Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T01:07:52.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Coherent structures in wall-bounded turbulence

Published online by Cambridge University Press:  14 March 2018

Javier Jiménez Open the ORCID record for Javier Jiménez [Opens in a new window]
Javier Jiménez*
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, 28040 Madrid, Spain Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: jimenez@torroja.dmt.upm.es
Get access Rights & Permissions [Opens in a new window]

Abstract

This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives.

‘Chance is the name we give to what we choose to ignore (Voltaire)’

JFM classification

Turbulent Flows: Turbulence control Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 842 , 10 May 2018 , P1
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.Google Scholar
Adrian, R. J., Meinhart, C. D. & Tomkins, C. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.Google Scholar
Adrian, R. J. & Moin, P. 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow. J. Fluid Mech. 190, 531–559.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. 2009 Estimation of turbulent convection velocities and corrections to Taylor’s approximation. J. Fluid Mech. 640, 5–26.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
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.Google Scholar
Antonia, R. A. & Atkinson, J. D. 1973 High-order moments of Reynolds shear stress fluctuations in a turbulent boundary layer. J. Fluid Mech. 58, 581–593.CrossRefGoogle Scholar
Arnold, V. I. 1983 Geometric Methods in the Theory of Ordinary Differential Equations. Springer.Google Scholar
Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality. Phys. Rev. A 38, 364–374.Google Scholar
Bergé, P., Pomeau, Y. & Vidal, C. 1984 Order within Chaos: Towards a Deterministic Approach to Turbulence. Wiley-Interscience.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.CrossRefGoogle Scholar
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497–504.Google Scholar
Brown, G. L. & Roshko, A. 1974 On the density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded shear flow. Phys. Fluids A 5, 774–777.CrossRefGoogle Scholar
Choi, H., Jeon, W.-P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139.CrossRefGoogle Scholar
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.Google Scholar
Corrsin, S.1958 Local isotropy in turbulent shear flow. NACA Research Memo. 58B11.Google Scholar
Cvitanović, P. 1988 Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61, 2729–2732.Google Scholar
Deardorff, J. W. 1970 A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453–480.CrossRefGoogle Scholar
Dong, S., Lozano-Durán, A., Sekimoto, A. & Jiménez, J. 2017 Coherent structures in statistically stationary homogeneous shear turbulence. J. Fluid Mech. 816, 167–208.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Earman, J. & Norton, J. D. 1998 The wrath of Maxwell’s demon. Part I. From Maxwell to Szilard. Stud. Hist. Phil. Mod. Phys. 29, 435–471.Google Scholar
Encinar, M. P. & Jiménez, J. 2016 Characterization of linear-like Orr bursts in fully turbulent channel flows. In Proc. Div. Fluid Dyn., p. L32.6. American Physical Society.Google Scholar
Farge, M. 1992 Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24, 395–457.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II: nonautonomous operators. J. Atmos. Sci. 53, 2041–2053.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.CrossRefGoogle Scholar
Farrell, B. F., Ioannou, P. J., Jiménez, J., Constantinou, N. C., Lozano-Durán, A. & Nikolaidis, M. 2016 A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 290–315.Google Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.Google Scholar
Gasquet, C. & Witomski, P. 1998 Fourier Analysis and Applications. Springer.Google Scholar
Gaster, M., Kit, E. & Wygnanski, I. 1985 Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech. 150, 23–39.Google Scholar
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.Google Scholar
Hall, P. & Sherwin, S. J. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Herbert, T. 1976 Periodic secondary motions in a plane channel. In Proc. 5th Intl Conf. Numerical Methods Fluid Dyn. (ed. de Vooren, A. I. V. & Zandbergen, P. J.), pp. 235–240. Springer.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18, 011702.Google Scholar
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511.Google Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 467–477.CrossRefGoogle Scholar
Hwang, Y., Willis, A. P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for Re 𝜏 up to 1000. J. Fluid Mech. 802, R1.Google Scholar
Jiménez, J. 1987a Bifurcations and bursting in two-dimensional Poiseuille flow. Phys. Fluids 30, 3644–3646.Google Scholar
Jiménez, J. 1987b Coherent structures and dynamical systems. In Proc. CTR Summer School, pp. 323–324. Stanford University.Google Scholar
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173–196.CrossRefGoogle Scholar
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.Google Scholar
Jiménez, J. 2013a How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.Google Scholar
Jiménez, J. 2013b Near-wall turbulence. Phys. Fluids 25, 101302.Google Scholar
Jiménez, J. 2015 Direct detection of linearized bursts in turbulence. Phys. Fluids 27, 065102.Google Scholar
Jiménez, J. 2016 Optimal fluxes and Reynolds stresses. J. Fluid Mech. 809, 585–600.Google Scholar
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.Google Scholar
Jiménez, J. & Kawahara, G. 2013 Dynamics of wall-bounded turbulence. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 221–269. Cambridge University Press.Google Scholar
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M. & Shiba, M. 2005 Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids 17, 015105.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.Google Scholar
Jiménez, J. & Simens, M. P. 2001 Low-dimensional dynamics of a turbulent wall flow. J. Fluid Mech. 435, 81–91.Google Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.Google Scholar
Kim, H. T., Kline, S. J. & Reynolds, W. C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133–160.Google Scholar
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.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.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 301–305; Reprinted in Proc. R. Soc. Lond. A 434, (1991), pp. 9–13.Google Scholar
Kraichnan, R. H. 1971 Inertial range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535.Google Scholar
