Recurrence of Travelling Waves in Transitional Pipe Flow

Abstract

The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface which separates initial conditions which uneventfully relaminarise and those which lead to a turbulent evolution. Evidence for recurrent travelling wave visits is found in both 5D and 10D long periodic pipes but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion over the travelling waves in which their individual properties are appropriately weighted. Rather, further dynamical structures such as periodic orbits need to be isolated and included in any such expansion.

37 pages, 27 figures

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Cited by in corpus (43)

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• Visualizing the geometry of state space in plane Couette flow
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• Asymmetric, helical and mirror-symmetric travelling waves in pipe flow
• Laminar-turbulent boundary in plane Couette flow
• Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos
• On the transient nature of localized pipe flow turbulence
• Relative periodic orbits form the backbone of turbulent pipe flow
• Revealing the state space of turbulent pipe flow by symmetry reduction
• Heteroclinic connections in plane Couette flow
• Relative periodic orbits in transitional pipe flow
• Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
• Highly-symmetric travelling waves in pipe flow
• Exact coherent states and connections to turbulent dynamics in minimal channel flow
• On the self-sustained nature of large-scale motions in turbulent Couette flow
• Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities
• From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow
• Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits
• Symmetry reduction in high dimensions, illustrated in a turbulent pipe
• Transition to turbulence in shear flows
• Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow
• Stable Manifolds and the Transition to Turbulence in Pipe Flow
• Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence
• Small scale exact coherent structures at large Reynolds numbers in plane Couette flow
• Suppression of turbulence and travelling waves in a vertical heated pipe
• On the stability of plane Couette-Poiseuille flow with uniform cross-flow
• Exact coherent structures and shadowing in turbulent Taylor-Couette flow
• Long wavelength instability of coherent structures in plane Couette flow
• Unstable Equilibria and Invariant Manifolds in Quasi-Two-Dimensional Kolmogorov-like Flow
• Surfing the edge: using feedback control to find nonlinear solutions
• Studying edge geometry in transiently turbulent shear flows
• Heteroclinic and Homoclinic Connections in a Kolmogorov-Like Flow
• Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised `edge' states
• Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow
• Observing a Dynamical Skeleton of Turbulence in Taylor-Couette Flow Experiments
• Localization in a spanwise-extended model of plane Couette flow
• Coherent structures in Dissipative Particle Dynamics simulations of the transition to turbulence in compressible shear flows
• Vortexons in axisymmetric Poiseuille pipe flows
• Stably stratified exact coherent structures in shear flow: the effect of Prandtl number
• Master-modes in 3D turbulent channel flow
• Groebner basis methods for stationary solutions of a low-dimensional model for a shear flow
• Coherent structures in fully-developed pipe turbulence
• The role of periodic orbits and bubbles of chaos during the transition to turbulence