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6 - Dynamics of Wall-Bounded Turbulence

Published online by Cambridge University Press:  05 February 2013

By
J. Jiménez  and
G. Kawahara
Edited by
Peter A. Davidson ,
Yukio Kaneda  and
Katepalli R. Sreenivasan
J. Jiménez
Affiliation:
Stanford University
G. Kawahara
Affiliation:
Osaka University
Peter A. Davidson
Affiliation:
University of Cambridge
Yukio Kaneda
Affiliation:
Aichi Institute of Technology, Japan
Katepalli R. Sreenivasan
Affiliation:
New York University
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Summary

Introduction

This chapter deals with the dynamics of wall-bounded turbulent flows, with a decided emphasis on the results of numerical simulations. As we will see, part of the reason for that emphasis is that much of the recent work on dynamics has been computational, but also that the companion chapter by Marusic and Adrian (2012) in this volume reviews the results of experiments over the same period.

The first direct numerical simulations (DNS) of wall-bounded turbulence (Kim et al., 1987) began to appear soon after computers became powerful enough to allow the simulation of turbulence in general (Siggia, 1981; Rogallo, 1981). Large-eddy simulations (LES) of wall-bounded flows had been published before (Deardorff, 1970; Moin and Kim, 1982) but, after DNS became current, they were de-emphasized as means of clarifying the flow physics, in part because doubts emerged about the effect of the poorly resolved near-wall region on the rest of the flow. Some of the work summarized below has eased those misgivings, and there are probably few reasons to distrust the information provided by LES on the largest flow structures, but any review of the physical results of numerics in the recent past necessarily has to deal mostly with DNS. Only atmospheric scientists, for whom the prospect of direct simulation remains remote, have continued to use LES to study the mechanics of the atmospheric surface layer (see, for example, Deardorff, 1973; Siebesma et al., 2003). A summary of the early years of numerical turbulence research can be found in Rogallo and Moin (1984) and Moin and Mahesh (1998).

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Ten Chapters in Turbulence , pp. 221 - 268
Publisher: Cambridge University Press
Print publication year: 2012

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References

Abe, H., Kawamura, H., and Matsuo, Y. 2004. Surface heat-flux fluctuations in a turbulent channel flow up to Reτ = 1020 with Pr = 0.025 and 0.71. Int. J. Heat Fluid Flow, 25, 404–419.Google Scholar
Adrian, R. J. 2007. Hairpin vortex organization in wall turbulence. Phys. Fluids., 19, 041301.Google Scholar
Adrian, R. J., Meinhart, C. D., and Tomkins, C. D. 2000. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech., 422, 1–54.Google Scholar
Afzal, N., and Yajnik, K. 1973. Analysis of turbulent pipe and channel flows at moderately large Reynolds number. J. Fluid Mech., 61, 23–31.Google Scholar
Artuso, R., Aurell, E., and Cvitanović, P. 1990a. Recycling of strange sets: I. Cycle expansions. Nonlinearity, 3, 325–359.Google Scholar
Artuso, R., Aurell, E., and Cvitanović, P. 1990b. Recycling of strange sets: II. Applications. Nonlinearity, 3, 361–386.Google Scholar
Aubry, N., Holmes, P., Lumley, J. L., and Stone, E. 1988. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192, 115–173.Google Scholar
Bakewell, H. P., and Lumley, J. L. 1967. Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids, 10, 1880–1889.Google Scholar
Barenblatt, G. I., Chorin, A. J., and Prostokishin, V. M. 2000. Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech., 410, 263–283.Google Scholar
