** **

**
Stability & Chaos
**

* By Dr Damien BIAU *
This course presents the modern tools available to analyse instabilities which develop in a flow.

By varying the flow parameters (the Reynolds number for example), sudden regime changes are observed: starting from the laminar (steady) state, periodic oscillations are observed, then more or less abruptly, the flow turns turbulent.

These changes are the consequence of the instabilities which appear successively above a critical threshold in the phase space of the flow considered. The aim is to develop the theory of these instabilities and their physical interpretation.

- Historical and Theoretical Presentation of the Problem. Examples of hydrodynamic instability (Kelvin-Helmoltz, Taylor-Couette, Rayleigh-Bénard...). Basics: Eigen values problem, adjunct operator, numerical solution, analytical examples.

- Linear Analysis of Local Stability: stability of parallel flows (pipe, canal) and non parallel flows (boundary layer, shear layer)

- Linear Analysis of Global Stability: confined and open flows, spatio-temporal development of perturbations

- Receptivity and Sensitivity of Perturbations.

- Weakly Non Linear Stability: non linear mechanisms in laminar-turbulent transition. Energetic theory. Asymptotic development around a linear threshold of small amplitude. Landau Equation. Bifurcations Classification.

- Coherent Structures in Turbulent Flows.