Dynamical systems and the transition to turbulence in linearly stable shear flows

Abstract

Plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures, which transiently show up in the temporal evolution of the turbulent flow. Here we summarize the evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow and on the coherent structures in pipe flow.

Introduction. Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds' paper

“…But with the nonlinear terms included, other persistent flow structures around which the turbulent flow organizes itself can exist, very much as in other systems where periodic orbits carry chaotic dynamics (Cvitanovic & Eckhardt 1991). Such structures will have to be three dimensional, and have been found by Faisst & Eckhardt (2003), , Pringle & Kerswell (2007) and Eckhardt et al (2008). A family of highly symmetric states is discussed by Pringle et al (2009).…”

Transient turbulence in Taylor-Couette flow

“…While the statistical approach of the past century has provided great insight, it has proven to have limited quantitative predictive power, even for the simplest of flows. Dynamical systems has emerged as a complementary technique and has shown promise in clarifying the physics of turbulence [1]. In this picture, turbulent flows are viewed as trajectories in a high-dimensional phase space that wander between unstable solutions that coexist with the laminar solution [2].…”

Small scale exact coherent structures at large Reynolds numbers in plane Couette flow

“…Many of them share relatively stable relations between their height, width, and downstream periodicity: if H denotes the height, then the width of the structures is about πH and the downstream wavelength is about 2πH. For plane Couette flow, the exact optimal relations are documented in (Clever & Busse 1997, Waleffe 2003, and the estimates for pipe flow are given in (Faisst & Eckhardt 2003, Eckhardt et al 2008, Pringle et al 2009). Similar results are available for plane Poiseuille flow, though the optimal wavelength described in (Zammert & Eckhardt 2016a) shows that there is some variability in the optimal ratios.…”

“…An approach to finding smaller structures is suggested by the behaviour of states in pipe flow (Faisst & Eckhardt 2003, Eckhardt et al 2008, Pringle et al 2009): as number of vortices along the circumference increases, they move closer to the walls and also their downstream wavelength decreases. Apparently, the vortices try to maintain the geometric relations as they become narrower.…”