This lecture was given by Prof. Peter J. Schmid, Imperial College London, UK in the framework of the von Karman Lecture Series on Machine Learning for Fluid Mechanics organized by the von Karman Institute and the Université libre de Bruxelles in February 2020.
Hydrodynamic stability theory has played a central role in the development of theoretical fluid mechanics. It has developed a great many mathematical and computational tools to probe the behavior of perturbation about a steady or quasi-steady flow state and has formed the foundation for
subsequent research fields, such as flow control. Starting in the beginning to determine the large-time stability of generic configurations, it has matured into a sophisticated set of tools applicable to largescale
systems and multi-scale simulations. During this process, many fundamental assumptions and limitations of early stability theory have been (and will continue to be) challenged and abandoned.
They will be replaced by a general and flexible numerical platform based on complex optimization, fast solvers and parallel algorithms. In this manner and form, modern hydrodynamic stability theory will assert its central position in the quantitative analysis of complex fluid systems.