Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T23:03:30.508Z Has data issue: false hasContentIssue false
  • English
  • Français

A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows

Published online by Cambridge University Press:  10 April 2014

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

Abstract

Packets of hairpin-shaped vortices and streamwise counter-rotating vortex pairs (CVPs) appear to be key structures during the late stages of the transition process as well as in low-Reynolds-number turbulence in wall-bounded flows. In this work we propose a robust model consisting of minimal flow elements that can produce packets of hairpins. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a two-dimensional (2D) wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable: the CVP modifies the base flow due to the induced velocity forming an inflection point in the base-flow velocity profile. Consequently, the 2D wavy vortex sheet is amplified, causing undulation of the CVP. The undulation is further enhanced as the wave continues to be amplified and eventually the CVP breaks down into several segments. The induced velocity generates highly localized patches of spanwise vorticity above the regions connecting two consecutive streamwise elements of the CVP. These patches widen with time and join with the streamwise vortical elements situated beneath them forming a packet of hairpins. The results of the unbounded (having no walls) model are compared with pipe and channel flow experiments and with a direct numerical simulation of a transition process in Couette flow. The good agreement in all cases demonstrates the universality and robustness of the model.

JFM classification

Instability: Instability Instability: Transition to turbulence Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 30 - 43
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

Acarlar, M. S. & Smith, C. R. 1987 A study of hairpin vortices in a laminar boundary layer. Part 1. Hairpin vortices generated by a hemisphere protuberance. J. Fluid Mech. 175, 1–41.Google Scholar
Adrian, R. J., Meinhart, C. D. & Tomkins, C. D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.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. & Cohen, J. 2007 Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow. J. Fluid Mech. 588, 189–215.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
Blackwelder, R. F. 1983 Analogies between transitional and turbulent boundary layers. Phys. Fluids 26, 2807–2815.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., Robinet, J. -C. & Bottaro, A. 2010a 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
Cherubini, S., Robinet, J. -C., Bottaro, A. & De Palma, P. 2010b Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J. Fluid Mech. 656, 231–259.Google Scholar
Chu, J. C. & Goldstein, D. B.2012 Investigation of turbulent wedge spreading mechanism with comparison to turbulent spots. AIAA Paper 2012-0751, Proceedings of the 50th AIAA ASM, Nashville, TN, pp. 1–15. DOI: 10.2514/6.2012-751.Google Scholar
Cohen, J., Karp, M. & Shukhman, I.2009 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., Shukhman, I. G., Karp, M. & Philip, J. 2010 An analytical-based method for studying the nonlinear evolution of localized vortices in planar homogenous shear flows. J. Comput. Phys. 229, 7765–7773.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
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.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. 2009 Nonlinear wave interactions from transient growth in plane-parallel shear flows. Eur. J. Mech. (B/Fluids) 28, 420–429.Google Scholar
Haidari, A. H. & Smith, C. R. 1994 The generation and regeneration of single hairpin vortices. J. Fluid Mech. 277, 135–162.Google Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.Google Scholar
Hof, B., de Lozar, A., Avila, M., Tu, X. & Schneider, T. M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327, 1491–1494.Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research, Report CTR-S88, pp. 193–208.Google Scholar
Hutchins, N., Hambleton, W. T. & Marusic, I. 2005 Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54.Google Scholar
Levinski, V. & Cohen, J. 1995 The evolution of a localized vortex disturbance in external shear flows. Part 1. Theoretical considerations and preliminary experimental results. J. Fluid Mech. 289, 159–182.Google Scholar
Malkiel, E., Levinski, V. & Cohen, J. 1999 The evolution of a localized vortex disturbance in external shear flows. Part 2. Comparison with experiments in rotating shear flows. J. Fluid Mech. 379, 351–380.Google Scholar
Marusic, I. 2001 On the role of large-scale structures in wall turbulence. Phys. Fluids 13, 735–743.Google Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329, 193–196.Google Scholar
Peixinho, J. & Mullin, T. 2007 Finite-amplitude thresholds for transition in pipe flow. J. Fluid Mech. 582, 169–178.Google Scholar
Philip, J.2009 The relationship between streaks and hairpin vortices in subcritical wall bounded shear flows. PhD thesis, Technion - Israel Institute of Technology.Google Scholar
Philip, J. & Cohen, J. 2010 Formation and decay of coherent structures in pipe flow. J. Fluid Mech. 655, 258–279.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
Rayleigh, L. 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 1, 57–72.Google Scholar
Rist, U.2012 Visualization and tracking of vortices and shear layers in the late stages of boundary-layer laminar-turbulent transition. AIAA paper 2012-0084, Proceedings of the 50th AIAA ASM, Nashville, TN, pp. 1–20. DOI: 10.2514/6.2012-84.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.Google Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–226.Google Scholar
Skote, M., Haritonidis, J. H. & Henningson, D. S. 2002 Varicose instabilities in turbulent boundary layers. Phys. Fluids 14, 2309–2323.Google Scholar
Strand, J. S. & Goldstein, D. B. 2011 Direct numerical simulations of riblets to constrain the growth of turbulent spots. J. Fluid Mech. 668, 267–292.Google Scholar
Suponitsky, V., Cohen, J. & Bar-Yoseph, P. Z. 2005 The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow. J. Fluid Mech. 535, 65–100.Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.Google Scholar
Theodorsen, T.1952 Mechanism of turbulence. In Proceedings of Second Midwestern Conf. on Fluid Mech. pp. 1–19.Google Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.Google Scholar
Zhou, J., Adrian, R. J. & Balachandar, S. 1996 Autogeneration of near-wall vortical structures in channel flow. Phys. Fluids 8, 288–291.Google Scholar
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.Google Scholar