Review
An overview of modeling and experiments of vortex-induced vibration of circular cylinders

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Abstract

This paper reviews the literature on the mathematical models used to investigate vortex-induced vibration (VIV) of circular cylinders. Wake-oscillator models, single-degree-of-freedom, force–decomposition models, and other approaches are discussed in detail. Brief overviews are also given of numerical methods used in solving the fully coupled fluid–structure interaction problem and of key experimental studies highlighting the nature of VIV.

Introduction

Vortex-induced vibration (VIV) occurs when shedding vortices (a von Kármán vortex street) exert oscillatory forces on a cylinder in the direction perpendicular to both the flow and the structure. The structure starts to oscillate due to these forces if it is not fixed. For fixed cylinders, the vortex-shedding frequency is related to the non-dimensional Strouhal number. The Strouhal number is defined as S=fvD/U, where fv is the predominant frequency of vortex shedding. U is the steady velocity of the flow, and D is the diameter of the cylinder. The Strouhal number is found to be nearly constant with a value of 0.2 for a large range of Reynolds numbers. This range is often called the subcritical range and spans the Reynolds number range of 300–2×105 [1].

For flow past cylinders that are free to vibrate, the phenomenon of synchronization or lock-in is observed. For low flow speeds, the vortex-shedding frequency fv will be the same as that of a fixed cylinder. This frequency is fixed by the Strouhal number. As the flow speed is increased, the shedding frequency approaches the vibration frequency of the cylinder f0. In this regime of flow speeds, the vortex-shedding frequency no longer follows the Strouhal relationship. Rather, the shedding frequency becomes “locked-in” to the oscillation frequency of the cylinder (i.e., f0fv). If the vortex-shedding frequency is close to the natural frequency of the cylinder fn, as is often the case, large body motions are observed within the lock-in regime (the structure undergoes near-resonance vibration).

It is also well known that a hysteresis behavior may exist in the amplitude variation and frequency capture depending on the approach to the resonance range—whether from a low velocity or from a high velocity [2]. As will be discussed later, the two branches of this hysteresis loop are associated with different vortex-shedding modes and transition between these branches is associated with a phase jump of 180 [3]. Shown in Fig. 1 is a typical response in the lock-in region of a freely vibrating circular cylinder with light damping. The hysteresis effect is clearly seen, with higher amplitudes achieved when the reduced velocity is increased over a certain range. Also seen is the lock-in phenomenon, with the vortex-shedding and body oscillation frequencies collapsing into a single frequency close to the natural frequency of the cylinder. The straight line S=0.198 is the line of constant Strouhal number.

The amplitude of the structural response during lock-in and the band of fluid velocities over which the lock-in phenomenon exists is strongly dependent on a reduced damping parameter expressing the ratio of the damping force to the excitation force. The Scruton number, Sc=4πmζ/ρD2, is but one of many representations for this reduced damping parameter found in the literature. As the reduced damping parameter increases, lock-in becomes characterized by a decreasing peak structural amplitude (see, for example, Fig. 1 of Ref. [4]) and occurs over a decreasing band of velocities (see, for example, Fig. 4 of Ref. [5]). It is also worth noting that different phenomena are seen in structures with high and low structure–fluid density ratios m*=m/ρD2, where m is the cylinder mass per unit length and ρ is the fluid density. For systems with high m*, the vortex-shedding frequency is entrained by the structural frequency. For systems with low m*, it is the fluid oscillation which sets the frequency and the entrainment frequency instead tends towards the shedding frequency fv.

While resonance in flow-induced in-line oscillations of circular cylinders is an important topic, especially for systems with small structural damping (or small reduced damping parameters, depending on the mass ratio), it is not considered as an independent topic in this review. Lock-in occurs when the in-line frequency approaches twice the Strouhal frequency (fv) and the amplitude of the alternating force (drag fluctuations) and the response of the cylinder are an order of magnitude smaller than those in the transverse direction [2]. Defining the reduced velocity as Vr=U/fnD, the most prominent feature of in-line oscillations of a rigid spring-mounted circular cylinder is the existence two excitation regions separated by Vr2: symmetric vortex shedding for Vr<2, and alternating vortex streets for Vr>2 [6].

