In this study, a CFD-based linear dynamics model combined with the direct Computational Fluid Dynamics/Computational Structural Dynamics (CFD/CSD) simulation method is utilized to study the physical mechanisms underlying frequency lock-in in vortex-induced vibrations (VIVs). An identification method is employed to construct the reduced-order models (ROMs) of unsteady aerodynamics for the incompressible flow past a vibrating cylinder at low Reynolds numbers ( $Re$ ). Reduced-order-model-based fluid–structure interaction models for VIV are also constructed by coupling ROMs and structural motion equations. The effects of the natural frequency of the cylinder, mass ratio and structural damping coefficient on the dynamics of the coupled system at $Re=60$ are investigated. The results show that the frequency lock-in phenomenon at low Reynolds numbers can be divided into two patterns according to different induced mechanisms. The two patterns are ‘resonance-induced lock-in’ and ‘flutter-induced lock-in’. When the natural frequency of the cylinder is in the vicinity of the eigenfrequency of the uncoupled wake mode (WM), only the WM is unstable. The dynamics of the coupled system is dominated by resonance. Meanwhile, for relatively high natural frequencies (i.e. greater than the eigenfrequency of the uncoupled WM), the structure mode becomes unstable, and the coupling between the two unstable modes eventually leads to flutter. Flutter is the root cause of frequency lock-in and the higher vibration amplitude of the cylinder than that of the resonance region. This result provides evidence for the finding of De Langre (J. Fluids Struct., vol. 22, 2006, pp. 783–791) that frequency lock-in is caused by coupled-mode flutter. The linear model exactly predicts the onset reduced velocity of frequency lock-in compared with that of direct numerical simulations. In addition, the transition frequency predicted by the linear model is in close coincidence with the amplitude of the lift coefficient of a fixed cylinder for high mass ratios. Therefore, it confirms that linear models can capture a significant part of the inherent physics of the frequency lock-in phenomenon.

Nonlinearities can be present in an aeroelastic system because of some aerodynamic factors that occur in transonic flight regimes or at a large angle of attack. The sources are shock wave motions and separated flows. Complex aeroelastic problems due to the aerodynamic nonlinearity can be studied using high-fidelity computational fluid dynamics (CFD) codes. However, the computational cost may be very high. Hence, this kind of problem is conveniently solved using the reduced-order model (ROM) for unsteady aerodynamic analysis. Many studies have been done using reduced-order modeling for aeroelastic analysis. However, most of the reduced-order aerodynamic models are dynamic linear models and have loads proportional to the structural motions. In the current paper, a nonlinear unsteady reduced-order aerodynamic model is constructed using the radial basis function neural network model. This kind of ROM is used to analyze the limit cycle oscillation (LCO) for two linear structural models with large shock motions in transonic flow. Unlike the input signals in the traditional design method, the signals of the self-excited vibration of the aeroelastic system are designed as the input signals in the current paper. Coupling the structural equations of motion and nonlinear aerodynamic ROM, the system responses are determined by time marching the governing equations using a hybrid linear multistep algorithm. Then, the LCO change with velocities (dynamic pressure) was analyzed. The two transonic aeroelastic examples show that both the structural responses and the LCO characteristics simulated using the nonlinear ROM agree well with those obtained using the direct CFD method. Moreover, the computational efficiency of the nonlinear ROM-based method is improved by one to two orders of magnitude compared with that of the direct CFD method.

Frequency lock-in can occur on a spring suspended airfoil in transonic buffeting flow, in which the coupling frequency does not follow the buffet frequency but locks onto the natural frequency of the elastic airfoil. Most researchers have attributed this abnormal phenomenon to resonance. However, this interpretation failed to reveal the root cause. In this paper, the physical mechanism of frequency lock-in is studied by a linear dynamic model, combined with the coupled computational fluid dynamics/computational structural dynamics (CFD/CSD) simulation. We build a reduced-order model of the flow using the identification method and unsteady Reynolds-averaged Navier–Stokes computations in a post-buffet state. A linear aeroelastic model is then obtained by coupling this model with a degree-of-freedom equation for the pitching motion. Results from the complex eigenvalue analysis indicate that the coupling between the structural mode and the fluid mode leads to the instability of the structural mode. The instability range coincides with the lock-in region obtained by the coupled CFD/CSD simulation. Therefore, the physical mechanism underlying frequency lock-in is caused by the linear coupled-mode flutter – the coupling between one structural mode and one fluid mode. This is different from the classical single-degree-of-freedom flutter (e.g. transonic buzz), which occurs in stable flows; the present flutter is in the unstable buffet flow. The response of the airfoil system undergoes a conversion from forced vibration to self-sustained flutter. The coupling frequency certainly should lock onto the natural frequency of the elastic airfoil.

Galloping is a type of fluid-elastic instability phenomenon characterized by large-amplitude low-frequency oscillations of the structure. The aim of the present study is to reveal the underlying mechanisms of galloping of a square cylinder at low Reynolds numbers ( $Re$ ) via linear stability analysis (LSA) and direct numerical simulations. The LSA model is constructed by coupling a reduced-order fluid model with the structure motion equation. The relevant unstable modes are first yielded by LSA, and then the development and evolution of these modes are investigated using direct numerical simulations. It is found that, for certain combinations of $Re$ and mass ratio ( $m^{\ast }$ ), the structure mode (SM) becomes unstable beyond a critical reduced velocity $U_{c}^{\ast }$ due to the fluid–structure coupling effect. The galloping oscillation frequency matches exactly the eigenfrequency of the SM, suggesting that the instability of the SM is the primary cause of galloping phenomenon. Nevertheless, the $U_{c}^{\ast }$ predicted by LSA is significantly lower than the galloping onset $U_{g}^{\ast }$ obtained from numerical simulations. Further analysis indicates that the discrepancy is caused by the nonlinear competition between the leading fluid mode (FM) and the SM. In the pre-galloping region $U_{c}^{\ast }<U^{\ast }U_{g}^{\ast }$ , mode competition is weakened because of the large difference in mode frequencies, and thereby no mode lock-in can happen. Consequently, galloping occurs, with the responses determined by the joint action of SM and FM. The unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable FM results in the high-frequency vortex shedding in the wake. The dynamic mode decomposition (DMD) technique is successfully applied to extract the coherent flow structures corresponding to SM and FM, which we refer to as the galloping mode and the von Kármán mode, respectively. In addition, we show that, due to the mode competition mechanism, the galloping-type oscillation completely disappears below a critical mass ratio. From these results, we conclude that transverse galloping of a square cylinder at low $Re$ is essentially a kind of single-degree-of-freedom (SDOF) flutter, superimposed by a forced vibration induced by the natural vortex shedding. Mode competition between SM and FM in the nonlinear stage can put off the onset of galloping, and can completely suppress the galloping phenomenon at relatively low $Re$ and low $m^{\ast }$ conditions.

The traditional method for obtaining aerodynamic parameters of airfoils by solving Navier–Stokes equations is a time-consuming computing task. In this article, a novel data-driven deep attention network (DAN) is proposed for reconstruction of incompressible steady flow fields around airfoils. To extract the geometric representation of the input airfoils, the grayscale image of the airfoil is divided into a set of patches, and these are input into the transformer encoder by embedding. The geometric parameters extracted from the transformer encoder, together with the Reynolds number, angle of attack, flow field coordinates, and distance field, are input into a multilayer perceptron to predict the flow field of the airfoil. Through analysis of a large number of qualitative and quantitative experimental results, it is concluded that the proposed DAN can improve the interpretability of the model while obtaining good prediction accuracy and generalization capability for different airfoils and flow-field states.