Introduction to hydrodynamic instability - II

"In this lecture, we introduce the method of normal modes as a foundational tool for analyzing hydrodynamic stability. We begin by examining normal modes in systems with discrete degrees of freedom and then extend the concept to continuous systems, supported by illustrative experimental observations. Subsequently, we demonstrate how disturbances or perturbations in various flow configurations can be represented in normal mode form to facilitate linear stability analysis. As a specific example, we briefly outline the formulation of a linear stability problem for plane Poiseuille flow, leading to the derivation of the Orr–Sommerfeld equation. We then present the Kelvin–Helmholtz instability and show how it can be analyzed using the Rayleigh equation. This is followed by a discussion on boundary layer instabilities, particularly those associated with Tollmien–Schlichting (T–S) waves. We highlight the nature of disturbances and flow conditions under which such instabilities arise. The lecture concludes with recommendations for further reading and study in the field of hydrodynamic stability."