Kravchenko, A. G., Moin, P. & Moser, R. D. 1996 Zonal embedded grids for numerical simulations of wall-bounded turbulent flows. J. Comput. Phys. 127, 412–423.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1958 Statistical Mechanics, 2nd edn. Addison-Wesley.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, 2nd edn. Addison-Wesley.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Re 𝜏 ≈ 5200. J. Fluid Mech. 774, 395–415.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 917–928.Google Scholar
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100–130.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014a Effect of the computational domain on direct simulations of turbulent channels up to Re 𝜏 = 4200. Phys. Fluids 26, 011702.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014b Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascade. J. Fluid Mech. 759, 432–471.CrossRefGoogle Scholar
Lu, S. S. & Willmarth, W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60, 481–511.CrossRefGoogle Scholar
Lumley, J. & Blossey, P. 1998 Control of turbulence. Annu. Rev. Fluid Mech. 30, 311–327.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.Google Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329, 193–196.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
McKeon, B. J. 2017 The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Mezić, I. 2013 Analysis of fluid flows via the spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378.Google Scholar
Mizuno, Y. & Jiménez, J. 2011 Mean velocity and length-scales in the overlap region of wall-bounded turbulent flows. Phys. Fluids 23, 085112.Google Scholar
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without walls. J. Fluid Mech. 723, 429–455.Google Scholar
Moin, P. & Kim, J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech. 118, 341–377.Google Scholar
Moin, P. & Moser, R. D. 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509.Google Scholar
Moisy, F. & Jiménez, J. 2004 Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 513, 111–133.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–286.Google Scholar
Orlandi, P. & Jiménez, J. 1994 On the generation of turbulent wall friction. Phys. Fluids 6, 634–641.Google Scholar
Örlü, R., Fiorini, T., Segalini, A., Bellani, G., Talamelli, A. & Alfredsson, P. H. 2017 Reynolds stress scaling in pipe flow turbulence – first results from CICLoPE. Phil. Trans. R. Soc. Lond. A 375, 20160187.Google Scholar
Orr, W. M. 1907 The stability or instability of the steady motions of a perfect liquid, and of a viscous liquid. Part I: a perfect liquid. Proc. R. Irish Acad. A 27, 9–68.Google Scholar
Osawa, K. & Jiménez, J.2018 Intense structures of different momentum fluxes in turbulent channels. J. Phys.: Conf. Ser. (to appear).Google Scholar
Pearson, K. 1901 On lines and planes of closest fit to systems of points in space. Phil. Mag. 6, 559–572.Google Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.Google Scholar
Piomelli, U. & Balaras, E. 2002 Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34, 349–374.Google Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2014 Turbulence statistics in Couette flow at high Reynolds number. J. Fluid Mech. 758, 327–343.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1986 Numerical Recipes. Cambridge University Press.Google Scholar
Prigogine, I. 1978 Time, structure, and fluctuations. Science 201, 777–785.Google Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flow. Phys. Fluids 21, 015109.Google Scholar
Pumir, A. 1996 Turbulence in homogeneous shear flows. Phys. Fluids 8, 3112–3127.Google Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow, with application to Malkus’ theory. J. Fluid Mech. 27, 253–272.Google Scholar
Richardson, L. F. 1920 The supply of energy from and to atmospheric eddies. Proc. R. Soc. Lond. A 97, 354–373.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.Google Scholar
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. NASA Tech. Memo 81315.Google Scholar
Rogers, M. M. & Moin, P. 1987 The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 176, 33–66.Google Scholar
Rowley, C. W. & Dawson, S. T. M. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387–417.CrossRefGoogle Scholar
Ruelle, D. 1978 Statistical Mechanics: Thermodynamic Formalism. Addison-Wesley.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.Google Scholar
Sekimoto, A., Dong, S. & Jiménez, J. 2016 Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows. Phys. Fluids 28, 035101.Google Scholar
Sekimoto, A. & Jiménez, J. 2017 Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow. J. Fluid Mech. 827, 225–249.Google Scholar
Shannon, C. E. & Weaver, W. 1949 The Mathematical Theory of Communication. University of Illinois Press.Google Scholar
Sillero, J.2014 High Reynolds numbers turbulent boundary layers. PhD thesis, U. Politécnica Madrid.Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 25, 105102.Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 26, 105109.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. Appl. Maths 45, 561–590.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.CrossRefGoogle Scholar
Sreenivasan, K. R. 1991 Fractals and multifractals in fluid turbulence. Annu. Rev. Fluid Mech. 23, 539–600.Google Scholar
Stauffer, D. & Aharony, A. 1994 Introduction to Percolation Theory. Taylor and Francis.Google Scholar
Stretch, D. D.1990 Automated pattern eduction from turbulent flow diagnostics. CTR Ann. Res. Briefs, pp. 145–157. Stanford University.Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Toh, S. & Itano, T. 2005 Interaction between a large-scale structure and near-wall structures in channel flow. J. Fluid Mech. 524, 249–262.Google Scholar
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97–120.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Tuerke, F. & Jiménez, J. 2013 Simulations of turbulent channels with prescribed velocity profiles. J. Fluid Mech. 723, 587–603.Google Scholar
Voltaire, F. 1994 Dictionnaire Philosophique: Atomes. Oxford University Press.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Wallace, J. M., Eckelman, H. & Brodkey, R. S. 1972 The wall region in turbulent shear flow. J. Fluid Mech. 54, 39–48.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wiener, N. 1961 Cybernetics. MIT Press.Google Scholar
Wu, X., Moin, P., Wallace, J. M., Skarda, J., Lozano-Durán, A. & Hickey, J.-P. 2017 Transitional–turbulent spots and turbulent–turbulent spots in boundary layers. Proc. Natl Acad. Sci. USA 114, E5292–E5299.CrossRefGoogle ScholarPubMed
Wu, J., Zhou, Y., Lu, X. & Fan, M. 1999 Turbulent force as a diffusive field with vortical forces. Phys. Fluids 11, 627–635.Google Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.Google Scholar