Fluid Mechanics Seminar:Thomas S. Eaves

Transitional phenomena are ubiquitous in fluid dynamics; they occur whenever there are multiple states in which a flow can reside. Frequently, we are able to investigate when and how a flow transitions from one state to another by performing a linear stability analysis and obtaining critical thresholds for various flow 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 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 solution to the Navier-Stokes equations which is linearly stable at every Reynolds number (Re), and yet sustained turbulent dynamics are observed in plane Couette flow for Re>318 and in pipe flow for Re>2040. 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. Other examples include yielded and unyielded states in ideal Bingham fluids, for which transition from an unyielded flow requires finite amplitude, and therefore inherently nonlinear, perturbations. 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.

Bio: Tom Eaves completed his BA & MMath followed a Ph.D in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge under the supervision of C.P. Caulfield. Since September, 2016 he is a postdoctoral research fellow in the Department of Mathematics at UBC, working with Neil Balmforth and Mark Martinez.