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Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number

Published online by Cambridge University Press:  05 January 2010

YONGYUN HWANG  and
CARLO COSSU
YONGYUN HWANG
Affiliation:
Laboratorie d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France
CARLO COSSU*
Affiliation:
Laboratorie d'Hydrodynamique (LadHyX), CNRS-École Polytechnique, F-91128 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

We compute the optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing at Re = 750. The equations for the coherent perturbations are linearized near the turbulent mean flow and include the associated eddy viscosity. The mean flow is found to be linearly stable but it has the potential to amplify steamwise streaks from streamwise vortices. The most amplified structures are streamwise uniform and the largest amplifications of the energy of initial conditions and of the variance of stochastic forcing are realized by large-scale streaks having spanwise wavelengths of 4.4h and 5.2h respectively. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realizations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing, related to the sensitivity to boundary conditions and artificial forcing, can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.7h). The optimal large-scale streaks are furthermore found proportional to the mean turbulent profile in the viscous sublayer and up to the buffer layer.

JFM classification

Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 333 - 348
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

del Álamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15, L41.CrossRefGoogle Scholar
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. 2001 On the breakdown of boundary layers streaks. J. Fluid Mech. 428, 29–60.CrossRefGoogle Scholar
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258–3269.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.CrossRefGoogle Scholar
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14, L57–L60.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids 5, 1390–1400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 b Stochastic forcing of the linearized Navier-Stokes equation. Phys. Fluids A 5, 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. J. Atmos. Sci. 53, 2025–2053.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237–250.CrossRefGoogle Scholar
Fontane, J., Brancher, P. & Fabre, D. 2008 Stochastic forcing of the Lamb–Oseen vortex. J. Fluid Mech. 613, 233–254.CrossRefGoogle Scholar
Fransson, J., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501.CrossRefGoogle ScholarPubMed
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed shear layers. Annu. Rev. Fluid Mech. 16, 365–424.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 b Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. A 365, 647–664.CrossRefGoogle ScholarPubMed
Jang, P. S., Benney, D. J. & Gran, D. L. 1986 On the origin of streamwise vortices in a turbulent boundary layer. J. Fluid Mech. 169, 109–123.CrossRefGoogle Scholar
Jiménez, J. 2007 Recent developments on wall-bounded turbulence. Rev. R. Acad. Cien. Ser. A Mat. 101, 187–203.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flow. J. Fluid Mech. 543, 145–83.CrossRefGoogle Scholar
Kim, K. C. & Adrian, R. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.CrossRefGoogle Scholar
Kitoh, O., Nakabayashi, K. & Nishimura, F. 2005 Experimental study on mean velocity and turbulence characteristics of plane Couette flow: low-Reynolds-number effects and large longitudinal vortical structures. J. Fluid Mech. 539, 199.CrossRefGoogle Scholar
Kitoh, O. & Umeki, M. 2008 Experimental study on large-scale streak structure in the core region of turbulent plane Couette flow. Phys. Fluids 20, 025107.CrossRefGoogle Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Komminaho, J., Lundbladh, A. & Johansson, A. V. 1996 Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Landahl, M. T. 1990 On sublayer streaks. J. Fluid Mech. 212, 593–614.CrossRefGoogle Scholar
Lee, M. J. & Kim, J. 1991 The structure of turbulence in a simulated plane Couette flow. In Eighth Symp. on Turbulent Shear Flow, pp. 5.3.1–5.3.6. Technical University of Munich, Munich, Germany.Google Scholar
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. In Proceedings of URSI-IUGG Colloquium on Atomspheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarsky, V. I.), pp. 139–154. Nauka.Google Scholar
Moin, P. & Kim, J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech. 118, 341–377.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21, 015109.CrossRefGoogle 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.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (02), 263–288.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Smith, J. 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
Tillmark, N. 1995 Experiments on transition and turbulence in plane Couette flow. PhD thesis, Department of Mechanics, Royal Institute of Technology (KTH), Stockholm.Google Scholar
Tillmark, N. & Alfredsson, H. 1994 Structures in turbulent plane Couette flow obtained from correlation measurements. In Advances in Turbulences V (ed. Benzi, R.), pp. 502–507. Kluwer.Google Scholar
Tomkins, C. D. & Adrian, R. J. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 37–74.CrossRefGoogle Scholar
Tomkins, C. D. & Adrian, R. J. 2005 Energetic spanwise modes in the logarithmic layer of a turbulent boundary layer. J. Fluid Mech. 545, 141–162.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 A new direction in hydrodynamic stability: beyond eigenvalues. Science 261, 578–584.CrossRefGoogle Scholar
Tsukahara, T., Iwamoto, K. & Kawamura, H. 2007 POD analysis of large-scale structures through DNS of turbulence Couette flow. In Advances in Turbulence XI (ed. Palma, J. M. L. M. & Silva Lopes, A.) pp. 245–247. Springer.CrossRefGoogle Scholar
Tsukahara, T., Kawamura, H. & Shingai, K. 2006 DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. J. Turbul. 7, 19.CrossRefGoogle Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95, 319–343.CrossRefGoogle Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519.CrossRefGoogle Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control. Prentice Hall.Google Scholar