Extreme vortex states and the hydrodynamic blow-up problem


Event details

Date 21.03.2016
Hour 11:0012:00
Speaker Prof. Bartosz Protas, McMaster University, Canada
Bio: Prof. Protas is an applied mathematician interested primarily in problems combining modelling, analysis and computation with the theory of optimization and control. He holds a Ph.D. degree awarded jointly by the Warsaw University of Technology and Universite Pierre et Marie Curie (PARIS VI) in 2000. After post-doctoral experience at the University of California, San Diego, he joined the Department of Mathematics and Statistics at McMaster University in Canada as SHARCNET Chair in Scientific Computing, and has been on the faculty there since then receiving promotion to the rank of Professor in 2015.
Dr. Protas' research is a blend of fundamental investigations in core problems of hydrodynamics and applied studies relevant to industrial fluid mechanics and heat transfer, thermodynamics and electrochemistry. He focuses on the use of advanced mathematical methods to develop efficient computational algorithms and models useful for studying complex physical and electrochemical processes. Dr. Protas has a successful track record of industrial collaborations.  A recipient of the Early Researcher Award (ERA), Dr. Protas has also held several visiting appointments at premier research universities in Europe and in Asia.  Two of his former Ph.D. students (Dr. Bukshtynov and Dr. Ayala) received the dissertation awards of the Canadian Applied and Industrial Mathematics Society (CAIMS) for the best doctoral theses in applied mathematics defended at a Canadian university, respectively, in 2012 and 2014.  Dr. Protas' responsibilities include directing the School of Computational Science and Engineering, one of McMaster's flagship interdisciplinary graduate programs, and serving on the Editorial Boards of the International Journal of Computer Mathematics (Section B) and the Journal of Computational Science.http://www.math.mcmaster.ca/bprotas
Category Conferences - Seminars
In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D (Lu & Doering, 2008). They are therefore intimately related to the question of singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem.

We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it allows one to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. 

In the presentation we will review a number of new results concerning 2D and 3D flows characterized by the maximum possible growth of, respectively, palinstrophy and enstrophy. It will be shown that certain types of initial data, such as the Taylor-Green vortex, which have been used in numerous computational studies of the blow-up problem are in fact a particular instance (corresponding to an asymptotic limit) of our family of extreme vortex states. We will present results comparing the growth of relevant quantities in high-resolution direct numerical simulations of the Navier-Stokes system obtained using our extreme vortex states and different initial data employed in earlier studies.

Practical information

  • Informed public
  • Free


  • Prof. François Gallaire


  • Prof. François Gallaire

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