Highly-symmetric travelling waves in pipe flow

Abstract

The recent theoretical discovery of finite-amplitude travelling waves in pipe flow has re-ignited interest in the transitional phenomena that Osborne Reynolds studied 125 years ago. Despite all being unstable, these waves are providing fresh insight into the flow dynamics. Here we describe two new classes of highly-symmetric travelling waves (possessing rotational, shift-&-reflect and mirror symmetries) and report a new family of mirror-symmetric waves which is the first found in pipe flow not to have shift-&-reflect symmetry. The highly-symmetric waves appear at lower Reynolds numbers than the originally-discovered non-mirror-symmetric waves found by Faisst & Eckhardt 2003 and Wedin & Kerswell 2004 and have much higher wall shear stresses. The first M-class comprises of the various discrete-rotationally-symmetric analogues of the mirror-symmetric wave found in Pringle & Kerswell (2007) and have a distinctive double layer structure of fast and slow streaks across the pipe radius. The second N-class has the more familiar separation of fast streaks to the exterior and slow streaks to the interior and looks the precursor to the class of non-mirror-symmetric waves already known.

16 pages, 8 figures, for Phil Trans theme issue on pipe flow

References in corpus (9)

• Travelling waves in pipe flow
• Recurrent motions within plane Couette turbulence
• Turbulence transition and the edge of chaos in pipe flow
• Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
• Transition in pipe flow: the saddle structure on the boundary of turbulence
• Asymmetric, helical and mirror-symmetric travelling waves in pipe flow
• Recurrence of Travelling Waves in Transitional Pipe Flow
• Relative periodic orbits in transitional pipe flow
• Statistical analysis of coherent structures in transitional pipe flow

Cited by in corpus (22)

• On coherent structure in wall turbulence
• Transition in pipe flow: the saddle structure on the boundary of turbulence
• Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos
• Relative periodic orbits form the backbone of turbulent pipe flow
• Revealing the state space of turbulent pipe flow by symmetry reduction
• The Openpipeflow Navier--Stokes Solver
• Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
• Spatially localized solutions of shear flows
• Exact coherent states and connections to turbulent dynamics in minimal channel flow
• Spatiotemporal perspective on the decay of turbulence in wall-bounded flows
• The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow
• Drag reduction in pipe flow by optimal forcing
• From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow
• Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow
• Small scale exact coherent structures at large Reynolds numbers in plane Couette flow
• Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence
• Long wavelength instability of coherent structures in plane Couette flow
• Complexity of the laminar-turbulent boundary in pipe flow
• Surfing the edge: using feedback control to find nonlinear solutions
• Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow
• Cartography of high-dimensional flows: A visual guide to sections and slices
• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow