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The engine behind (wall) turbulence: perspectives on scale interactions

Published online by Cambridge University Press:  24 March 2017

B. J. McKeon
B. J. McKeon*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
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Abstract

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulent Flows
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 817 , 25 April 2017 , P1
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© 2017 Cambridge University Press 

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References

Adrian, R. J. 2007 Vortex organization in wall turbulence. Phys. Fluids 19, 041301.Google Scholar
Afzal, N. 1984 Mesolayer theory for turbulent flows. AIAA J. 22, 437–439.CrossRefGoogle Scholar
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Baars, W. J., Talluru, K. M., Hutchins, N. & Marusic, I. 2015 Wavelet analysis of wall turbulence to study large-scale modulation of small scales. Exp. Fluids 56, 188.Google Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis and application to the determination of coherent structures in a turbulent backward facing step flow. J. Fluid Mech. 798, 485–504.CrossRefGoogle Scholar
Bourguignon, J.-L.2013 Models of turbulent pipe flow. PhD thesis, California Institute of Technology.Google Scholar
Bourguignon, J.-L., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2014 Compact representation of wall-bounded turbulence using compressive sampling. Phys. Fluids 26, 015109.CrossRefGoogle Scholar
Bradshaw, P. & Koh, Y. M. 1981 A note on Poisson’s equation for pressure in a turbulent flow. Phys. Fluids 24 (771), 241–258.CrossRefGoogle Scholar
Brandt, L. 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. (B/Fluids) 47, 80–96.CrossRefGoogle Scholar
Butler, K. & Farrell, B. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5, 774–777.Google Scholar
Candes, E. J. & Wakin, M. B. 2008 An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30.Google Scholar
Cess, R. D.1958 A study of the literature on heat transfer in turbulent tube flow. Tech. Rep. 8-0529-R24. Westinghouse Research.Google Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.CrossRefGoogle Scholar
Chernyshenko, S. I. & Baig, M. F. 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99–131.CrossRefGoogle Scholar
Chernyshenko, S. I., Cicca, G. M., Iollo, A., Smirnov, A. V., Sandham, N. D. & Hu, Z. W. 2006 Analysis of data on the relation between eddies and streaky structures in turbulent flows using the placebo method. Fluid Dyn. 41 (5), 772–783.Google Scholar
Cheung, L. C. & Zaki, T. 2014 An exact representation of the nonlinear triad interaction terms in spectral space. J. Fluid Mech. 748, 175–188.Google Scholar
Chung, D. & McKeon, B. J. 2010 Large-eddy simulation investigation of large-scale structures in a long channel flow. J. Fluid Mech. 661, 341–364.Google 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
Dennis, D. & Nickels, T. 2011a Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180–217.CrossRefGoogle Scholar
Dennis, D. & Nickels, T. 2011b Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures. J. Fluid Mech. 673, 218–244.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn, Cambridge Mathematical Libraries. Cambridge University Press.CrossRefGoogle Scholar
Duggleby, A., Ball, K. S., Pail, M. R. & Fischer, P. F. 2007 Dynamical eigenfunction decomposition of turbulent pipe flow. J. Turbul. 8 (43), 1–24.Google Scholar
Duvvuri, S.2016 Non-linear scale interactions in a forced turbulent boundary layer. PhD thesis, California Institute of Technology.CrossRefGoogle Scholar
Duvvuri, S. & McKeon, B. J. 2015 Triadic scale interactions in a turbulent boundary layer. J. Fluid Mech. 767, R4.Google Scholar
Duvvuri, S. & McKeon, B. J. 2016 Non-linear interactions isolated through scale synthesis in experimental wall turbulence. Phys. Rev. Fluids 1 (3), 032401(R).Google Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Scheider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366, 1297–1315.Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Farrell, B. & Ioannou, J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 2600–2609.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237–250.Google 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.Google Scholar
Farrell, B. F. & Ioannou, P. J.2014 Statistical state dynamics: a new perspective on turbulence in shear flow. arXiv:1412.8290v1.Google Scholar
Farrell, B. F., Ioannou, P. J., Jiménez, J., Constantinou, N. C., Lozano-Duran, A. & Nikolaidis, M. A. 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
Fosas de Pando, M., Schmid, P. J. & Sipp, D. 2015 Nonlinear model-order reduction for oscillator flows using POD-DEIM. In Proc. IUTAM 14, pp. 329–336. Elsevier.Google Scholar
Ganapathisubramani, B., Hutchins, N., Monty, J. O., Chung, D. & Marusic, I. 2012 Amplitude and frequency modulation in wall turbulence. J. Fluid Mech. 712, 61–91.Google Scholar
