Thursday, 4 June 2020

Speaker: Prof. Tobias Schneider

Emergent Complexity in Physical Systems, Ecole Polytechnique Fédérale de Lausanne, Switzerland

From patterns in turbulence to the buckling of shells - the role of unstable invariant solutions in nonlinear mechanics

Abstract

The transition to turbulence of fluid flows is ubiquitous, arising in our every-day experience when we ride a bicycle or take off in an airplane. Despite this ubiquity, the laminar-turbulent transition in wall-bounded flows is one of the least understood phenomena in fluid mechanics. During transition, the flow may self-organize into patterns with regular spatial and temporal structure, whose origins remain unexplained. A canonical flow exhibiting a large variety of complex spatio-temporal flow patterns is thermal convection in a fluid layer between two parallel plates kept at different temperature and inclined against gravity. We study the dynamics of the so-called inclined layer convection (ILC) system, using a fully nonlinear dynamical systems approach based on a state space analysis of the governing equations. Exploiting the computational power of our highly parallelized numerical continuation tools (www.channelflow.ch), we construct a large set of invariant solutions of ILC and discuss their bifurcation structure. We show that unstable equilibria, travelling waves, periodic orbits and heteroclinic orbits form dynamical networks that support moderately complex chaotic dynamics.

The introduced nonlinear dynamical systems methods centered around invariant solutions are not only revolutionizing our understanding of fluid turbulence but they may also help explain complex behaviour in other intrinsically nonlinear mechanical systems. We will specifically argue that unstable elastic equilibria control when thin-walled cylindrical shells such as rocket walls or soda cans buckle and collapse. This may open avenues towards predicting the notoriously imperfect-sensitive load-carrying capacity of shell structures without prior knowledge of the shell's defects.

Biography

Tobias Schneider is an assistant professor in the School of Engineering at EPFL. He received his doctoral degree in theoretical physics in 2007 from the University of Marburg in Germany working. He then joined Harvard University as a postdoctoral fellow. In 2012, Tobias returned to Europe to establish an independent research group at the Max-Planck Institute for Dynamics and Self-Organization in Goettingen. Since 2014, he is working at EPFL, where he heads the 'Emergent Complexity in Physical Systems' laboratory. Tobias' research is focused on nonlinear mechanics with emphasis on spatial turbulent-laminar patterns in fluid flows transitioning to turbulence. His lab combines dynamical systems and pattern-formation theory with large-scale computer simulations. His group develops computational tools and continuation methods for studying the bifurcation structure of nonlinear differential equations such as those describing the flow of a fluid or the elastic response of loaded shells.

Notes

by Lucas Benoit-Maréchal.

Symmetry-reduced Direct Numerical Simulations (DNS) of ILC give access to unstable states not captured by the linear and weakly nonlinear theories, such as the periodic orbit of crawling rolls, and show invariant solutions are transiently visited by the chaotic dynamics.

A bifurcation analysis is conducted via parametric continuation for a number of inclination angles, revealing many different invariant solutions which can be interpreted as "building blocks" of the complex full nonlinear dynamics.

The same framework can be applied to general nonlinear problems outside fluid mechanics such as cylinder buckling. A linearly stable point in the state space is surrounded by a basin of attraction, whose boundary separates perturbations that trigger instability from the ones that do not.

Invariant solutions (edge states) embedded in the basin boundary have a single unstable direction, which allows to characterize the basin of attraction as a function of load using edge-tracking techniques (based on initial conditions).