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The engine behind (wall) turbulence: perspectives on scale interactions. (English) Zbl 1383.76239
Summary: 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.

MSC:
76F40 Turbulent boundary layers
Software:
channelflow; Eigtool
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[1] Adrian, R. J., Vortex organization in wall turbulence, Phys. Fluids, 19, (2007) · Zbl 1146.76307
[2] Afzal, N., Mesolayer theory for turbulent flows, AIAA J., 22, 437-439, (1984) · Zbl 0545.76060
[3] Del Álamo, J. C.; Jiménez, J., Linear energy amplification in turbulent channels, J. Fluid Mech., 559, 205-213, (2006) · Zbl 1095.76021
[4] Baars, W. J.; Talluru, K. M.; Hutchins, N.; Marusic, I., Wavelet analysis of wall turbulence to study large-scale modulation of small scales, Exp. Fluids, 56, 188, (2015)
[5] Beneddine, S.; Sipp, D.; Arnault, A.; Dandois, J.; Lesshafft, L., 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, (2016) · Zbl 1422.76070
[6] Bourguignon, J.-L.2013 Models of turbulent pipe flow. PhD thesis, California Institute of Technology.
[7] Bourguignon, J.-L.; Sharma, A. S.; Tropp, J. A.; Mckeon, B. J., Compact representation of wall-bounded turbulence using compressive sampling, Phys. Fluids, 26, (2014)
[8] Bradshaw, P.; Koh, Y. M., A note on Poisson’s equation for pressure in a turbulent flow, Phys. Fluids, 24, 771, 241-258, (1981)
[9] Brandt, L., The lift-up effect: the linear mechanism behind transition and turbulence in shear flows, Eur. J. Mech. (B/Fluids), 47, 80-96, (2014) · Zbl 1297.76073
[10] Butler, K.; Farrell, B., Optimal perturbations and streak spacing in wall-bounded turbulent shear flow, Phys. Fluids A, 5, 774-777, (1993)
[11] Candes, E. J.; Wakin, M. B., An introduction to compressive sampling, IEEE Signal Process. Mag., 25, 21-30, (2008)
[12] Cess, R. D.1958 A study of the literature on heat transfer in turbulent tube flow. Tech. Rep. 8-0529-R24. Westinghouse Research.
[13] Chakraborty, P.; Balachandar, S.; Adrian, R. J., On the relationships between local vortex identification schemes, J. Fluid Mech., 535, 189-214, (2005) · Zbl 1071.76015
[14] Chernyshenko, S. I.; Baig, M. F., The mechanism of streak formation in near-wall turbulence, J. Fluid Mech., 544, 99-131, (2005) · Zbl 1083.76031
[15] Chernyshenko, S. I.; Cicca, G. M.; Iollo, A.; Smirnov, A. V.; Sandham, N. D.; Hu, Z. W., Analysis of data on the relation between eddies and streaky structures in turbulent flows using the placebo method, Fluid Dyn., 41, 5, 772-783, (2006) · Zbl 1200.76109
[16] Cheung, L. C.; Zaki, T., An exact representation of the nonlinear triad interaction terms in spectral space, J. Fluid Mech., 748, 175-188, (2014) · Zbl 1309.76101
[17] Chung, D.; Mckeon, B. J., Large-eddy simulation investigation of large-scale structures in a long channel flow, J. Fluid Mech., 661, 341-364, (2010) · Zbl 1205.76146
[18] Cossu, C.; Pujals, G.; Depardon, S., Optimal transient growth and very large scale structures in turbulent boundary layers, J. Fluid Mech., 619, 79-94, (2009) · Zbl 1156.76400
[19] Dennis, D.; Nickels, T., Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 1. Vortex packets, J. Fluid Mech., 673, 180-217, (2011) · Zbl 1225.76009
[20] Dennis, D.; Nickels, T., Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures, J. Fluid Mech., 673, 218-244, (2011) · Zbl 1225.76034
[21] Drazin, P. G.; Reid, W. H., Hydrodynamic Stability, (2004), Cambridge University Press · Zbl 1055.76001
[22] Duggleby, A.; Ball, K. S.; Pail, M. R.; Fischer, P. F., Dynamical eigenfunction decomposition of turbulent pipe flow, J. Turbul., 8, 43, 1-24, (2007) · Zbl 1273.76192
[23] Duvvuri, S.2016 Non-linear scale interactions in a forced turbulent boundary layer. PhD thesis, California Institute of Technology.
[24] Duvvuri, S.; Mckeon, B. J., Triadic scale interactions in a turbulent boundary layer, J. Fluid Mech., 767, (2015)
[25] Duvvuri, S.; Mckeon, B. J., Non-linear interactions isolated through scale synthesis in experimental wall turbulence, Phys. Rev. Fluids, 1, 3, (2016)
[26] Eckhardt, B.; Faisst, H.; Schmiegel, A.; Scheider, T. M., Dynamical systems and the transition to turbulence in linearly stable shear flows, Phil. Trans. R. Soc. Lond. A, 366, 1297-1315, (2008)
[27] Ellingsen, T.; Palm, E., Stability of linear flow, Phys. Fluids, 18, 487-488, (1975) · Zbl 0308.76030
[28] Farrell, B.; Ioannou, J., Stochastic forcing of the linearized Navier-Stokes equations, Phys. Fluids, 5, 11, 2600-2609, (1993) · Zbl 0809.76078
[29] Farrell, B. F.; Ioannou, P. J., Perturbation structure and spectra in turbulent channel flow, Theor. Comput. Fluid Dyn., 11, 237-250, (1998) · Zbl 0926.76057
[30] Farrell, B. F.; Ioannou, P. J., Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow, J. Fluid Mech., 708, 149-196, (2012) · Zbl 1275.76125
[31] Farrell, B. F. & Ioannou, P. J.2014 Statistical state dynamics: a new perspective on turbulence in shear flow. arXiv:1412.8290v1.
[32] Farrell, B. F.; Ioannou, P. J.; Jiménez, J.; Constantinou, N. C.; Lozano-Duran, A.; Nikolaidis, M. A., 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, (2016) · Zbl 1383.76225
[33] Flores, O.; Jiménez, J., Hierarchy of minimal flow units in the logarithmic layer, Phys. Fluids, 22, (2010)
[34] Fosas De Pando, M., Schmid, P. J. & Sipp, D.2015Nonlinear model-order reduction for oscillator flows using POD-DEIM. In Proc. IUTAM 14, pp. 329-336. Elsevier.
[35] Ganapathisubramani, B.; Hutchins, N.; Monty, J. O.; Chung, D.; Marusic, I., Amplitude and frequency modulation in wall turbulence, J. Fluid Mech., 712, 61-91, (2012) · Zbl 1275.76138
[36] Gayme, D. F.; Mckeon, B. J.; Papachristodolou, A.; Bamieh, B.; Doyle, J. C., Streamwise constant model of turbulence in plane Couette flow, J. Fluid Mech., 665, 99-119, (2010) · Zbl 1225.76149
[37] Gibson, J. F.; Halcrow, J.; Cvitanović, P., Visualizing the geometry of state space in plane Couette flow, J. Fluid Mech., 611, 107-130, (2008) · Zbl 1151.76453
[38] Gómez Carrasco, F.; Blackburn, H.; Rudman, M.; Mckeon, B.; Luhar, M.; Moarref, R.; Sharma, A., On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow, Phys. Fluids, 26, (2014)
[39] Gómez Carrasco, F.; Blackburn, H.; Rudman, M.; Sharma, A.; Mckeon, B., A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator, J. Fluid Mech., 798, R2, (2016)
[40] Gómez Carrasco, F.; Blackburn, H.; Rudman, M.; Sharma, A.; Mckeon, B., Streamwise-varying steady transpiration control in turbulent pipe flow, J. Fluid Mech., 796, 588-616, (2016)
[41] Halko, N.; Martinsson, P. G.; Tropp, J. A., Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 2, 217-288, (2011) · Zbl 1269.65043
[42] Hall, P.; Sherwin, S. J., Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures, J. Fluid Mech., 661, 178-205, (2010) · Zbl 1205.76085
[43] Hall, P.; Smith, F. T., On strongly nonlinear vortex/wave interactions in boundary-layer transition, J. Fluid Mech., 227, 641-666, (1991) · Zbl 0721.76027
[44] Hamilton, J. M.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032
[45] Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, (1996), Cambridge University Press · Zbl 0923.76002
[46] Hoyas, S.; Jiménez, J., Scaling of the velocity fluctuations in turbulent channels up to Re_𝜏 = 2003, Phys. Fluids, 18, 1, (2006)