Gayme, D. F., McKeon, B. J., Papachristodolou, A., Bamieh, B. & Doyle, J. C. 2010 Streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99–119.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.Google Scholar
Gómez Carrasco, F., Blackburn, H., Rudman, M., McKeon, B., Luhar, M., Moarref, R. & Sharma, A. 2014 On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow. Phys. Fluids 26, 101703.Google Scholar
Gómez Carrasco, F., Blackburn, H., Rudman, M., Sharma, A. & McKeon, B. 2016a A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, R2.Google Scholar
Gómez Carrasco, F., Blackburn, H., Rudman, M., Sharma, A. & McKeon, B. 2016b Streamwise-varying steady transpiration control in turbulent pipe flow. J. Fluid Mech. 796, 588–616.CrossRefGoogle Scholar
Halko, N., Martinsson, P. G. & Tropp, J. A. 2011 Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53 (2), 217–288.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
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.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
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18 (1), 011702.Google Scholar
Hussain, A. K. M. F. & Reynolds, W. C. 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241–258.Google Scholar
Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365, 647–664.Google Scholar
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.Google Scholar
Hwang, Y. & Cossu, C. 2010a 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
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.Google Scholar
Jacobi, I.2012 Structure of the turbulent boundary layer under static and dynamic roughness perturbation. PhD thesis, California Institute of Technology.Google Scholar
Jacobi, I. & McKeon, B. J. 2011 Dynamic roughness-perturbation of a turbulent boundary layer. J. Fluid Mech. 688, 258–296.Google Scholar
Jacobi, I. & McKeon, B. J. 2013 Phase relationships between large and small scales in the turbulent boundary layer. Exp. Fluids 54, 1481.Google Scholar
Jeun, J., Nichols, J. W. & Jovanović, M. R. 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.CrossRefGoogle Scholar
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.CrossRefGoogle Scholar
Jiménez, J. 2015 Direct detection of linearized bursts in turbulence. Phys. Fluids 27, 065102.CrossRefGoogle 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.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.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, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.Google Scholar
Klewicki, J. C. 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. Trans. ASME J. Fluids Engng 132, 094001.Google Scholar
Klewicki, J. C., Fife, P., Wei, T. & McMurtry, P. 2007 A physical model of the turbulent boundary layer consonant with the mean momentum balance structure. Phil. Trans. R. Soc. Lond. A 365, 823–839.Google Scholar
Landahl, M. 1967 A wave-guide model for turbulent shear flow. J. Fluid Mech. 29 (3), 441–459.Google Scholar
Lehew, J., Guala, M. & McKeon, B. J. 2011 A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer. Exp. Fluids 51 (4), 997–1012.CrossRefGoogle Scholar
Lu, L. & Papadakis, G. 2014 An iterative method for the computation of the response of linearised Navier–Stokes equations to harmonic forcing and application to forced cylinder wakes. Intl J. Numer. Meth. Fluids 74 (11), 794–817.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014a On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis. J. Fluid Mech. 751, 38–70.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014b Opposition control within the resolvent analysis framework. J. Fluid Mech. 749, 597–626.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2015 A framework for studying the effect of compliant surfaces on wall turbulence. J. Fluid Mech. 768, 415–441.CrossRefGoogle Scholar
Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116, 214501.CrossRefGoogle ScholarPubMed
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. 2012 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18 (1), 405–432.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures of turbulent boundary layers. J. Fluid Mech. 628, 311–337.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical layer model for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
McKeon, B. J., Sharma, A. S. & Jacobi, I.2010 Predicting structural and statistical features of wall turbulence. arXiv:1012-0426.Google Scholar
McKeon, B. J., Sharma, A. S. & Jacobi, I. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25, 031301.Google Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186, 178–197.CrossRefGoogle 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
Moarref, R., Jovanović, M. R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26, 051701.CrossRefGoogle Scholar
