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Theme of the Summer School 

Scientific context :
Most actual flows are anisotropic (rotating turbulence, sheared turbulence, wall regions etc.). A key characteristic of such flows is the loss of three-dimensionality, which leads in particular to the formation of coherent structures and waves, a typical signature of anisotropy present down to very small scales. In these contexts, the classical theory of turbulence, essentially based on the local isotropy hypothesis, is not applicable. While retaining the main tools and methods used in the analysis of isotropic turbulence (spectral methods, statistical and dynamical analysis), new developments can be done in order to interpret the anisotropy of flows at all scales (both large and small), and thus to assess the limits of the existing theory and thus to progress in the understanding of anisotropic turbulence.

Objective:
The goal of this school is to convey the core knowledge of the advanced understanding of anisotropic turbulence to researchers in different fields, based on the expertise of the speakers invited to the school. For this, we anticipate a series of comprehensive courses, with extensions in specialized conferences.

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Outline programme:

The fundamental courses are naturally organized along the following three axes: 

1) Morphology of anisotropic flows
The first question pertains to the morphology of anisotropic flows, i.e. the local and instantaneous structure of the flow (or a region of the flow). The flows which will be presented are: pure rotating flows (open or confined); rotating turbulence with mean shear (as in rotating channels); the dynamics of stratified fluids, and magneto-hydrodynamic flows (MHD). 

2) Statistical description of anisotropic flows
a)      Spectral space description
Fourier space (anisotropic decomposition along two directions: axial and radial) and transport energy equation in spectral space. The general anisotropic description is generated by divergence-free modes; the energy spectrum is generalized towards including four real terms (decomposition in ‘directivity-polarization-helicity’), associated with generalized energy transfers, via dynamical equations. In the axisymmetric case, the above terms depend on two components of the wavevector (radial and axial). 

b)      Physical space description
The physical space is divided into contributions, based on fields filtered at a given scale (increments). For instance, transport equations for the kinetic energy at a given scale can be expressed in the framework of axisymmetric homogeneous turbulence, in which a number of scalar functions are needed, which could be found from either experiments or numerical simulations.  

3) The statistics and dynamics of coupled fields
Be it in spectral or physical space, the generalization of the statistics and dynamics of coupled fields is available, and is not restricted to the fluctuation velocity: advected scalar, active or passif, advected vector (scalar gradient, magnetic field in MHD). A last part of the program will be devoted to the mixing of passive scalar in axisymmetric flows. The mixing at Sc=1 (gas within gas, important for combustion applications, for instance), and high Schmidt number cases (liquids, industrial processes) will be discussed. 

Advanced specialized courses: 

1)      A specific question concerns the presence of coherent structures (vortices), their size, persistence and dynamics in rotating flows. The accurate knowledge of the dynamics of these structures in close relation with the surrounding flow, opens the possibility to apply some control to them (active or passive). The surrounding flow can be simple, or populated with anisotropic waves that interact with the vortices and modulate, or even dominate (wave turbulence), the classical turbulent flow. These waves are present in the turbulence in rotating fluids, stratified flows, or conducting ones (in the MHD context). 

2)      The statistical description of anisotropic flows also requires to relate closely the spectral and the physical space statistics, a topic which will be discussed. Transport equations will be presented for the velocity field, but also for the vorticity field accessible via acoustic scattering techniques, or 3D PIV measurements. 

3)      We shall also consider more complex flows such as the axisymmetric jet, with or without adverse flow current (in which the shear region at the outer boundaries of the jet exhibit strong recirculations), recirculating turbulence in a mixing reactor. These flows can incorporate a strong anisotropy linked to the inhomogeneities. 

In this last context, we will emphasize the practical importance of the specific modeling issues raised, with a careful treatment of the flows (and mixing) with rotation or swirl in a confined geometry, for instance in the combustion chamber with a swirl-enhanced burner. Many factors (not necessarily decoupled) are potentially at the origin of flame instabilities in such a chamber. Among them, one identifies: the shedding of organized vortex structures in the recirculation zone; the interaction between heat production and acoustics; spatial inhomogeneities and partial premixing between the fuel and the gas; … A conclusion emerges: the instabilities in a reacting flow are linked to complex phenomena, and their study generally requires to focus on the influence of one given parameter. Focusing on the role played by turbulent structures, their dynamics (stability/instability), is of prime importance in this study. We will specifically look at this dynamics in a simplified flow, including rotation or a recirculation, in that bringing a few elements for understanding the intricate puzzle of the phenomenology of reacting flows with rotation.

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Organisation: 

The duration of the school will be 14 days. The daily timetable is twice 3 hours, with a break devoted to discussions after each 90 minutes session, and a mid-day break favorable to direct exchanges between taught and teachers, and between researchers themselves. The lessons will be completed with the presentation of focused examples which elucidate some aspects of the theory. The contexts of the presentation will be made available to the participants on the Web site of the School. Organizational details are left to the responsibility of each professor in charge of a session, in particular as regards the diffusion of the lessons after the School.

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Audience:

Primarily - PhD students, post-doctoral researchers, researchers in the following domains : fluid mechanics, physics, geophysics and astrophysics.

Additionally - Engineers and researchers from industry, applied mathematicians.

No pre-requisite knowledge is expected from the participants, other than that of base knowledge in fluid mechanics and turbulence.