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Self-sustained states at Kolmogorov microscale

Published online by Cambridge University Press:  28 September 2015

Kengo Deguchi*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
Email address for correspondence: [email protected]

Abstract

It is shown theoretically how the scaling of coherent structures in shear flows changes their asymptotic development at large Reynolds number. Based on the theory a family of nonlinear self-sustained states at Kolmogorov microscale is numerically identified on the laminar–turbulent boundary of shear flows. Theoretically and numerically the states connect to known asymptotic states existing at larger scale. The asymptotically very small amplitude of the new states may explain why strongly sheared, linearly stable laminar flows can cause a turbulent transition by small disturbances. The numerically obtained Kolmogorov-scale solutions can be used to describe the theoretically minimal self-sustained structures appearing in various shear flows.

Type
Rapids
Copyright
© 2015 Cambridge University Press 

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References

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.CrossRefGoogle ScholarPubMed
Blackburn, H. M., Hall, P. & Sherwin, S. 2013 Lower branch equilibria in Couette flow: the emergence of canonical stats for arbitrary shear flows. J. Fluid Mech. 726, R2.CrossRefGoogle Scholar
Brand, E. & Gibson, J. F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.Google Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137153.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014a Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. Lond. A 372, 20130352, 1–19.Google Scholar
Deguchi, K. & Hall, P. 2014b The high Reynolds number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99112.Google Scholar
Deguchi, K. & Hall, P. 2014c Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602625.CrossRefGoogle Scholar
Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 5885.Google Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25, 084103.CrossRefGoogle Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 2561.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, E. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.CrossRefGoogle 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), 15941598.CrossRefGoogle ScholarPubMed
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703716.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.Google Scholar
Lemoult, G., Aider, J.-L. & Wesfield, J. E. 2012 Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow. Phys. Rev. E 85, 025303.CrossRefGoogle ScholarPubMed
Morrison, J. F., McKeon, B. J., Jiang, W. & Smits, A. J. 2004 Scaling of the streamwise velocity component in turbulent pipe flow. J. Fluid Mech. 508, 99131.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.Google Scholar
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731751.Google Scholar
Sommerfeld, A. 1908 Ein Beitrag zur hydrodynamischen Erlärung der turbulenten Flüssigkeitsbewegungen. Atti del 4. Congr. Internat. Math. Roma 3, 116124.Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states: transition and control. Phys. Rev. Lett. 98, 204501.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171534.Google Scholar
Willis, A. P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514540.CrossRefGoogle Scholar
Zammert, S. & Eckhardt, B. 2014 Streamwise and doubly-localised periodic orbits in plane Couette flow. J. Fluid Mech. 761, 348359.CrossRefGoogle Scholar