CISM-ECCOMAS International Summer School on "Coherent Structures in Unsteady Flows: Mathematical and Computational Methods"

June 3, 2019 — June 7, 2019


  • George Haller (ETH Zürich, Institute for Mechanical Systems, Zürich, Switzerland)

Turbulence is sometimes described as the last unsolved problem of classical physics. While significant advances have been made towards solving this problem, the available results are largely confined to asymptotic scaling laws for non-dimensional descriptors of homogeneous, isotropic turbulence. In contrast, turbulent flows observed in nature and technology tend to display coherent structures, rendering the hypotheses of homogeneity and isotropy invalid. Coherence in unsteady flows has been studied in four main contexts:
- Eulerian coherent vortices (ECV): spatial patterns in the instantaneous velocity field.
- Lagrangian coherent structures (LCS): persistent spatial patterns in tracer distributions.
- Objective Eulerian coherent structures (OECS): the instantaneous limits of LCSs.
- Exact Coherent States (ECS): persistent temporal patterns in the velocity field.

ECV are well studied in fluid dynamics but have no universally accepted definition or detection algorithm. This is largely due to the inherent dependence of this concept on the observers and to the lack of an experimentally verifiable ground truth. In contrast, LCSs, OECSs and ECS have observer-independent features,  and hence definitions and algorithms aimed at their identification are experimentally verifiable. Going beyond a popular-level understanding of these three coherence notions, however, requires a solid command of higher mathematical and computational concepts.

The objective of this course is to provide an opportunity for junior researches to obtain a first-hand introduction to contemporary coherent structure detection techniques and computational algorithms. The lectures will be delivered by recognized experts who have participated directly in the development and applications of these methods. The list of speakers is intentionally broad, ranging from mathematicians through engineers to physical oceanographers. The scope of the course is similarly broad, covering necessary fundamentals from dynamical systems, continuum mechanics, fluid mechanics, probability, operator theory, stability theory for the Navier-Stokes equations, as well as related computational algorithms. Of the specific coherence concepts, the lectures will cover Lagrangian and Eulerian coherent structures, coherent diffusion barriers, coherent sets and exact coherent states in turbulence.

This course is addressed to doctoral students, postdocs, and young scientists wishing to learn the basics of the field. Given the diversity in the expertise of the speakers and the broad range of the applications, the participants are expected to gain substantial insight into this exciting and quickly involving area of research. They will also have access to Matlab and high-performance Fortran codes implementing the main techniques. Finally, participants will have a chance to briefly introduce a poster summarizing their own related research.


See also