Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T08:39:13.734Z Has data issue: false hasContentIssue false
  • English
  • Français

Tracking stages of transition in Couette flow analytically

Published online by Cambridge University Press:  09 May 2014

Michael Karp  and
Jacob Cohen
Michael Karp
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
Jacob Cohen*
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: aerycyc@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The current study focuses on a transition scenario in which the linear transient growth mechanism is initiated by four decaying normal modes. It is shown that the four modes, the initial structure of which corresponds to counter-rotating vortex pairs, are sufficient to capture the transient growth mechanism. More importantly, it is demonstrated that the kinetic energy growth of the initial disturbance is not the key parameter in this transition mechanism. Rather, it is the ability of the transient growth process to generate an inflection point in the wall-normal direction and consequently to make the flow susceptible to a three-dimensional disturbance leading to transition to turbulence. Because of the minimal number of modes participating in the transition process, it is possible to follow its earlier key stages analytically and to compare them with the results of direct numerical simulation. This procedure reveals the role of various flow parameters during the transition, such as the difference between symmetric and antisymmetric transient growth scenarios. Moreover, it is shown that the resulting modified base flow of the linear process is not sufficient to produce a significant localized maximum of the base-flow vorticity (i.e. a ‘strong’ inflection point), and it is only due to nonlinear effects that the base flow becomes unstable with respect to an infinitesimal three-dimensional disturbance. Finally, the physical mechanism during key stages of transition is well captured by the analytical expressions. Furthermore, the vortex dynamics during these stages is very similar to the model proposed by Cohen, Karp & Mehta (J. Fluid Mech., vol. 747, 2014, pp. 30–43) according to which streamwise variation of the initial counter-rotating vortex pair is required to generate concentrated spanwise vorticity, which together with the lift-up by the induced velocity and shear of the base flow generates packets of hairpins.

JFM classification

Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 748 , 10 June 2014 , pp. 896 - 931
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.Google Scholar
Asai, M., Minagawa, M. & Nishioka, M. 2002 The instability and breakdown of a near-wall low-speed streak. J. Fluid Mech. 455, 289–314.Google Scholar
Ben-Dov, G., Levinski, V. & Cohen, J. 2003 On the mechanism of optimal disturbances: the role of a pair of nearly parallel modes. Phys. Fluids 15, 1961–1972.Google Scholar
Biau, D. & Bottaro, A. 2004 Transient growth and minimal defects: two possible initial paths of transition to turbulence in plane shear flows. Phys. Fluids 16, 3515–3529.Google Scholar
Brandt, L. & de Lange, H. C. 2008 Streak interactions and breakdown in boundary layer flows. Phys. Fluids 20, 024107.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Cherubini, S. & De Palma, P. 2013 Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716, 251–279.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82, 066302.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011a Edge states in a boundary layer. Phys. Fluids 23, 051705.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011b The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221–253.Google Scholar
Cohen, J., Karp, M. & Mehta, V. 2014 A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows. J. Fluid Mech 747, 30–43.CrossRefGoogle Scholar
Cohen, J., Karp, M. & Shukhman, I.2009a The formation of packets of hairpins in shear flows. Proceedings of Global Flow Instability and Control IV, Crete, Greece. www.cfm.upm.es/resources/Crete-IV—Summary.pdf.Google Scholar
Cohen, J., Philip, J. & Ben-Dov, G. 2009b Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows. Phil. Trans. R. Soc. A 367, 509–527.Google Scholar
Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83–114.Google Scholar
Dauchot, O. & Daviaud, F. 1995 Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids 7, 335–343.CrossRefGoogle Scholar
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water in pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82, 026316.Google Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25, 084103.Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108, 044501.Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Gavarini, M. I., Bottaro, A. & Nieuwstadt, F. T. M. 2004 The initial stage of transition in pipe flow: role of optimal base-flow distortions. J. Fluid Mech. 517, 131–165.Google Scholar
Gibson, J. F.2012 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep. University of New Hampshire. Channelflow.org.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Han, G., Tumin, A. & Wygnanski, I. 2000 Laminar–turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. Part 2. Late stage of transition. J. Fluid Mech. 419, 1–27.Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Report CTR-S88, pp. 193–208. Centre for Turbulence Research.Google Scholar
Kreiss, G., Lundbladh, A. & Henningson, D. S. 1994 Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech. 270, 175–198.Google Scholar
Lundbladh, A., Henningson, D. S. & Reddy, S. C. 1994 Threshold amplitudes for transition in channel flows. In Transition, Turbulence and Combustion (ed. Hussaini, M. Y. et al. ), vol. I, pp. 309–318. Kluwer.CrossRefGoogle Scholar
Marquillie, M., Ehrenstein, U. & Laval, J.-P. 2011 Instability of streaks in wall turbulence with adverse pressure gradient. J. Fluid Mech. 681, 205–240.Google Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number $10^7$ . J. Comput. Phys. 186, 178–197.Google Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.Google Scholar
O’Sullivan, P. L. & Breuer, K. S. 1994 Transient growth in circular pipe flow. I. Linear disturbances. Phys. Fluids 6, 3643–3651.CrossRefGoogle Scholar
Patel, V. C. & Head, M. R. 1969 Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech. 38, 181–201.Google Scholar
Philip, J., Svizher, A. & Cohen, J. 2007 Scaling law for a subcritical transition in plane Poiseuille flow. Phys. Rev. Lett. 98, 154502.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.Google Scholar
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415–443.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.Google Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 1, 57–72.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53, 15–47.Google Scholar
Reshotko, E. & Tumin, A. 2001 Spatial theory of optimal disturbances in a circular pipe flow. Phys. Fluids 13, 991–996.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Tillmark, N. & Alfredsson, P. H. 1992 Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89–102.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59, 281–335.CrossRefGoogle Scholar
Zikanov, O. Y. 1996 On the instability of pipe Poiseuille flow. Phys. Fluids 8, 2923–2932.Google Scholar