Vortex-induced vibration occurs when shedding vortices exert oscillatory forces on the cylinder systems. The forces are induced by vortices shed alternately from the edges of a bluff body. The time varying non-uniform pressure distribution around the object resulting from the vortex shedding creates structural vibrations in both IL and CF di-rections. VIV is of practical interest to many fields of engineering. For example, it can cause vibrations in heat exchanger tubes; it influences the dynamics of riser tubes bring-ing oil from the seabed to the surface; it is important to the design of civil engineerbring-ing structures such as bridges and chimney stacks, as well as to the design of marine and land vehicles; and it can cause large-amplitude vibrations of tethered structures in the ocean [89]. In the aeroelastic applications where the fluid is air, the mass ratio m∗ is of order 100 (m∗=mass of oscillating structure/displaced fluid mass). In the hydroelstastic applications in water, the mass ratiom∗ is of order 1 or 10 [90].
The practical significance of VIV has led to a large number of fundamental studies which can be divided by two periods. The first period was approximately from Feng’s work in 1968 and it mainly focused on 1-degree of freedom (1-DOF) VIV study (i.e. only the transverse/CF direction). The second period was approximately from Jauvtis and Williamson’s 2-DOF VIV study [90, 91]. Nowadays, researchers are mainly concerned with 2-DOF VIV investigation and show great interests in 2-DOF VIV study at small mass ratio and damping ratio conditions. VIV study can also be classified into two branches: the wake vortex dynamics and VIV induced structural response.
2.5.1 Wake formation
For fixed cylinders, the vortex shedding frequency fs is related to the non-dimensional Strouhal numberS and follows the following relationship:
S =fsDo/v, (2.17)
where v is the coming flow velocity, Do is the diameter of the cylinder structure. The Strouhal numberShas been found to be nearly constant 0.2 for a large range of Reynolds number from 300 to 2×105 which is called subcritical range.
For a cylinder free to vibrate, when the shedding frequency approaches the vibration fre-quency of the cylinder, the vortex-shedding frefre-quency no longer follows equation (2.17) and synchronized (locked-in) the vibration frequency. This condition always occurs when the shedding frequency is close the natural frequency of the cylinder. Williamson and Roshko [92] investigated a whole series of synchronization regions including the funda-mental lock-in region experifunda-mentally in an amplitude-wavelength plane up to amplitudes of five diameters.
2.5.2 Griffin plot
Skop and Griffin derived a parameterSG
SG= 2π3S2(m∗ζ), (2.18)
from a VIV response analysis involving a van der Pol equation in 1973, where ζ is the structural damping ratio. After extensive complications of many different investigations, the peak CF response versus SG was drawn by Griffin et al. Then the classical log-log-form Griffin plot became widely used for CF peak response predictions. An updated version of Griffin plot was given in [93]. Despite the extensive use of this plot by engineers, researchers (e.g. Sarpkaya, Bearman) have began to question the validity and stated that the peak response might not be a unique function of the combinedm∗ζ.
2.5.3 1-DOF VIV study
There has been a great deal of work concerned with bodies restrained to move in the direction transverse to the free stream. The shedding of vortices from cylindrical bluff objects and the ensuing vibrations have been well documented for one DOF cases refer-ring to [94–96].
Feng’s study at high mass ratio (m∗ = 320) demonstrated that when the oscillation frequency fv coincides with the vortex formation frequency fs, the resonance of the structure will occur over a range of normalised velocityVr (approximately from 5 to 8)
Vr =v/fNDo, (2.19)
where v is the incoming velocity, fn is the natural frequency, Do is the external diam-eter. For the classical Feng-type response at high m∗ζ, there exist only two branches, namely the initial and lower branch with apparently a hysteresis between the two modes as shown in Figure 2.2. In the initial branch,the vortex wake comprises a “2S” mode,
Figure 2.2: The classical Feng-type VIV response exhibits two branches. Az is the CF amplitude; H: hysteretic mode transition
representing 2 single vortices formed per cycle. The lower branch comprises the “2P”
mode, whereby 2 vortex pairs are formed per cycle [89].