[47] Hussain, A. K. M. F.; Reynolds, W. C., The mechanics of an organized wave in turbulent shear flow, J. Fluid Mech., 41, 241-258, (1970)
[48] Hutchins, N.; Marusic, I., Large-scale influences in near-wall turbulence, Phil. Trans. R. Soc. Lond. A, 365, 647-664, (2007) · Zbl 1152.76421
[49] Hwang, Y., Statistical structure of self-sustaining attached eddies in turbulent channel flow, J. Fluid Mech., 767, 254-289, (2015)
[50] Hwang, Y.; Cossu, C., Amplification of coherent streaks in the turbulent Couette flow: an input – output analysis at low Reynolds number, J. Fluid Mech., 643, 333-348, (2010) · Zbl 1189.76191
[51] Hwang, Y.; Cossu, C., Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow, J. Fluid Mech., 664, 51-73, (2010) · Zbl 1221.76104
[52] Jacobi, I.2012 Structure of the turbulent boundary layer under static and dynamic roughness perturbation. PhD thesis, California Institute of Technology.
[53] Jacobi, I.; Mckeon, B. J., Dynamic roughness-perturbation of a turbulent boundary layer, J. Fluid Mech., 688, 258-296, (2011) · Zbl 1241.76032
[54] Jacobi, I.; Mckeon, B. J., Phase relationships between large and small scales in the turbulent boundary layer, Exp. Fluids, 54, 1481, (2013)
[55] Jeun, J.; Nichols, J. W.; Jovanović, M. R., Input – output analysis of high-speed axisymmetric isothermal jet noise, Phys. Fluids, 28, 4, (2016)
[56] Jiménez, J., Cascades in wall-bounded turbulence, Annu. Rev. Fluid Mech., 44, 27-45, (2012) · Zbl 1388.76089
[57] Jiménez, J., Direct detection of linearized bursts in turbulence, Phys. Fluids, 27, (2015)
[58] Jiménez, J.; Kawahara, G.; Simens, M. P.; Nagata, M.; Shiba, M., Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions, Phys. Fluids, 17, (2005) · Zbl 1187.76248
[59] Jiménez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, 213-240, (1991) · Zbl 0721.76040
[60] Jovanović, M. R.; Bamieh, B., Componentwise energy amplification in channel flows, J. Fluid Mech., 534, 145-183, (2005) · Zbl 1074.76016
[61] Kawahara, G.; Uhlmann, M.; Van Veen, L., The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech., 44, 203-225, (2012) · Zbl 1352.76031
[62] Kim, J.; Lim, J., A linear process in wall-bounded turbulent shear flows, Phys. Fluids, 12, 8, 1885-1888, (2000) · Zbl 1184.76284
[63] Klewicki, J. C., Reynolds number dependence, scaling, and dynamics of turbulent boundary layers, Trans. ASME J. Fluids Engng, 132, (2010)
[64] Klewicki, J. C.; Fife, P.; Wei, T.; Mcmurtry, P., A physical model of the turbulent boundary layer consonant with the mean momentum balance structure, Phil. Trans. R. Soc. Lond. A, 365, 823-839, (2007) · Zbl 1152.76407
[65] Landahl, M., A wave-guide model for turbulent shear flow, J. Fluid Mech., 29, 3, 441-459, (1967) · Zbl 0147.46005
[66] Lehew, J.; Guala, M.; Mckeon, B. J., A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer, Exp. Fluids, 51, 4, 997-1012, (2011)
[67] Lu, L.; Papadakis, G., 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, (2014)
[68] Luhar, M.; Sharma, A. S.; Mckeon, B. J., On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis, J. Fluid Mech., 751, 38-70, (2014)
[69] Luhar, M.; Sharma, A. S.; Mckeon, B. J., Opposition control within the resolvent analysis framework, J. Fluid Mech., 749, 597-626, (2014)
[70] Luhar, M.; Sharma, A. S.; Mckeon, B. J., A framework for studying the effect of compliant surfaces on wall turbulence, J. Fluid Mech., 768, 415-441, (2015)
