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Exact coherent states in channel flow

Published online by Cambridge University Press:  08 January 2016

D. P. Wall  and
M. Nagata
D. P. Wall
Affiliation:
Department of Aeronautics and Astronautics, Nippon Bunri University, Oita-shi, Oita 870-0397, Japan
M. Nagata*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China
*
†Email address for correspondence: nagata.masato.45x@st.kyoto-u.ac.jp
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Abstract

Three spatially extended travelling wave exact coherent states, together with one spanwise-localised state, are presented for channel flow. Two of the extended flows are derived by homotopy from solutions to the problem of channel flow subject to a spanwise rotation investigated by Wall & Nagata (J. Fluid Mech., vol. 727, 2013, pp. 523–581). Both these flows are asymmetric with respect to the channel centreplane, and feature streaky structures in streamwise velocity flanked by staggered vortical structures. One of these flows features two streak/vortex systems per spanwise wavelength, while the other features one such system. The former substantially reduces the value of the lowest Reynolds number at which channel flow solutions ,other than the basic flow, are known to exist down to 665. The third flow has, in contrast, half-turn rotational symmetry about a streamwise axis through a point on the channel centreplane, and is found to be the flow from which one of the asymmetric flows bifurcates in a symmetry-breaking bifurcation. This flow is found to exist on an isolated bifurcation branch, whose upper and lower branches both lie on the boundary basin separating initial conditions that lead to turbulent events, and those that directly decay back to laminar flow. The structure of this flow, in which the disturbance to the basic flow is concentrated in a core region in a spanwise period, allowed the derivation of a corresponding spanwise-localised flow, which is also discussed.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 444 - 468
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Copyright
© 2016 Cambridge University Press 

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References

Abe, M.2009 Nonlinear solutions in plane MHD Poiseuille flow. Masters thesis, Kyoto University, Kyoto, Japan.Google Scholar
Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502.Google Scholar
Byrd, R. H., Lu, P., Nocedal, J. & Zhu, C. 1995 A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208.Google Scholar
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.Google Scholar
Deguchi, K. & Nagata, M. 2010 Travelling hairpin shaped fluid vortices in plane Couette flow. Phys. Rev. E 82, 056325.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Ehrenstein, U. & Koch, W. 1991 Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Gibson, J. F.2013 Channelflow: a spectral Navier–Stokes simulator in $\text{C}++$ . Tech. Rep. University of New Hampshire.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.Google 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
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.Google Scholar
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114501.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan. 70, 703–716.Google Scholar
Jimenez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.Google Scholar
Kerswell, R. R. 2005 Recent Progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.Google Scholar
Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. & Deguchi, K. 2013 Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech. 735, 1–11.CrossRefGoogle Scholar
Nishioka, M. & Asai, M. 1985 Some observations of the subcritical transition in plane Poiseuille flow. J. Fluid Mech. 150, 441–450.Google Scholar
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731–751.CrossRefGoogle Scholar
Patel, V. C. & Head, M. R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181–201.Google Scholar
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97–115.CrossRefGoogle Scholar
Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99, 074502.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.Google Scholar
Sengupta, T. K. 2012 Instabilities of Flows and Transition to Turbulence. CRC Press.Google Scholar
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 67–76.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95, 319–343.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Wall, D. P. & Nagata, M. 2006 Nonlinear secondary flow through a rotating channel. J. Fluid Mech. 564, 25–55.Google Scholar
Wall, D. P. & Nagata, M. 2013 Three-dimensional exact coherent states in rotating channel flow. J. Fluid Mech. 727, 533–581.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
Zammert, S. & Eckhardt, B. 2014a Periodically bursting edge states in plane Poiseuille flow. Fluid Dyn. Res. 46 (4), 041419.Google Scholar
Zammert, S. & Eckhardt, B. 2014b Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.Google Scholar