Bradshaw, P. 1967. Inactive motions and pressure fluctuations in turbulent boundary layers. J. Fluid Mech., 30, 241–258.Google Scholar
Brown, G. L., and 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., and Farrell, B. F. 1993. Optimal perturbations and streak spacing in wall-bounded shear flow. Phys. Fluids A, 5, 774–777.Google Scholar
Cherhabili, A., and Ehrenstein, U. 1997. Finite-amplitude equilibrium states in plane Couette flow. J. Fluid Mech., 342, 159–177.Google Scholar
Christensen, K. T., and Adrian, R. J. 2001. Statistical evidence of hairpin vortex packets in wall turbulence. J. Fluid Mech., 431, 433–443.Google Scholar
Christiansen, F., Cvitanović, P., and Putkaradze, V. 1997. Spatiotemporal chaos in terms of unstable recurrent patterns. Nonlinearity, 10, 55–70.Google Scholar
Clever, R. M., and Busse, F. H. 1992. Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech., 234, 511–527.Google Scholar
Clever, R. M., and Busse, F. H. 1997. Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech., 344, 137–153.Google Scholar
Cvitanović, . 1987. Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett., 61, 2729–2732.Google Scholar
Darcy, H. 1854. Recherches expérimentales rélatives au mouvement de l'eau dans les tuyeaux. Mém. Savants Etrang. Acad. Sci. Paris, 17, 1–268.Google Scholar
Dauchot, O., and Daviaud, F. 1995. Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids, 7, 335–343.Google Scholar
Deardorff, J. W. 1970. A numerical study of three-dimensional turbulent channel flow at large Reynolds number. J. Fluid Mech., 41, 453–480.Google Scholar
Deardorff, J. W. 1973. The use of subgrid transport equations in a three-dimensional model of atmospheric turbulence. J. Fluid Eng., 95, 429–438.Google Scholar
deGraaff, D. B., and Eaton, J. K. 2000. Reynolds number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech., 422, 319–346.Google Scholar
del Álamo, J. C., and Jiménez, J. 2003. Spectra of very large anisotropic scales in turbulent channels. Phys. Fluids, 15, L41–L44.Google Scholar
del Álamo, J. C., and Jiménez, J. 2006. Linear energy amplification in turbulent channels. J. Fluid Mech., 559, 205–213.Google Scholar
del Álamo, J. C., Jiménez, J., Zandonade, P., and Moser, R. D. 2004. Scaling of the energy spectra of turbulent channels. J. Fluid Mech., 500, 135–144.Google Scholar
del Álamo, J. C., Jiménez, J., Zandonade, P., and Moser, R. D. 2006. Self-similar vortex clusters in the logarithmic region. J. Fluid Mech., 561, 329–358.Google Scholar
Duguet, Y., Pringle, C. C., and Kerswell, R. R. 2008. Relative periodic orbits in transitional pipe flow. Phys. Fluids, 20, 114102.Google Scholar
Duguet, Y., Schlatter, P., and Henningson, D. S. 2009. Localized edge states in plane Couette flow. Phys. Fluids, 21, 111701.Google Scholar
Eckhardt, B., Scheider, T. M., Hof, B., and Westerweel, J. 2007. Turbulence transition in pipe flow. Ann. Rev. Fluid Mech., 39, 447–468.Google Scholar
Ehrenstein, U., and Koch, W. 1991. Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech., 228, 111–148.Google Scholar
Ehrenstein, U., Nagata, M., and Rincon, F. 2008. Two-dimensional nonlinear plane Poiseuille–Couette flow homotopy revisited. Phys. Fluids, 20, 064103.Google Scholar
Faisst, H., and Eckhardt, B. 2003. Traveling waves in pipe flow. Phys. Rev. Lett., 91, 224502.Google Scholar
Feller, W. 1971. An Introduction to Probability theory and its Applications. third edn. Vol. 1. Wiley. pg. 446–448.
Flores, O., and Jiménez, J. 2005. Linear dynamics of turbulent structures in the log layer. Pages LR–1 of: Proc. Div. Fluid Dyn.Am. Phys. Soc.
