The three-dimensional flow field past an I-shaped dual-step cylinder has been obtained by numerical integration of the Navier–Stokes equations at Reynolds number ($Re_D$) 150. The I-shaped cylinder consisted of two large-diameter (D) cylinders with a small-diameter (d) cylinder in between. With a view to exploring the vortex dynamics and structural loads, simulations were performed for eight different lengths $l$ of the small cylinder, varied from $l/D=10$ to 0.2 for a fixed diameter ratio $D/d=2$. When the length of the small cylinder is sufficiently large, the wake behind the I-shaped cylinder is similar to the wake behind the single-step cylinder (Tian et al., J. Fluid Mech., vol. 891, 2020, A24). As the small cylinder length decreases, the enhanced interactions between the two steps make the present wake deviate from the wake of the single-step cylinder, leading to four different wake modes distinguished by different combinations of vortex cells. The physical formation mechanisms were analysed in terms of the vortex dynamics. Besides the wake flow, the streamwise vortices around the I-shaped step cylinder were also investigated. A pair of edge vortices and a junction vortex were identified for $l/D \geq ~1$. When the gap between the two steps becomes too small, $l/D \leq ~0.2$, the junction vortex disappears, and only a pair of edge vortices exists. Varying the distance between the two steps strongly affects the structural loads (drag and lift) along the I-shaped cylinder. The dependence of the loads on $l/D$ was readily explained by the different wake modes.

Flow field in the near wake of a short-finite circular cylinder at $L/D=1.0$ with an angle of attack between 0$^\circ$–15$^\circ$, where the transition from the non-reattaching flow to the reattaching flow appears, is investigated in wind tunnel tests with a supportless condition. Stereo particle image velocimetry measurements were applied to the experiments at the Reynolds number of $3.46\times 10^4$, and velocity fields in the near wake were obtained. The data was mainly analysed using spectral proper orthogonal decomposition. Characteristic large-scale wake structures of recirculation bubble pumping and large-scale vortex shedding were identified in the near wake of the cylinder regardless of the angle of attack. The phase difference of expansion and contraction of the recirculation flow appears in the recirculation bubble pumping at $\alpha \neq 0^\circ$. On the other hand, the eigenfunctions of velocity fluctuations at the vortex shedding frequency show a similar spatial pattern regardless of $\alpha$. Frequency analyses of wake position calculated from the reconstructed velocity field clarified that peak frequency is different between two in-plane directions when $\alpha \neq 0^\circ$. In addition, three vortex shedding patterns (anticlockwise/clockwise circular and flapping) are identified not only at $\alpha =0^\circ$ but also $\alpha \neq 0^\circ$. The feature of wake position in the radial direction for each pattern is observed regardless of the angle of attack. The relationship between the recirculation bubble pumping and the wake position in the radial direction is apparent in the non-reattaching flow but is weaker with $\alpha$ in the reattaching flow.

The concept of vortex lock-in for a single circular cylinder in an oscillating flow, induced through acoustic forcing, is revisited. Multiple cylinder diameters are investigated over a Reynolds number range between 500 and 7200. The lock-in behaviour is investigated quantitatively through hot-wire anemometry and planar particle image velocimetry measurements. The results corroborate previous findings describing the frequency range over which vortex lock-in occurs. It is found that the cylinder location in a standing wave (pressure node or velocity node) had a significant influence on the lock-in behaviour. A novel scaling which captures the onset of vortex lock-in is proposed which demonstrates that the Strouhal number is important in predicting the amplitude of the velocity fluctuations required to induce lock-in. Velocity fields also reveal the existence of bimodal vortex shedding during lock-in. This is confirmed using snapshot proper orthogonal decomposition which demonstrates that symmetric and alternate shedding modes are simultaneously present during lock-in and that symmetric shedding is inherent to the near wake region only. Reduced-order reconstruction of the instantaneous velocity fields confirmed that features associated with the forcing frequency control the shear layer roll-up up to $x/d=2.1$ while the influence of the asymmetric mode is simply to skew the trajectory of the vortex pair. Since vortex roll-up and the cylinder wake ends at $x/d=2.1$, the emergence of spectral content at $0.5f_e$ is attributed to a wavelength doubling measured between the vortical structures in the flow field.

