Summary
In plane Couette flow, the incompressible fluid between two plane parallel walls is driven by the motion of those walls. The laminar solution, in which the streamwise velocity varies linearly in the wall-normal direction, is known to be linearly stable at all Reynolds numbers (Re). Yet, in both experiments and computations, turbulence is observed for Re ≳ 360.
In this article, we show that for certain threshold perturbations of the laminar flow, the flow approaches either steady or traveling wave solutions. These solutions exhibit some aspects of turbulence but are not fully turbulent even at Re = 4,000. However, these solutions are linearly unstable and flows that evolve along their unstable directions become fully turbulent. The solution approached by a threshold perturbation could depend upon the nature of the perturbation. Surprisingly, the positive eigenvalue that corresponds to one family of solutions decreases in magnitude with increasing Re, with the rate of decrease given by Re α with α ≈ −0.46.
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References
D. Acheson. Elementary Fluid Dynamics. Oxford University Press, Oxford, 1990.
K.H. Bech, N. Tillmark, P.H. Alfredsson, and H.I. Andersson. An investigation of turbulent plane Couette flow at low Reynolds number. Journal of Fluid Mechanics, 286:291–325, 1995.
A. Cherhabili and U. Eherenstein. Finite-amplitude equilibrium states in plane Couette flow. Journal of Fluid Mechanics, 342:159–177, 1997.
J.F. Gibson, J. Halcrow, and P. Cvitanovic. Visualizing the geometry of state space in plane Couette flow. Journal of Fluid Mechanics, 2008. To appear. Available at http://www. arxiv.org: 0705. 3957.
G. Kawahara. Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Physics of Fluids, 17:art. 041702, 2005.
S.J. Kline, W.C. Reynolds, F.A. Schraub, and P.W. Rundstadler. The structure of turbulent boundary layers. Journal of Fluid Mechanics, 30:741–773, 1967.
G. Kreiss, A. Lundbladh, and D.S. Henningson. Bounds for threshold amplitudes in subcritical shear flows. Journal of Fluid Mechanics, 270:175–198, 1994.
M. Lagha, T.M. Schneider, F. De Lillo, and B. Eckhardt. Laminar-turbulent boundary in plane Couette flow. 2007. preprint.
A. Lundbladh, D.S. Henningson, and S.C. Reddy. Threshold amplitudes for transition in channel flows. In M.Y. Hussaini, T.B. Gatski, and T.L. Jackson, editors, Turbulence and Combustion. Kluwer, Holland, 1994.
A.S. Monin and A.M. Yaglom. Statistical Fluid Mechanics. The MIT Press, Cambridge, 1971.
M. Nagata. Three dimensional finite amplitude solutions in plane Couette flow: bifurcation from infinity. Journal of Fluid Mechanics, 217:519–527, 1990.
M. Nagata. Three-dimensional traveling-wave solutions in plane Couette flow. Physical Review E, 55:2023–2025, 1997.
S.A. Orszag and L.C. Kells. Transition to turbulence in plane Poiseuille and plane Couette flow. Journal of Fluid Mechanics, 96:159–205, 1980.
T.M. Schmicgcl and B. Eckhardt. Fractal stability border in plane Coucttc flow. Physical Review letters, 79:5250, 1997.
T.M. Schneider, B. Eckhardt, and J.A. Yorke. Turbulence transition and edge of chaos in pipe flow. Physical Review Letters, 99:034502, 2007.
L.N. Trefethen, A.E. Trefethen, S.C. Reddy, and T.A. Driscoll. Hydrodynamic stability without eigenvalues. Science, 261:578–584, 1993.
D. Viswanath. Recurrent motions within plane Couette turbulence. Journal of Fluid Mechanics, 580:339–358, 2007.
F. Waleffe. Three-dimensional coherent states in plane shear flows. Physical Review Letters, 81:4140–4143, 1998.
F. Waleffe. Exact coherent structures in channel flow. Journal of Fluid Mechanics, 435:93–102, 2001.
F. Waleffe. Homotopy of exact coherent structures in plane shear flows. Physics of Fluids, 15:1517–1534, 2003.
J. Wang, J.F. Gibson, and F Waleffe. Lower branch coherent states in shear flows: transition and control. Physical Review Letters, 98:204501, 2007.
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Viswanath, D. (2008). The Dynamics of Transition to Turbulence in Plane Couette Flow. In: Munthe-Kaas, H., Owren, B. (eds) Mathematics and Computation, a Contemporary View. Abel Symposia, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68850-1_6
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DOI: https://doi.org/10.1007/978-3-540-68850-1_6
Publisher Name: Springer, Berlin, Heidelberg
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