A practical review on linear and nonlinear global approaches to flow instabilities

This paper aims at reviewing linear and nonlinear approaches to study the stability of fluid flows. We provide a concise but self-contained exposition of the main concepts and specific numerical methods designed for global stability studies, including the classical linear stability analysis, the adjoint-based sensitivity, and the most recent nonlinear developments. Regarding numerical implementation, a number of ideas making resolution particularly efficient are discussed, including mesh adaptation, simple shift-invert strategy instead of the classical Arnoldi algorithm, and a simplification of the recent nonlinear self-consistent (SC) approach proposed by Mantic-Lugo et al. (2014, "Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake," Phys. Rev. Lett., 113(8), p. 084501). An open-source software implementing all the concepts discussed in this paper is provided. The software is demonstrated for the reference case of the two-dimensional (2D) flow around a circular cylinder, in both incompressible and compressible cases, but is easily customizable to a variety of other flow configurations or flow equations.