it to vibrate. Indeed, elastically mounted structures near resonance develop flow-induced oscillations by extracting energy from the flow. In turn, the oscillations of the structure modify the flow and give rise to a coupled nonlinear interaction.

In general, the VIV response of a circular cylinder in uniform flow is dependent on the Reynolds number, the mass ratio, the damping ratio and the reduced velocity. The Reynolds number is defined as Re=UDν, where U is the free stream velocity, D is the cylinder diameter and ν is the kinematic viscosity of the fluid. The mass ratio is defined as m=4mπρD2 where m and ρ are the mass per unit length and the fluid density, respectively. The damping ratio is the ratio of damping to the critical damping, given by ζ=c2km, where k is the spring stiffness. Finally, the reduced velocity as defined by Sumer et al. (2006) is the ratio of the wavelength of the cylinder trajectory to its diameter and is given by Ur=UfsD, alternatively, it can be thought of as a non-dimensional structural oscillation period, hence (VIV) resonance occurs when this matches the shedding period, typically when Ur5 for a circular cylinder.

Most past studies have focused on one-degree-of-freedom cross-flow VIV of a circular cylinder. Vortex shedding occurs due to the presence of two separating shear layers that subsequently roll up alternately into low pressure discrete vortical structures inducing oscillation in a direction transverse to the free stream. If the vortex-shedding frequency is close to the natural frequency of the cylinder, large oscillation amplitudes occur. Indeed, such a response can occur over a wide range of reduced velocities (Bearman, 1984). This reduced velocity range over which the structure undergoes near-resonant vibration is referred to as the lock-in range. The amplitude of cylinder vibration can undergo jumps as the reduced velocity is varied, which give rise to different response branches. For a low mass-damping parameter (product of mass ratio and damping ratio, mζ), Khalak and Williamson (1999) experimentally found three response branches: the initial, upper and lower branches as the ratio of vortex-shedding frequency to structure natural frequency was varied (fyfn). At higher mζ, they found only two different branches: the initial and the lower branch. These branches are identified by the jumps in the amplitude response. The maximum vibration amplitude was observed to be close to one cylinder diameter (D) in the upper branch, with a relatively lower amplitude of oscillation occurring in the lower branch. A 2S vortex shedding pattern (two single vortices shed per cycle, i.e. a von-Karman street-type wake) in the initial branch, a 2P mode ( two vortex pairs of opposite sign shed per cycle of vibration) in the lower branch and a 2P mode (similar to 2P except that vortex pairs in one of the half cycles convect away from in front of the body) in the upper branch (Williamson and Roshko, 1988). At low mζ, the mode change between initial and upper response branches involves hysteresis, whereas intermittent switching of modes occurs at the upper and lower branch transition. Both the transition jumps also show jumps in amplitude and frequency. The upper-to-lower branch transition is also characterized by a jump from 0° to 180° in the phase difference between lift force and displacement signals, whereas the phase difference is 0° for both the initial and the upper branch. The peak vibration amplitude is dependent on the mass-damping ratio, mζ, whereas the synchronization regime (measured by the range of reduced velocity Ur) is primarily determined by m. Khalak and Williamson (1997) showed that for high mζ, the frequency of cylinder oscillation ( fy ) was close to structural natural frequency (fn). At low mζ, fy is higher than fn in the synchronization regime, yet fy remains below the vortex-shedding frequency, fv. In the lock-in or synchronization region, the frequency of cylinder vibration was found to be same as the vortex-shedding frequency, i.e. fvfy.