This study conducts experimental investigations into wake-induced vibration (WIV) of a circular cylinder placed downstream of an oscillating cylinder. Surprisingly, it is observed that the previously identified WIV phenomenon, characterized by a sustained increase in amplitude at higher reduced velocities, does not occur when the upstream cylinder oscillates at large amplitudes. Instead, a different phenomenon, which we refer to as the ‘wake-captured vibration’, becomes dominant. The experiments reveal a negative correlation between the vortex-induced vibration amplitude response of the upstream cylinder and the WIV amplitude response of the downstream cylinder. Through a quasi-steady and linear instability analysis, the study demonstrates that the previously proposed wake-displacement mechanism may not be applicable for predicting the cylinder WIV response in the wake of an oscillating body. This is because the lift force gradients across the wake, measured through stationary cylinder experiments, decrease significantly when the upstream cylinder vibrates at higher amplitudes. Consequently, actively controlled vibration experiments are conducted to systematically map the hydrodynamic properties of the downstream cylinder vibrating in the wake of an oscillating cylinder. The findings align with observations from free-vibration experiments, and help to explain the amplitude and frequency response of WIV. Additionally, wake visualization through particle image velocimetry is conducted to provide further insights into the complex wake and vortex–body interactions.

This paper presents a numerical study on the flow around two tandem circular cylinders beneath a free surface at a Reynolds number of $180$. The free-surface effects on the wake dynamics and hydrodynamic forces are investigated through a parametric study, covering a parameter space of gap ratios from $0.20$ to $2.00$, spacing ratios from $1.50$ to $4.00$ and Froude numbers from $0.2$ to $0.8$. A jet-like flow accompanied by a shear layer of positive vorticity separating from the free surface is formed in the wake at small gap ratios, which significantly alters the wake pattern through its dynamic behaviours. At shallow submergence depths, the three-dimensional wake transitions from mode B to mode A as the distance between the cylinders increases. As submergence depth increases, the wavy deformation of the primary vortex cores disappears in the wake, and the flow transitions to a two-dimensional state. Higher Froude numbers can extend the effect of the free surface to deeper submergence depths. The critical spacing ratio tends to be larger at higher Froude numbers. Furthermore, the free-surface deformation is examined. The free-surface profile typically comprises a hydraulic jump immediately ahead of the upstream cylinder, trapped waves in the vicinity of the two tandem cylinders and well-defined travelling waves on the downstream side. The frequencies of the waves cluster around the vortex shedding frequency, indicating a close association between the generation of waves and the vortex shedding process.

Investigations are conducted on the effect of wall proximity on the flow around a cylinder under an axial magnetic field, using the electrical potential probe technology to measure the velocity of liquid metal flow. The study focused on the impact of the inlet velocity of the fluid, the magnetic field and wall proximity on the characteristics of velocity fields, particularly on the vortex-shedding mode. Based on different magnitudes of the magnetic field and the distance from the cylinder to the duct wall, three types of vortex-shedding modes are identified, (I) shear layer oscillation state, (II) quasi-two-dimensional vortex-shedding states and (III) transition of the magnetohydrodynamic to hydrodynamic Kármán street. The transitions between these modes are analysed in detail. The experimental results show that the weak wall-proximity effect leads to the formation of the Kármán vortex street, while a reverse Kármán vortex street and secondary vortices emerge under a strong wall-proximity effect. It is noticed that the Kelvin–Helmholtz instability drives vortex shedding under regime I, leading to an increase in the Strouhal number (St) with stronger magnetic fields. Additionally, under a strong axial magnetic field, the wall-proximity effect (‘Shercliff layer effect’) promotes the instability of shear layers on both sides of the cylinder. These unique coupling effects are validated by variations in modal coefficients and energy proportions under different vortex-shedding regimes using the proper orthogonal decomposition method.

