Skip to main content

The minimal seed of turbulent transition in the boundary layer

  • S. Cherubini (a1), P. De Palma (a1), J.-C. Robinet (a2) and A. Bottaro (a3)

This paper describes a scenario of transition from laminar to turbulent flow in a spatially developing boundary layer over a flat plate. The base flow is the Blasius non-parallel flow solution; it is perturbed by optimal disturbances yielding the largest energy growth over a short time interval. Such perturbations are computed by a nonlinear global optimization approach based on a Lagrange multiplier technique. The results show that nonlinear optimal perturbations are characterized by a localized basic building block, called the minimal seed, defined as the smallest flow structure which maximizes the energy growth over short times. It is formed by vortices inclined in the streamwise direction surrounding a region of intense streamwise disturbance velocity. Such a basic structure appears to be a robust feature of the base flow since it is practically invariant with respect to the initial energy of the perturbation, the target time, the Reynolds number and the dimensions of the computational domain. The minimal seed grows very rapidly in time while spreading, and it triggers nonlinear effects which bring the flow to turbulence in a very efficient manner, through the formation of a turbulence spot. This evolution of the initial optimal disturbance has been studied in detail by direct numerical simulations. Using a perturbative formulation of the Navier–Stokes equations, each linear and nonlinear convective term of the equations has been analysed. The results show the fundamental role of the streamwise inclination of the vortices in the process. The nonlinear coupling of the finite amplitude disturbances is crucial to sustain such streamwise inclination, as well as to generate dislocations within the flow structures, and local inflectional velocity distributions. The analysis provides a picture of the transition process characterized by a sequence of structures appearing successively in the flow, namely, vortices, hairpin vortices and streamwise streaks. Finally, a disturbance regeneration cycle is conceived, initiated by the fast nonlinear amplification of the minimal seed, providing a possible scenario for the continuous regeneration of the same fundamental flow structures at smaller space and time scales.

Corresponding author
Email address for correspondence:
Hide All
1. Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.
2. Berlin, S., Lundbladh, A. & Henningson, D. S. 1994 Spatial simulations of oblique transition in a boundary layer. Phys. Fluids 6, 13 , 1949.
3. Berlin, S., Wiegel, M. & Henningson, D. S. 1999 Numerical and experimental investigations of oblique boundary layer transition. J. Fluid Mech. 393, 2357.
4. Biau, D. & Bottaro, A. 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc. A 367, 529544.
5. Bottaro, A. 1990 Note on open boundary conditions for elliptic flows. Numer. Heat Transfer B 18, 243256.
6. Bottaro, A. 1993 On longitudinal vortices in curved channel flow. J. Fluid Mech. 251, 627660.
7. Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in a boundary layers subject to free stream turbulence. J. Fluid Mech. 517, 167198.
8. Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2010a Rapid path to transition via nonlinear localized optimal perturbations. Phys. Rev. E 82, 066302.
9. Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2011 Edge states in a boundary layer. Phys. Fluids 23, 051705.
10. Cherubini, S., Robinet, J.-Ch., Bottaro, A. & De Palma, P. 2010b Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J. Fluid Mech. 656, 231259.
11. Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
12. Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12, 120130.
13. Corino, E. R. & Brodkey, R. S. 1969 A visual investigation of the wall region in turbulent flow. J. Fluid Mech. 37, 1.
14. Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82, 026316.
15. Duguet, Y., Schlatter, P. & Henningson, D. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21, 111701.
16. Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition of pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
17. Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
18. Farrell, B. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 20932102.
19. Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243.
20. Guo, Y. & Finlay, W. H. 1991 Splitting, merging and wave length selection of vortices in curved and/or rotating channel flow due to Eckhaus instability. J. Fluid Mech. 228, 661691.
21. Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305, 15941598.
22. Hunt, J. C. R., Wray, A. & Moin, P. 1988 Eddies, stream, and convergence zones in turbulent flows. Report CTR-S88, Center for Turbulence Research.
23. Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185212.
24. Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
25. Kline, S. J., Reynold, W. C., Schraub, F. & Rundstander, P. W. 1967 The structure of turbulent boundary layer flows. J. Fluid Mech. 30, 741773.
26. Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.
27. Luchini, P. 2000 Reynolds number indipendent instability of the Blasius boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289309.
28. Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429443.
29. Mellibovsky, F., Meseguer, A., Schneider, T. M. & Eckhardt, B. 2009 Transition in localized pipe-flow turbulence. Phys. Rev. Lett. 103, 054502.
30. Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time steppers. J. Fluid Mech. 650, 181214.
31. Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Non-equilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.
32. Nagata, M. 1990 Three-dimensional finite amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519527.
33. Polak, E. & Ribière, G. 1969 Note sur la convergence de directions conjugées. Rev. Française Automat. Informat. Rech. Opér. 16, 3543.
34. 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.
35. Reddy, S. C., Schmid, P., Baggett, P. & Henningson, D. S. 1998 On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269303.
36. Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601.
37. Saric, W. S., Reed, H. J. & Kerschen, E. 2002 Boundary-layer receptivity to free stream disturbances. Annu. Rev. Fluid Mech. 34, 291319.
38. Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Fows. Springer.
39. Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.
40. Schneider, T. M., Gibson, J. F. & Burke, J. 2010a Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.
41. Schneider, T. M., Marinc, D. & Eckhardt, B. 2010b Localised edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441451.
42. Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.
43. Singer, B. A. 1996 Characteristics of a young turbulent spot. Phys. Fluids 8, 509512.
44. Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402414.
45. Viswanath, D. & Cvitanovic, P. 2009 Stable manifolds and the transition to turbulence in pipe flow. J. Fluid Mech. 627, 215.
46. Waleffe, F. 1995 Transition in shear flows: nonlinear normality versus non-normal linearity. Phys. Fluids 7, 30603066.
47. Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.
48. Waleffe, F. 1998 Three-dimensional states in plane shear flow. Phys. Rev. Lett. 81, 41404143.
49. Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
50. Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subject to free stream turbulence Part I: boundary layer structure and receptivity. J. Fluid Mech. 281, 193218.
51. Willmarth, W. W. & Lu, S. S. 1972 Structure of the Reynolds stress near the wall. J. Fluid Mech. 55, 65.
52. Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally-zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 541.
53. Zuccher, S., Luchini, P. & Bottaro, A. 2004 Algebraic growth in a blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. Eur. J. Mech. (B/Fluids) 513, 135160.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 81 *
Loading metrics...

Abstract views

Total abstract views: 252 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd March 2018. This data will be updated every 24 hours.