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Nonlinear dynamics of large-scale coherent structures in turbulent free shear layers

Published online by Cambridge University Press:  16 December 2015

Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Nankai, Tianjin 300072, PR China Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Xiuling Zhuang
Affiliation:
Department of Mechanics, Tianjin University, Nankai, Tianjin 300072, PR China
*
Email address for correspondence: x.wu@ic.ac.uk

Abstract

Fully developed turbulent free shear layers exhibit a high degree of order, characterized by large-scale coherent structures in the form of spanwise vortex rollers. Extensive experimental investigations show that such organized motions bear remarkable resemblance to instability waves, and their main characteristics, including the length scales, propagation speeds and transverse structures, are reasonably well predicted by linear stability analysis of the mean flow. In this paper, we present a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation is based on the triple decomposition of the instantaneous flow into a mean field, coherent fluctuations and small-scale turbulence but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion. The system is closed by employing a gradient type of model for the time- and phase-averaged Reynolds stresses of fine-scale turbulence. In the high-Reynolds-number limit, the nonlinear non-equilibrium critical-layer theory for laminar-flow instabilities is adapted to turbulent shear layers by accounting for (1) the enhanced non-parallelism associated with fast spreading of the mean flow, and (2) the influence of small-scale turbulence on coherent structures. The combination of these factors with nonlinearity leads to an interesting evolution system, consisting of coupled amplitude and vorticity equations, in which non-parallelism contributes the so-called translating critical-layer effect. Numerical solutions of the evolution system capture vortex roll-up, which is the hallmark of a turbulent mixing layer, and the predicted amplitude development mimics the qualitative feature of oscillatory saturation that has been observed in a number of experiments. A fair degree of quantitative agreement is obtained with one set of experimental data.

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Papers
Copyright
© 2015 Cambridge University Press 