[71] Marston, J. B.; Chini, G. P.; Tobias, S. M., Generalized quasilinear approximation: application to zonal jets, Phys. Rev. Lett., 116, (2016)
[72] Marusic, I.; Mathis, R.; Hutchins, N., Predictive model for wall-bounded turbulent flow, Science, 329, 193-196, (2010) · Zbl 1226.76015
[73] Marusic, I.; Monty, J. P.; Hultmark, M.; Smits, A. J., On the logarithmic region in wall turbulence, J. Fluid Mech., 716, (2012) · Zbl 1284.76206
[74] Maslowe, S. A., Critical layers in shear flows, Annu. Rev. Fluid Mech., 18, 1, 405-432, (1986) · Zbl 0634.76046
[75] Mathis, R.; Hutchins, N.; Marusic, I., Large-scale amplitude modulation of the small-scale structures of turbulent boundary layers, J. Fluid Mech., 628, 311-337, (2009) · Zbl 1181.76008
[76] Mckeon, B. J.; Sharma, A. S., A critical layer model for turbulent pipe flow, J. Fluid Mech., 658, 336-382, (2010) · Zbl 1205.76138
[77] Mckeon, B. J., Sharma, A. S. & Jacobi, I.2010 Predicting structural and statistical features of wall turbulence. arXiv:1012-0426.
[78] Mckeon, B. J.; Sharma, A. S.; Jacobi, I., Experimental manipulation of wall turbulence: a systems approach, Phys. Fluids, 25, (2013)
[79] Meseguer, A.; Trefethen, L. N., Linearized pipe flow to Reynolds number 107, J. Comput. Phys., 186, 178-197, (2003) · Zbl 1047.76565
[80] Metzger, M. M.; Klewicki, J. C., A comparative study of near-wall turbulence in high and low Reynolds number boundary layers, Phys. Fluids, 13, 692-701, (2001) · Zbl 1184.76364
[81] Moarref, R.; Jovanović, M. R.; Sharma, A. S.; Tropp, J. A.; Mckeon, B. J., A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization, Phys. Fluids, 26, (2014)
[82] Moarref, R., Park, J. S., Sharma, A. S., Willis, A. P., Graham, M. & Mckeon, B. J.2015Approximation 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.
[83] Moarref, R.; Sharma, A. S.; Tropp, J. A.; Mckeon, B. J., Model-based scaling and prediction of the streamwise energy intensity in high-Reynolds number turbulent channels, J. Fluid Mech., 734, 275-316, (2013) · Zbl 1294.76181
[84] Monty, J. P.; Hutchins, N.; Ng, H. C. H.; Marusic, I.; Chong, M. S., A comparison of turbulent pipe, channel and boundary layer flows, J. Fluid Mech., 632, 431-442, (2009) · Zbl 1183.76036
[85] Nagata, M., Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, J. Fluid Mech., 217, 519-527, (1990)
[86] Natrajan, V. K.; Wu, Y.; Christensen, K. T., Spatial signatures of retrograde spanwise vortices in wall turbulence, J. Fluid Mech., 574, 155-167, (2007) · Zbl 1108.76319
[87] Park, J. S.; Graham, M. D., Exact coherent states and connections to turbulent dynamics in minimal channel flow, J. Fluid Mech., 782, 430-454, (2015) · Zbl 1381.76097
[88] Perry, A. E.; Henbest, S.; Chong, M. S., A theoretical and experimental study of wall turbulence, J. Fluid Mech., 195, 163-199, (1986) · Zbl 0597.76052
[89] Pope, S. B., Turbulent Flows, (2000), Cambridge University Press · Zbl 0966.76002
[90] Reynolds, O., 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, (1883) · JFM 16.0845.02
[91] Reynolds, W. C.; Hussain, A. K. M. F., The mechanics of an organized wave in shear flow. Part 3. Theoretical models and comparisons with experiment, J. Fluid Mech., 54, 263-288, (1972)
[92] 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.
[93] Rosenberg, K. & Mckeon, B. J.2016Modeling 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.
[94] Saxton-Fox, T. & Mckeon, B. J.2016Scale 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.