Flows over a disc are studied in a wind tunnel over incidence angles between $0^\circ \text { and }36^\circ$, a Reynolds number of $2.7 \times 10^4$ and rotational speed ratios of $0\unicode{x2013}10$. Smoke-wire visualization, particle image velocimetry and hot-film anemometry are employed. Two vortex shedding modes originating from the upstream surface of the disc are observed. The first is dominant at incidence angles up to ${\sim }21^\circ$. Beyond $21^\circ$, the second mode dominates. It appears as a soliton on the vortices and has a shedding frequency nearly twice that of the first at the highest incidence angle. The Strouhal number monotonically increases with incidence angle from approximately 0.2 to 0.4. Spectral analysis of the hot-film measurements confirms the findings from flow visualization experiments. Flows over the spinning disc generally mimic the stationary disc flows; however, centrifugal forces lead to cross-stream instability features that may be attributed to spiral wave instabilities intrinsic to the boundary layers in rotating flows. Velocity measurements are used to construct streamline patterns to compare with the smoke streaklines. The unsteadiness of the flows results in large variances. Mean strain rates are extracted from velocity data, where the fixed disc case at normal incidence compares well with theoretical predictions. The unsteady boundary layer thickness over the fixed disc, however, is approximately twice that predicted by theory for steady flow, stemming from the dominance of large unsteady vortices. Limited comparisons are made of the Strouhal numbers from experiments and numerical calculations in the literature.

The penetration of a spherical vortex into turbulence is studied theoretically and experimentally. The characteristics of the vortex are first analysed from an integral perspective that reconciles the far-field dipolar flow with the near-field source flow. The influence of entrainment on the vortex drag force is elucidated, extending the Maxworthy (J. Fluid Mech., vol. 81, 1977, pp. 465–495) model to account for turbulent entrainment into the vortex movement and vortex penetration into an evolving turbulent field. The physics are explored numerically using a spherical vortex (initial radius $R_0$, speed $U_{v0}$), characterised by a Reynolds number $Re_0(=2R_0U_{v0}/\nu$, where $\nu$ is the kinematic viscosity) of 2000, moving into decaying homogeneous turbulence (root-mean-square $u_0$, integral scale $L$) with turbulent intensity $I_t=u_0/U_{v0}$. When the turbulence is absent ($I_t=0$), a wake volume flux leads to a reduction of vortex impulse that causes the vortex to slow down. In the presence of turbulence ($I_t> 0$), the loss of vortical material is enhanced and the vortex speed decreases until it is comparable to the local turbulent intensity and quickly fragments, penetrating a distance that scales as $I_t^{-1}$. In the experimental study, a vortex ($Re_0\sim 1490\unicode{x2013}5660$) propagating into a statistically steady, spatially varying turbulent field ($I_{ve}=0.02$ to 0.98). The penetration distance is observed to scale with the inverse of the turbulent intensity. Incorporating the spatially and temporally varying turbulent fields into the integral model gives a good agreement with the predicted trend of the vortex penetration distance with turbulent intensity and insight into its dependence on the structure of the turbulence.

Wave-assisted propulsion (WAP) systems directly convert wave energy into thrust using elastically mounted hydrofoils. The wave conditions as well as the design of the hydrofoil drive the fluid–structure interaction of the hydrofoil and, consequently, its performance. We employ simulations using a sharp-interface immersed boundary method to examine the effect of three key parameters on the flow physics, the fluid–structure interaction as well as thrust performance of these systems – the stiffness of the torsional spring, the location of the pitch axis and the Strouhal number. We demonstrate the utility of ‘maps’ of energy exchange between the flow and the hydrofoil system, as a way to understand and predict these characteristics. The force-partitioning method (FPM) is used to decompose the pressure forces into interpretable components and to quantify the mechanisms associated with thrust generation. Based on the results from FPM, a phenomenological model for the thrust generated by the WAP foil is presented. The parameters associated with this model are estimated based on data from over 450 distinct simulations. The predictions of the model are compared with the simulations and the use of this model for guiding WAP design is discussed.

