An overview of vortex-induced vibration (VIV) of bridge decks

Abstract

A brief overview of vortex-induced vibration (VIV) of circular cylinders is first given as most of VIV studies have been focused on this particular bluff cross-section. A critical literature review of VIV of bridge decks that highlights physical mechanisms central to VIV from a renewed perspective is provided. The discussion focuses on VIV of bridge decks from wind-tunnel experiments, full-scale observations, semi-empirical models and computational fluids dynamics (CFD) perspectives. Finally, a recently developed reduced order model (ROM) based on truncated Volterra series is introduced to model VIV of long-span bridges. This model captures successfully salient features of VIV at “lock-in” and unlike most phenomenological models offers physical significance of the model kernels.

Keywords

vortex-induced vibration (VIV) / Volterra series / bridge

Introduction

Flow around bluff bodies typically separates resulting in shedding of vortices that impact the body with some periodicity. The fluctuating pressures around the bluff body generated by the periodic vortex shedding result in the fluctuations in the cross-wind force characterized by a dominant frequency described by the Strouhal number S_t = fD/U, where f is the dominate frequency component of the vortex-induced force; D is the front projection area of the structure cross-section per unit length; U is the oncoming flow velocity. The Strouhal number is a function of structural geometry and Reynolds numbers R_e = UB/ν, where B is the characteristic size of the structure and ν is the kinematic viscosity of the media around the structure. As the frequency of vortex-induced forces approaches the modal frequency of the bluff body, “lock-in” a range of wind velocity in which the appearance and disappearance of complex wave-forms is observed [1], emerges with relatively large associated response. Due to nonlinear fluid-structure interactions, the vortex-induced vibration (VIV) exhibits limit cycle oscillations (LCOs). Although VIV response does not always result in catastrophic failures, it seriously impacts the fatigue life and a loss of desired functionality in the case of buildings.

It should be noted that the focus of most studies in the literature has been on the VIV of circular cylinders. On the one hand, the findings reveal some fundamental mechanisms of VIV as the circular cylinder is an important and representative bluff body. While on the other hand, VIV of bridge decks, as a typical bluff body of structural engineering significance characterized by a significantly long afterbody, exhibits substantially different flow features from circular cylinders. For example, there are often fixed separation points for bridge decks instead of moving separation points for circular cylinders, which vary with Reynolds numbers and body oscillation frequency and amplitude. Besides, the torsional force induced by the wake vortices may be important for bridge-deck-like bluff bodies as the structural geometry features a long afterbody. The angle of wind incidence also becomes an important parameter of VIV of bridge decks due to asymmetric bridge deck cross-section. It is well known that the Kármán “vortex street” resulting from the flow separation is the main physical origin of VIV for circular cylinders. The underlying VIV mechanisms of bridge-deck-like bluff bodies may belong to one of the following four types, i.e., the leading-edge vortex shedding, the impinging leading-edge vortices, the trailing-edge vortex shedding and the vortex shedding alternating between edges [2].

VIV of circular cylinders

Since the pioneering work of Strouhal [3] and Rayleigh [4], a large number of studies have been carried out on the fundamental mechanisms around VIV with a focus on circular sections. A chronological development of the subject is well documented in several reviews (e.g., Sarpkaya [5]; Bearman [6]; Williamson [7]; Billah [8]; Sarpkaya [9]; Williamson and Govardhan [10]). A brief overview of VIV is provided here as a primer for VIV of bridge decks.

Fundamental studies of VIV

von Kármán and Rubach [11] characterized the famous “vortex streets” phenomenon with the Kármán constant K= h/a, where h is the transverse distance between the two rows of vortices and a is the longitudinal space period, as indicated in Fig. 1.

There are two vortex sheets at each side of the body stemming from the boundary layer separation, which roll up into the “vortex streets” in the wake. The Kármán “vortex streets” were deduced in ideal situations, e.g., the nonviscous fluid and the concentration of vorticity. Actually, in a viscous flow, in order to keep the moment of vorticity, which is equal to the mean vorticity per unit length multiplied by h, as a constant, the transverse distance h has to increase with time since the mean vorticity decreases with time due to the interaction of the two vortex rows (two separating shear layers) with vorticities of opposite sign [12,13].

Gerrard [14] on the other hand, revealed the underlying physical mechanism of vortex formation in the wake of bluff bodies. Central to Gerrard’s [14] study is the formation region in which the irrotational flow from outside the wake crosses the wake axis. This flow is divided into three parts, as presented in Fig. 2. One part is entrained by the growing vortex on the opposite side; the second part is entrained by the shear layer upstream of the vortex; the third part reverses to the interior of the formation region.