At small m∗ζ (typically at m∗ = 5 to m∗ = 10), there is a higher-amplitude mode appears between the initial branch and lower branch as shown in Figure 2.3, namely the Upper branch which also comprises a “2P” mode of vortex formation. The transition between initial branch and upper branch is hysteretic, while the transition between up-per branch and lower branch involves instead an intermittent switching as shown in [97].
Figure 2.3: 1-DOF VIV response at smallm∗ζ exhibits two branches. Az is the CF amplitude; H: hysteretic mode transition; I: intermittent mode switch
2.5.4 2-DOF VIV study
At first, researchers investigated the 2-DOF VIV, assuming the mass ratios and/or the natural frequencies are different in the IL and CF direction. Sarpkaya [98] performed experiments at various natural frequencies and stated that the results from CF-only ex-periments were not expected to remain valid for 2-DOF VIV. Motived by these findings, Jauvtis and Williamson [90, 91] performed 2-D VIV study at the same mass ratio and natural frequencies in the IL and CF directions.
In most practical cases, cylindrical structures has the same mass ratio and natural frequency in the IL and CF directions. And it is unusual for cylinders only to have flexibility in the CF direction and generally a cylinder is free to vibrate due to vortex shedding in all directions normal to its axis [99]. Although the IL peak amplitudes are less than those in the CF direction, the vortex induced vibrations involved significant IL motion. Once the vortex is shed, a fluctuating drag is generated, and thus the IL frequency is twice that for the CF direction. IL motion is of considerable importance [100, 101] and IL vortex-induced curvatures may cause as much as CF curvature [102].
After 2-D experimental studies of a designed pendulum apparatus, Jauvtis and Williamson [90] concluded that the freedom to oscillate in the IL direction affected the CF vibration surprisingly little which held true to the low mass ratios of at least m∗ = 5. Jauvtis and Williamson [91] concluded when the mass ratio was reduced below 6, there was a
remarkable jump in the amplitude response (the peak of the dimensionless CF ampli-tude reached 1.5), corresponding to a new high-ampliampli-tude mode of response named the super-upper branch. And there is a corresponding vortex formation mode, namely the 2T mode which comprises 2 triplets of vortices per cycle.
Williamson and Govardhan [103, 104] summarized these fundamental results and dis-coveries concerning vortex-induced vibration, many of which were related to the push to explore very low mass and damping. However, it is not clear that if these recent in-vestigations have direct relevance to offshore engineering applications dealing with long flexible cylinders, especially those prone to vibrate at high mode numbers [105].
2.5.5 Parametric study
The vibrations caused by vortices generated by the flow past a structure depends on several factors [106]. An important requirement in VIV research is to simplify the problem to consider only those variables that have a significant effect [99]. Parametric study highlighting some key parameters (e.g. mass ratio, boundary conditions, and structural nonlinearity et al.) has been performed by investigators (e.g. [107, 108]).
It is worthwhile to note that Govardhan and Williamson [109] have deduced a critical mass ratio at small mass-damping values which agrees very well with a wide set of experimental data. Below this critical value, the lower branch can never be reached and ceases to exist. The effect of end conditions on the vortex induced vibration of cylinders was investigated by Morse et al. in [110].
Displacements variation along the span
The effects of the spanwise variation of the displacements along the cylinder system have been rarely accounted for [111], although it is important considering that the external-fluid-induced forces vary along the span discussed in detail in [112]. Furthermore, the case of the flexible cantilever cannot be considered equivalent to the elastically restrained rigid cylinder referring to [113]. And remember that the largest response amplitude of transverse vibrations occurred at the bottom part of the free hanging pipe for all top excitation frequencies and amplitudes in [88].
Structural nonlinearity
The influences of nonlinearity in structural quantities are significantly important and
which contained a nonlinear structural spring-mass-damper oscillator and a 2-D VIV wake oscillator model. They omitted the mean drag force, introduced cubic terms with calibrated coefficients to describe the geometrical coupling of the IL and CF motions, and included quadric terms to allow for the wake-cylinder interactions. The jump phe-nomenon in IL and/or CF VIVs might disappear if the structural nonlinearity terms were neglected.