We present an experimental investigation aimed at understanding the effects of surface roughness on the time-mean drag coefficient ($\bar {C}_{D}$) of finite-span cylinders ($\text {span/diameter} = \text {aspect ratio}$, $0.51 \le AR \le 6.02$) freely rolling without slip on an inclined plane. While lubrication theory predicts an infinite drag force for ideally smooth cylinders in contact with a smooth surface, experiments yield finite drag coefficients. We propose that surface roughness introduces an effective gap ($G_{eff}$) resulting in a finite drag force while allowing physical contact between the cylinder and the plane. This study combines measurements of surface roughness for both the cylinder and the plane panel to determine a total relative roughness ($\xi$) that can effectively describe $G_{eff}$ at the point of contact. It is observed that the measured $\bar {C}_{D}$ increases as $\xi$ decreases, aligning with predictions of lubrication theory. Furthermore, the measured $\bar {C}_{D}$ approximately matches combined analytical and numerical predictions for a smooth cylinder and plane when the imposed gap is approximately equal to the mean peak roughness ($R_p$) for rough cylinders, and one standard deviation peak roughness ($R_{p, 1\sigma }$) for relatively smooth cylinders. As the time-mean Reynolds number ($\overline {Re}$) increases, the influence of surface roughness on $\bar {C}_{D}$ decreases, indicating that wake drag becomes dominant at higher $\overline {Re}$. The cylinder aspect ratio ($AR$) is found to have only a minor effect on $\bar {C}_{D}$. Flow visualisations are also conducted to identify critical flow transitions and these are compared with visualisations previously obtained numerically. Variations in $\xi$ have little effect on the cylinder wake. Instead, $AR$ was observed to have a more pronounced effect on the flow structures observed. The Strouhal number ($St$) associated with the cylinder wake shedding was observed to increase with $\overline {Re}$, while demonstrating little dependence on $AR$.

Turbulent flow around curved tandem cylinders has been studied for the first time, by means of direct numerical simulation. The convex configuration was used, with a nominal gap ratio of $L/D = 3$ and a Reynolds number of 3900. Along the span, the flow regimes vary from alternating overshoot/reattachment to co-shedding. Three distinct Strouhal numbers coexist in the flow that are tied directly to different tandem cylinder flow regimes. This result differs substantially from convex curved tandem cylinders at a transitional Reynolds number, where only a single dominant frequency is found. All regimes exhibit some degree of instability, so that the flow can be considered multistable. A mode switch from alternating overshoot/reattachment to symmetric reattachment is found. Complex interactions are observed between the primary instability, the shear layer instability and the flow mode alterations. As opposed to previous investigations with single and tandem straight cylinders in the subcritical flow regime, our results indicate that there may be direct feedback from the primary instability to the shear layer instability. The downdraft region in the gap exhibits slow meandering, and may travel upstream and amplify the shear layer instability, causing early transition in the gap shear layer. This downdraft is governed by the slow modulations of the vortex formation region in the lower gap, meaning that the vortex dynamics of this region may indirectly influence the shear layer instability higher up in the gap.