Recently, the VIV area has benefitted from development in innovative experimental procedures and increasing computational capability, such as Particle image velocimetry (PIV) and the direct numerical simulation (DNS) techniques. A more detailed behavior of the wake vortices, e.g., the different wake patterns such as 2S (two single vortices per cycle of motion), 2P (two vortex pairs per cycle of motion) and P+ S modes, have been observed, as presented in Fig. 3. The relevant underlying physical mechanisms have been deduced, which have been presented in Williamson and Govardhan [10].

Modeling VIV of circular cylinders

Increasing availability of experimental data has confirmed some general features of VIV, such as the self-excited and self-limiting vibration at “lock-in,” the frequency of wake vortices entrained by the structural frequency at “lock-in”, the nonlinear hysteresis behavior and the frequency demultiplication. The vortex-induced forces and the attendant structural response (especially in the “lock-in” region in the cross-wind direction) can be estimated utilizing empirical models, developed over the years based on experimental observations. It should be noted that, apart from the fluctuating transverse force, the fluctuating in-line force is also induced by the wake vortices with roughly double the dominant frequency-component and with a lower intensity. Most of the models reviewed in this paper are based on the assumption of full correlation of the forces spaced along the spanwise direction of the structure.

Bishop and Hassan [1,15] systematically carried out a series of experiments to establish a solid foundation for the development of VIV mathematical models based on the experimental observations. The experimental results of the stationary circular cylinder with typical features of the mean and fluctuating drag and lift coefficients were presented. Similar to the (1/S_t)–Re curve, these aerodynamic coefficients are also characterized by critical positions in terms of Reynolds number as these plots exhibit major changes in trends. Besides, the experimental observations suggested that it may be appropriate to utilize a conventional mechanical oscillator for simulating the circular cylinder wake behavior. Whereas in the case of oscillating circular cylinders experimental results show “hysteresis loop” and frequency demultiplication phenomena. Besides, the experimental results showed that there was a sudden change in the force amplitude and the phase difference between the force and the structural motion at a certain “critical” frequency. Furthermore, these experimental observations suggested that the wake behavior behind the circular cylinder could be simulated using a nonlinear self-excited oscillator (wake oscillator).

Stimulated by the suggestion in Bishop and Hassan’s [1,15] study, Hartlen and Currie [16] simulated the VIV with a second-order linear mechanical oscillator and a second-order nonlinear wake oscillator which was coupled with the structural motion. The governing equations of their model were substantially derived based on the experimental data. It was shown that the proposed wake oscillator was a Von der Pol type oscillator which captured typical features of VIV at “lock-in”, such as the self-excited and self-limiting properties, the Strouhal relationship and the interaction between the wake and the structural motion. This phenomenological approach based wake oscillator, however, failed to offer a clear and relevant physical significance of its empirical parameters. Skop and Griffin [17] improved the model by assigning physical meaning to each empirical parameter, which showed added advantages as it demonstrated good predictive capability of VIV. Typically, both empirical parameters h and G in the improved wake oscillator simply showed a logarithmic relationship with the mass-damping ratio (represented by the parameter ζ/μ in their study, where ζ means the mechanical damping coefficient and μ is the mass ratio of the displaced mass of fluid to the mass of the structure), as shown in Fig. 4. Based on the similar idea, Landl [18] also improved Hartlen and Currie’s model by introducing a fifth-order aerodynamic damping term.

Iwan and Blevins [19] presented a model based on basic fluid-mechanics principles and the momentum conservation theory which directly related parameters in their model to physical constants of a VIV system. The identification of the model parameters was basically based on stationary body and forced-vibration, which suggested a underlying assumption that fluid memory effects on VIV system was significantly weak. Employing a quite different approach compared to Hartlen and Currie’s scheme, this model, however, also showed a Vol de Pol type nonlinear behavior. Similarly, Dowell [20] offered another self-consistent nonlinear VIV model based on basic fluid mechanics considerations, which was verified using Jones et al.’s [21] experimental results at high Reynolds number.

The models based on the wake-oscillator concept obtained significant success as applied to practical engineering problems. Birkhoff [12], on the other hand, presented an attractive observation that the behavior of wake swinging from side to side was similar to the tail of a swimming fish. The impetus of Birkhoff’s concept resulted in several interesting nonlinear models [22,23]. Additionally, Basu and Vickery [24] proposed another type of VIV model under the framework of linear random-vibration theory. In their study, the vortex-induced effects were simulated with a narrow-band random force while the motion-induced effects were considered using a nonlinear damping term which could capture the amplitude-dependent and Reynolds number effects. The concept of vortex-induced stochastic aerodynamic force and of motion-induced deterministic aeroelastic force was further developed by d’Asdia et al. [25] through correlating the along-wind and cross-wind aerodynamic forces.

VIV of bridge decks

Basic understanding of VIV of bridge decks has been improving during the past several decades as it has been a subject of focused wind-tunnel experiments, full-scale observations, and development of semi-empirical models and computational fluid dynamics (CFD) schemes. A review focusing on VIV of bridge decks highlighting physical mechanisms with a renewed perspective is provided. Only a selected few studies highlighting the progress in the VIV of bridge decks are included for the sake of brevity.

