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Universal continuous transition to turbulence in a planar shear flow

Published online by Cambridge University Press:  04 July 2017

Matthew Chantry ,
Laurette S. Tuckerman Open the ORCID record for Laurette S. Tuckerman [Opens in a new window]  and
Dwight Barkley
Matthew Chantry
Affiliation:
Atmospheric, Oceanic and Planetary Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, France Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Laurette S. Tuckerman*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, France Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Dwight Barkley
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
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Abstract

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

JFM classification

Instability: Instability Instability: Transition to turbulence
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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References

Avila, K.2013 Shear flow experiments: characterizing the onset of turbulence as a phase transition. PhD thesis, Georg-August University School of Science.Google Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.Google Scholar
Barkley, D 2011 Simplifying the complexity of pipe flow. Phys. Rev. E 84, 016309.Google ScholarPubMed
Barkley, D 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.Google Scholar
Bergé, P., Pomeau, Y. & Vidal, C. 1998 L’espace Chaotique. Hermann Éd. des Sciences et des Arts.Google Scholar
Bottin, S. & Chaté, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6, 143–155.CrossRefGoogle Scholar
Bottin, S., Daviaud, F., Manneville, P. & Dauchot, O. 1998 Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett. 43, 171–176.Google Scholar
Chantry, M., Tuckerman, L. S. & Barkley, D. 2016 Turbulent–laminar patterns in shear flows without walls. J. Fluid Mech. 791, R8.Google Scholar
Chaté, H. & Manneville, P. 1988 Spatio-temporal intermittency in coupled map lattices. Physica D 32, 409–422.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2010 Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129.CrossRefGoogle Scholar
Grassberger, P. 1982 On phase transitions in Schlögl’s second model. Z. Phys. B 47, 365–374.CrossRefGoogle Scholar
Henkel, M., Hinrichsen, H., Lübeck, S. & Pleimling, M. 2008 Non-equilibrium Phase Transitions. vol. 1. Springer.Google Scholar
Janssen, H.-K. 1981 On the nonequilibrium phase transition in reaction–diffusion systems with an absorbing stationary state. Z. Phys. B 42, 151–154.Google Scholar
Kanazawa, T., Shimizu, M. & Kawahara, G.2017 Periodic solutions representing the origin of turbulent bands in channel flow. Presented at KITP Conference: Recurrence, Self-Organization, and the Dynamics of Turbulence, 9–13 January 2017, Kavli Institute for Theoretical Physics. http://online.kitp.ucsb.edu/online/transturb-c17/kawahara/.Google Scholar
Kaneko, K. 1985 Spatiotemporal intermittency in coupled map lattices. Prog. Theor. Exp. Phys. 74, 1033–1044.Google Scholar
Lemoult, G., Shi, L., Avila, K., Jalikop, S. V., Avila, M. & Hof, B. 2016 Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys. 12, 254–258.Google Scholar
Lübeck, S. 2004 Universal scaling behavior of non-equilibrium phase transitions. Intl J. Mod. Phys. B 18, 3977–4118.Google Scholar
Manneville, P. 2016 Transition to turbulence in wall-bounded flows: where do we stand? Bull. JSME 3, 15–00684.Google Scholar
Marcus, P. S. & Lee, C. 1998 A model for eastward and westward jets in laboratory experiments and planetary atmospheres. Phys. Fluids 10, 1474–1489.CrossRefGoogle Scholar
Paranjape, C., Vasudevan, M., Duguet, Y. & Hof, B.2017 Transition to turbulence in channel flow. Presented at KITP Conference: Recurrence, Self-Organization, and the Dynamics of Turbulence, 9–13 January 2017, Kavli Institute for Theoretical Physics. http://online.kitp.ucsb.edu/online/transturb-c17/hof/rm/jwvideo.html.Google Scholar
Pedlosky, J 2012 Geophysical Fluid Dynamics. Springer.Google Scholar
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11.Google Scholar
Prigent, A., Grégoire, G., Chaté, H. & Dauchot, O. 2003 Long-wavelength modulation of turbulent shear flows. Physica D 174, 100–113.Google Scholar
Rolf, J., Bohr, T. & Jensen, M. H. 1998 Directed percolation universality in asynchronous evolution of spatiotemporal intermittency. Phys. Rev. E 57, R2503.Google Scholar
Sano, M. & Tamai, K. 2016 A universal transition to turbulence in channel flow. Nat. Phys. 12, 249–253.Google Scholar
Shih, H.-Y., Hsieh, T.-L. & Goldenfeld, N. 2016 Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys. 12, 245–248.Google Scholar
Suri, B., Tithof, J., Mitchell, R. Jr., Grigoriev, R. O. & Schatz, M. F. 2014 Velocity profile in a two-layer Kolmogorov-like flow. Phys. Fluids 26, 053601.CrossRefGoogle Scholar
Takeuchi, K. A., Kuroda, M., Chaté, H. & Sano, M. 2007 Directed percolation criticality in turbulent liquid crystals. Phys. Rev. Lett. 99, 234503.CrossRefGoogle ScholarPubMed
Takeuchi, K. A., Kuroda, M., Chaté, H. & Sano, M. 2009 Experimental realization of directed percolation criticality in turbulent liquid crystals. Phys. Rev. E 80, 051116.Google ScholarPubMed
Xiong, X., Tao, J., Chen, S. & Brandt, L. 2015 Turbulent bands in plane-Poiseuille flow at moderate Reynolds numbers. Phys. Fluids 27, 041702.CrossRefGoogle Scholar

Chantry et al. supplementary movie

Evolution of turbulence fraction in a domain of streamwise and spanwise size $2560h imes 2560h$. Left (black box): $Re=173.808$, very near $Re_c=173.80$. Right (red box): $Re=173.888$, above $Re_c$. Middle: Power law $F \sim t^{-0.4505}$ (blue line). For $Re\approx Re_c$ (black curve), the power law is followed throughout the evolution shown. For $Re>Re_c$ (red curve), after an initial power-law decay, the turbulence fraction saturates at a finite value.

Download Chantry et al. supplementary movie(Video)
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