Helical symmetric turbulence in annular pipe flow and its related conservation laws

2D and 3D turbulence exhibit key differences. To bridge these differences, we examine helically symmetric flows, which are described by the reduced helical system (r,ξ), with ξ =az+bφ, and the constants a, b obey a²+b²>0, while (r,φ,z) denote cylindrical coordinates. Helically symmetric flows are sometimes called 2 ½ D flows, as they live on the 2D (r,ξ) manifold, while they have a 3D velocity vector (u^r, u^ξ, u^η). Here the velocity component u^η is locally parallel to the helix. As to whether u^η is zero or not, these helical flows possess infinitely many generalised helicity conservation laws (CL) for u^η≠0 and helical enstrophy for u^η=0. Matsukawa and Tsukahara (2025) conducted direct numerical simulations (DNS) of the transition processes in annular pipe flows, such as the Taylor–Couette–Poiseuille flow (TCP). The Taylor vortex and wavy Taylor vortex flows transition from their original forms to helical-shaped localized turbulence as the axial pressure gradient increases. We observe a close connection between the helical CL and the DNS. For this, helical CL were applied to DNS data from TCP flow by transforming the data into a helical coordinate framework and filtering the small-scale turbulent fluctuations to enable symmetry reduction. We find that there is a helical symmetry on the large scales, while this is broken on small scales.