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 引用本文: 袁昊, 寇家庆, 张伟伟. 流体力学预解分析方法研究进展. 力学学报, 待出版.
Yuan Hao, Kou Jiaqing, Zhang Weiwei. Research progress of resolvent analysis in fluid mechanics. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Yuan Hao, Kou Jiaqing, Zhang Weiwei. Research progress of resolvent analysis in fluid mechanics. Chinese Journal of Theoretical and Applied Mechanics, in press.

## RESEARCH PROGRESS OF RESOLVENT ANALYSIS IN FLUID MECHANICS

• 摘要: 线性稳定性分析长期以来是揭示复杂流动机理的重要手段, 通过求解流动线性化算子的特征值问题获得流动的直接模态及其频率、增长率等信息. 然而其仅能描述系统微幅扰动随时间的指数发展, 无法捕捉系统受迫和响应特征. 预解分析从线性输入输出动力学系统出发, 通过提取流动系统受谐波激励的强迫/响应模态及其增益, 捕捉系统关于不同频率扰动的受迫模式和能量放大效应. 该方法建立了流动对外激励的空间敏感性和对应响应的空间模式分析的统一框架, 对复杂流体动力学问题的分析、建模和控制有很强的应用潜力. 文章针对预解分析方法展开了全面综述: 首先介绍了预解分析理论框架、实现难点与改进方法, 讨论了预解增益和模态的物理意义; 同时, 从基础假设、数学理论、算法流程以及物理含义等方面对比了线性稳定性分析和预解分析算法, 并给出了两者在一定条件下的联系; 进一步展示了预解分析在揭示流动机理、建立低维模型以及指导流动控制等方面的研究成果; 最后通过Ginzburg–Landau方程和方柱绕流问题, 展示了预解分析在动力学系统特征提取上的应用潜力. 在此基础上, 针对现有研究的不足和困难, 讨论了预解分析方法在改进算法、非线性系统分析、流动控制等方面的未来研究方向.

Abstract: Linear stability analysis has long been an important method to reveal the complex flow mechanism. By solving the eigenvalue problem of the linearized operator of the Navier-Stokes equations, the direct mode of the flow and its frequency, growth rate and other information can be obtained. However, it can only describe the exponential time dependence under small perturbations, and can't capture the forcing-response characteristics of the system. Based on the linear input-output dynamic system, resolvent analysis extracts the forcing / response modes and their gains of the flow system excited under harmonics, and captures the forcing types and energy amplification to system disturbances across multiple frequencies. This approach establishes a unified framework for the spatial sensitivity of flow to external excitation and the spatial modal analysis of corresponding response, and has potential applications to the analysis, modeling and control of complex flow problems. This review gives a general introduction to resolvent analysis. Firstly, the theoretical framework of resolvent analysis, its existing challenges and improved algorithms are introduced, and the physical significance of resolvent modes and gains are discussed. At the same time, the linear stability analysis and resolvent analysis are compared from the aspects of basic assumptions, mathematical theory, algorithmic process and physical meaning. The relationship between these two algorithms under certain conditions is also given. Furthermore, research progresses in revealing flow mechanism, constructing reduced-order models and designing flow control laws based on resolvent analysis are demonstrated. The application potential of the resolvent analysis in the feature extraction of the dynamic system will be shown by two cases: the Ginzburg–Landau equation and the flow past a square cylinder. Based on these, in view of the limitations and difficulties of the existing research, the future research direction of resolvent analysis is discussed in the aspects of improved algorithms, nonlinear system analysis and flow control.

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