Vortex-induced vibrations (VIV) in wind turbine towers present a well-known challenge for wind turbine manufacturers (Veers et al., 2023). Numerous cases of excessive and often “unexpected” vibrations have been reported during the installation and commissioning phases – prior to electrification – as well as when turbines are parked or idling at low speeds. These vibrations are frequently attributed to fluctuating wind loads acting on the cylindrical tower, resulting from vortex shedding in its wake. It is noted that in some cases, vortex shedding originating from the blades can also induce unsteady aerodynamic loads and lead to VIV in the wind turbine structure (Horcas et al., 2022; Zou et al., 2015), but such vibrations are not within the scope of the present study. During VIV, substantial vibration amplitudes can develop within seconds across various components of the wind turbine, ultimately leading to limit cycle oscillations.

When the vortex shedding frequency approaches a natural frequency of a tower mode, a dynamic interaction occurs between the unsteady flow and the tower's elastic response – known as lock-in. The lock-in phenomenon is experienced by the turbine as a synchronization between its structural vibration frequency and the periodic aerodynamic excitation caused by the flow around the tower. VIV due to tower vortex shedding can occur under several conditions: (a) on the bare tower, before the rotor–nacelle assembly (RNA) is installed, (b) on partially assembled RNAs – with none, one, or two blades mounted, or (c) on the fully assembled turbine when the rotor is parked or rotating slowly in idling mode. It is important to note that tower vortex shedding can occur under all inflow conditions. However, during normal operation, rotor rotation enhances wake mixing, thereby disrupting coherent vortex structures and mitigating VIV.

For typical onshore and offshore wind turbines, once the rotor–nacelle assembly (RNA) is installed, the vortex shedding frequency of the tower often aligns with the first bending mode frequency of the structure at low to moderate wind speeds, typically in the range of 5–15 ms-1. In contrast, when the RNA is not yet installed, the lock-in phenomenon tends to occur at higher wind speeds. In addition, onshore wind turbine manufacturers sometimes modify tower designs – particularly in terms of height and diameter – to better harness site-specific wind resources. For example, taller towers are often used at low-wind sites to access stronger wind flows at higher altitudes. In such cases, where towers can be up to 50% taller than standard configurations, the vortex shedding frequency may approach the second bending mode frequency of the tower, typically at wind speeds between 15–20 ms-1. In both scenarios – whether due to the absence of the RNA or the use of tall towers – the elevated lock-in wind speeds result in larger aerodynamic excitation forces, which in turn cause more severe structural vibrations. A common feature in all these cases is that during lock-in, high-amplitude oscillations develop in the crossflow direction. In the case of the standalone tower, this is the direction with the lowest aerodynamic damping, as the damping of the longitudinal direction is substantially enhanced by the action of the tower drag force. The situation is slightly different in the case the RNA is installed, because the aerodynamic damping of both tower vibration directions is affected by the aerodynamic loading of the rotor, which depends on the pitch setting of the blades. In parked or idling operation, the rotor-induced damping is relatively small in both directions (Wang et al., 2017). As a result of the crossflow vibrations of the tower, fatigue damage is greatly accelerated, which potentially compromises the structural integrity of the wind turbine.

The prediction of wind turbine loads and vibration amplitudes during tower lock-in events primarily relies on the engineering framework established by the Eurocode standard (European Committee for Standardization (CEN), 2005; Manolas et al., 2022). According to this standard, the effects of vibrations induced by vortex shedding are represented as a periodic inertial force distribution along the height of the tower, acting perpendicular to the wind direction. This force is dependent on the tower's linear mass distribution, the frequency and mode shape of the excited structural mode, and the maximum vibration displacement amplitude, which can be determined using two alternative approaches provided in the Eurocode standard.

Engineering models in the literature are commonly classified according to how they represent the fluid forcing term, which in turn determines the degrees of freedom in the mathematical formulation (Fernandez-Aldama et al., 2024). In this work, we specifically distinguish between models based on whether they incorporate an explicit oscillator equation for the lift force.

A first class of alternative engineering models explored in the literature incorporates additional nonlinear aeroelastic force components into the sinusoidally varying lift force induced by vortex shedding around the cylindrical tower. These additional terms typically include a component in phase with the structural motion (aerodynamic stiffness) and/or a nonlinear component in phase with the motion velocity (aerodynamic damping). For a rigid elastically supported cylinder, such models are commonly expressed as single-equation formulations, where the nonlinear stiffness and/or damping forces are introduced as external excitations into the one-degree-of-freedom dynamic motion equation. Representative models of this type include those proposed in the literature (Basu and Vickery, 1983; Magnus et al., 2013; Scanlan, 1981). Although the aforementioned models can effectively predict the maximum vibration amplitude occurring during lock-in, they are unable to provide detailed information of the distribution of the self-excitation power within the lock-in region if constant values of the defining parameters of the model are considered. An improvement to the above shortcoming has been suggested in Kurniawati et al. (2025) and Livanos et al. (2024) by providing an experimentally calibrated map of the variation in the model's defining parameter as a function of the wind speed and the response amplitude. A critical review of the above models can be found in Lupi et al. (2018). In the above reference, prediction improvements are proposed for one-equation models, based on the results from wind tunnel experiments on a rigid cylinder undergoing imposed periodic motions.

