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Recurrent motions within plane Couette turbulence

Published online by Cambridge University Press:  21 May 2007

D. VISWANATH
D. VISWANATH*
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
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Abstract

The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low-speed fluid away from the wall, has been studied experimentally, theoretically and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. Five new three-dimensional solutions of turbulent plane Couette flow are produced, one of which is periodic while the other four are relative periodic. Each of these five solutions demonstrates the breakup and re-formation of near-wall coherent structures. Four of our solutions are periodic, but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic, but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. It is argued that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the breakup and re-formation of coherent structures. A new periodic solution of plane Couette flow is also computed that could be related to transition to turbulence.

The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. This computationally demanding requirement is addressed with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas – namely the Newton–Krylov iteration and the locally constrained optimal hook step. Each of the six solutions is accompanied by an error estimate.

Dynamical principles are discussed that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier–Stokes equation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 580 , 10 June 2007 , pp. 339 - 358
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Acarlar, M. S. & Smith, C. R. 1987 A study of hairpin vortices in a laminar boundary layer. J. Fluid Mech. 175, 1–83.CrossRefGoogle Scholar
Ascher, U. M., Ruuth, S. J. & Spiteri, R. J. 1997 Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Maths 25, 151–167.CrossRefGoogle Scholar
Bech, K. H., Tillmark, N., Alfredsson, P. H. & Andersson, H. I. 1995 An investigation of turbulent plane Couette flow at low Reynolds number. J. Fluid Mech. 286, 291–325.CrossRefGoogle Scholar
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G. 2005 Chaos: Classical and Quantum. ChaosBook.org, Niels Bohr Institute, Copenhagen.Google Scholar
Dennis, J. E. & Schnabel, R. B. 1996 Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, art. 224502.CrossRefGoogle ScholarPubMed
Gibson, J. F. 2002 Dynamical systems models of wall-bounded, shear-flow turbulence. PhD thesis, Cornell University.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Hof, B., van, Doorne, C. W. H. Westerweel, C. W. H. et al. . 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flows. Science 305, 1594–1598.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems, and Symmetry. Cambridge Univeristy Press.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 701–714.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2005 Interaction between large-scale structure and near-wall structures in channel flow. J. Fluid Mech. 524, 249–262.Google Scholar
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M. & Shiba, M. 2005 Characterization of near-wall turbulence in terms of equilibrium and bursting solutions. Phys. Fluids 17, 015105.CrossRefGoogle Scholar
Katok, A. & Hasselblatt, B. 1995 Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press.CrossRefGoogle Scholar
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
Kelley, C. T. 2003 Solving Nonlinear Equations with Newton's Method. SIAM, Philadelphia.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Lorenz, E. N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics. MIT Press.Google Scholar
Moser, J. 1973 Stable and Random Motions in Dynamical Systems. Princeton University Press.Google Scholar
Sancheź, J., Net, M., Garćia-Archilla, B. & Simó, C. 2004 Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201, 13–33.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Smith, C. R. & Metzler, S. P. 1983 The study of hairpin vortices in a laminar boundary layer. J. Fluid Mech. 129, 27–54.CrossRefGoogle Scholar
Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. SIAM, Philadelphia.CrossRefGoogle Scholar
Viswanath, D. 2001 Global errors of numerical ODE solvers and Lyapunov's theory of stability. IMA J. Numer. Anal. 21, 387–406.CrossRefGoogle Scholar
Viswanath, D. 2003 Symbolic dynamics and periodic orbits of the Lorenz attractor. Nonlinearity 16, 1035–1056.CrossRefGoogle Scholar
Viswanath, D. 2004 The fractal property of the Lorenz attractor. Physica D 190, 115–128.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar