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Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow

Published online by Cambridge University Press:  15 August 2012

Brian F. Farrell  and
Petros J. Ioannou
Brian F. Farrell
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
Petros J. Ioannou*
Affiliation:
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, 15784, Greece
*
†Email address for correspondence: pjioannou@phys.uoa.gr
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Abstract

Streamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

JFM classification

Instability: Parametric instability Instability: Transition to turbulence Turbulent Flows: Turbulent transition
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 149 - 196
Copyright
Copyright © Cambridge University Press 2012

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References

1. Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.CrossRefGoogle Scholar
2. Alfredsson, P. H. & Matsubara, M. 1996 Streaky structures in transition. In Proc. Transitional Boundary Layers in Aeronautics (ed. Henkes, R. A. W. M. & van Ingen, J. L. ). pp. 373–386. Royal Netherlands Academy of Arts and Sciences. Elsevier.Google Scholar
3. Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.CrossRefGoogle Scholar
4. Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.CrossRefGoogle Scholar
5. Bakas, N. A. & Ioannou, P. J. 2011 Structural stability theory of two-dimensional fluid flow under stochastic forcing. J. Fluid Mech. 682, 332–361.CrossRefGoogle Scholar
6. Bakchinov, A. A., Katasonov, M. M. & Kozlov, V. V. 1997 Experimental study of localized disturbances and their development in a flat plate boundary layer. Preprint No. l-97, ITAM, Russian Academy of Sciences, Novosibirsk, Russia (in Russian).Google Scholar
7. Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258–3269.CrossRefGoogle Scholar
8. Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
9. Benney, D. J. 1960 A nonlinear theory for oscillations in a parallel flow. J. Fluid Mech. 10 (02), 209–236.CrossRefGoogle Scholar
10. Benney, D. J. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70 (02), 1–19.CrossRefGoogle Scholar
11. Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539–575.CrossRefGoogle Scholar
12. Berlin, S. & Henningson, D. S. 1999 A nonlinear mechanism for receptivity of free stream disturbances. Phys. Fluids 11 (02), 3749–3760.CrossRefGoogle Scholar
13. Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 23–57.CrossRefGoogle Scholar
14. Brandt, L., Henningson, D. S. & Ponziani, D. 2002 Weakly nonlinear analysis of boundary layer receptivity to free stream disturbances. Phys. Fluids 14, 1426–1441.CrossRefGoogle Scholar
15. Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
16. Butler, K. M. & Farrell, B. F 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids 4, 1637–1650.CrossRefGoogle Scholar
17. Chapman, S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.CrossRefGoogle Scholar
18. Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14 (8), 57–60.CrossRefGoogle Scholar
19. Cossu, C., Brandt, L., Bagheri, S. & Henningson, D. S. 2011 Secondary threshold amplitudes for sinuous streak breakdown. Phys. Fluids 23 (7), 074103.CrossRefGoogle Scholar
20. DelSole, T. 2004 Stochastic models of quasi-geostrophic turbulence. Surv. Geophys. 25, 107–194.CrossRefGoogle Scholar
21. DelSole, T. & Farrell, B. F. 1996 The quasi-linear equilibration of a thermally mantained stochastically excited jet in a quasi-geostrophic model. J. Atmos. Sci. 53, 1781–1797.2.0.CO;2>CrossRefGoogle Scholar
22. Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.CrossRefGoogle Scholar
23. Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.CrossRefGoogle Scholar
24. Farrell, B. F. & Ioannou, P. J. 1993a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids 5, 1390–1400.CrossRefGoogle Scholar
25. Farrell, B. F. & Ioannou, P. J. 1993b Perturbation growth in shear flow exhibits universality. Phys. Fluids 5, 2298–2300.CrossRefGoogle Scholar
26. Farrell, B. F. & Ioannou, P. J. 1993c Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50, 4044–4057.2.0.CO;2>CrossRefGoogle Scholar