Wind-tunnel experimental results

Nakamura and Mizota [26] investigated spring-mounted rectangular prisms with various aspect ratios under smooth wind conditions. They noted that, at zero incidence of wind flow, the boundary layers separated at two front edges with no downstream reattachment for the models with the aspect ratios of 1∶1 and 1∶2 while the separated shear layers reattach somewhere on the side faces for the model with an aspect ratio of 1∶4. Their experimental results in terms of base pressures and lift force coefficients suggested an enhancement in vortex-induced effects with decreasing aspect ratio of length (along-wind direction) to width (cross-wind direction) of rectangular prisms. Whereas the decreasing aspect ratio of the spanwise length to the size of the cross-section weakened vortex-induced forces [27,28]. All models with different aspect ratios presented VIV with a notable “lock-in” region. In addition, Nakamura and Mizota [26] explained the physical mechanisms of “lock-in” phenomenon. The abrupt phase change of the unsteady lift force, which closely correlated with the phase change of the near-wake velocity field, resulted in negative aerodynamic damping near the “lock-in” range and led to large vortex-induced motion amplitudes. Similar observations have been reported for circular cylinders as discussed in the preceding section [1,15].

Komatsu and Kobayashi [29] carried out a series of experiments of various cross-sections, such as the L-shaped, T-shaped, H-shaped and rectangular cylinders with various aspect ratios. Both free and forced harmonic oscillations were used together with smoke-wire flow visualization technique. They noted that Kármán vortex street (generated from the trailing edge) and motion-induced vortices (generated from the leading edge) were the two main sources of VIV of bluff prismatic cylinders. They also reported an empirical formula for the critical wind velocity corresponding to the maximum VIV response, which is a linear function of the aspect ratio of bluff cylinders. Shiraishi and Matsumoto [30] investigated a series of cross-sections such as rectangular, H-shaped, trapezoidal and hexagonal sections with various aspect ratios in both heaving and torsional degrees of freedom, including the consideration of various flow angles of attack. Three generation mechanisms of VIV were classified, i.e., separated vortices from leading edge due to structural motion, secondary vortices at trailing edge due to structural motion and separated vortices from trailing edge due to Kármán vortex shedding. Five simplified bridge deck cross-sections with various aspect ratios were investigated to guide the deck optimization for reducing VIV response. The results indicated that bridge decks with long afterbody are more effective in mitigating VIV. While the emergence of multi-“lock-in” regions was attributed to the “frequency demultiplication” phenomenon (Van der Pol [31]) by most of the researchers, Shiraishi and Matsumoto [30] claimed that the smaller-peak oscillation in the lower wind velocity region resulted from the arrival of the separated vortices from the leading edge to the trailing edge after n cycles of heaving motion (where n is a natural number) or after (2n-1)/2 cycles of torsional motion. Accordingly, the onset critical reduced wind velocity for the smaller-peak oscillation of the torsional mode was [2V_cr/(2n-1)] where Vcr was the onset critical reduced wind velocity for the larger-peak oscillation. Zhang et al.’s [32] experimental data, as shown in Fig. 5, sheds a different view point, where the lower critical reduced wind velocity for the smaller-peak oscillation of the torsional mode was noted to be around V_cr/2.

The secondary vortex was also observed by Nakamura and Nakashima [33] and it was attributed to being a characteristic of the elongated sharp-edged bluff bodies (e.g., typical bridge decks) since no similar phenomenon was observed for oscillating circular cylinders [34]. In the study by Nakamura and Nakashima [33], both stationary and oscillating experiments of bluff prisms with elongated H-shaped, ⊢-shaped and rectangular cross-sections were carried out in a wind tunnel and in a water tank with a hydrogen-bubble visualization technique. They noted that VIV occurred even if a splitter plate, which prevented the interaction between shear layers, was inserted in the wake. Hence, both the Kármán vortices (double-layer flow instability) and the impinging-shear-layer instability (single-layer flow instability) were believed to be the contributing mechanism for VIV response.

Matsumoto et al. [35] investigated the interaction between Kármán and motion-induced vortices and the effects of turbulence on VIV for a rectangular cylinder with an aspect ratio of 4∶1 and a hexagonal box girder with/without handrails. On the one hand, the experimental results indicated that VIV of both models was essentially excited by the motion-induced vortices which were mitigated by the presence of Kármán vortices. While on the other hand, in order to explain the physical mechanism of “frequency entrainment” in the “lock-in” region, Sarpkaya and Schoaff [36] emphasized the role of structural motion in enhanceing vortex strength.