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References

Alper, A. & Liu, J. T. C. 1978 On the interactions between large-scale structure and fine-grained turbulence in a free shear layer. Part II: The development of spatial interactions in the mean. Proc. R. Soc. Lond. A 359, 497523.Google Scholar
Antonia, R. A., Browne, L. W. B., Rajagopalan, S. & Chambers, A. J. 1983 On the organized motion of a turbulent plane jet. J. Fluid Mech. 134, 4966.CrossRefGoogle Scholar
Antonia, R. A., Chambers, A. J., Britz, D. & Browne, L. W. B. 1986 Organized structures in a turbulent plane jet: topology and contribution to momentum and heat transport. J. Fluid Mech. 172, 211229.Google Scholar
Batt, R. G. 1975 Some measurements on the effect of tripping the two-dimensional shear layer. AIAA J. 13, 245247.Google Scholar
Bechert, D. W. & Pfizenmaier, E. 1975 On the amplification of broadband jet noise by a pure tone excitation. J. Sound Vib. 43, 581587.Google Scholar
Benney, D. J. & Bergeron, F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.CrossRefGoogle Scholar
Bishop, K. A., Ffowcs Williams, J. E. & Smith, W. 1971 On the noise sources of the unsuppressed high-speed jet. J. Fluid Mech. 50, 2131.Google Scholar
Browand, F. K. & Troutt, T. R. 1980 A note on spanwise structure in the two-dimensional mixing layer. J. Fluid Mech. 97, 771781.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775816.Google Scholar
Cantwell, B. J. 1981 Organized motion in turbulent flow. Annu. Rev. Fluid Mech. 13, 457515.Google Scholar
Cavalieri, A. V. G., Rodriguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559592.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1994 Nonlinear spatial evolution of helical disturbances to an axial jet. J. Fluid Mech. 281, 371402.Google Scholar
Cohen, J., Marasli, B. & Levinski, V. 1994 The interaction between the mean flow and coherent structures in turbulent mixing layers. J. Fluid Mech. 260, 8194.Google Scholar
Cowley, S. J. 1985 Pulsatile flow through distorted channels: low-Strouhal-number and translating critical-layer effect. Q. J. Mech. Appl. Maths 38 (4), 589619.CrossRefGoogle Scholar
Cowley, S. J. & Wu, X. 1994 Asymptotic approaches to transition modelling. In Progress in Transition Modelling, AGARD Report 793, Ch. 3, pp. 138.Google Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397413.Google Scholar
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 547591.Google Scholar
Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78, 535560.Google Scholar
Dziomba, B. & Fiedler, H. E. 1985 Effect of initial conditions on two-dimensional free shear layers. J. Fluid Mech. 152, 419442.Google Scholar
Estevadeordal, J. & Kleis, S. J. 2002 Influence of vortex-pairing location on the three-dimensional evolution of plane mixing layers. J. Fluid Mech. 462, 4377.Google Scholar
Fiedler, H. E. & Mensing, P. 1985 The plane turbulent shear layer with periodic excitation. J. Fluid Mech. 150, 281309.Google Scholar
Gaster, M., Kit, E. & Wygnanski, I. 1985 Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech. 150, 2339.Google Scholar
Goldstein, M. E. 1995 The role of nonlinear critical layers in boundary-layer transition. Phil. Trans. R. Soc. Lond. A 352, 425442.Google Scholar
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97120.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295330.Google Scholar
Goldstein, M. E. & Leib, S. J. 1988 Nonlinear roll-up of externally excited shear layers. J. Fluid Mech. 191, 481515.CrossRefGoogle Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97128.Google Scholar
Haberman, R. 1972 Critical layers in parallel shear flows. Stud. Appl. Maths 51, 139161.Google Scholar
Haynes, P. H. & Cowley, S. J. 1986 The evolution of an unsteady translating nonlinear Rossby-wave critical layer. Geophys. Astrophys. Fluid Dyn. 35, 155.Google Scholar
Ho, C. M. & Huerre, P. 1990 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365422.Google Scholar
Hussain, A. K. M. F. 1983 Coherent structures – reality and myth. Phys. Fluids 26, 28162850.Google Scholar
Hussain, A. F. M. F. & Clark, A. R. 1981 On the coherent structure of the axisymmetric mixing layer: a flow-visualization study. J. Fluid Mech. 104, 263294.Google Scholar
Hussain, A. F. M. F. & Hasan, M. A. Z. 1985 Turbulence suppression in free turbulent shear flows under controlled excitation. Part 2. Jet-noise reduction. J. Fluid Mech. 105, 159168.CrossRefGoogle Scholar
Hussain, A. F. M. F. & Reynolds, W. C. 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241258.CrossRefGoogle Scholar
Hussain, A. F. M. F. & Thompson, C. A. 1980 Controlled symmetric perturbation of the plane jet: an experimental study in the initial region. J. Fluid Mech. 100, 397431.Google Scholar
Hussain, A. F. M. F. & Zaman, K. B. M. 1981 The preferred mode of the axisymmetric jet. J. Fluid Mech. 110, 3971.Google Scholar