A second class of engineering models investigated in the literature involves representing the aerodynamic excitation – specifically the lift force – through the solution of a second-order oscillator equation. A representative example of this type is the Hartlen and Currie model (Hartlen and Currie, 1970). For a rigid elastically supported cylinder, these models are formulated as coupled systems of two equations (and thus two state variables): the first is the structural dynamic equation of motion, and the second is the oscillator equation that governs the evolution of the lift force. The primary advantage of this class of models is that, even for constant values for its defining parameters, in addition to predicting the maximum vibration amplitude during lock-in, they are also capable of capturing the distribution profile of vibration amplitudes and response frequencies as a function of wind velocity within the lock-in region.

In the present study, a newly calibrated VIV model based on the Hartlen–Currie lift-oscillator equation is proposed, with the aim of serving as a predictive tool for wind turbine tower lock-in events that can be readily integrated into comprehensive, state-of-the-art aeroelastic design frameworks. The constants of the oscillator equation are identified through an optimization procedure designed to accurately reproduce the maximum oscillation amplitude of rigid elastically supported, untapered cylinders at the dimensionless wind velocity where this peak amplitude occurs. Subsequently, model constants and variations are introduced across the full range of dimensionless wind velocities to enable the model to reasonably capture the overall amplitude profile within the lock-in region. Target values for the maximum amplitude and corresponding dimensionless wind velocity are derived from two wind tunnel experiments (Belloli et al., 2012; Feng, 1968) conducted on rigid elastically supported cylinders with different Scruton numbers.

The calibrated oscillator equation is then integrated into NTUA's in-house multibody aeroelastic code, hGAST (Bagherpour et al., 2018; Manolas, 2016; Manolas et al., 2020; Panteli et al., 2022; Riziotis and Voutsinas, 1997), for the prediction of wind turbine tower loads and vibrations in elastic cantilevered cylinders. The lift equation is used to compute the lift force distribution along the correlation length, defined according to the Eurocode, for a cylindrical untapered wind turbine tower. Model predictions are validated against wind tunnel measurements (Lupi et al., 2021) of vibration amplitudes for an cylindrical untapered tower with a clamped base and a free top end, undergoing vortex-induced vibrations (VIV) at the natural frequency of its first bending mode. The results show that the model (calibrated based on measurements on rigid elastically mounted cylinders) successfully captures the maximum vibration amplitude and the maximum response amplitude wind speed at lock-in. To accurately estimate the range of wind velocities over which lock-in occurs, across a range of Scruton (Sc) numbers, the model needs further calibration based on observations from experimental data of clamped vibrating cylinders in their bending modes. It should be noted that vortex-induced vibrations in higher-order modes (second tower bending mode), as well as the influence of ambient turbulence on the development of the lock-in phenomenon, are not addressed in the present study and are identified as subjects for future research.

The paper is organized as follows. In Sect. 2, first, the lift-oscillator model and its parameter calibration procedure are introduced (Sect. 2.1). Next, the calibrated model's performance is assessed against state-of-the-art semi-empirical tools for vortex-induced vibration (VIV) prediction (Sect. 2.2); the integration of the lift-oscillator model into the existing multi-body solver is then described (Sect. 2.3). In Sect. 3, a comparison between the model predictions and experimental measurements on cantilever flexible cylinder are presented, whereas Sect. 4 provides the conclusions of the present work.

The present work is structured into the following methodological steps: a.

The two constants of the Hartlen–Currie lift-oscillator model are calibrated using data from two wind tunnel experiments conducted on rigid elastically mounted, untapered cylinders subjected to crossflow vortex-induced vibrations (VIV). The calibration aims at determining the constant values that yield the target maximum oscillation amplitude at the specific dimensionless velocity where this amplitude occurs within the lock-in region, and across different values of the Sc number. This is achieved through an optimization process using the COBYLA algorithm, as implemented in the “scipy.optimize” module of the SciPy library (Virtanen et al., 2020). For dimensionless velocities other than the one corresponding to the peak amplitude, a simplified parameter identification approach is employed by assuming a linear decay of the parameters from their values at maximum resonance. This follows the reasoning of Tamura (2020), who argues that parameter selection should not rely solely on curve fitting but must also respect the physical characteristics of the phenomenon.