According to the results in Matsumoto et al. [35], the effects of turbulence on VIV is rather complicated. While the turbulence stabilizes the VIV of rectangular cylinders, its effects on the hexagonal box girder depended on several other features such as the solidity ratio of the handrail and the Scruton number. The effects of turbulence on the motion-induced force is uncertain due to the limited understanding of this issue (e.g., Scanlan [37]; Haan and Kareem [38]; Kareem and Wu [39]). Whereas, one deterministic effect of turbulence is that it increases vorticity diffusion and thus reduces vortex strength. Similar observation has been advanced by Sarpkaya [5]. Besides, experimental and field observations indicate that turbulence decreases the spanwise correlation, which concomitantly decreases VIV response of the structure.

Suppose the motion-induced vortices are not affected by turbulence, one possible approach to examine the final effects of turbulence on VIV (stabilization or destabilization), which is illustrated in Fig. 6, may depend on the relative intensity of Kármán vortices compared to the motion-induced vortices. If Kármán vortices dominate VIV, then the turbulence decreases structural response [5]; on the other hand, if the motion-induced vortices dominated VIV, then the turbulence may increase the structural response as the enhancement of structural response due to weakening mitigative effects of Kármán vortices on the motion-induced vortices is significant. More experimental data sets are necessary to validate this conjecture.

Larose et al. [40] investigated VIV of Stonecutters bridge deck (twin-box) based on a 1∶20 scaled sectional model. Two distinct “lock-in” regions were observed for the torsional mode while there was only one well-defined “lock-in” region for the heaving mode. The Reynolds number effects on VIV of the bridge deck was examined in their study. It appeared that the bridge deck appendages (e.g., longitudinal guide vanes and maintenance gantry rails) were less efficient in mitigating VIV response at low Reynolds number as compared to at high Reynolds number.

Larsen et al. [41] studied VIV of Storebælt (Great Belt East) bridge deck (single-box) using a large scaled model (1∶30) with a high Reynolds number (around 500000, bridge deck width based). On the one hand, the experimental results indicated that, in the high Reynolds number range, the Strouhal number was slightly higher at Reynolds number around 500000 compared to that at the Reynolds number around 100000. While on the other hand, in the low Reynolds number range, Sarpkaya [42] successfully predicted the maximum amplitudes of vortex-induced forces at Reynolds number between 300 and 1000, measured by Griffin and Koopmann [43], utilizing the data measured over the Reynolds number range from 5000 to 25000. This successful prediction suggested that the Reynolds number may not be an important parameter for oscillating circular cylinders. Larsen et al. [41] also investigated in detail the mitigating effects of guide vanes on VIV utilizing this large-scale bridge sectional model.

Diana et al. [44] observed VIV of a multi-box bridge deck in detail in the wind tunnel based on Messina Strait suspension bridge. The experimental results presented a strong dependence on the VIV amplitude, which indicated significant nonlinear behavior. Two “lock-in” ranges were identified for the heaving mode with the bridge deck model in their study. Besides, the bridge deck shape was optimized to mitigate VIV response based on experimental observations.

Recently, Zhang et al. [32] investigated VIV of Xihoumen bridge deck (twin-box) with various scaled models. The experimental results indicated that the “lock-in” emerges at higher and wider-range of damping ratio with larger amplitude and wider wind velocity range for low Reynolds number cases compared to for high Reynolds number cases. This suggested that the predicted response based on experimental results of scaled model at low Reynolds number was over-conservative. Similar conclusion was obtained by Schewe and Larsen [45]. Besides, the effects of guide vanes on VIV was studied at various wind angles of attack. It was shown that the guide vanes mitigate VIV response for the zero angle of attack case effectively. If the attack angle changes (±3°), the guide vanes on the one hand mitigate VIV response at high Reynolds number cases while it enhances VIV response for low Reynolds number cases. Two distinct “lock-in” regions were observed for the torsional mode in their study while one well-defined “lock-in” region was identified for the heaving mode.

Full-scale observations

Smith [46] presented field measurements of Wye Bridge, a cable-stayed box girder bridge with a 235 m span. The records indicated that the large amplitude response occurred when the wind velocity (1min mean value) was in the range of 7–8 m/s, the wind direction was approximately perpendicular to the bridge axis, the wind inclination was in the range of -5°-0°, and the turbulence intensity was low (around 5.0%). This large amplitude vibration was considered as VIV since the vibration frequency was close to the natural frequency of the bridge first bending mode. The critical wind speed for VIV was close to the prediction based on the wind-tunnel sectional model while the amplitude of the response was significantly lower than that derived from the wind-tunnel results. The full-scale observations supported the wind-tunnel based argument that the predicted response based on the experimental results of scaled model at low Reynolds number was over-conservative.

Kumarasena et al. [47,48] investigated the wind-induced motion of Deer Isle Bridge with field observations, wind-tunnel experiments and finite element analyses. On-site data showed that VIV occurred at a wind velocity of 9.11 m/s, with a high response level. Kumarasena et al. [48] pointed out that the ordinary vortex shedding of the stationary deck was affected significantly by the turbulence even with a low intensity level as it changed from being the narrow-banded (periodic) to the broad-banded.

Owen et al. [49] monitored Kessorck Bridge with an “open cross-section” and with a 240 m main span. The observation indicated that VIV occurred when the wind velocity was in the range of 22–25 m/s. Specifically, a 3-min duration of the large amplitude motion of the bridge deck in first mode, which was excited by the vortex shedding, was observed based on the one-hour records. The observations indicated that the emergence of VIV was very sensitive to many factors such as the wind velocity, wind direction and turbulence effects. “Lock-in” phenomenon was identified using the field measurements, where the influence of turbulence was diminished. In addition, Owen et al. [49] pointed out that it needed a certain time to build up the large amplitude response of VIV after the corresponding vortex shedding occurred. The torsional VIV was identified utilizing the theory developed by Shiraishi and Matsumoto [30].

To decrease the intense VIV response observed on Rio-Niterói Bridge with a steel twin-box bridge deck, which had been shut down at occasions when the wind velocity approached 14 m/s, Ronaldo et al. [50] designed passive and active control devices which significantly decreased (the level of peak-to-peak amplitude is around 4.2 cm) or ceased VIV from a much larger intensity observed on site (the level of peak-to-peak amplitude is around 50 cm).

Larsen et al. [51] monitored wind-induced behavior of the full-scale Storebælt suspension bridge. The observations indicated that the unacceptable wind-induced oscillations occurred as the wind velocity reached the range of 4–12 m/s with a direction perpendicular to the bridge axis and with a rather low turbulence intensity. The field measurements showed that the large amplitude oscillations were almost sinusoidal motions with a primary frequency component, which confirmed the phenomenon as VIV. The observed critical wind velocity of VIV on site had a good agreement with the result based on the scaled section model and scaled taut strip model, but slightly higher compared to that derived from the full-scale aeroelastic model test. The guide vanes were designed and installed to mitigate VIV of Storebælt suspension bridge based on the field measurements. Excellent effectiveness was achieved so that VIV ceased and the flutter stability was also improved. Frandsen [52] also investigated the Storebælt suspension bridge on site. VIV was found at a low wind velocity (around 8 m/s) with the wind direction almost normal to the bridge axis and with the low turbulence intensity. The “lock-in” phenomenon lasted about two hours based on a 4.5 h record with the peak-to-peak amplitude around 0.7 m. The time histories of pressure on the deck surface and the accelerations of the bridge deck were simultaneously recorded and indicated high correlation in the “lock-in” range.

Fujino and Yoshida [53] carried out full-scale measurements on a ten-span continuous steel box-girder bridge of the Trans-Tokyo Bay Highway Crossing (the measuring tool was installed on the center with the longest span of 240 m). VIV was observed under the prevailing winds (13–18 m/s for the first mode VIV and around 23 m/s for the second mode VIV) with the wind direction almost normal to the bridge axis (within±20°) and with low turbulence intensity. VIV response of the scaled model in wind tunnel and that of field observation were in good agreement with respect to the amplitude and wind velocity range of the “lock-in”. Whereas, it should be noted that the “equivalent” full-scale amplitudes based on wind-tunnel results were simply obtained utilizing the nondimensional amplitude of scaled model multiplied by the actual bridge deck size, which may be questionable in light of nonlinearity in VIV.

Recently, Li et al. [54] studied VIV of a suspension bridge with a twin-box girder and with a 1650 m center span. VIV was observed in the wind velocity range of 6–10 m/s with the direction almost perpendicular to the bridge axis and with low turbulence intensity. Time-frequency S transform technique was utilized to identify the vortex-shedding pattern around the bridge deck. It was identified that the structural motion increased the vortex-shedding intensity. Similar observations have been reported for circular cylinders as discussed in the preceding section [36]. Hence, the vortex shedding in the “lock-in” stage not only occurred in the gap between the two bridge decks and at the tail end of the downstream deck section as in the early stage of VIV but also occurred at the tail end of the upstream deck section and around the entire lower surface of the downstream deck. Based on the spanwise installed pressure sensors, it was observed that the spanwise correlation remained constant, with relatively small value as compared to Wilkinson’s data [55]. Besides, the estimated Strouhal number based on the field measurements remained constant within the Reynolds number range of 1.74 × 10⁷ – 2.40 × 10⁷ (bridge deck width based).

It should be noted that the in-line VIV is not reported during all the VIV investigations (experiments and full-scale observations) discussed above.

Semi-empirical models

The concept of wake oscillator has been utilized to model vortex-induced effects since the early work of Bishop and Hassan [1,15]. In this scheme, VIV is simulated by a system of two coupled equations, in which the structural motion is modeled by a second-order linear mechanical oscillator excited by the vortex-induced force modeled by a second-order nonlinear wake oscillator (Vol del Pol type oscillator) coupled with the structural motion (Hartlen and Currie [16]). The models rely on several experimentally derived parameters needed to characterize the system and to describe coupling between the linear mechanical and the nonlinear wake oscillators. The experimental procedures involved to identify these parameters need to be sophisticated and care should be exercised as considerable element of uncertainty exists in such experiments (e.g., Sarpkaya [5]).

