The role of periodic orbits and bubbles of chaos during the transition to turbulence

Abstract

Starting with turbulence that explores a wide region in phase space, we discover several relative periodic orbits (RPOs) embedded within a subregion of the chaotic turbulent saddle. We also extract directly from simulation, several travelling waves (TWs). These TWs together with the RPOs are unstable states and are believed to provide the skeleton of the chaotic saddle. Earlier studies have shown that such invariant solutions can help to explain wall bounded shear flows, and a finite subset of them are expected to dominate the dynamics (Faisst & Eckhardt 2003; Pringle & Kerswell 2007; Hof et al. 2004). The introduction of symmetries is typically necessary to facilitate this approach. Applying only the shift-reflect symmetry, the geometry is less constrained than previous studies in pipe flow. A 'long-period' RPO is identified that is only very weakly repelling. Turbulent trajectories are found to frequently approach and frequently shadow this orbit. In addition the orbit characterises a resulting 'bubble' of chaos, itself a saddle, deep within the turbulent sea (Kreilos et al. 2014). We explicitly analyse the merger of the two saddles and show how it results in a considerable increase of the total lifetime. Both exits and entries to the bubble are observed, as the stable manifolds of the inner and outer saddles intertwine. We observe that the typical lifetime of the turbulence is influenced by switches between the inner and outer saddles, and is thereby dependent on whether or not it 'shadows' or 'visits' the vicinity of the long-period RPO (Cvitanovic et al. 2014). These observations, along with comparisons of flow structures, show that RPOs play a significant role in structuring the dynamics of turbulence.