J. Fluid Mech. (2017), vol. 816, pp. 719–745. c  Cambridge University Press 2017 doi:10.1017/jfm.2017.103 719 Nonlinear mode interactions in a counter-rotating split-cylinder flow Paloma Gutierrez-Castillo 1 and Juan M. Lopez 1, † 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (Received 11 November 2016; revised 9 February 2017; accepted 10 February 2017) The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number Re based on the rotation rate and radius, imparts the system with an O(2) symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The O(2) symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber m (the value of m depending on the cylinder aspect ratio Γ ) are the primary modes of instability as Re and Γ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a m = 2 steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved. Key words: bifurcation, nonlinear dynamical systems, nonlinear instability 1. Introduction Flows driven between differentially rotating disks have a long history due to their relevance to many processes in nature and technology, as well as being of fundamental interest (Batchelor 1951; Stewartson 1953; Rott & Lewellen 1966; Zandbergen & Dijkstra 1987; Lopez 1998; Launder, Poncet & Serre 2010). The fundamental issue concerning the nature of the bulk flow between differentially rotating disks depends critically on the lateral conditions at large radii, and for enclosed systems, the sidewall boundary condition plays a major role in determining the bulk flow, due to the induced meridional flow. For the most part, these types of † Email address for correspondence: juan.m.lopez@asu.edu available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2017.103 Downloaded from https:/www.cambridge.org/core. Arizona State University Libraries, on 13 Mar 2017 at 21:16:11, subject to the Cambridge Core terms of use,