2.5.6 VIV prediction model
There are basically two approaches to predict the VIV induced dynamic response of a cylinder system. The first is a CFD based procedure to solve the Navier-Stoker equa-tions. Yamamoto et al. [116] employed an Euler-Bernoulli equation as the structural model and adopted the Discrete Vortex Method (DVM) to evaluate the hydrodynamic forces. The mathematical model was solved by finite difference method to investigate the hydoelastic interactions that took place between oscillating flexible cylinders and fluid forces. The other approach is the semi-approach in which a flexible riser is usu-ally modelled as a beam and the hydrodynamic force coefficients are calibrated from experiments.
Lift force
Based on the formulation of the fluid force applied on the solid in the direction of lift, Pa¨ıdoussis, et al. [117] classified the current VIV models into three types according to the calculations of VIV induced forceF.
Type A: Forced system models, whereF is independent of CF displacement, and there-fore only depends on time.
Type B: Fluidelastic system models, where F depends on the CF displacement. The dependence may include all time derivatives, integrals and even time delays.
Type C: Coupled system models, where F depends on another variable related to the wake dynamics, say q, the evolution of which depends on CF displacement.
Actually, Type A is a particularly simply case of types B and C. Type B is also a particular case of type C.
Drag force
In 2-DOF VIV study, to model the external fluid force acting in the IL direction, the drag coefficient can be regarded to be composed of a mean drag component and a fluctuating component [118].
The coupling between IL and CF motions
It is worthwhile noting that Vandiver and Jong [119] demonstrated the existence of a quadratic relationship between IL and CF motions under both lock-in and non-lock-in conditions, by means of the bispectral analysis of cable strumming experimental data.
Moreover, for a single long flexible circular cylinder, pin-jointed at its ends, as the ten-sion is increased, it starts to behave like a cable, or a tenten-sion dominated structure. In this case, IL motions are strongly coupled to the CF motion [105]. Combined IL and CF VIVs have studied experimentally at MIT (see [120]) and NTNU (see [121]).
When a cylinder vibrates, the drag force and lift force that result from vortex shed-ding do not respectively coincide with the y (IL direction) and z(CF direction) axes.
Based on the model of the relative motions of the cylinder as shown in Figure 2.4. Wang
Figure 2.4: Illustration of one cross-section of a cylinder in a cross flow and the fluid forces exerted on the vibrating cylinder
et al. [111] proposed the fluid force coefficientcy(t) andcz(t) in the IL and CF directions:
cy(t) = cD(t)cosθ−cL(t)sinθ, (2.20) cz(t) = cD(t)cosθ+cL(t)sinθ, (2.21)
where cD and cL are drag and lift coefficients. The iteration process started from the situation in which the cylinder is stationary. For a stationary cylinder,
cD(t) = cDs+cDvsin(2ωst+φD), (2.22) cL(t) = cLvsin(ωst+φL). (2.23) wherecDsis the mean drag coefficient,ωsis the vortex shedding angular frequency,CDv
is the amplitude of VIV induced drag coefficient,CLv is the amplitude of VIV induced lift coefficient. Note is made here that the author has proposed a new model for the coupling effect of IL and CF motions which is described in detail in Chapter 3.
2.5.7 State of art
One large scaled experiment was carried out at Delf Hydraulics in the Delta Flume [122]. Blind predictions employing 11 different numerical models (e.g. VIVA; VIVANA;
SHEAR7; ABAVIV) were performed and compared with the experimental measurements in [123]. Barely satisfactory agreements were observed while empirical models were more successful at predicting CF displacements and curvatures than current codes based on CFD. Predictions of vortex induced IL displacements were provided only by the CFD-based codes, and average ratios (the ratio between simulation results and experimental data) were between 135% − 175%. Therefore, VIV is far from fully understood and requires more efforts.