27. Farrell, B. F. & Ioannou, P. J. 1993d Stochastic forcing of perturbation variance in unbounded shear and deformation flows. J. Atmos. Sci. 50, 200–211.2.0.CO;2>CrossRefGoogle Scholar
28. Farrell, B. F. & Ioannou, P. J. 1993e Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5, 2600–2609.CrossRefGoogle Scholar
29. Farrell, B. F. & Ioannou, P. J. 1994 Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett. 72, 1118–1191.CrossRefGoogle ScholarPubMed
30. Farrell, B. F. & Ioannou, P. J. 1995 Stochastic dynamics of the midlatitude atmospheric jet. J. Atmos. Sci. 52, 1642–1656.2.0.CO;2>CrossRefGoogle Scholar
31. Farrell, B. F. & Ioannou, P. J. 1996a Generalized stability. Part I. Autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
32. Farrell, B. F. & Ioannou, P. J. 1996b Generalized stability. Part II. Non-autonomous operators. J. Atmos. Sci. 53, 2041–2053.2.0.CO;2>CrossRefGoogle Scholar
33. Farrell, B. F. & Ioannou, P. J. 1998a Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 215–227.CrossRefGoogle Scholar
34. Farrell, B. F. & Ioannou, P. J. 1998b Turbulence supresssion by active control. Phys. Fluids 8, 1257–1268.CrossRefGoogle Scholar
35. Farrell, B. F. & Ioannou, P. J. 1999 Perturbation growth and structure in time dependent flows. J. Atmos. Sci. 56, 3622–3639.2.0.CO;2>CrossRefGoogle Scholar
36. Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118.2.0.CO;2>CrossRefGoogle Scholar
37. Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 3652–3665.CrossRefGoogle Scholar
38. Farrell, B. F. & Ioannou, P. J. 2008a Formation of jets by baroclinic turbulence. J. Atmos. Sci. 65, 3353–3375.CrossRefGoogle Scholar
39. Farrell, B. F. & Ioannou, P. J. 2008b The stochastic parametric mechanism for generation of surface water waves by wind. J. Phys. Oceanogr. 38, 862–879.CrossRefGoogle Scholar
40. Farrell, B. F. & Ioannou, P. J. 2009 A stochastic structural stability theory model of the drift wave-zonal flow system. Phys. Plasmas 16, 112903.CrossRefGoogle Scholar
41. Gayme, D. F., McKeon, B. J., Papachristodoulou, A., Bamieh, B. & Doyle, J. C. 2010 A streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99–119.CrossRefGoogle Scholar
42. Gibson, J., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
43. Halcrow, J., Gibson, J. F., Cvitanovic, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365–376.CrossRefGoogle Scholar
44. Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
45. Hamilton, K., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
46. Hernon, D., Walsh, E. J. & McEligot, D. M. 2007 Experimental investigation into the routes to bypass transition and the shear-sheltering phenomenon. J. Fluid Mech. 591, 461–479.CrossRefGoogle Scholar
47. Hoepffner, J. & Brandt, L. 2008 Stochastic approach to the receptivity problem applied to bypass transition in boundary layers. Phys. Fluids 20, 024108.CrossRefGoogle Scholar
48. Hogberg, M., Bewley, T. R. & Henningson, D. S. 2003 Relaminarization of turbulence using linear state-feedback control. Phys. Fluids 15, 3572–3575.CrossRefGoogle Scholar
49. Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
50. Hwang, Y. & Cossu, C. 2010a Amplification of coherent structures in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
51. Hwang, Y. & Cossu, C. 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
52. Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
53. Jang, P. S., Benney, D. J. & Gran, R. L. 1986 On the origin of streamwise vortices in a turbulent boundary layer. J. Fluid Mech. 169, 109–123.CrossRefGoogle Scholar
54. Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
55. Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
56. Jovanovic, M. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
57. Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
58. Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
59. Kim, J., Kline, S. J. & Reynolds, W. C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layers. J. Fluid Mech. 50, 133–160.CrossRefGoogle Scholar
60. Kim, J. & Lim, J. 2000 A linear process in wall bounded turbulent shear flows. Phys. Fluids 12, 1885–1888.CrossRefGoogle Scholar
61. Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
