Centre modes in pipe flow

Abstract

Abstract In two previous papers, Ozcakir, Tanveer, Hall, & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) and Ozcakir, Hall & Tanveer (2019, Nonlinear exact coherent structures in pipe flow and their instabilities, J. Fluid Mech., 868, 341–368) investigated numerically and asymptotically high Reynolds number exact coherent structures in pipe flow. It was found that, in addition to the structures described by the vortex–wave interaction theory by Hall & Smith (1991, On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech., 227, 641–666), there exists vortical structures localized near the centre of the pipe with a core of size $O(Re^{-1/4})$ convected downstream at a speed that deviates from the pipe centreline speed by $O(Re^{-1/2})$, where $Re$ is the Reynolds number. In the finite Reynolds number calculations by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328), asymptotic state was referred to as a nonlinear viscous core state (NVC). However the reduced asymptotic equations were not solved and only limited confirmation of the theory was found numerically. Here, in order to conclusively confirm the existence of the NVC state we first describe direct numerical calculations on the asymptotically reduced $Re>>1$ equations for such state states. The results are then compared in detail to the finite $Re$ calculations up-to $Re=10^6$; the latter regime is at much higher values of the Reynolds number than those reported in Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328). The results are found to be in excellent agreement with the finite $Re$ calculations in a region between $Re=10^5$ and $10^6$, thereby confirming that the structure observed by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) is indeed a finite Reynolds number realization of an asymptotic NVC state.