Flores, O., and Jiménez, J. 2006. Effect of wall-boundary disturbances on turbulent channel flows. J. Fluid Mech., 566, 357–376.Google Scholar
Flores, O., and Jiménez, J. 2010. Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids, 22, 071704.Google Scholar
Flores, O., Jiménez, J., and del Álamo, J. C. 2007. Vorticity organization in the outer layer of turbulent channels with disturbed walls. J. Fluid Mech., 591, 145–154.Google Scholar
Foias, C., Manley, O., Rosa, R., and Temam, R. 2001. Navier–Stokes Equations and Turbulence. Cambridge University Press.
Frederickson, P., Kaplan, J. L., Yorke, E. D., and Yorke, J. A. 1983. The Lyapunov dimension of strange attractors. J. Diffl. Equat., 49, 185–207.Google Scholar
Gaster, M., Kit, E., and Wygnanski, I. 1985. Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech., 150, 23–39.Google Scholar
Gibson, J. F., Halcrow, J., and Cvitanović, P. 2008. Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech., 611, 107–130.Google Scholar
Gibson, J. F., Halcrow, J., and Cvitanović, P. 2009. Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech., 638, 243–266.Google Scholar
Guckenheimer, J., and Holmes, P. 1986. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd ed. Springer Verlag.
Hagen, G. H. L. 1839. Über den Bewegung des Wassers in engen cylindrischen Röhren. Poggendorfs Ann. Physik Chemie, 46, 423–442.Google Scholar
Halcrow, J., Gibson, J. F., and Cvitanović, P. 2009. Heteroclinic connections in plane Couette flow. J. Fluid Mech., 621, 365–376.Google Scholar
Hamilton, J. M., Kim, J., and Waleffe, F. 1995. Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech., 287, 317–348.Google Scholar
Ho, C. H., and Huerre, P. 1984. Perturbed free shear layers. Ann. Rev. Fluid Mech., 16, 365–424.Google Scholar
Holmes, P., Lumley, J. L., and Berkooz, G. 1996. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1st ed. Cambridge University Press.
Hoyas, S., and Jiménez, J. 2006. Scaling of the velocity fluctuations in turbulent channels up to Reτ = 2003. Phys. Fluids, 18, 011702.Google Scholar
Hoyas, S., and Jiménez, J. 2008. Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids, 20, 101511.Google Scholar
Hutchins, N., and Marusic, I. 2007. Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech., 579, 467–477.Google Scholar
Itano, T., and Generalis, S. C. 2009. Hairpin vortex solution in planar Couette flow: A tapestry of knotted vortices. Phys. Rev. Lett., 102, 114501.Google Scholar
Itano, T., and Toh, S. 2001. The dynamics of bursting process in wall turbulence. J. Phys. Soc. Jpn., 70, 703–716.Google Scholar
Jiménez, J. 1987. Coherent structures and dynamical systems. Pages 323–324 of: Proc. CTR Summer Program 1987. Stanford University.
Jiménez, J. 1998. The largest scales of turbulence. Pages 137–154 of: CTR Ann. Res. Briefs. Stanford University.