To simplify the parameter identification procedure, Scanlan [56] proposed an explicit function for the vortex-induced force for application to VIV of bridge decks. In other words, the VIV of bridge decks was simulated with a single ordinary differential equation, where the vortex-induced force was explicitly expressed as the sum of the self-excited terms (induced by structural motion) and the forced terms (induced by vortex shedding). The Scanlan’s proposal is given by

(1)m\left(\ddot{y}+2\xi {\omega }_0\dot{y}+{\omega }_0^{2}y\right)={\displaystyle \frac{1}{2}}\rho U^2\left(2D\right)\left[Y_1\left(K\right)\left(1-ϵ{\displaystyle \frac{Y^2}{D^2}}\right){\displaystyle \frac{\dot{y}}{U}}+Y_2\left(K\right){\displaystyle \frac{y}{D}}+{\displaystyle \frac{1}{2}}{C}_L\left(K\right)\sin \left(\omega t\right)\right],

where m is the mass per unit span length; y is the displacement of the cross-wind degree of freedom; ζ is the mechanical damping ratio to critical; ω₀ is the mechanical circular frequency; ρ is the air density; U is oncoming mean wind velocity; D is the cross-wind dimension of the structure; K= ωD/U is the reduced frequency of vortex shedding (where ω is the circular frequency of vortex shedding determined using the Strouhal relationship outside the “lock-in” or “frequency entrainment” behavior inside the “lock-in”); Y₁, ϵ, Y₂ and C_L are parameters in this VIV model, which could be identified based on the experimental observation.

As shown in Eq. (1), for Scanlan’s VIV model: 1) there is no direct interplay between the vortex-induced and motion-induced effects; 2) consideration of the nonlinear motion-induced effects with a nonzero value of ϵ results in the nonlinear VIV system; 3) the self-limiting feature was obtained with the nonlinear damping depended on the vibration amplitude instead of the phase change in the external vortex-induced force. It is possible that the Scanlan’s VIV model overweighed the motion-induced effects, in light of Sarpkaya’s [5] observation that VIV was a forced oscillation with self-excited character. Another feature of Scanlan’s VIV model was that the LCO amplitude was only dependent on a single parameter, the mass-damping parameter. Although Sarpkaya [42] has demonstrated that, for structures with small values of the mass-damping parameter, the nondimensional mass and damping affect the structural response independently. For structures with large values of the mass-damping parameter (a normal situation for the bridge deck) the structural response depends on the combination parameter of the mass ratio and damping, which is represented effectively by the Scruton number [57]. Griffin and Ramberg [58] showed that, for similar values of the mass-damping parameter, the maximum LCO amplitudes are nearly equal and occur at similar ranges of the reduced wind velocity, as indicated in Fig. 7, where Y is the cross-flow amplitude and n is the natural frequency of the structure.

In contrast to Skop and Griffin’s model [17], in which both empirical parameters simply show a logarithm relationship with the mass-damping ratio as shown in Fig. 4, the empirical parameters Y₁ and ϵ in Scanlan’s model change significantly and irregularly with the mechanical damping ratio (and therefore the mass-damping ratio) [59] as shown in Fig. 8, where each symbol indicates a difference bridge deck. Hence, Scanlan’s model seems to have rather limited predictive capability for VIV of bridge decks with various mass-damping ratios. It should be noted that, since Scanlan’s model is a nonlinear one, the prototype results are not simply obtained by scaling model results. Often to such scaling issues from wind-tunnel to full-scale predictions are not reported.

Goswami et al. [60] further developed Scanlan’s single-degree-of-freedom (SDOF) VIV model based on Billah’s [8] coupled wake oscillator model. Larsen [61] generalized Scanlan’s VIV model by introducing a power multiplier ν to modify the curvature of the predicted steady-state response versus the Scruton number. The so-called generalized VIV model presents a downward curvature, which is consistent with the experimental results and is opposite to the predictions of Scanlan’s model. Parameter identification schemes based on either the steady-state response or the transient response are given for this generalized VIV model. Recently, Diana [44] proposed a new numerical model for VIV of bridge decks by representing the fluid-structure interaction with a second-order nonlinear mechanical system coupled with the structural motion through linear and cubic relationships.

Computational fluid dynamics simulations

With the burgeoning growth in computational capability, CFD is becoming a powerful tool for the analysis of wind-induced effects on the structures. Hence, it is a promising approach to simulate VIV of bridge decks. Most of the literature on the numerical simulation of VIV of bridge decks, with relatively large Reynolds number, is limited to 2D domain.