This survey paper is concerned with vortex shedding from bodies in unsteady flow due either to time dependent motion of the body in a still fluid or unsteady motion of the fluid about a fixed body. The fluid is treated as incompressible, and the main emphasis is on starting flows and oscillatory flows. Much of the discussion describes 2D flow around sections of long or slender bodies. The first part of the paper covers the inviscid flow scaling of the forces induced by vortex shedding in time dependent flows which drive the shedding. This is followed by application of Wu’s impulse integral of the moment of vorticity to predict the forces induced by vortex shedding from a body in both inviscid and viscous flows. Vortex shedding phenomena involving small amplitude, high-frequency oscillatory flow such as vortex-induced vibration (VIV) and fluid-structure interaction (FSI) are not included in this discussion as in these cases the unsteady flow controls rather than drives the vortex shedding and they are well covered elsewhere.

The second part of the paper describes a vortex force mapping (VFM) method derived by considering the Lamb–Gromyko formulation for the pressure contribution which allows the integral of the vorticity field to be restricted to regions which are not far from the body. It is applied to both inviscid and viscous flows. The section finishes with discussion of application of the VFM to the calculation of forces induced on bodies from flow field measurements, such as particle image velocimetry (PIV).

This study explores the implementation of an online control strategy based on dynamic mode decomposition in the context of flow control. The investigation is conducted mainly with a fixed Reynolds number of $Re = 100$, focusing on the flow past a circular cylinder constrained between two walls to mitigate vortex shedding. The control approach involves the activation of two synthetic jets on the cylinder through blowing and suction. Velocity fluctuations in the wake, specifically in the $x$-direction, are harnessed to ascertain the mass flow rate of the jets using the linear quadratic regulator and online dynamic mode decomposition. The study systematically assesses the control performance across various configurations, including different values of the input penalty factor ${\boldsymbol {R}}$, varying numbers of probes and distinct probe arrangement methods. The synthetic jets prove effective in stabilising the separation bubble, and their interaction with the unsteady wake leads to a notable reduction in drag force, its fluctuations and the amplitude of the lift force. Specifically, the mean and standard deviation of the drag coefficient witness reductions of $7.44\,\%$ and $96.67\,\%$, respectively, and the standard deviation of the lift coefficient experiences an impressive reduction of $85.18\,\%$. The robustness of the proposed control method has also been tested on two more complicated cases, involving unsteady incoming flows with multiple frequency components. Comparatively, the methodology employed in this paper yields results akin to those obtained through deep reinforcement learning in terms of control effectiveness. However, a noteworthy advantage lies in the substantial reduction of computational resource consumption, highlighting the efficiency of the proposed approach.

Free transverse oscillations of an elastically mounted circular cylinder with low mass-damping are studied with a focus on the effects of the cylinder end condition on structural oscillations and vortex shedding. While the top end of the cylinder pierces the free surface of a water channel, the lower end is changed to have three end conditions: an attached endplate, an endplate unattached from the cylinder at varying gaps and no endplate. All three response branches are examined, with the reduced velocity sweeping $2 \le {U^\ast } \le 15$ (corresponding to a Reynolds number range of $2000 \le Re \le 14\;700$). Although the cylinder oscillations are unaffected by the end condition in the initial and upper branches, they show significant dependency on the end condition in the lower branch. When the endplate is attached or unattached with a small gap, the upper-to-lower branch transition occurs with a sudden decrease in oscillation amplitude, after which the lower branch maintains a near-constant amplitude. For larger gaps or no endplate, with increasing reduced velocity, the oscillation amplitude decreases gradually from its peak without any discernible sign of transition between the upper and lower branches. The three-dimensional effects of the gap are the basis for these differences in oscillation. Atop the strong tip vortex, a low-magnitude streamwise velocity region develops downstream of the cylinder, which delays upper-to-lower branch transition over the cylinder span that sees this low-velocity region. With increasing gap, this low-velocity region and the delay in transition spread over larger spanwise extents, forcing the overall body to oscillate at larger amplitudes.