Jiménez, J. 2003. Computing high-Reynolds number flows: Will simulations ever substitute experiments?J. of Turbulence, 22.Google Scholar
Jiménez, J. 2004. Turbulent flows over rough walls. Ann. Rev. Fluid Mech., 36, 173–196.Google Scholar
Jiménez, J. 2007. Recent developments in wall-bounded turbulence. Rev. R. Acad. Cien. Serie A, Mat., 101, 187–203.Google Scholar
Jiménez, J. 2012. Cascades in wall-bounded turbulence. Ann. Rev. Fluid Mech., 44, 27–45.Google Scholar
Jiménez, J., and Hoyas, S. 2008. Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech., 611, 215–236.Google Scholar
Jiménez, J., and Moin, P. 1991. The minimal flow unit in near-wall turbulence. J. Fluid Mech., 225, 221–240.Google Scholar
Jiménez, J., and Moser, R. D. 2007. What are we learning from simulating wall turbulence?Phil. Trans. Roy. Soc.A, 365, 715–732.Google Scholar
Jiménez, J., and Pinelli, A. 1999. The autonomous cycle of near wall turbulence. J. Fluid Mech., 389, 335–359.Google Scholar
Jiménez, J., and Simens, M. P. 2001. Low-dimensional dynamics in a turbulent wall flow. J. Fluid Mech., 435, 81–91.Google Scholar
Jiménez, J., Wray, A. A., Saffman, P. G., and Rogallo, R. S. 1993. The structure of intense vorticity in isotropic turbulence. J. Fluid Mech., 255, 65–90.Google Scholar
Jiménez, J., del Álamo, J. C., and Flores, O. 2004. The large-scale dynamics of near-wall turbulence. J. Fluid Mech., 505, 179–199.Google Scholar
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M., and Shiba, M. 2005. Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids, 17, 015105.Google Scholar
Kato, S., and Yamada, M. 2003. Unstable periodic solutions embedded in a shell model turbulence. Phys. Rev.E, 68, 025302(R).Google Scholar
Kawahara, G. 2005. Laminarization of minimal plane Couette flow: Going beyond the basin of attraction of turbulence. Phys. Fluids, 17, 041702.Google Scholar
Kawahara, G. 2009. Theoretical interpretation of coherent structures in near-wall turbulence. Fluid Dynamics Research, 41, 064001.Google Scholar
Kawahara, G., and Kida, S. 2001. Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech., 449, 291–300.Google Scholar
Kawahara, G., Jiménez, J., Uhlmann, M., and Pinelli, A. 2003. Linear instability of a corrugated vortex sheet–a model for streak instability. J. Fluid Mech., 483, 315–342.Google Scholar
Keefe, L., Moin, P., and Kim, J. 1992. The dimension of attractors underlying periodic turbulent Poiseuille flow. J. Fluid Mech., 242, 1–29.Google Scholar
Kerswell, R. R. 2005. Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity, 18, R17–R44.Google Scholar
Kerswell, R. R., and Tutty, O. R. 2007. Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech., 584, 69–102.Google Scholar
Kida, S. 1985. Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan, 54, 3132–2136.Google Scholar
Kim, J. 1985. Turbulence structures associated with the bursting event. Phys. Fluids, 28, 52–58.Google Scholar
Kim, J., and Hussain, F. 1993. Propagation velocity of perturbations in channel flow. Phys. FluidsA, 5, 695–706.Google Scholar
Kim, J., Moin, P., and Moser, R. D. 1987. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech., 177, 133–166.Google Scholar
Kim, K.C., and Adrian, R. J. 1999. Very large-scale motion in the outer layer. Phys. Fluids, 11, 417–422.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A., and Runstadler, P. W. 1967. 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 a very large Reynolds numbers. Dokl. Akad. Nauk. SSSR, 30, 301–305. Reprinted in Proc. Roy. Soc. London. A 434, 9–13 (1991).Google Scholar
Lee, S-H., and Sung, H. J. 2007. Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall. J. Fluid Mech., 584, 125–146.Google Scholar
Lee, S-H., and Sung, H. J. 2011. Very-large-scale motions in a turbulent boundary layer. J. Fluid Mech., 673, 80–120.Google Scholar
Lozano-Durán, A., and Jiménez, J. 2010. Time-resolved evolution of the wall-bounded vorticity cascade. Pages EB–3 of: Proc. Div. Fluid Dyn. Am. Phys. Soc.
Marusic, I., and Adrian, R. J. 2012. The eddies and scales of wall-turbulence. This volume.
