# Workshop on Extreme Events and Criticality in Fluid Mechanics: Computations and Analysis

January 25 - 29, 2016, The Fields Institute

## Overview

The key mathematical model describing fluid flows is the Navier-Stokes system of nonlinear partial differential equations (PDEs) representing the conservation of mass and momentum in a viscous incompressible fluid. Its complexity for fully three dimensional flows has for a long time defied mathematicians, computational scientists and physicists. However, the last decade or so has witnessed a number of research efforts which began to shed new light on some unresolved open problem of theoretical fluid mechanics. Some examples are:

- the discovery of "edge states'' demarcating laminar and turbulent states in certain shear flows, such as pipe and channel flow, followed by their experimental observation,
- the construction of "minimal seeds'', i.e. finite-sized perturbation that lead to subcritical transitions in flows with edge states,
- the computation of an increasing variety of invariant solutions to the Navier-Stokes equations such as periodic orbits, invariant tori and connecting orbits, which together can form the "skeleton'' of turbulence
- the identification of localized vortex states in extremely violent events as possible precursors for singularity formation.

A common feature in all of these investigations is that, in addition to an accurate solution of the underlying PDEs, they rely on an innovative use of computational techniques allowing one to identify the initial data with some very special properties. For the different problems mentioned above, these correspond to initial data lying on the manifolds separating laminar states from turbulent ones (the so-called ``edge states''), states leading to unstable spatio-temporally complex periodic motions, or the initial data generating the most singular flow evolution (as quantified by the deterioration of certain measures of regularity). The computational approaches required to robustly identify such special initial data typically involve solution of constrained variational optimization problems or fixed-points problems for the underlying discretized PDEs. The structure of these problems is more often than not quite complicated and may suffer from stiffness, nonsmoothness as well as ill-posedness.

While certain progress has already been achieved along these lines, much remains to be done. The scientific community working on these problems, which is rather decentralized, is at a point where methodological advances are needed to warrant new breakthroughs. In the proposed thematic program we will therefore focus on the following open and emerging problems:

- reliable solution of large-scale, possibly nonsmooth, variational optimization and fixed-point problems for PDE systems arising in fluid mechanics applications,
- development of novel modeling and simulation techniques to describe the organization of fluid motion at high Reynolds numbers and on large domains
- discretization of PDE problems in the presence of potential singularities,
- optimal ways to computationally search for extreme behaviour (variational approaches, "instanton formulations'', Monte Carlo, approaches based on complexified equations, etc.),
- approaches to studying the existence and regularity of solutions to flow problems using computer-assisted proofs and rigorous computations.

One of the main objectives of the proposed thematic program will be to make some headway with the open problems identified above. There are natural connections between some of these issues and the open problems relevant in the context of the second and third main theme of the program offering valuable opportunities for exchange of ideas.

## Schedule

09:10 |
The stages of transition in pipe flow
Dwight Barkley, University of Warwick |

10:30 to 11:30 |
Bypass transition in parallel and non-parallel boundary layer flows
Yohann Duguet, CNRS |

11:30 |
Bifurcation structure of plane Couette flow with the Smagorinsky model
Eiichi Sasaki, Osaka University |

14:00 to 15:00 |
Seeding the edge dynamically: Using nonlinear adjoints and Koopman modes to investigate the unstable manifold of the edge state in plane Couette flow
Colm-cille Caulfield, University of Cambridge |

15:00 to 16:00 |
Mathematical models of anomalous enstrophy dissipation and enstrophy cascade in 2D turbulence
Takashi Sakajo, Kyoto University |

16:00 |
Two-dimensional flow on compact surfaces
David Dritschel, University of St Andrews |

09:00 |
The interplay between theory and computation in the study of 3D Euler equations
Thomas Hou, Caltech |

10:30 to 11:30 |
Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows
Miguel Bustamante, University College Dublin |

11:30 |
Dynamical scale-invariance for the Navier-Stokes equations in critical spaces
Koji Ohkitani, University of Sheffield |

14:00 |
Helicity annihilation in simulated trefoil reconnection
Robert Kerr, University of Warwick |

15:25 to 16:00 |
Extreme events in turbulent Rayleigh-Benard convection
Joerg Schumacher, TU Ilmenau |

16:00 |
Instantons and Burgers Turbulence --- Recent Progress
Tobias Schaefer, College of Staten Island |

09:00 |
Wall to wall optimal transport
Charles Doering, University of Michigan |

10:30 to 11:30 |
Large Rayleigh-Number Coherent Convective Solutions of the 2D Oberbeck--Boussinesq and Darcy--Oberbeck--Boussinesq
Greg Chini, University of New Hampshire |

11:30 |
Optimal heat transfer enhancement in plane Couette flow
Genta Kawahara, Osaka University |

09:00 |
A Universal Transition to Turbulence in Channel Flow
Masaki Sano, The University of Tokyo |

10:30 to 11:30 |
Exact coherent structures in the transition to turbulence in plane Poiseuille flow
Bruno Eckhardt, Philipps-Universitat Marburg |

11:30 |
Transition in shear flows without walls
Laurette Tuckerman, CNRS |

14:00 to 15:00 |
No Title Specified
Björn Hof, IST Austria |

15:00 to 16:00 |
Self-similarity in variable-viscosity jets
Luminita Danaila, CORIA |

16:00 |
Experiments on the organisation of turbulent pipe flow
David Dennis, University of Liverpool |

09:00 to 10:00 |
Computer-assisted proofs for periodic orbits of PDEs
Jean-Philippe Lessard, Université Laval |

10:00 to 11:00 |
Rigorous computational dynamics for three pattern formation problems
Jan Bouwe van den Berg, VU Amsterdam |

11:00 |
Existence and regularity of rotating global solutions for active scalars
Javier Gomez-Serrano, Princeton University |

14:00 to 15:00 |
Some rigorous results on interfacial problems through a quasi-solution approach
Saleh Tanveer, Ohio State University |

15:00 |
Part 1: Reduced Basis method for a bifurcation study of a thermal convection problem and Part 2: Reduced basis approximation for convection dominated problems
Yvon Maday, Université Pierre et Marie Curie Paris 6 |