62. Komminaho, J., Lundbladh, A. & Johansson, A. 1996 Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285.CrossRefGoogle Scholar
63. Kreiss, G., Lundbladh, A. & Henningson, D. S. 1994 Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech. 270 (1), 175–198.CrossRefGoogle Scholar
64. Kurian, T. & Fransson, J. H. M. 2009 Grid-generated turbulence revisited. Fluid Dyn. Res. 41, 1–32.CrossRefGoogle Scholar
65. Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243.CrossRefGoogle Scholar
66. Laval, J.-P., Dubrulle, B. & McWilliams, J. C. 2003 Langevin models of turbulence: renormalization group, distant interaction algorithms or rapid distortion theory? Phys. Fluids 15, 1327–1339.CrossRefGoogle Scholar
67. Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
68. Lumley, J. L. 1967 The structure of inhomogeneous turbulence. In Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarskii, V. I. ). pp. 166–178. Nauka.Google Scholar
69. Marston, J. B. 2010 Statistics of the general circulation from cumulant expansions. Chaos 20, 041107.CrossRefGoogle ScholarPubMed
70. Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 1955–1966.CrossRefGoogle Scholar
71. Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
72. 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.CrossRefGoogle Scholar
73. Nagata, M. 1990 Three-dimensional travelling-wave solutions in plane Couette flow. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
74. Nagata, M. 1997 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. Phys. Rev. E 55, 2023–2025.CrossRefGoogle Scholar
75. Pedlosky, J. & Thomson, J. 2003 Baroclinic instability of time-dependent currents. J. Fluid Mech. 490, 189–215.CrossRefGoogle Scholar
76. Poulin, F. J., Flierl, G. R. & Pedlosky, J. 2003 Parametric instability in oscillatory shear flows. J. Fluid Mech. 481, 329–353.CrossRefGoogle Scholar
77. Poulin, F. J., Flierl, G. R. & Pedlosky, J. 2010 The baroclinic adjustment of time-dependent shear flows. J. Phys. Oceanogr. 40, 1851–1865.CrossRefGoogle Scholar
78. Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous shear flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
79. 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.CrossRefGoogle Scholar
80. Schlatter, P., Brandt, L., De Lange, H. C. & Henningson, D. S. 2008 On streak breakdown in bypass transition. Phys. Fluids 20 (10), 101505.CrossRefGoogle Scholar
81. Schmid, P. J. & Henningson, D. S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids A: Fluid Dyn. 4, 1986–1989.CrossRefGoogle Scholar
82. Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
83. Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
84. Sirovich, L., Ball, K. S. & Keefe, L. R. 1990 Plane waves and structures in turbulent channel flow. Phys. Fluids A 2, 2217–2226.CrossRefGoogle Scholar
85. Smith, C. R. & Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27–54.CrossRefGoogle Scholar
86. Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 1633–1656.CrossRefGoogle Scholar
87. Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct numerical simulation. Astrophys. J. 727, 127.CrossRefGoogle Scholar
88. Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
89. Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining proccess. Stud. Appl. Maths 95, 319–343.CrossRefGoogle Scholar
90. Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids A 9, 883–900.CrossRefGoogle Scholar
91. Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
92. Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
93. Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
94. Westin, K. J. A., Bakchinov, A. A., Kozlov, V. V. & Alfredsson, P. H. 1998 Experiments on localized disturbances in a flat plate boundary layer. Part 1. The receptivity and evolution of a localized free stream disturbance receptivity and evolution of a localized free stream disturbance. Eur. J. Mech. (B/Fluids) 17, 823–846.CrossRefGoogle Scholar
95. Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.CrossRefGoogle Scholar
96. 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.CrossRefGoogle Scholar
97. Zhang, Y. & Held, I. M. 1999 A linear stochastic model of a GCM’s midlatitude storm tracks. J. Atmos. Sci. 56, 3416–3435.2.0.CO;2>CrossRefGoogle Scholar