Mathis, R., Hutchins, N., and Marusic, I. 2009. Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech., 628, 311–337.Google Scholar
Meinhart, C. D., and Adrian, R. J. 1995. On the existence of uniform momentum zones in a turbulent boundary layer. Phys. Fluids, 7, 694–696.Google Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. M., and Eckhardt, B. 2009. Transition in localized pipe flow turbulence. Phys. Rev. Lett., 103, 054502.Google Scholar
Moin, P., and Kim, J. 1982. Numerical investigation of turbulent channel flow. J. Fluid Mech., 118, 341–377.Google Scholar
Moin, P., and Mahesh, K. 1998. Direct numerical simulation: A tool in turbulence research. Ann. Rev. Fluid Mech., 30, 539–578.Google Scholar
Moin, P., and Moser, R. D. 1989. Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech., 200, 471–509.Google Scholar
Morrison, W. R. B., Bullock, K. J., and Kronauer, R. E. 1971. Experimental evidence of waves in the sublayer. J. Fluid Mech., 47, 639–656.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
Nagata, M. 1997. Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev.E, 55, 2023–2025.Google Scholar
Oberlack, M. 2001. A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech., 427, 299–328.Google Scholar
Obukhov, A. M. 1941. On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk. SSSR, 32, 22–24.Google Scholar
Okino, S., Nagata, M., Wedin, H., and Bottaro, A. 2010. A new nonlinear vortex state in square-duct flow. J. Fluid Mech., 657, 413–429.Google Scholar
Österlund, J. M., Johansson, A. V., Nagib, H. M., and Hites, M. 2000. A note on the overlap region in turbulent boundary layers. Phys. Fluids, 12, 1–4.Google Scholar
Pinelli, A., Uhlmann, M., Sekimoto, A., and Kawahara, G. 2010. Reynolds number dependence of mean flow structure in square duct turbulence. J. Fluid Mech., 644, 107–122.Google Scholar
Prigent, A., Grégorie, G., Chaté, H., Dauchot, O., and van Saarloos, W. 2002. Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett., 89, 014501.Google Scholar
Pringle, C. C., and Kerswell, R. R. 2007. Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett., 99, 074502.Google Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S., and Henningson, D. S. 1998. On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech., 365, 269–303.Google Scholar
Reynolds, O. 1894. On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. Roy. Soc. London, 186, 123–164. Papers, ii, 535.Google Scholar
Richardson, L. F. 1920. The supply of energy from and to atmospheric eddies. Proc. Roy. Soc.A, 97, 354–373.Google Scholar
Rogallo, R. S. 1981. Numerical experiments in homogeneous turbulence. Tech. Memo 81315. NASA.
Rogallo, R. S., and Moin, P. 1984. Numerical simulations of turbulent flows. Ann. Rev. Fluid Mech., 16, 99–137.Google Scholar
Saiki, Y., and Yamada, M. 2009. Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems. Phys. Rev.E, 79, 015201(R).Google Scholar
Saiki, Y., and Yamada, M. 2010. Unstable periodic orbits embedded in a continuous time dynamical system – time averaged properties. RIMS Kokyuroku, 1713, 111–123.
Schlatter, P., and Örlü, R. 2010. Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech., 659, 116–126.Google Scholar
Schlatter, P., Örlü, R., Li, Q., Fransson, J.H.M., Johansson, A.V., Alfredsson, P. H., and Henningson, D. S. 2009. Turbulent boundary layers up to Reθ = 2500 through simulation and experiments. Phys. Fluids, 21, 05702.Google Scholar
Schmiegel, A. 1999. Transition to turbulence in linearly stable shear flows. Ph.D. thesis, Faculty of Physics, Philipps-Universität Marburg.
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F., and Eckhardt, B. 2008. Laminar-turbulent boundary in plane Couette flow. Phys. Rev.E, 78, 037301.Google Scholar
Schneider, T. M., Gibson, J. F., and Burke, J. 2010. Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett., 104, 104501.Google Scholar
Schoppa, W., and Hussain, F. 2002. Coherent structure generation in near-wall turbulence. J. Fluid Mech., 453, 57–108.Google Scholar
She, Z.-S. 1993. Constrained Euler system for Navier–Stokes turbulence. Phys. Rev. Lett., 70, 1255–1258.Google Scholar
Siebesma, A. P., Bretherton, C. S., Brown, A., A., Chlond, Cuxart, J., Duynkerke, P. G., Jiang, H. L., Khairoutdinov, M., Lewellen, D., Moeng, C. H., Sanchez, E., Stevens, B., and Stevens, D. E. 2003. A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219.Google Scholar
Siggia, E. D. 1981. Numerical study of small scale intermittency in three-dimensional turbulence. J. Fluid Mech., 107, 375–406.Google Scholar
Sillero, J. A., Borrell, G., Gungor, A. G., Jiménez, J., Moser, R.D., and Oliver, T. A. 2010. Direct simulation of the zero-pressure-gradient boundary layer up to Reθ = 6000. Pages EB–4 of: Proc. Div. Fluid Dyn. Am. Phys. Soc.
