Exact coherent structures in pipe flow: travelling wave solutions

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Three-dimensional travelling wave solutions are found for pressure-driven fluid flow through a circular pipe. They consist of three well-defined flow features - streamwise rolls and streaks which dominate and streamwise-dependent wavy structures. The travelling waves can be classified by the m-fold rotational symmetry they possess about the pipe axis with m = 1, 2, 3, 4, 5 and 6 solutions identified. All are born out of saddle-node bifurcations with the lowest corresponding to m = 3 and traceable down to a Reynolds number (based on the mean velocity) of 1251. The new solutions are found using a constructive continuation procedure based upon key physical mechanisms thought generic to wall-bounded shear flows. It is believed that the appearance of these new alternative solutions to the governing equations as the Reynolds number is increased is a necessary precursor Io the turbulent transition observed in experiments

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Theoretical or Mathematical/ bifurcation flow instability laminar flow laminar to turbulent transitions pipe flow rotational flow shear flow turbulence/ exact coherent structures pipe flow three-dimensional travelling wave solutions pressure-driven fluid flow circular pipe streamwise rolls streamwise streaks streamwise-dependent wavy structures m-fold rotational symmetry saddle-node bifurcations Reynolds number mean velocity constructive continuation procedure wall-bounded shear flows turbulent transition/ A4760 Flows in ducts, channels, and conduits A4715F Stability of laminar flows A4725 Turbulent flows, convection, and heat transfer A4730 Rotational flow, vortices, buoyancy and other flows involving body forces A4752 Chaos and fractals in flow A0547 Nonlinear dynamical systems and bifurcations