Analysis of Fluid Systems: Stability, Receptivity, SensitivityLecture notes from the FLOW NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

Abstract

This article presents techniques for the analysis of fluid systems. It adopts an optimizationbased point of view, formulating common concepts such as stability and receptivity in terms of a cost functional to be optimized subject to constraints given by the governing equations. This approach differs significantly from eigenvaluebased methods that cover the timeasymptotic limit for stability problems or the resonant limit for receptivity problems. Formal substitution of the solution operator for linear timeinvariant systems results in the matrix exponential norm and the resolvent norm as measures to assess the optimal response to initial conditions or external harmonic forcing. The optimizationbased approach can be extended by introducing adjoint variables that enforce governing equations and constraints. This step allows the analysis of far more general fluid systems, such as timevarying and nonlinear flows, and the investigation of wavemaker regions, structural sensitivities, and passive control strategies.