[95] Schmid, P. J., Nonmodal stability theory, Annu. Rev. Fluid Mech., 39, 129-162, (2007) · Zbl 1296.76055
[96] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, (2001), Springer · Zbl 0966.76003
[97] Schoppa, W.; Hussain, F., Coherent structure generation in near-wall turbulence, J. Fluid Mech., 453, 57-108, (2002) · Zbl 1141.76408
[98] 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.
[99] 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.
[100] Sharma, A. S.; Mckeon, B. J., On coherent structure in wall turbulence, J. Fluid Mech., 728, 196-238, (2013) · Zbl 1291.76173
[101] Sharma, A. S.; Mezić, I.; Mckeon, B. J., On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations, Phys. Rev. Fluids, 1, 3, (2016)
[102] Sharma, A. S.; Moarref, R.; Mckeon, B. J., Scaling and interaction of self-similar modes in models of high-Reynolds number wall turbulence, Phil. Trans. R. Soc. Lond. A, 375, (2089) · Zbl 1404.76129
[103] Sharma, A. S.; Moarref, R.; Mckeon, B. J.; Park, J. S.; Graham, M.; Willis, A. P., Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence, Phys. Rev. E, 93, (2016)
[104] De Silva, C. M.; Hutchins, N.; Marusic, I., Uniform momentum zones in turbulent boundary layers, J. Fluid Mech., 786, 309-331, (2016) · Zbl 1381.76106
[105] Sirovich, L.; Ball, K. S.; Keefe, L. R., Plane waves and structures in turbulent channel flow, Phys. Fluids A, 2, 12, 2217-2226, (1990)
[106] Smits, A. J.; Mckeon, B. J.; Marusic, I., High Reynolds number wall turbulence, Annu. Rev. Fluid Mech., 43, 353-375, (2011) · Zbl 1299.76002
[107] Sreenivasan, K. R.1988A 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.
[108] Talluru, K. M.; Baidya, R.; Hutchins, N.; Marusic, I., Amplitude modulation of all three velocity components in turbulent boundary layers, J. Fluid Mech., 746, (2014) · Zbl 1416.76065
[109] Tennekes, H.; Lumley, J. L., A First Course in Turbulence, (1999), MIT · Zbl 0285.76018
[110] Theodorsen, T.1952Mechanism of turbulence. In Proc. 2nd Midwestern Conference on Fluid Mech., pp. 1-19. Ohio State University.
[111] Thomas, V. L.; Farrell, B. F.; Ioannou, P. J.; Gayme, D. F., A minimal model of self-sustaining turbulence, Phys. Fluids, 27, (2015)
[112] 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.
[113] Townsend, A. A., The Structure of Turbulent Shear Flow, (1956), Cambridge University Press · Zbl 0070.43002
[114] Townsend, A. A., Equilibrium layers and wall turbulence, J. Fluid Mech., 11, 97-120, (1961) · Zbl 0127.42602
[115] Townsend, A. A., The Structure of Turbulent Shear Flow, (1976), Cambridge University Press · Zbl 0325.76063
[116] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators, (2005), Princeton University Press · Zbl 1085.15009
[117] Trefethen, L. N.; Trefethen, A. E.; Reddy, S.; Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science, 261, 5121, (1993) · Zbl 1226.76013
[118] Waleffe, F., The nature of triad interactions in homogeneous turbulence, Phys. Fluids, 4, 2, 350-363, (1991) · Zbl 0745.76027
[119] Waleffe, F., Transition in shear flows: non-linear normality versus non-normal linearity, Phys. Fluids, 7, 3060-3066, (1995) · Zbl 1026.76528
[120] Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 4, 883-900, (1997)
[121] Waleffe, F., Exact coherent structures in channel flow, J. Fluid Mech., 435, 93-102, (2001) · Zbl 0987.76034
[122] Waleffe, F., Homotopy of exact coherent structures in plane shear flows, Phys. Fluids, 15, 6, 1517-1534, (2003) · Zbl 1186.76556
[123] Wu, X.; Moin, P., A direct numerical simulation study on the mean velocity characteristics in pipe flow, J. Fluid Mech., 608, 81-112, (2008) · Zbl 1145.76393
[124] Zare, A.; Jovanović, M. R.; Georgiou, T. T., Colour of turbulence, J. Fluid Mech., 812, 636-680, (2017) · Zbl 1383.76303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.