The calibrated Hartlen–Currie model (calibration based on rigid elastically supported cylinder) is validated against an experimental setup consisting of a cantilevered cylinder exposed to low Reynolds number flow (Re≈ 20 000), replicating the configuration of Lupi et al. (2021). In the experiments, different cylinder models are used for testing different Sc conditions, which are also replicated in the simulations presented herein. The main geometric and structural parameters of the cylinder models tested in the wind tunnel are summarized in Table 1. The simulations are conducted employing calibration parameters obtained from Belloli et al. (2012), corresponding to the experimental dataset that exhibited the closest agreement with Lupi's cantilever tower experiment regarding the values Umax⁡∗ and its variation with Sc. Differences are only expected to be marginal in the case the calibration parameters obtained from Feng (1968) are considered.

As shown in Figs. 9b, c and 10, the model accurately reproduces the maximum tip displacement for the two higher Sc values (Sc=5.18 and Sc=7.02). Comparing the Eurocode and the Blevins model highlights the Eurocode's behaviour in overpredicting the VIV response, whereas the opposite is observed with the Blevins model when they are used for a real-world cantilever structure. The calibrated Hartlen–Currie model as implemented in hGAST gives more accurate predictions albeit it is also conservative at the lowest Sc tested.

More specifically, for Sc=2.94, it overpredicts the displacement by approximately 55%. The wind tunnel experimental results presented in Fig. 3 indicate that, for Sc≈ 3, the maximum vibration amplitude in the VIV response of a rigid elastically supported cylinder lies in the range of 0.4–0.5, which is significantly higher than the corresponding values reported in Lupi's experiment for the same Sc. This clearly demonstrates that the present results are governed by the experimental dataset used for model calibration. Furthermore, as indicated by the γ coefficient in Blevins' model (see Appendix B), which exceeds unity in the case of a cantilever beam, the maximum VIV tip amplitudes of a cantilever beam are expected to be higher than those of a rigid elastically supported cylinder. In addition, the predicted lock-in region is well captured for Sc=2.94, whereas it is considerably underpredicted for higher Sc values. This outcome also reflects the influence of the rigid-cylinder measurements used for calibration, which show a narrower lock-in region that decreases with increasing Sc. In contrast, for the cantilever case, increasing Sc primarily reduces the maximum displacement amplitude, while the lock-in range appears to remain essentially unchanged.

The underprediction of the lock-in region highlights a limitation: calibrations based solely on rigid-cylinder data are sufficient for estimating maximum amplitudes but inadequate for realistic cantilever elastic structures, such as wind turbine towers. The lock-in zone width depends on the rate at which the defining parameters vary towards zero from their calibrated value at Umax⁡∗. To explore this sensitivity, two additional parameter ranges were tested. The baseline calculation used U∗∈[5.5,9] (in accordance with Belloli's wind tunnel tests), while two alternative ranges of U∗∈[4.5,10] and U∗∈[5,10] are also considered. Figure 9b compares the model response across the lock-in range for these different parameter zones. The model consistently links the predicted lock-in width to the span of the chosen parameter range. Specifically, a faster increase or decrease in the calibration constants shifts the predicted maximum VIV response to higher reduced velocities and produces a much steeper slope, failing to capture the gradual buildup of oscillations. In contrast, the decaying part of the oscillations (at higher U∗ values) appears largely unaffected by the choice of parameter range.

The study presents the calibration and validation of a predictive engineering model for vortex-induced vibrations (VIV) in wind turbine towers, based on the Hartlen–Currie lift-oscillator formulation. The governing equations of the lift-oscillator model are expressed in a non-dimensional form to facilitate the calibration of the model's defining parameters. In this formulation, calibration is required only for specific sets of the problem's characteristic dimensionless parameters.

Model parameters are calibrated using experimental data obtained from rigid elastically mounted, untapered cylinders, enabling accurate prediction of vibration amplitudes and the wind speeds Umax⁡∗ at which maximum VIV amplitudes occur, associated with lock-in phenomena. Integration of the calibrated model into a multibody aeroelastic solver demonstrates its capability to simulate VIV responses in elastic cantilevered cylinders, representing configurations relevant to real wind turbine towers. Comparisons with wind tunnel measurements show that the proposed model successfully predicts the maximum vibration amplitudes and the maximum response velocity of VIV. However, it tends to underestimate the width of the lock-in velocity range, particularly at higher Scruton numbers. This discrepancy does not indicate a shortcoming of the model itself, as it can be mitigated by refining parameter calibration using direct VIV wind tunnel measurements of the cantilever tower.

Furthermore, both the calibration and validation of the model rely on low-Reynolds-number wind tunnel data, significantly lower than those expected under full-scale wind turbine operating conditions, due to the lack of high-Reynolds-number experimental data. This does not constitute a limitation of the current work, as the model can be readily updated when such high-Reynolds-number data become available.

The present approach focuses on the first bending mode, while neglecting higher-order modes and the influence of ambient turbulence – factors identified as directions for future research. Overall, the calibrated Hartlen–Currie model provides a robust predictive tool that can be integrated into comprehensive aeroelastic design frameworks, enhancing the reliability and fatigue assessment of wind turbine towers subjected to vortex shedding excitation.