This course provides an introduction to the fundamental principles and analytical techniques of instability theory as applied to fluid mechanics. The course examines several canonical types of flow instabilities, elucidating the underlying physical mechanisms governing their development. Linear stability analysis is formulated through the method of normal modes, with particular emphasis on local instability characteristics—including temporal, spatial, and spatio-temporal growth—in parallel base flows, illustrated through the example of a free shear layer and boundary layers. The concepts of transient growth and optimal perturbations are introduced and analyzed for Poiseuille flow. The distinction between convective and absolute instability is discussed, highlighting the physical interpretation of wave packet evolution and its implications for the onset of self-sustained oscillations in open flows. This course provides an introduction to the fundamental principles and analytical techniques of instability theory as applied to fluid mechanics. The course examines several canonical types of flow instabilities, elucidating the underlying physical mechanisms governing their development. Linear stability analysis is formulated through the method of normal modes, with particular emphasis on local instability characteristics—including temporal, spatial, and spatio-temporal growth—in parallel base flows, illustrated through the example of a free shear layer and boundary layers. The concepts of transient growth and optimal perturbations are introduced and analyzed for Poiseuille flow. The distinction between convective and absolute instability is discussed, highlighting the physical interpretation of wave packet evolution and its implications for the onset of self-sustained oscillations in open flows. The stability of non-parallel steady flows, such as the cylinder wake, is subsequently characterized through global eigenmode analysis, linking local instability features to the emergence of global oscillatory behavior. The concept of Structural Sensitivity is presented and discussed to analyze localization. Theoretical lectures are complemented by integrated exercise sessions, in which students will use MATLAB/Python together with FreeFem++ on their own computers to gain numerical and computational experience with the theoretical concepts presented in class. Assessment is based on a final group project. Bibliography: F. Charru: "Hydrodynamic Instabilities", Cambridge University Press, 2011. P. G. Drazin: “Introduction to hydrodynamic stability, Cambridge University Press”

Guest Lecture: Dr. Flavio Giannetti is a Full Professor of Fluid Dynamics (ING/IND-06) in the Department of Industrial Engineering at the University of Salerno. He holds a Bachelor’s degree in Mathematics from the University of Siena, followed by advanced postgraduate training and a Ph.D. in Applied Mathematics and Theoretical Physics from the University of Cambridge, where his research focused on nonlinear stability, boundary layer receptivity, and flow instabilities. After a postdoctoral appointment at the University of Salerno on fluid-dynamic instabilities and transition to turbulence, he progressed through the academic ranks from Assistant Professor to Associate Professor, obtaining the Italian National Scientific Qualification for Full Professor in 2017. He was appointed Full Professor in 2023. Program Outline 1. Introduction to Dynamical Systems (1 h) Classification of fixed points and bifurcations. Basic concepts of stability and transition to instability in low-dimensional systems. 2. Governing Equations and Linearization (1 h) Review of the Navier–Stokes equations and their linearization about a base flow. Perturbation formulation and energy considerations. 3. Asymptotic Stability and Normal Mode Analysis (2 h) Definition of asymptotic stability; derivation and interpretation of normal modes. Introduction to numerical methods for eigenvalue problems in hydrodynamic stability. 4. Stability of Parallel and Quasi-Parallel Flows (2 h) Analysis of shear flows and boundary-layer flows. Inviscid and viscous instability mechanisms. 5. Non-Modal Stability and Transient Growth (1 h) Non-normal operators, transient energy amplification, and example of Poiseuille flow. 6. Types of Instabilities: Spatial and Spatio-Temporal Frameworks (2 h) Spatial versus spatio-temporal stability theory. Convective and absolute instabilities; criteria for distinguishing instability types and their physical implications. 7. Global Stability and Structural Sensitivity (1 h) Stability of non-parallel flows and global mode analysis. Connections to weakly non-parallel concepts. Structural sensitivity, mode localization, and examples of short- and long-range instabilities. Bibliography: 1) F. Charru: "Hydrodynamic Instabilities", Cambridge University Press, 2011. 2) P. G. Drazin: “Introduction to hydrodynamic stability, Cambridge University Press” Supplementary Resources: https://basilisk.fr/sandbox/easystab/M2DET/Instabilities.mdher materials

In presenza

Sviluppo di project work in team

P.D.2-2 - Marzo

Monday, 2 March 2026, from 9:00 a.m. to 12:30 p.m., at Ferrari Classroom, 2nd floor, DIMEAS Tuesday, 3 March 2026, from 9:00 a.m. to 12:30 p.m., at Ferrari Classroom, 2nd floor, DIMEAS Wednesday, 4 March 2026, from 9:00 a.m. to 12:30 p.m., at Ferrari Classroom, 2nd floor, DIMEAS