The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow

Abstract

The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can co-exist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate the origin of the first localized solution in a modulational symmetry-breaking Hopf bifurcation from a known global travelling wave that has 2-fold rotational symmetry about the pipe axis. Similar behaviour is found for a global wave of 3-fold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that all global solutions should be expected to lead to more realistic localised counterparts through such bifurcations, which provides a constructive route for their generation.

5 pages, 6 figures

References in corpus (3)

• Turbulence transition and the edge of chaos in pipe flow
• Relative periodic orbits in transitional pipe flow
• Highly-symmetric travelling waves in pipe flow

Cited by in corpus (28)

• The rise of fully turbulent flow
• Ecological collapse and the emergence of traveling waves at the onset of shear turbulence
• The Openpipeflow Navier--Stokes Solver
• Exact coherent states and connections to turbulent dynamics in minimal channel flow
• Linear instability of viscoelastic pipe flow
• Koopman mode expansions between simple invariant solutions
• Streamwise and doubly-localised periodic orbits in plane Poiseuille flow
• Symmetry reduction in high dimensions, illustrated in a turbulent pipe
• Transition to turbulence in shear flows
• Transition to magnetorotational turbulence in Taylor--Couette flow with imposed azimuthal magnetic field
• Suppression of turbulence and travelling waves in a vertical heated pipe
• Heteroclinic path to spatially localized chaos in pipe flow
• Exact coherent states with hairpin-like vortex structure in channel flow
• Surfing the edge: using feedback control to find nonlinear solutions
• Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow
• Self-sustainment of coherent structures in counter-rotating Taylor-Couette flow
• Homoclinic snaking in plane Couette flow: bending, skewing, and finite-size effects
• Geometry of transient chaos in streamwise-localized pipe flow turbulence
• An adaptive preconditioner for steady incompressible flows
• Streamwise decay of localized states in channel flow
• Localization in a spanwise-extended model of plane Couette flow
• Analysis and modeling of localized invariant solutions in pipe flow
• Transition in the asymptotic suction boundary layer over a heated plate
• The origin of localized snakes-and-ladders solutions of plane Couette flow
• Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits
• Large-scale streak instabilities of transitional channel flow
• Complexity of localised coherent structures in a boundary-layer flow
• The role of periodic orbits and bubbles of chaos during the transition to turbulence