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The onset of transient turbulence in minimal plane Couette flow

Published online by Cambridge University Press:  10 January 2019

Julius Rhoan T. Lustro Open the ORCID record for Julius Rhoan T. Lustro [Opens in a new window] ,
Genta Kawahara Open the ORCID record for Genta Kawahara [Opens in a new window] ,
Lennaert van Veen Open the ORCID record for Lennaert van Veen [Opens in a new window] ,
Masaki Shimizu Open the ORCID record for Masaki Shimizu [Opens in a new window]  and
Hiroshi Kokubu Open the ORCID record for Hiroshi Kokubu [Opens in a new window]
Julius Rhoan T. Lustro
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Genta Kawahara*
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Lennaert van Veen
Affiliation:
Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario L1H 7K4, Canada
Masaki Shimizu
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Hiroshi Kokubu
Affiliation:
Graduate School of Science, Kyoto University, Kitashirakawa Oiwake, Sakyo, Kyoto 606-8502, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The onset of transient turbulence in minimal plane Couette flow has been identified theoretically as homoclinic tangency with respect to a simple edge state for the Navier–Stokes equation, i.e., the gentle periodic orbit (the lower branch of a saddle-node pair) found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291–300). The first tangency of a pair of distinct homoclinic orbits to this periodic edge state has been discovered at Reynolds number $Re\equiv Uh/\unicode[STIX]{x1D708}=Re_{T}\approx 240.88$ ($U$, $h$, and $\unicode[STIX]{x1D708}$ being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively). At $Re>Re_{T}$ a Smale horseshoe appears on the Poincaré section through transversal homoclinic points to generate a transient chaos that eventually relaminarises. In numerical experiments a sustaining chaos, which is a consequence of period-doubling cascade stemming from the upper branch of another saddle-node pair of periodic orbits, is observed in a narrow range of the Reynolds number, $Re\approx 240.40$–240.46. At the upper edge of this $Re$ range it is found that the chaotic set touches the lower branch of this pair, i.e., another edge state. The corresponding chaotic attractor is replaced by a chaotic saddle at $Re\approx 240.46$, and subsequently this saddle touches the gentle periodic edge state on the boundary of the laminar basin at the tangency Reynolds number $Re=Re_{T}$. After this crisis on the boundary of the laminar basin, for $Re>Re_{T}$, chaotic transients that eventually relaminarise can be observed.

JFM classification

Nonlinear Dynamical Systems: Chaos Instability: Transition to turbulence
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , R2
Copyright
© 2019 Cambridge University Press 

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