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Three-dimensional exact coherent states in rotating channel flow

Published online by Cambridge University Press:  28 June 2013

D. P. Wall  and
M. Nagata
D. P. Wall*
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, University of Kyoto, Kyoto 606-8501, Japan
M. Nagata
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, University of Kyoto, Kyoto 606-8501, Japan
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 727 , 25 July 2013 , pp. 533 - 581
Copyright
©2013 Cambridge University Press 

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Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G1 flow shown in figure 5. Two main vortices can be seen with each vortex strongest when the other is weakest.

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Video 17.7 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G1 flow shown in figure 5. Two main vortices can be seen with each vortex strongest when the other is weakest.

Download Wall and M. Nagata supplementary movie(Video)
Video 3.2 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G3 flow shown in figure 6. The vortex structures oscillate across a full spanwise wavelength, moving from close to the channel walls towards the centreline (z = 0) of the channel.

Download Wall and M. Nagata supplementary movie(Video)
Video 18.2 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G3 flow shown in figure 6. The vortex structures oscillate across a full spanwise wavelength, moving from close to the channel walls towards the centreline (z = 0) of the channel.

Download Wall and M. Nagata supplementary movie(Video)
Video 3.4 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G5 flow shown in figure 6. The aligned main vortices exhibit almost zero spanwise movement while the vortices aligned either side of the pulses appear periodically near the z = 1 wall.

Download Wall and M. Nagata supplementary movie(Video)
Video 4.4 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G5 flow shown in figure 6. The aligned main vortices exhibit almost zero spanwise movement while the vortices aligned either side of the pulses appear periodically near the z = 1 wall.

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Video 757.6 KB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G6 flow shown in figure 9. The flow in each spanwise half-wavelength resembles that of the G1 superharmonic flow.

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Video 9.4 MB

Wall and M. Nagata supplementary movie

Movie of velocity vectors in spanwise -wall normal plane for G6 flow shown in figure 9. The flow in each spanwise half-wavelength resembles that of the G1 superharmonic flow.

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Video 1.5 MB

Wall and M. Nagata supplementary movie

Movie 5: Movie of velocity vectors in spanwise -wall normal plane for G7 flow shown in figure 9. Both staggered vortices (appearing in the centre of the half-wavelength intervals 0 < y′ < 0.5 and 0.5 < y′ < 1) and aligned vortices (appearing adjacent to y′ = 0, 0.5, 1.0) can be observed.

Download Wall and M. Nagata supplementary movie(Video)
Video 10.7 MB

Wall and M. Nagata supplementary movie

Movie 5: Movie of velocity vectors in spanwise -wall normal plane for G7 flow shown in figure 9. Both staggered vortices (appearing in the centre of the half-wavelength intervals 0 < y′ < 0.5 and 0.5 < y′ < 1) and aligned vortices (appearing adjacent to y′ = 0, 0.5, 1.0) can be observed.

Download Wall and M. Nagata supplementary movie(Video)
Video 1.7 MB

Wall and M. Nagata supplementary movie

Movie 6: Movie of velocity vectors in spanwise -wall normal plane for G10 flow shown in figure 9. Six or four main streamwise vortices can be observed at any point in time depending on whether a low-speed pulse is respectively present or not.

Download Wall and M. Nagata supplementary movie(Video)
Video 18.1 MB

Wall and M. Nagata supplementary movie

Movie 6: Movie of velocity vectors in spanwise -wall normal plane for G10 flow shown in figure 9. Six or four main streamwise vortices can be observed at any point in time depending on whether a low-speed pulse is respectively present or not.

Download Wall and M. Nagata supplementary movie(Video)
Video 3.3 MB

Wall and M. Nagata supplementary movie

Movie 7: Movie of velocity vectors in spanwise -wall normal plane for G11 flow shown in figure 12. Aligned vortices periodically appear and disappear with the appearance and disappearance of low-speed pulses.

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Video 15.4 MB

Wall and M. Nagata supplementary movie

Movie 7: Movie of velocity vectors in spanwise -wall normal plane for G11 flow shown in figure 12. Aligned vortices periodically appear and disappear with the appearance and disappearance of low-speed pulses.

Download Wall and M. Nagata supplementary movie(Video)
Video 2.9 MB

Wall and M. Nagata supplementary movie

Movie 8: Movie of velocity vectors in spanwise -wall normal plane for G13 flow shown in figure 12. The two-layer vortex structure of this solution can be observed.

Download Wall and M. Nagata supplementary movie(Video)
Video 17.4 MB

Wall and M. Nagata supplementary movie

Movie 8: Movie of velocity vectors in spanwise -wall normal plane for G13 flow shown in figure 12. The two-layer vortex structure of this solution can be observed.

Download Wall and M. Nagata supplementary movie(Video)
Video 3.3 MB

Wall and M. Nagata supplementary movie

Movie 9: Movie of velocity vectors in spanwise -wall normal plane for G14 flow shown in figure 12. The layer of vortices closest to the z = 1 wall in the two-layer structure is invariant in the streamwise direction.

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Video 13.9 MB

Wall and M. Nagata supplementary movie

Movie 9: Movie of velocity vectors in spanwise -wall normal plane for G14 flow shown in figure 12. The layer of vortices closest to the z = 1 wall in the two-layer structure is invariant in the streamwise direction.

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Video 2.5 MB

Wall and M. Nagata supplementary movie

Movie 10: Movie of velocity vectors in spanwise -wall normal plane for G15 flow shown in figure 12. A quadruple-layer streamwise vortex structure can be observed featuring weak inner and stronger outer layers.

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Video 19.4 MB

Wall and M. Nagata supplementary movie

Movie 10: Movie of velocity vectors in spanwise -wall normal plane for G15 flow shown in figure 12. A quadruple-layer streamwise vortex structure can be observed featuring weak inner and stronger outer layers.

Download Wall and M. Nagata supplementary movie(Video)
Video 4.1 MB