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Advanced Fluid Mechanics: transition to turbulence & turbulence - Applications to thermoconvection, aerodynamics & wind energy

ECTS Credits: 3

Duration: 21 hours + 9 h Erasmus & Fondation

Semester: S9

Persons in charge:

Emmanuel Plaut, professor Uni. of Lorraine - 

Joachim Peinke, professor Uni. of Oldenburg -

Keywords: Nonlinear physics, instabilities, bifurcations, statistical and stochastic modeling, wind energy

Prerequisites: Statistics, Fluid mechanics


Introduction to the theory of bifurcations, in the context of the transition to spatio-temporal complexity or turbulence in fluid systems. Advanced turbulence modeling. Importance of small scale turbulence in the context of Wind Energy.


In the first part of this course (sessions 1-6), given by EP, 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 Boussinesq approximation, the thermal buoyancy, 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...


   In the second part of this course (sessions 7-9), given by JP, statistical and stochastic modeling of turbulence is presented, focusing on the universal structure of small scale turbulence as well as on applications to Wind Energy. The small scale statistics of turbulent flows is reviewed: cascade, power spectra, intermittency corrections, extreme events... For Wind Energy, the characterization of wind conditions is presented: the IEC 61400 norm is described, and `wind gusts' are discussed. Experimental methods to investigate the impact of turbulence on the wind energy conversion, like sensors and wind tunnel with (active) grids, are presented. Finally, methods are developed, to handle the turbulent dynamical aspects of the energy conversion of a wind turbine. Topics are power output for a single turbine and a farm, as well as monitoring fatigue.


The last session corresponds to a test with programming, EP will help and validate one or more steps.


Web page of the module: .

A funding of the ERASMUS program and of the Fondation Mines Nancy permits the involvement of JP, who has been recently the president of the European Academy of Wind Energy.



Transition to Turbulence:

  • understand basic thermoconvection phenomena, in particular, the Rayleigh-Bénard system: onset through an instability and supercritical bifurcation, notion of secondary instabilities, ...
  • understand the fundamentals of the transition to turbulence in Open Shear Flows: subcritical bifurcation, strong secondary instabilities, ...
  • be able to perform, with some guidance, a linear stability analysis, either with formal calculations or a numerical spectral method
  • understand the bases of the weakly nonlinear analysis, up to the computation of amplitude equations; be able to perform such a weakly nonlinear analysis with some guidance

Wind Energy and Turbulence:

  • knowledge of quantities that characterize turbulence (turbulent energy, length scales, ...)
  • be able to extract from a velocity time series mean and standard deviation values, a PDF of the velocity and of the velocity increments, understand the notion of intermittency and extreme events
  • understand the importance of the wind speed characteristics for Wind Energy
  • be able to construct the IEC power curve from real data



  •  Written test
  • Continous assessment
  • Oral presentation
  • Project
  • Written report
  • Aucune étiquette