The majority of the numerical studies have been performed at low Re using two-dimensional simulations. Blackburn and Henderson (1996) presented simulation results at Re=250, and showed a lower maximum amplitude of approximately 0.6D. They also observed a chaotic response over a range of fnfv and only the regular 2S mode, implying that the 2P mode need not necessarily be associated with the large amplitude response plateau in the lock-in regime. The branching behaviour was numerically investigated by Leontini et al. (2006b) at Re=200 who found two synchronous response branches that resembled the upper and lower branches, supported by instantaneous amplitude rather than peak amplitude. They also suggested that the branching at higher Re was not the product of three-dimensionality, which was contrary to Govardhan and Williamson (2000) who implied that the upper branch does not occur at low Re for two-dimensional flow. Pan et al. (2007) and Guilmineau and Queutey (2004) performed simulations for the low mass-damping case, with motion constrained to transverse oscillations to a free stream by employing a two-dimensional Reynolds-averaged Navier–Stokes (RANS) model based on the Shear-Stress Transport (SST) kω turbulence model. They were able to get the initial and lower branches but the results did not match the upper branch found experimentally. CFD studies of VIV of a circular cylinder have also been conducted using three-dimensional numerical models. Wang et al. (2017) investigated two-degree-of-freedom VIV of a circular cylinder with varying in-line to cross-flow natural frequency ratios at low mass-ratio (m=2) at Re=500. Gsell et al. (2016) simulated 2DOF VIV of a circular cylinder at Re=3900 through direct numerical simulation of the 3D Navier–Stokes equations. Notably, the maximum transverse amplitude and structural response compared well with experimental observations. Pastrana et al. (2018) modelled VIV using large-eddy simulation (LES) at subcritical Reynolds numbers (Re=3900,5300,11000).

The effects of damping on the vibration response is important as damped VIV can be harnessed for converting flow energy into electrical energy. A cylinder undergoing VIV has kinetic/potential energy that can be extracted using a power transducer such as an electromagnetic generator (Soti et al., 2017). In the past, the effects of damping on the response behaviour of the system were explored by Vickery and Watkins (1964) and Scruton (1963), who reported peak amplitudes in air and water against a mass-damping parameter (mζ). In early studies, Feng (1968) reported the effects of damping on both the amplitude and frequency response. They used an electromagnetic eddy-current-based damper for applying different damping values to the system of circular and D-shaped cylinders in a wind tunnel. Due to the high mass-damping parameter, the amplitude was small and there were no discontinuities; two-branch response behaviour (the initial and the lower branches) was observed. Recently, Klamo et al. (2005) studied the damped system experimentally using a controlled magnetic eddy current technique to provide variable damping, and reported that the maximum amplitudes (Ay,max) depend not only on damping but also on Reynolds number. Subsequently, Klamo et al. (2006) studied the effects of controlled damping on the amplitude and frequency response, and showed that a VIV system transitions from a two-branch to three-branch response as damping is varied from high to low. They also observed three distinct branches for the frequency response, analogous to that seen for the three amplitude branches. They reported hysteresis between the lower branch and the desynchronized region at low Reynolds numbers. Blevins and Coughran (2009) experimentally measured the steady-state response of the elastically supported cylinder for six values of damping, for distinct Reynolds numbers from Re [170,150000]. They observed monotonic decay of maximum amplitude along the transverse direction that was a function of damping for a constant Re. Soti et al. (2018) studied experimentally the effect of damping on the VIV response at mass ratio 3 for Reynolds numbers of 1200 to 11 000. Unlike previous studies, they relied on the frequency response for branch identification and argued for the presence of the upper branch for a much larger value of mass-damping then previously reported.

All the aforementioned work was done for a circular cylinder elastically supported by a spring–damper system in parallel. In the spring–damper parallel system (also referred to as the Kelvin-Voigt model Findley and Davis, 2013), the amplitude response decreases with increasing damping due to an increase in the dissipation of the mechanical energy of cylinder by the damper. de Lima et al. (2018) have proposed the use of a 1-DOF viscoelastically-mounted cylinder in fluid flow at Reynolds number 10 000 to suppress the vibrations induced by vortex shedding. Importantly, the frequency and temperature play a significant effect on viscoelastic properties, and were considered in that investigation.