Simens, M.P., Jiménez, J., Hoyas, S., and Mizuno, Y. 2009. A high-resolution code for turbulent boundary layers. J. Comput. Phys., 228, 4218–4231.Google Scholar
Smith, C. R., and 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.Google Scholar
Spalart, P. R., Coleman, G. N., and Johnstone, R. 2008. Direct numerical simulation of the Ekman layer: A step in Reynolds number, and cautious support for a log law with a shifted origin. Phys. Fluids, 20, 101507.Google Scholar
Swearingen, J. D., and 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., and Lumley, J. L. 1972. A First Course in Turbulence. MIT Press.
Toh, S., and Itano, T. 2003. A periodic-like solution in channel flow. J. Fluid Mech., 481, 67–76.Google Scholar
Toh, S., and 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 ed. Cambridge University Press.
Uhlmann, M., Kawahara, G., and Pinelli, A. 2010. Traveling-waves consistent with turbulence-driven secondary flow in a square duct. Phys. Fluid, 22, 084102.Google Scholar
van Veen, L., and Kawahara, G. 2011. Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett., 107, 114501.Google Scholar
van Veen, L., Kawahara, G., and Matsumura, A. 2011. On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comp., 33, 25–44.Google Scholar
van Veen, L., Kida, S., and Kawahara, G. 2006. Periodic motion representing isotropic turbulence. Fluid Dynamics Research, 38, 19–46.Google Scholar
Vincent, A., and Meneguzzi, M. 1991. The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech, 225, 1–25.Google Scholar
Viswanath, D. 2007. Recurrent motions within plane Couette turbulence. J. Fluid Mech., 580, 339–358.Google Scholar
Viswanath, D. 2009. The critical layer in pipe flow at high Reynolds number. Phil. Trans. Roy. Soc.A, 367, 561–576.Google Scholar
Waleffe, F. 1997. On a self-sustaining process in shear flows. Phys. Fluids, 9, 883–900.Google Scholar
Waleffe, F. 1998. Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett., 81, 4140–4143.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, 1517–1534.Google Scholar
Wang, J., Gibson, J., and Waleffe, F. 2007. Lower branch coherent states in shear flows: Transition and control. Phys. Rev. Lett., 98, 204501.Google Scholar
Wedin, H., and Kerswell, R. R. 2004. Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech., 508, 333–371.Google Scholar
Wedin, H., Bottaro, A., and Nagata, M. 2009. Three-dimensional traveling waves in a square duct. Phys. Rev.E, 79, 065305(R).Google Scholar
Willis, A. P., and Kerswell, R. R. 2008. Coherent structures in localized and global pipe turbulence. Phys. Rev. Lett., 100, 124501.Google Scholar
Willis, A. P., and Kerswell, R. R. 2009. Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states. J. Fluid Mech., 619, 213–233.Google Scholar
Wosnik, M., Castillo, L., and George, W. K. 2000. A theory for turbulent pipe and channel flows. J. Fluid Mech., 421, 115–145.Google Scholar
Wu, X., and Moin, P. 2010. Transitional and turbulent boundary layer with heat transfer. Phys. Fluids., 22, 085105.Google Scholar
Wygnanski, I. J., and Champagne, F. H. 1973. On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech., 59, 281–335.Google Scholar
Zhou, J., Adrian, R. J., S., Balachandar, and Kendall, T. M. 1999. Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech., 387, 353–396.Google Scholar

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