Dynamic stall at low Reynolds numbers, $Re \sim O(10^4)$, exhibits complex flow physics with co-existing laminar, transitional and turbulent flow regions. Current state-of-the-art stall onset criteria use parameters that rely on flow properties integrated around the leading edge. These include the leading edge suction parameter or $LESP$ (Ramesh et al., J. Fluid Mech., vol. 751, 2014, pp. 500–538) and boundary enstrophy flux or $BEF$ (Sudharsan et al., J. Fluid Mech., vol. 935, 2022, A10), which have been found to be effective for predicting stall onset at moderate to high $Re$. However, low-$Re$ flows feature strong vortex-shedding events occurring across the entire airfoil surface, including regions away from the leading edge, altering the flow field and influencing the onset of stall. In the present work, the ability of these stall criteria to effectively capture and localize these vortex shedding events in space and time is investigated. High-resolution large-eddy simulations for an SD7003 airfoil undergoing a constant-rate, pitch-up motion at two $Re$ (10 000 and 60 000) and two pitch rates reveal a rich variety of unsteady flow phenomena, including instabilities, transition, vortex formation, merging and shedding, which are described in detail. While stall onset is reflected in both $LESP$ and $BEF$, local vortex-shedding events are identified only by the $BEF$. Therefore, $BEF$ can be used to identify both dynamic stall onset and local vortex-shedding events in space and time.

Acoustic resonance is an important factor that contributes to aeroengine compressor failure. In this study, a plane cascade of compressor blades was designed to reproduce acoustic resonance via a low-speed wind tunnel test. A high-frequency hot-wire, microphone and strain gauge were used to synchronously measure the fluid, acoustic and structural parameters. We analysed the variation in the amplitude and frequency of the multi-field parameters with increasing mean flow velocity and explored the multi-field interaction mechanism that induces the acoustic resonance of the plane cascade. The plane cascade effectively reproduced the acoustic resonance phenomenon. The first-order acoustic-mode frequency of the plane cascade flow duct, second-order torsional vibration mode frequency of the blade and shedding mode frequency of the tip clearance leakage vortex were equal under acoustic resonance. The fluid, acoustic and structural fields showed a strong interaction effect, achieving the maximum blade vibration amplitude and causing fatigue cracks of torsional vibration at the blade root. The frequency lock-in region of the compressor plane cascade was divided into an ‘acoustic–structure’ interaction region, a ‘fluid–acoustic–structure’ interaction region and a first-order acoustic-mode dominant region with increasing mean flow velocity, which demonstrates an interesting phenomenon in which the fluid–acoustic–structure modes compete: acoustic mode > blade vibration mode > vortex shedding mode. The results demonstrate a unique approach to the study of acoustic resonance that provides insight into the acoustic resonance mechanism in a cascade of compressor blades.

A two-dimensional body that moves suddenly in a viscous fluid can instantly generate vortices at its sharp edges. Recent work using inviscid flow theory, based on the Birkhoff–Rott equation and the Kutta condition, predicts that the ‘starting vortices’ generated by the sharp and straight edges of a body – i.e. the vortices formed immediately after motion begins – can be one of three distinct self-similar types. We explore the existence of these starting vortices for a flat plate and two symmetric Joukowski aerofoils immersed in a viscous fluid, using high-fidelity direct numerical simulations (DNS) of the Navier–Stokes equations. A lattice Green's function method is employed and simulations are performed for chord Reynolds numbers ranging from 5040 to 45 255. Vortices generated at the leading and trailing edges of the flat plate show agreement with the derived inviscid theory, for which a detailed assessment is reported. Agreement is also observed for the two symmetric Joukowski aerofoils, demonstrating the utility of the inviscid theory for arbitrary bodies. While this inviscid theory predicts an abrupt transition between the starting-vortex types, DNS shows a smooth transition. This behaviour occurs for all Reynolds numbers and is related to finite-time effects – there is a maximal time for which the (self-similar) starting vortices exist. We confirm the inviscid prediction that the leading-edge starting vortex of a flat plate can be suppressed dynamically. This has implications for the performance of low-speed aircraft such as model aeroplanes, micro air vehicles and unmanned air vehicles.

Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations are conducted for flow past rectangular cylinders with various cross-sectional aspect ratios. The primary focuses are the interactions between the 2-D wake transitions in the spanwise vortex street (with distance downstream) and the 3-D wake transitions in the streamwise vortices, and the influence of both 2-D and 3-D wake transitions on the hydrodynamic forces on the cylinder. The 2-D wake transitions generally move upstream with increasing Reynolds number and decreasing aspect ratio. The corresponding reasons are explained. The 2-D wake transitions emerging close to the cylinder may directly alter the hydrodynamic forces on the cylinder, e.g. the Strouhal number, time-averaged drag coefficient and root-mean-square lift coefficient. By using specifically designed numerical cases to decompose the effects of the two 2-D transitions, it is found that the first 2-D transition from the primary to the two-layered vortex street results in reductions in the hydrodynamic forces, while the second 2-D transition to the secondary vortex street results in increases in the forces. The reduction/increase in the hydrodynamic forces becomes more significant when the transition location moves closer to the cylinder. The physical mechanisms for the influence on the hydrodynamic forces are elucidated. The upstream movement of the 2-D wake transitions also induces complex interactions between the 2-D and 3-D wake transitions (which also depends on the type of the 3-D mode). Correspondingly, the 3-D hydrodynamic forces may be governed by both 2-D and 3-D wake transitions (and their mutual influence).

The actuator line method (ALM) is a commonly used technique to simulate slender lifting and dragging bodies such as wings or blades. However, the accuracy of the method is significantly reduced near the tip. To quantify the loss of accuracy, translating wings with various aspect and taper ratios are simulated using several methods: wall-resolved Reynolds-averaged Navier–Stokes (RANS) simulations, an advanced ALM with two-dimensional (2-D) mollification of the force, a lifting line method, a mollified lifting line method and a vortex lattice method. Significant differences in the lift and drag distributions are found on the part of the wing where the distance to the tip is smaller than approximately 3 chords and are identified to arise from both the forces mollification and the uneven induced velocity along the chord. Correction functions acting on the lift coefficient and effective angle of attack near the wing tip are then derived for rectangular wings of various aspect ratios. They are then also applied to wings of various taper ratios using the ‘effective dimensionless distance to the tip’ as the main parameter. The application of the correction not only leads to a much improved lift distribution, but also to a more consistent drag distribution. The correction functions are also obtained for various mollification sizes, as well as for ALM with three-dimensional (3-D) mollification. These changes mostly impact the correction for the effective angle of attack. Finally, the correction is applied to simulations of the NREL Phase VI wind turbine, leading to an enhanced agreement with the experimental data.

Vortex-induced vibrations and galloping of an elastically mounted square cylinder are investigated for cylinder mass ratio m* = 2–50, damping ratio ζ = 0–1.0, mass-damping ratio m*ζ = 0–50 and flow reduced velocity Ur = 1–80. We home in on the effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on the critical reduced velocity Urc marking the onset of galloping, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added mass ratio and $m_{ae}^\ast $ is the effective added mass ratio. Vibration responses, forces, vibration frequencies and added mass ratios are studied and discussed. The different branches of vortex-induced vibrations have different dependencies of $m_{ae}^\ast $ on Ur. The $m_{ae}^\ast $ in the initial branch is positive and drops rapidly with Ur, but that in the lower branch is negative and declines gently. In the galloping regime, $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping, declining slightly with increasing Ur. Our results and prediction equations show that when ζ = 0, Urc is independent of m* for m* ≥ 5, albeit slightly higher for m* = 3. The latter is ascribed to mode competition. When ζ > 0, Urc linearly increases with increasing ζ. Detailed analysis substantiates that m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ does not serve as the unique criterion to predict the galloping occurrence. Here, we propose a new combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta$ in the relationship between galloping onsets and structural properties, which successfully scales all data of Urc at different m* and ζ values.