The engineering implications of VIV have been well documented in the literature. Structures such as tall buildings, chimneys, stacks and long-span bridges develop pronounced vibrations when exposed to fluid flow. See Refs. [7], [8], [9], [10] for studies focusing on the VIV of these structures. The length and higher flexibility of some of these structures further aggravate the problem. In offshore applications, VIV of long slender structures such as pipelines, risers, tendons, and spar platforms challenge engineering designers [11]. Some examples of fundamental studies on the nature of the VIV of marine structures are included in Refs. [12], [13], [14], [15], [16]. Extensive research has also been done in the area of VIV assessment [17], [18], [19] and suppression [20], [21].

In this review, both experimental and theoretical investigations of the fundamental aspects of vortex-induced vibration of circular cylinders are discussed in some detail. The authors’ goal has been to be thorough without being exhaustive. The main focus of this review is on the semi-empirical models used to predict the response of the cylinder to the forces from the flow. These models are not rigorous and generally provide minimal insights into the flow field. To understand the flow effect on a structure, it is important that the actual flow field be described. Consequently, a secondary focus of this review is to discuss the flow characteristics around the cylinder. The flow field generated by flow separation around a body is a very complex fluid dynamics problem. However, much progress has been made toward the understanding of flow around bluff bodies. This is especially true in the field of computational fluid dynamics (CFD), and in keeping with the primary focus of this review, only selected papers highlighting this progress have been included.

While many reviews of the subject have been written in the past [2], [5], [22], [23], [24], [25], a more contemporary review paper focusing on semi-empirical models is needed. The importance of such a paper follows from the fact that while VIV continues to be the subject of intensive research efforts and is quickly evolving, many of these simplified models continue to be used today. Among their attractions is the fact that they can be used in higher Reynolds number flows than CFD models and they have been solved in both the time and frequency domains. In addition, an alternative new method for the modelling of VIV is presented. The method is based on the variational principles of mechanics and leads to a more fundamental (without ad hoc assumptions) derivation of the equations of motion, yet remains inexorably linked to physical data. Experimental data help to verify the model predictions, thus leading to the most advantageous model framework.

Section snippets

Experimental studies

There are innumerable experimental studies on the vortex-induced vibration of bluff bodies, especially circular cylinders. These studies have examined a multitude of phenomena, from vortex shedding from a stationary bluff body to vortex shedding from an elastic body. The vibration caused by vortices generated by the flow past a structure depends on several factors. The correlation of the force components, the shedding frequency, the Reynolds number, the material damping and structural stiffness

Semi-empirical models

In this review, every attempt has been made to preserve the notation of the governing equations as given in the references. This facilitates the reader's ability to correspond between this review and a given paper. Work with structures undergoing vortex-induced vibration can be classified into three main types. The first class consists of wake–body (wake–oscillator) coupled models, in which the body and the wake oscillations are coupled through common terms in equations for both. The second

Numerical methods

Numerical methods are an alternative way to solve the fully coupled problem of VIV of bluff bodies. For flow-induced vibration, four basic issues should be considered in any numerical simulation: modelling of the flow field, modelling of the structural vibration, modelling of the fluid–structure interaction, and data analysis [76].

The flow field behind a stationary cylinder and the flow field behind a cylinder forcibly oscillated at a specified amplitude and frequency (the forced vibration

Concluding remarks

A variety of issues concerning the vortex-induced vibration of circular cylinders have been discussed. Selected papers highlighted the influence of vortex-induced unsteady forces on the cylinder, including the phase of the forces relative to the body motion. The phenomenon of lock-in has been discussed and the factors that influence the response of the cylinder (mass and damping) have been listed. The mathematical modeling of vortex-induced oscillations, using nonlinear oscillators and

Acknowledgements

This work is supported by the Office of Naval Research Grant No. N00014-97-1-0017. We would like to thank our program manager Dr. Thomas Swean for his interest and financial support. We also thank Dr. Timothy Wei for providing some of the figures.

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