Highly symmetric travelling waves in pipe flow

The recent theoretical discovery of finite-amplitude travelling waves (TWs) in pipe flow has reignited 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. We describe two new classes of TWs, which, while possessing more restrictive symmetries than previously found TWs of Faisst & Eckhardt (2003 Phys. Rev. Lett. 91, 224502) and Wedin & Kerswell (2004 J. Fluid Mech. 508, 333 371), seem to be more fundamental to the hierarchy of exact solutions. They exhibit much higher wall shear stresses and appear at notably lower Reynolds numbers. The first M-class comprises the various discrete rotationally symmetric analogues of the mirror-symmetric wave found in Pringle & Kerswell (2007 Phys. Rev. Lett. 99, 074502), and have a distinctive double-layered 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 like the precursor to the class of non-mirror-symmetric waves already known.