Ten lectures on turbulence: 3. Chaos, solitons, and turbulence

Abstract

Abstract Turbulence can be interpreted as a high-dimensional nonlinear dynamical process governed by the Navier–Stokes equations, in which ordered motion evolves toward chaotic behavior through instability, nonlinear interaction, and formation of coherent structures. Classical achievements in dynamic systems such as the strange attractors, delayed embedding, and Lyapunov exponent established a powerful framework for understanding the transition from periodic or quasi-periodic motion to chaos. Besides, nonlinear systems can generate solitons or low-dimensional organized coherent structures. The present work connects these ideas with recent progress on soliton-like coherent structures and examines their role in hypersonic boundary-layer transition through a Mach 6 sharp-cone case study. Phase-space reconstruction shows a continuous deformation from nearly periodic circular orbits to distorted elliptical trajectories and finally to irregular chaotic motion. The spatial distribution of largest Lyapunov exponent exhibits two characteristic regions of enhanced chaotic divergence: one associated with high-frequency second-mode saturation below the sonic line, and the other associated with the low-frequency first-mode evolution between the critical layer and the sonic line. Lagrangian-based material-surface tracking reveals that soliton-like coherent structures appear in both of the two regions, which can organize hot-streak formation and promote vortex formation. These results suggest that the nonlinear dynamics of hypersonic transition is deterministic in which the chaotic divergence and localized coherent structures develop together. We provide a natural bridge between mathematical ideas on the nonlinear wave organization and physical discoveries of coherent structures in hypersonic transitional flows, and points toward nonlinear dynamical analysis as a promising framework for future studies of turbulence generation.