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Three-dimensional exact coherent states in rotating channel flow. (English) Zbl 1291.76342
Summary: Three-dimensional exact, finite-amplitude solutions are presented for the problem of channel flow subject to a system rotation about a spanwise axis. The solutions are of travelling wave form, and may bifurcate as tertiary flows from the two-dimensional streamwise-independent secondary flow, or as secondary flows directly from the basic flow. For the tertiary flows, we consider solutions of spanwise superharmonic and subharmonic type. We distinguish flows on the basis of symmetry, originating eigenmode and major solution branch, and thus identify 15 distinct flows: 5 superharmonic tertiary, 5 subharmonic tertiary and 5 secondary flows. The tertiary flows all feature a single layer of vortical structures in the spanwise-wall-normal plane, the secondary flows feature single-, double-, triple- or quadruple-layer flow structures in this plane. All flows feature low-speed streamwise-orientated streaks in the streamwise velocity component and/or pulses of low-speed streamwise velocity. The streaks may be sinusoidal or varicose. Sinusoidal streaks are flanked by staggered streamwise vortices, varicose streaks and pulses are flanked by aligned vortices. A comparison with previous simulation and experimental studies finds that the simplest three-dimensional flows observed previously correspond to superharmonic tertiary flows bifurcating from the upper branch of the secondary flow. The mean absolute vorticity of the present flows is also considered. A flattening of the profile of this vorticity is observed in the central region of the channel for two-dimensional secondary and many of the three-dimensional flows, with two-step profiles also observed. This phenomenon is attributed to mixing of the vorticity across zones of the channel in which streamwise vortex structures exist, and is demonstrated by a two-dimensional model. The phenomenon appears to be distinct to that observed in fully turbulent rotating channel flows.

MSC:
76U05 General theory of rotating fluids
76E07 Rotation in hydrodynamic stability
76D05 Navier-Stokes equations for incompressible viscous fluids
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