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Transition to turbulence in thermoconvection

and aerodynamics

Credits: 2 ECTS

Duration: 21 hours

Semester: S9

Person in charge: Emmanuel Plaut, professor

Keywords: nonlinear phenomena, stability, instability, bifurcation theory

Prerequisites: fluid dynamics elementary course, numerical methods

Objectives: learn relevant nonlinear phenomena in hydrodynamics with heat transfers (thermoconvection) and aerodynamics; learn some methods to model these phenomena.

Program and contents:

The transition to spatio-temporal complexity and turbulence in fluid dynamics, which is intrinsically nonlinear, is studied by focusing on two families of systems. This is also an occasion to enrich the knowledge and know-how of the students in general fluid mechanics. The families of systems studied are:

  1. Natural convection or `thermoconvection'

    The emphasis is on the `Rayleigh-Bénard' configuration in extended geometry, where convection rolls set in under the influence of a vertical downward temperature gradient, through an instability. This is an occasion to introduce the thermal buoyancy, the Oberbeck - Boussinesq approximation the methods of the linear and weakly nonlinear stability analyses, and to evidence a supercritical pitchfork bifurcation that leads to increased heat transfers. The secondary instabilities are also briefly discussed. The chaos is introduced both with the (historical) Lorenz model and the (more realistic) example of the Rayleigh-Bénard convection in a square cell, where chaotic large-scale flow reversals occur. Other geometries and systems are also briefly discussed, for instance, the `differential heating' configuration, where the basic temperature gradient is horizontal, therefore, thermoconvection sets in directly, as it is often the case for heating in buildings.

  2. Open shear flows

    The emphasis is on the Tollmienn-Schlichting waves that set in through an instability of channel flows. In this different context, the linear and weakly nonlinear stability analyses already introduced are performed now with numerical computations (spectral method), to evidence a subcritical Hopf bifurcation. The further transition to turbulence is also briefly discussed, for channel flows, and also boundary layer flows and airfoils. Openings concerning aerodynamics and wind energy are finally presented.

    Importantly, the stability analyses methods and the theory of bifurcations (or `catastrophes') introduced here are, in fact, relevant for any nonlinear deterministic model; applications also exist in other domains of mechanical engineering, in physics, etc...

Please check the web page of this module on : it sketches the planning of this module, gives the lecture notes and instructions, etc... In particular, you will use Mathematica to perform formal (symbolic) and numerical computations on your laptop.

Abilities : 


Description and operational vocabulary


Chaos. The physics of the transition to turbulence in channel flows, boundary layer flows and airfoils


Linear vs nonlinear terms - linear vs nonlinear effects. Supercritical vs subcritical bifurcations. The physics of thermoconvection and thermal buoyancy


Linear and weakly nonlinear stability analyses. Numerical spectral method


Linear and weakly nonlinear stability analyses. Numerical spectral method




  • Written test
  • Continuous assessment
  • Oral presentation
  • Project
  • Written report
  • No labels