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In recent years, the data-driven turbulence model has attracted widespread concern in fluid mechanics. The existing approaches modify or supplement the original turbulence model by machine learning based on the experimental/numerical data, in order to augment the capability of the present turbulence models. Different from the previous researches, this paper directly reconstructs a mapping function between the turbulent eddy viscosity and the mean flow variables by neural networks and completely replaces the original partial differential equation model. On the other hand, compared with the machine learning models for the low Reynolds (Re) number flows based on direct numerical simulation data, high Reynolds number flows around airfoils present the apparent scaling effects and strong anisotropy, which induce large challenges in accuracy and generalization capability for the machine learning algorithm. We mainly concentrate on the high Reynolds number turbulent flows around the airfoils and take the results calculated by the computational fluid dynamics solver with the Spallart-Allmaras (SA) model as training data to construct a high-dimensional data-driven network model based on machine learning. The radial basis function neural network and the auxiliary optimization methods are adopted, and the individual models are built separately for the flow fields of the near-wall region, wake region, and far-field region. The training data in this paper is extracted from only three subsonic flow fields of NACA0012 airfoil. The data-driven turbulence model can be applied to various airfoils and flow states, and the predicted eddy viscosity, lift/drag coefficients, and skin friction distributions are all in good agreement with the results of the original SA model. This research demonstrates the promising prospect of machine learning methods in future studies about turbulence modeling.

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 $U_{c}^{\ast }<U^{\ast }<U_{g}^{\ast }$, the FM quickly reaches the nonlinear saturation state and then inhibits the development of the SM, thus postponing the occurrence of galloping. When $U^{\ast }>U_{g}^{\ast }$, 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.

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It is well known that the occurrence of vortex-induced vibration (VIV) at a subcritical Reynolds number, which is lower than 47, is induced by fluid-structure interaction. However, for the free flow, this phenomenon disappears at a Reynolds number about 20. The current study provides an explanation to the disappearance of the VIV by capturing the evolution of the purely fluid modes at Reynolds numbers between 12 and 55. To ensure accurate mode extraction, the dynamic mode decomposition technique is utilized. Results show that the stable von Kármán vortex shedding mode exists in a subcritical flow regime but becomes less distinct as the Reynolds number decreases. When the Reynolds number is lower than 18, this mode nearly vanishes, characterizing the lower boundary of fluid-structure instability.

Transonic buffet is a phenomenon of aerodynamic instability with shock wave motions which occurs at certain combinations of Mach number and mean angle of attack, and which limits the aircraft flight envelope. The objective of this study is to develop a modelling method for unstable flow with oscillating shock waves and moving boundaries, and to perform model-based feedback control of the two-dimensional buffet flow by means of trailing-edge flap oscillations. System identification based on the ARX algorithm is first used to derive a linear model of the input–output dynamics between the flap rotation (the control input) and the lift and pitching moment coefficients (system outputs). The model features a pair of unstable complex-conjugate poles at the characteristic buffet frequency. An appropriate reduced-order model (ROM) with a lower dimension is further obtained by a balanced truncation method that keeps the pair of unstable poles in the unstable subspace but truncates the dynamics in the stable subspace. Based on this balanced ROM, two kinds of feedback control are designed by pole assignment and linear quadratic methods respectively. These independent designs, however, result in similar suboptimal static output feedback control laws. When introduced in numerical simulations, they are both able to completely suppress the buffet instability. Furthermore, the resulting controllers are even able to stabilize buffet flows with nonlinear disturbances and in off-design flow conditions, thus implying their robustness. The analysis of the feedback control laws indicates that parameters (frequency and phase) corresponding to the ‘anti-resonance’ of the linear input–output model are vital for optimal control. The best performance is obtained when the control operates close to the ‘anti-resonance’, which is supported by the optimal frequency and the phase of the open-loop control as well as by the optimal phase of the closed-loop control.

This study proposes an improvement in the performance of reduced-order models (ROMs) based on dynamic mode decomposition to model the flow dynamics of the attractor from a transient solution. By combining higher order dynamic mode decomposition (HODMD) with an efficient mode selection criterion, the HODMD with criterion (HODMDc) ROM is able to identify dominant flow patterns with high accuracy. This helps us to develop a more parsimonious ROM structure, allowing better predictions of the attractor dynamics. The method is tested in the solution of a NACA0012 airfoil buffeting in a transonic flow, and its good performance in both the reconstruction of the original solution and the prediction of the permanent dynamics is shown. In addition, the robustness of the method has been successfully tested using different types of parameters, indicating that the proposed ROM approach is a tool promising for using in both numerical simulations and experimental data.

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