Hussain, A. F. M. F. & Zaman, K. B. M. 1985 An experimental study of organized motions in the turbulent plane mixing layer. J. Fluid Mech. 159, 85104.Google Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173195.Google Scholar
Kitsios, V., Cordier, L., Bonnet, J.-P., Ooi, A. & Soria, D. J. 2010 Development of a nonlinear eddy-viscosity closure for the triple-decomposition stability analysis of a turbulent channel. J. Fluid Mech. 664, 74107.Google Scholar
Liu, J. T. C. 1989 Cohenrent structures in transitional and turbulent free shear flows. Annu. Rev. Fluid Mech. 21, 285315.Google Scholar
Liu, J. T. C. & Merkine, L. 1976 On the interactions between large-scale structure and fine-grained turbulence in a free shear flow. Proc. R. Soc. Lond. A 352, 213247.Google Scholar
Oberleithner, K., Rukes, L. & Soria, J. 2014 Mean flow stability analysis of oscillating jet experiments. J. Fluid Mech. 757, 132.Google Scholar
Mankbadi, R. & Liu, J. T. C. 1981 A study of large-scale coherent structures and fine-grained turbulence in a round jet. Phil. Trans. R. Soc. Lond. A 298, 541602.Google Scholar
Marasli, B., Champagne, F. H. & Wygnanski, I. 1989 Modal decomposition of velocity signals in a plane, turbulent wake. J. Fluid Mech. 198, 255273.Google Scholar
Marasli, B., Champagne, F. H. & Wygnanski, I. 1991 On linear evolution of unstable disturbances in a plane turbulent wake. Phys. Fluids A 3 (4), 665674.Google Scholar
Marasli, B., Champagne, F. H. & Wygnanski, I. 1992 Effect of travelling waves on the growth of a plane turbulent wake. J. Fluid Mech. 235, 511528.Google Scholar
Meyer, T. R., Dutton, J. C. & Lucht, R. P. 2006 Coherent structures and turbulent molecular mixing in gaseous planar shear layers. J. Fluid Mech. 558, 179205.CrossRefGoogle Scholar
Nygaard, K. & Glezer, A. 1994 The effect of phase variations and cross-shear on vortical structures in a plane mixing layer. J. Fluid Mech. 276, 2159.Google Scholar
Oster, D. & Wygnanski, I. 1982 The forced mixing layer between parallel streams. J. Fluid Mech. 123, 91130.Google Scholar
Redekopp, L. G. 1977 On the theory of solitary Rossby wave. J. Fluid Mech. 82, 725745.Google Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechaniscs of an organised wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54, 263288.Google Scholar
Roshko, A. 1976 Structure of turbulent shear flows: a new look. AIAA J. 14 (10), 13491357.Google Scholar
Suzuki, T. 2013 Coherent noise sources of a subsonic round jet investigated using hydrodynamic and acoustic phased-microphone arrays. J. Fluid Mech. 730, 659698.Google Scholar
Suzuki, T. & Colonius, T. 2006 Instability waves in a subsonic round jet detected using a near-field phased microphone array. J. Fluid Mech. 565, 197226.Google Scholar
Tam, C. K. W. & Burton, D. E. 1984 Sound generated by instability waves of supersonic flow. Part 2. Axisymmetric jets. J. Fluid Mech. 138, 273295.Google Scholar
Weisbrot, I., Einav, S. & Wygnanski, I. 1982 The nonunique rate of spread of the two dimensional mixing layer. Phys. Fluids 25, 16911693.Google Scholar
Weisbrot, I. & Wygnanski, I. 1988 On coherent structures in a highly excited mixing layer. J. Fluid Mech. 195, 137159.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237255.Google Scholar
Wu, X. 2005 Mach wave radiation of nonlinearly evolving supersonic instability modes in shear layers. J. Fluid Mech. 523, 121159.Google Scholar
Wu, X. & Huerre, P. 2009 Low-frequency sound radiated by a nonlinear modulated wavepacket of helical modes on a subsonic circular jet. J. Fluid Mech. 637, 173211.Google Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear instability of shear flows to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681720.Google Scholar
Wu, X. & Tian, F. 2012 Spectral broadening and flow randomization in free shear layers. J. Fluid Mech. 706, 431469.Google Scholar
Wu, X. & Zhou, H. 1989 Linear instability of turbulent boundary layer as a mechanism for the generation of large scale coherent structures. Chin. Sci. Bull. 34 (20), 16851688; (English Edition).Google Scholar
Wygnanski, I., Champagne, F. H. & Marasli, B. 1986 On the large-scale structures in two-dimensional, small-deficit, turbulent wakes. J. Fluid Mech. 168, 3171.Google Scholar
Wygnanski, I., Oster, D., Fiedler, H. & Dziomba, B. 1979 On the perseverance of a quasi-two-dimensional eddy-structure in a turbulent mixing layer. J. Fluid Mech. 93, 325335.Google Scholar
Wygnanski, I. & Petersen, R. A. 1987 Coherent motion in excited free shear flows. AIAA J. 25, 201213.Google Scholar
Wygnanski, I. & Weisbrot, I. 1988 On the pairing process in an excited plane turbulent mixing layer. J. Fluid Mech. 195, 161173.Google Scholar
Zaman, K. B. M. & Hussain, A. F. M. F. 1980 Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J. Fluid Mech. 101, 449491.Google Scholar