Fujiwara et al. [62] investigated VIV of bridge decks with three types of elastically-mounted edge-beam cross-sections in 2D domain using direct integration of Navier-Stokes equations (height based Reynolds number range was 2100–4000). The incident wind had a 5° angle of attack. The numerical displacement results were compared to that of scaled model in the wind tunnel. The agreement between numerical and experimental results varied with the shape of the deck cross-section.

Nomura [63] investigated VIV of a Tacoma-like bridge deck (thin-H section of aspect ratio 5) in 2D domain based on the finite element method using the arbitrary Lagrangian-Eulerian formulation. Two sudden increases in the computed oscillation amplitudes and lift coefficients were identified around Reynolds number of 1000 and 2400. Typical flow pattern around the bridge deck were reported to be similar to the experimental results by Nakamura and Nakashima [33].

Lee et al. [64] studied VIV of two bridge decks, namely the Namehae bridge deck which represents a streamlined cross-section and the Seohae bridge deck which represents a bluff cross-section, using Reynolds-averaged Navier-Stokes (RANS) equations. The bridge deck remained stationary during the CFD simulation under the assumption that the structural motion would not significantly affect the flow domain if the cross-wind vibration amplitude was within 10% of the structural size. Good agreement was achieved between the drag and lift coefficients of CFD simulations and those of wind-tunnel tests for the Namehae bridge. For the Seohae bridge, both the computational and experimental VIV amplitudes showed dual displacement peaks.

Sarwar and Ishihara [65] investigated VIV of a rectangular and a box girder cross-section using three-dimensional (3D) large-eddy simulation (LES) approach. Both forced and free oscillations of the structure in the flow were simulated with the sliding mesh technique. The computational simulation of forced oscillations for the rectangular section obtained the “lock-in” region in which an abrupt change of the phase angle of the unsteady lift force, as an important physical mechanism of VIV, was detected. The computational response of the free oscillations for the rectangular section presented two “lock-in” regions, showing good agreement with the experimental results. Besides, aerodynamic mechanisms of countermeasuring VIV of bridge decks were studied using pressure distributions. Typically, they found the fairing had negative while the double flaps had positive effects on VIV.

Recently, the use of discrete-vortex method (DVM) to study VIV has gained attention. The development of DVM to calculate the vortex shedding of bluff bodies is based on Abernathy and Kronauer’s pioneering work [13] in 2D domain. Sarpkaya and Schoaff [36,66] developed the DVM comprehensively based on the interactions between the potential flow and the boundary layer around the stationary and oscillating circular cylinder. Larsen and Walther [67–69] applied DVM to the computational bridge engineering for the aeroelastic analysis with 2D viscous incompressible flow while VIV of bridge decks using DVM has also been studied by Frandsen [70] and Morgenthal [71].

Volterra series based model for VIV of bridge decks

In the absence of an analytical treatment of the flow around a stationary or an oscillating structure, a closed-form representation of the VIV phenomenon remains mathematical intractable. In lieu of this, semi-empirical models have been advanced to model VIV of bridge decks, but due to their phenomenological origin, their accuracy of their predictions is not ensured. In the experimental area, VIV studies, with scaled models, is limited to low Reynolds numbers, especially for the decks of super long-span bridges. A promising approach involves CFD, however, this has its own limitation at this juncture stemming from lack of robust turbulence models for engineering applications and high demand on computational resources. To close the gap between a reliable numerical simulation and the need to have predictive model for practical applications, reduced order models (ROMs) traditionally offer predictions with high fidelity with reduced computational effort. As there are much lower-order degrees of freedom involved in an ROM compared to CFD, it can be tailored to meet the demands placed by the fundamental physics of the application (e.g., Raveh [72]; Lucia et al. [73]). Among various ROMs, Volterra series based model, which is a form of Taylor series with memory effects, has the promise of effectively modeling the VIV system. The complex mapping rules (static linear/nonlinear relationships) and time lag (fluid memory effects) between the aerodynamic/aeroelastic inputs and outputs, the hallmark of VIV, can be represented by the superposition of scaled and time shifted fundamental responses, i.e., convolution. Since each feature is captured elegantly by the Volterra Series that makes it an ideal candidate for ROM modeling of a VIV system.

