Printemps- Eté
Niveau Master 2 3 ECTS - En anglais
Cours en option pour le parcours de Soft matter and biological physics M2
Enseignant(s) Alexandros ALEXAKIS ( ENS-PSL ) Bérengère DUBRULLE ( Université Paris-Saclay )
Contact - Secrétariat de l’enseignement

Turbulent flows are present all around us and are crucial in fields such as aeronautics, industry, meteorology, astrophysics, climate. 

We know their equation of motions (the Navier-Stokes equations). Yet, several theoretical and practical obstacles preclude the building of a complete theory, and impose to resort to modern and original tools of mathematics and physics, such as weak formalism, non-equilibrium physics, wavelets or multi-fractals. In this class, we will first explain the various difficulties associated with turbulence theory, and describe several tools and approaches that enable to draw a modern picture of turbulence.

For this purpose, we will navigate between theory and practice, using data from real experiments or numerical simulations to put into practice what we have learned.


• Introduction and examples
• Mathematical and Statistical tools
• Kolmogorov theory and beyond (Millenium prize, Weak-Karman Howarth)
• Intermittency description (wavelets, multifractal, Onsager conjecture)
• Turbulence in 2D (cascades and conservation laws)
• Turbulence modelling (coarse-graining, stochastic modelling, shell model, EDQNM)
• Dissipation and irreversibility
• Convection, MHD flows, rotating and stratified flows


Basic notion of Fluid Mechanics and probability theory are required.


The final exam will be based upon an homework. RoomL367 9am to 1pm/ Room LE115 1pm to 3pm.

  • Frisch, Turbulence, Cambridge
  • Landau & Lifschitz, Fluid Mechanics, Pergamon
  • Bohr, Jensen, Paladin, Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge
  • Dubrulle, Beyond Kolmogorov, JFM Perspectives, 2019.