Exact coherent structures in pipe flow: travelling wave solutions

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 to the turbulent transition observed in experiments.