Informations sur les exposés

The second order (van Dyke 1962) introduces the displacement thickness and clarifies the slip/ non slip boundary condition. 

The boundary layer is sensible to adverse pressure gradients which produce separation. In the classical framework this gives a singularity (Goldstein 1948). 

We show how the singularity of boundary layer separation is removed by the "Triple Deck" theory (Neiland 1969 Stewartson et al 1969 and Messiter 1969). Next we present Interactive Boundary Layer theory which is a weak form of triple deck. In IBL the body is increased by the size of the displacement thickness, changing the ideal fluid response, this allows to capture separation. With turbulent hypothesis (that we will remind), IBL with turbulent closure was an efficient technique to compute flows around airfoils up to the beginning of this century (Drela, Cebeci, Le Balleur...). 

Boundary layer was invented for applications in  aerodynamics. We will remind that many other flows like flows in arteries, granular avalanches flows, or flows in rivers may be solved with the   Prandtl system with extra/ adapted  boundary condition corresponding to the chosen phenomena (we call those equations RNSP).

For example we present the hydraulic jump. It is classical with Saint-Venant Bélanger description, and it may be reinterpreted with boundary layer equations (leading to boundary layer separation).

 

Lopez-Ruiz : Surface roughness has been identified as an essential parameter in fluid flow since the nineteenth century, but its effects on fluid dynamics are not fully understood. This talk regards the impact of coastal rough topography on oceanic circulation at the mesoscale. We study a singular perturbation problem from meteorology known as the single-layered quasi-geostrophic model. Assuming the rough coasts do not present a particular structure, the governing boundary layer equations are defined in infinite domains with not-decaying boundary data. Additionally, the eastern boundary layer exhibits convergence issues far from the boundary. In this regime, we establish the well-posedness of the boundary layer profiles in Kato spaces by adding ergodicity properties and using pseudo-differential analysis. We construct an approximate solution to the original problem and show convergence results.

Gerard Varet : Many  hydrodynamic instabilities take place near a solid boundary at high Reynolds number. This reflects into the mathematical theory of the classical Prandtl model for the boundary layer: it exhibits high frequency instabilities, limiting its well-posedness  to infinite regularity (Gevrey) spaces. After reviewing shortly this fact, we will turn to the Triple Deck model, an improvement of the Prandtl system that is commonly accepted to be more stable. We will show that this is actually wrong, and that the recent result of analytic well-posedness by Iyer and Vicol is more or less optimal. 

This is based on a joint work with Helge Dietert. 

Iyer :  In this talk I will discuss a recent work which proves stability of Prandtl's boundary layer in the vanishing viscosity limit. The result is an asymptotic stability result of the background profile in two senses: asymptotic as the viscosity tends to zero and asymptotic as x (which acts a time variable) goes to infinity. In particular, this confirms the lack of the "boundary layer separation" in certain regimes which have been predicted to be stable. This is joint work w. Nader Masmoudi (Courant Institute, NYU).