Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T19:32:58.207Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alfredsson, P. H. & Persson, H. 1989 Instabilities in channel flow with system rotation. J. Fluid Mech. 202, 543–557.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
DiPrima, R. C. & Swinney, H. L. 1985 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. Swinney, H. L. & Gollub, J. P.), Topics in Applied Physics, vol. 45. pp. 139–180. Springer.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Finlay, W. H. 1990 Transition to oscillatory motion in rotating channel flow. J. Fluid Mech. 215, 209–227.Google Scholar
Finlay, W. H. 1992 Transition to turbulence in a rotating channel. J. Fluid Mech. 237, 73–99.Google Scholar
Finlay, W. H., Keller, J. B. & Ferziger, J. H. 1988 Instability and transition in curved channel flow. J. Fluid Mech. 194, 417–456.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.Google Scholar
Hopfinger, E. J. & Linden, P. F. 1990 The effect of background rotation on fluid motions: a report on Euromech 245. J. Fluid Mech. 211, 417–435.Google Scholar
Iida, O., Fukudome, K., Iwata, T. & Nagano, Y. 2010 Low Reynolds number effects on rotating turbulent Poiseuille flow. Phys. Fluids 22, 085106.Google Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.Google Scholar
Kristoffersen, R. & Andersson, H. I. 1993 Direct simulations of low-reynolds-number turbulent flow in a rotating channel. J. Fluid Mech. 256, 163–197.Google Scholar
Lesieur, M., Yaglom, A. M. & David, F. (Eds) 2002 New Trends in Turbulence. Springer.Google Scholar
Lezius, D. K. & Johnstone, J. P. 1976 The structure and stability of turbulent boundary layers in rotating channel flow. J. Fluid Mech. 77, 153–175.Google Scholar
Matsubara, M. & Alfredsson, P. H. 1998 Secondary instability in rotating channel flow. J. Fluid Mech. 368, 27–50.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. 1998 Tertiary solutions and their stability in rotating plane Couette flow. J. Fluid Mech. 358, 357–378.Google Scholar
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328.CrossRefGoogle Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97–115.Google Scholar
Tsukahara, T., Tillmark, N. & Alfredsson, P. H. 2010 Flow regimes in a plane Couette flow with system rotation. J. Fluid Mech. 648, 5–33.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Wall, D. P. & Nagata, M. 2006 Nonlinear secondary flow through a rotating channel. J. Fluid Mech. 564, 25–55.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Yanase, S. & Kaga, Y. 2004 Zero-mean-absolute-vorticity state and vortical structures in rotating channel flow. J. Phys. Soc. Jpn 73, 1419–1422.CrossRefGoogle Scholar
Yang, K. S. & Kim, J. 1991 Numerical investigation of instability and transition in rotating plane Poiseuille flow. Phys. Fluids A 3, 633–641.Google Scholar

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 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.

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

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

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

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

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

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

Download Wall and M. Nagata supplementary movie(Video)
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