RESEARCH PROGRESS OF RESOLVENT ANALYSIS IN FLUID MECHANICS

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.