Scientific Computation and Applied & Industrial Mathematics: Tom Eaves

Transitional phenomena are ubiquitous in fluid dynamics and other nonlinear systems; they occur whenever there are multiple states in which a system can reside. Frequently, we are able to investigate when and how a system transitions from one state to another by performing a linear stability analysis and obtaining critical thresholds for various parameters beyond which our original state becomes "unstable". However, there are numerous examples for which such an approach does not work. Perhaps the most widely studied scenarios in fluid mechanics for which a linear stability analysis fails to predict transition are the canonical homogeneous shear flows of plane Couette flow and pipe flow. Both of these flows have a laminar (quiescent) solution to the Navier-Stokes equations which is linearly stable at all flow rates, and yet sustained turbulent dynamics are observed in plane Couette flow and in pipe flow for sufficiently rapid flow. Such systems are "two-state" systems for which both the laminar flow and turbulence co-exist as (locally) stable solutions to the Navier-Stokes equations. Recent developments in "generalised nonlinear stability theory" (Pringle & Kerswell, 2010) allow us to find minimal perturbation amplitudes, in a nonlinear sense, to transition between two (linearly) stable flow states. However, the full interpretation of the results of nonlinear stability theory is possible only when interpreting fluid flows in the language of dynamical systems. Drawing from the recent focus of interpreting turbulence in terms of coherent structures rather than statistics, Eaves & Caulfield (2015) interpreted the minimal thresholds for transition to turbulence in statically stable density-stratified plane Couette flow with a focus on coherent structures, and demonstrated that the effect of stratification has an unexpectedly significant impact on the transition scenario. In this talk, I will outline the methodology behind nonlinear stability theory, explain what turbulence is from a dynamical systems point of view, and outline how these two ideas were utilised in my thesis work on stratified shear flow. I will conclude with a brief overview of ongoing work and other extensions to this rapidly developing field of research.