However, recent studies by Mishra et al. (2019) on the Standard Linear Solid model (SLS) of viscoelasticity motivated us to use the two springs and damper system as shown in Fig. 1. They have simulated the fluid–structure interaction (FSI) of a viscoelastic thin plate attached to the lee side of the cylinder for Re=100. The displacement amplitude was found to be a non-monotonic function of structural damping. The SLS spring–damper system behaves similarly to the parallel spring–damper system up-to critical damping ratio (ζc). However, on a further increase of damping ratio, the amplitude response does continue to decrease. It will be shown that the higher damping ratio (ζ>ζc) response is similar to that at lower damping ratio (ζ>ζc). The non-monotonic vibration amplitude response with damping ratio is summarized in a “Griffin Plot” presented later in the paper. To the knowledge of the authors, the dynamical behaviour of an SLS spring–damper system at low mass ratios has received little or no attention in the literature, yet it provides an effective means of tuning the amplitude response for VIV suppression or energy extraction applications.

The layout of this paper is as follows. In Section 2, the governing equations for the structure (SLS spring–damper system), fluid flow, and coupling of the flow and structural solvers are provided. The numerical approach to solve this coupled system is also briefly discussed. In Section 3, a mathematical model of a simple spring-dashpot model of SLS is presented. This model helps to predict the complex phenomenon of VIV for the SLS system. The simulation results are given in Section 4 as a function of the two governing parameters of the SLS model. These parameters are: (a) damping ratio ζ, and (b) the ratio of spring constants of the two springs (R). The influence of these parameters on the amplitude of the vibration is been discussed. The branching based on the vibration amplitude response and maximum lift force, which leads to the beginning of the upper-type branch at low Re is also discussed.

Section snippets

Governing equations

In the present work, a circular cylinder of diameter D is placed in a free-stream flow. The cylinder mounted vertically on viscoelastic support, as shown in Fig. 2, is free to oscillate in the transverse direction, perpendicular to flow. The flow is assumed two-dimensional (2D) based on the considered Reynolds number. The fluid is assumed to be incompressible and viscous, while the motion of the cylinder behaves as a spring–mass–damper system as depicted in Fig. 1. Fluid flow is modelled in the

A simplified mathematical model of a flexibly mounted cylinder

Experiments of Khalak and Williamson (1999) have shown that in the lock-in region at dynamic steady state, both the displacement and the fluid force have nearly sinusoidal forms and oscillate at the same frequency fy. In the region where the cylinder oscillation frequency is synchronized with the periodic induced force, the transverse displacement and lift coefficient are given by Y=Asin(2πfyt)andCL=CL0sin(2πfyt+ϕ).Here A and CL0 represent the non-dimensional amplitude of cylinder and

Results and discussion

In this section, the effect of damping (ζ) and spring stiffness ratio (R) on the vortex-induced vibration of a circular cylinder will be discussed. There are five independent parameters in the study: mass ratio (m), damping ratio (ζ), spring-stiffness ratio (R), reduced velocity (Ur) and Reynolds number (Re). The reduced velocity is defined as Ur=U(fsD)=1fs, where U is the free-stream velocity. Also, fs and fs(=fsDU), respectively, are the dimensional and non-dimensional natural

Conclusions

The effect of a viscoelastic-type structural support on the response of vortex-induced vibration (VIV) of a circular cylinder has been investigated numerically using a spectral-element based FSI solver, for mass ratio m=2.546 at Re=150. A survey of the literature indicates that the effect of this type of SLS spring–damper system on VIV has received little attention, even through the mechanism allows further control on enhancing or suppressing the VIV response. The spring–damper system used to

CRediT authorship contribution statement

Rahul Mishra: Conceptualization, Software, Investigation, Writing - original draft . Atul Soti: Software, Writing - review & editing. Rajneesh Bhardwaj: Writing - review & editing. Salil S. Kulkarni: Writing - review & editing. Mark C. Thompson: Software, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Support from computing time allocations from the Pawsey Supercomputing Centre with funding from the Australian Government is strongly acknowledged. R.B. gratefully acknowledges the financial support of a CSR grant from Portescap Inc., India, and of an internal grant from Industrial Research and Consultancy Centre (IRCC), IIT Bombay .

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