Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Abstract

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

Invariant solutions of minimal large-scale structures in turbulent channel flow for up to 1000

“…The solution trajectories of these simulations are maintained roughly around 20I l and 20D l in the I-D plane (figure 5b), the values an order of magnitude larger than those at low Reynolds numbers (e.g. Kawahara & Kida 2001;Gibson et al 2008;Willis et al 2013Willis et al , 2016. This is essentially due to the high Reynolds number considered in the present study, which leads to the energy input to the system, substantially larger than that at low Reynolds numbers.…”

“…bursting, Flores & Jiménez 2010;Hwang & Bengana 2016). In this respect, extending the present approach to computation of the relative periodic orbits would be a fruitful path to follow towards more accurate modelling of the large-scale dynamics (Kawahara & Kida 2001;Willis et al 2013Willis et al , 2016. Finally, it should be mentioned that the self-sustaining energycontaining motions in wall-bounded turbulent flows at high Reynolds numbers appear in a self-similar form throughout the entire logarithmic region Hwang 2015), as originally hypothesized by Townsend (1976) (i.e.…”

“…Together with unstable periodic and/or relative periodic orbits (e.g. Kawahara & Kida 2001), these invariant solutions have been shown to form a skeleton for solution trajectories in phase space (Gibson et al 2008;Willis et al 2013Willis et al , 2016, and their understanding from a dynamical systems viewpoint has been at the heart of recent advancement in the understanding of bypass transition and low-Reynolds-number turbulence.…”

Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow

“…This method has been adopted for one-dimensional PDE and three-dimensional turbulent pipe [24,31]. Both in the laboratory and co-moving frames, the time evolution of Ω(x, t) is shown in FIG.2 and FIG.3.…”

Nonlinear waves with a threefold rotational symmetry in pipe flow: influence of a strongly shear-thinning rheology

“…It would be interesting to search for waves that are specifically non-Newtonian. For this purpose, one could start with direct numerical simulations of transitional or turbulent pipe flow of non-Newtonian fluids (we note in this context the work of Rudman et al 2004) and use methods similar to the ones used by Duguet et al (2008b); Avila et al (2013); Willis et al (2016).…”