Earlier models by Skop and Griffin [17] and Ehsan and Scanlan [59] were based on Van der Pol type equation with the approximation of slowly varying parameters that suggested classification of VIV as a weakly nonlinear system. This feature suggests the use of truncated Volterra series with finite terms for VIV modeling. The second-order Volterra series has been selected to simulate VIV of bridge decks by Wu and Kareem [74]. For a nonlinear system modeled with the second-order Volterra series, the response y(t) under an arbitrary input x(t) could be represented as follows [75]:

(2)y(t)={\displaystyle {\int }_{\mathrm{0}}^t{h}_{\mathrm{1}}(t-\tau )}x(\tau )\mathrm{d}\tau +{\displaystyle {\int }_{\mathrm{0}}^t{\displaystyle {\int }_{\mathrm{0}}^t{h}_{\mathrm{2}}(t-{\tau }_{\mathrm{1}},t-{\tau }_{\mathrm{2}})}x({\tau }_{\mathrm{1}})x({\tau }_{\mathrm{2}})}\mathrm{d}{\tau }_{\mathrm{1}}\mathrm{d}{\tau }_{\mathrm{2}},

where the steady-state term h₀ is neglected due to the focus on the dynamic response of the VIV system; h₁ represents the first-order kernel which describes the linear behavior of the system; h_n the higher-order terms which indicates the nonlinear behavior existing in the system. Based on earlier work by Rugh [75], a generalized impulse-function-based kernel identification scheme is developed, where the kernels of the second-order Volterra system could be expressed as [76]

(3)h_{\mathrm{1}}(t-\tau )={\displaystyle \frac{\mathrm{1}}{{\alpha }^{\mathrm{2}}-\alpha }}({\alpha }^{\mathrm{2}}y[\delta (t-\tau )]-y[\alpha \delta (t-\tau )]),

(4)h_{\mathrm{2}}(t-{\tau }_{\mathrm{1}},t-{\tau }_{\mathrm{2}})={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\kappa }_{\mathrm{1}}{\kappa }_{\mathrm{2}}}}[y[{\kappa }_{\mathrm{1}}\delta (t-{\tau }_{\mathrm{1}})+{\kappa }_{\mathrm{2}}\delta (t-{\tau }_{\mathrm{2}})]-y[{\kappa }_{\mathrm{1}}\delta (t-{\tau }_{\mathrm{1}})]-y[{\kappa }_{\mathrm{2}}\delta (t-{\tau }_{\mathrm{2}})]],

where δ(t) represents the Dirac delta function (unit-impulse function); y[δ(t)] indicates the unit-impulse response; α, κ₁ and κ₂ are selected constants.

Figure 9 shows the identified first- and second-order kernels (time interval Δt = 0.002 s) of VIV of a rectangular prism. As shown in the figure, the magnitude of the second-order kernel is several orders smaller than that of the first-order kernel. To demonstrate that the truncated Volterra series based model is able to simulate the “lock-in” behavior, the response at “lock-in” of this VIV system is obtained utilizing the identified Volterra kernels and compared to the reference response obtained using the fourth-order Runge-Kutta scheme, as shown in Fig. 10. The linear approximation response is based on the first-order kernel only while the nonlinear approximation represents the response obtained by utilizing up to the second-order Volterra series. As presented in Fig. 10, the linear approximation response shows notable discrepancy as compared to the reference results while there is indiscernible difference between the nonlinear approximation response and the reference data. This observation indicates that the second-order kernel is necessary and sufficient to simulate the “lock-in” phenomenon of a VIV system.

Closing remarks

A critical overview of the literature on vortex-induced vibration (VIV) of bridge decks is presented based on wind-tunnel experiments, full-scale observations, semi-empirical models and computational fluid dynamics (CFD). Several physical mechanisms surrounding VIV of bridge decks are elucidated in a new light, such as the complicated role of turbulence and Reynolds number effects on VIV. Conventional Scanlan’s model (Van de Pol type) for the VIV of bridge decks is examined in detail. In addition, the Volterra series based nonlinear model is introduced to simulate VIV of bridge decks. The kernel identification scheme based on the impulse function input is utilized to identify the first- and second-order kernels of Volterra series. It is demonstrated that the VIV system at “lock-in” could be accurately modeled by truncated Volterra series.

A selected summary of VIV of bridge decks based on the review presented in this study is given here.

1) The decreasing aspect ratio of length (along-wind direction) to width (cross-wind direction) of a prismatic cross-section enhances the vortex-induced effects; whereas a decreasing aspect ratio of the spanwise length to the size of the cross-section weakens the vortex-induced forces.

2) For the bluff body with a large aspect ratio, the smaller-peak oscillation in the lower wind velocity region results from the arrival of the separated vortices from the leading edge and advancing to the trailing edge.

3) The final effects of turbulence on VIV (stabilization or destabilization) may depend on the relative intensity of the Kármán vortices as compared to the motion-induced vortices.

4) Reynolds numbers effect on VIV may be more important in the high Reynolds number range as compared to the low Reynolds number range.

5) The predicted VIV response based on scaled experiments at low Reynolds number is over-conservative when compared to the full-scale observations.

6) Many bridge decks are prone to VIV in a relatively low wind velocity range for wind direction almost perpendicular to the bridge axis and low turbulence intensity.

7) Scanlan’s VIV model seems to have rather limited predictive capability for VIV of bridge decks with various mass-damping ratios.

8) The second-order kernel of the Volterra series based model is necessary and sufficient to simulate the “lock-in” phenomenon of a VIV system.

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