IMPA - O Instituto de Matemática Pura e Aplicada

We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. We prove related rigidity results characterizing the round spheres among Zoll spheres. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its 1-width in Urysohn's quantitative dimension theory. This is joint work with Lucas Ambrozio and Rafael Montezuma.
 

Despite the intrinsic complexity of turbulent flows, large-scale coherent structures are present in turbulence, and are related to quantities of interest, such as aerodynamic drag and sound radiation. This presentation shows methods to model such coherent structures, and approaches to control flows in order to reduce drag or noise radiation. Linear models will be considered first, with coherent structures modelled as dominant responses from the linearised Navier-Stokes system. Such dominant modes compare favourably with coherent structures educed from experimental or numerical data, and allow the proposition of changes to the system aiming at drag or noise reduction. Non-linear reduced-order models (ROMs) are then obtained by the Galerkin projection of the full governing equations onto modes obtained from the linearised system. Dynamical systems so obtained typically have dimensions of O(10) to O(10^3) degrees of freedom, and quantitative agreement with reference statistics is obtained for a number of canonical flow configurations. State estimation from a few sensors is explored using an extended Kálmán filter coupled with ROMs, reproducing accurately numerical results. Finally, a ROM for turbulent Couette flow is used to devise a control method aiming at turbulence suppression. The obtained strategy is seen to relaminarise turbulence in direct numerical simulations, showing that ROMs so obtained have promising applications in optimisation and control.