Moarref, R., Park, J. S., Sharma, A. S., Willis, A. P., Graham, M. & McKeon, B. J. 2015 Approximation of the exact traveling wave solutions in wall-bounded flows using resolvent modes. In International Symposium on Turbulence and Shear Flow Phenomena (TSFP-9), p. 3A–5, 1–6.Google Scholar
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling and prediction of the streamwise energy intensity in high-Reynolds number turbulent channels. J. Fluid Mech. 734, 275–316.Google Scholar
Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Natrajan, V. K., Wu, Y. & Christensen, K. T. 2007 Spatial signatures of retrograde spanwise vortices in wall turbulence. J. Fluid Mech. 574, 155–167.Google Scholar
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.Google Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 195, 163–199.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in shear flow. Part 3. Theoretical models and comparisons with experiment. J. Fluid Mech. 54, 263–288.Google Scholar
Rosenberg, K., Duvvuri, S., Luhar, M., McKeon, B. J., Barnard, C., Meloy, J. & Sheplak, M.2016 Deterministic wall turbulence? Phase relationships between velocity, wall-pressure, and wall-shear-stress in a forced turbulent boundary layer. AIAA Paper 2016-4396.Google Scholar
Rosenberg, K. & McKeon, B. J. 2016 Modeling turbulent channel flow via the resolvent framework and optimization. In Proceedings of the XXIV International Conference on Theoretical and Applied Mechanics, 21–26 August 2016, Montreal, Canada.Google Scholar
Saxton-Fox, T. & McKeon, B. J. 2016 Scale interactions and 3D critical layers in wall-bounded turbulent flows. In Proceedings of the XXIV International Conference on Theoretical and Applied Mechanics, 21–26 August 2016, Montreal, Canada.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.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.Google Scholar
Sharma, A., Moarref, R., Luhar, M., Goldstein, D. & McKeon, B. J.2014 Effects of a gain-based optimal forcing on turbulent channel flow. AIAA Paper 2014-1450.Google Scholar
Sharma, A. S. & McKeon, B. J.2013a Closing the loop: an explicit calculation of the nonlinearity in the resolvent formulation of wall turbulence. In 43rd AIAA Fluid Dynamics Conference. AIAA 2013-3118.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2013b On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Sharma, A. S., Mezić, I. & McKeon, B. J. 2016a On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier–Stokes equations. Phys. Rev. Fluids 1 (3), 032402(R).Google Scholar
Sharma, A. S., Moarref, R. & McKeon, B. J. 2089 Scaling and interaction of self-similar modes in models of high-Reynolds number wall turbulence. Phil. Trans. R. Soc. Lond. A 375, 20160089.Google Scholar
Sharma, A. S., Moarref, R., McKeon, B. J., Park, J. S., Graham, M. & Willis, A. P. 2016b Low-dimensional representations of exact coherent states of the Navier–Stokes equations from the resolvent model of wall turbulence. Phys. Rev. E 93, 021102(R).Google Scholar
de Silva, C. M., Hutchins, N. & Marusic, I. 2016 Uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 786, 309–331.Google Scholar
Sirovich, L., Ball, K. S. & Keefe, L. R. 1990 Plane waves and structures in turbulent channel flow. Phys. Fluids A 2 (12), 2217–2226.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. 1988 A unified view of the origin and morphology of the turbulent boundary layer structure. In Turbulence Management and Relaminarisation; Proceedings of the IUTAM Symposium, Bangalore, India, January 13–23, 1987 (A89-10154 01-34), pp. 37–61. Springer.Google Scholar
Talluru, K. M., Baidya, R., Hutchins, N. & Marusic, I. 2014 Amplitude modulation of all three velocity components in turbulent boundary layers. J. Fluid Mech. 746, R1.Google Scholar
Tennekes, H. & Lumley, J. L. 1999 A First Course in Turbulence. MIT.Google Scholar
Theodorsen, T. 1952 Mechanism of turbulence. In Proc. 2nd Midwestern Conference on Fluid Mech., pp. 1–19. Ohio State University.Google Scholar
Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27, 105104.CrossRefGoogle Scholar
Towne, A., Colonius, T., Jordan, P., Cavalieri, A. V. & Bres, G, A.2015 Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet. AIAA Paper 2015-2217.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.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. Cambridge University Press.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 5121.Google Scholar
Waleffe, F. 1991 The nature of triad interactions in homogeneous turbulence. Phys. Fluids 4 (2), 350–363.Google Scholar
Waleffe, F. 1995 Transition in shear flows: non-linear normality versus non-normal linearity. Phys. Fluids 7, 3060–3066.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 1517–1534.Google Scholar
Wu, X. & Moin, P. 2008 A direct numerical simulation study on the mean velocity characteristics in pipe flow. J. Fluid Mech. 608, 81–112.CrossRefGoogle Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.Google Scholar