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Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow

Published online by Cambridge University Press:  22 September 2010

YONGYUN HWANG  and
CARLO COSSU
YONGYUN HWANG*
Affiliation:
Laboratorie d'Hydrodynamique (LadHyX), CNRS–École Polytechnique, F-91128 Palaiseau, France
CARLO COSSU
Affiliation:
CNRS–Institut de Mécanique des Fluides de Toulouse (IMFT), Allée du Professeur Camille Soula, F-31400 Toulouse, France Département de Mécanique, École Polytechnique, F-91128 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

The linear response to stochastic and optimal harmonic forcing of small coherent perturbations to the turbulent channel mean flow is computed for Reynolds numbers ranging from Reτ = 500 to 20000. Even though the turbulent mean flow is linearly stable, it is nevertheless able to sustain large amplifications by the forcing. The most amplified structures consist of streamwise-elongated streaks that are optimally forced by streamwise-elongated vortices. For streamwise-elongated structures, the mean energy amplification of the stochastic forcing is found to be, to a first approximation, inversely proportional to the forced spanwise wavenumber while it is inversely proportional to its square for optimal harmonic forcing in an intermediate spanwise wavenumber range. This scaling can be explicitly derived from the linearized equations under the assumptions of geometric similarity of the coherent perturbations and of logarithmic base flow. Deviations from this approximate power-law regime are apparent in the pre-multiplied energy amplification curves that reveal a strong influence of two different peaks. The dominant peak scales in outer units with the most amplified spanwise wavelength of λz ≈ 3.5h, while the secondary peak scales in wall units with the most amplified λz+ ≈ 80. The associated optimal perturbations are almost independent of the Reynolds number when, respectively, scaled in outer and inner units. In the intermediate wavenumber range, the optimal perturbations are approximatively geometrically similar. Furthermore, the shape of the optimal perturbations issued from the initial value, the harmonic forcing and the stochastic forcing analyses are almost indistinguishable. The optimal streaks corresponding to the large-scale peak strongly penetrate into the inner layer, where their amplitude is proportional to the mean-flow profile. At the wavenumbers corresponding to the large-scale peak, the optimal amplifications of harmonic forcing are at least two orders of magnitude larger than the amplifications of the variance of stochastic forcing and both increase with the Reynolds number. This confirms the potential of the artificial forcing of optimal large-scale streaks for the flow control of wall-bounded turbulent flows.

JFM classification

Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 51 - 73
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

del Álamo, J. & 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
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
Andersson, P., Berggren, M. & Henningson, D. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.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
Cess, R. D. 1958 A survey of the literature on heat transfer in turbulent tube flow. Res. Rep. 8-0529-R24. Westinghouse.Google 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. & Brandt, L. 2004 On Tollmien–Schlichting waves in streaky boundary layers. Eur. J. Mech. B/ Fluids 23, 815–833.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.Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487.Google 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.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1994 Variance maintained by the stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett. 72, 1188–1191.CrossRefGoogle ScholarPubMed
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
Fransson, J., Brandt, L., Talamelli, A. & Cossu, C. 2004 Experimental and theoretical investigation of the non-modal growth of steady streaks in a flat plate boundary layer. Phys. Fluids 16, 3627–3638.CrossRefGoogle Scholar
Fransson, J., Brandt, L., Talamelli, A. & Cossu, C. 2005 Experimental study of the stabilisation of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17, 054110.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
De Graaff, D. & Eaton, J. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.CrossRefGoogle Scholar
Guala, M., Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–541.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., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003. Phys. Fluids 18, 011702.CrossRefGoogle Scholar
Hunt, J. C. R. & Morrison, J. F. 2000 Eddy structure in turbulent boundary layer. Eur. J. Mech. B 19, 673–694.Google 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.Google Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Iuso, G., Onorato, M., Spazzini, P. G. & di Cicca, G. M. 2002 Wall turbulence manipulation by large-scale streamwise vortices. J. Fluid Mech. 473, 23–58.CrossRefGoogle Scholar
Jiménez, J. 1998 The largest scales of turbulent wall flows. In CTR Annual Research Briefs, pp. 943–945. Center for Turbulence Research, Stanford University.Google 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.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flow. J. Fluid Mech. 543, 145–83.Google 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.Google 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.Google 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.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid. Mech. 628, 311–337.Google Scholar
McKeon, B. J. 2008 A model for ‘dynamic’ roughness in turbulent channel flow. In Proceedings of the Summer Program, pp. 399–410. Center for Turbulence Research, Stanford University.Google Scholar
Metzger, M. M. & Klewicki, J. C. 2001 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13, 692–701.Google Scholar
Moffatt, H. K. 1967 The interaction of turbulence with a strong wind shear. In Proceedings URSI-IUGG International Colloquium on Atmospheric Turbulence and Radio Wave Propagation, pp. 139–154. Nauka.Google Scholar
Morrison, W. R. B. & Kronauer, R. E. 1969 Structural similarity for fully developed turbulence in smooth tubes. J. Fluid Mech. 39, 117–141.Google Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Pujals, G., Cossu, C. & Depardon, S. 2010 a Forcing large-scale coherent streaks in a zero pressure gradient turbulent boundary layer. J. Turbulence 11 (25), 1–13.Google Scholar
Pujals, G., Depardon, S. & Cossu, C. 2010 b Drag reduction of a 3D bluff body using coherent streamwise streaks. Exp. Fluids (in press).Google 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. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Math. 53, 1547.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 (2), 263–288.CrossRefGoogle Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow, with application to Malkus's theory. J. Fluid Mech. 27 (2), 253–272.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.CrossRefGoogle Scholar
Schmid, P. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schoppa, S. & Hussain, F. 1998 A large-scale control strategy for drag reduction in turbulent boundary layers. Phys. Fluids 10, 1049.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
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
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
Townsend, A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google 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.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95, 319–343.Google Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB Differentiation Matrix Suite. ACM Trans. Math. Softw. 26, 465–519.Google Scholar
Willis, A. P., Hwang, Y. & Cossu, C. 2009 Optimal harmonic forcing and drag reduction in pipe flow. Bull. Am. Phys. Soc. 54 (19), 227.Google Scholar
Willis, A. P., Hwang, Y. & Cossu, C. 2010 Optimally amplified large-scale streaks and drag reduction in the turbulent pipe flow. Phys. Rev. E (submitted).Google Scholar
Zhou, K., Doyle, J. & Glover, K. 1996 Robust and Optimal Control. Prentice-Hall.Google Scholar