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On the mixing length eddies and logarithmic mean velocity profile in wall turbulence

  • Michael Heisel (a1) (a2), Charitha M. de Silva (a3), Nicholas Hutchins (a4), Ivan Marusic (a4) and Michele Guala (a1) (a2)...

Abstract

Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterize surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.

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Corresponding author

Email address for correspondence: heise070@umn.edu

References

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Adrian, R. J., Meinhart, C. D. & Tomkins, C. D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 154.
Baars, W. J., Hutchins, N. & Marusic, I. 2017 Self-similarity of wall-attached turbulence in boundary layers. J. Fluid Mech. 823, R2.
Bautista, J. C. C., Ebadi, A., White, C. M., Chini, G. P. & Klewicki, J. C. 2019 A uniform momentum zone–vortical fissure model of the turbulent boundary layer. J. Fluid Mech. 858, 609633.
Brutsaert, W. 2013 Evaporation into the Atmosphere: Theory, History and Applications, vol. 1. Springer Science & Business Media.
Chauhan, K., Philip, J., de Silva, C. M., Hutchins, N. & Marusic, I. 2014 The turbulent/non-turbulent interface and entrainment in a boundary layer. J. Fluid Mech. 742, 119151.
Cheng, C., Li, W., Lozano-Durán, A. & Liu, H. 2019 Identity of attached eddies in turbulent channel flows with bidimensional empirical mode decomposition. J. Fluid Mech. 870, 10371071.
Eisma, J., Westerweel, G., Ooms, G. & Elsinga, G. E. 2015 Interfaces and internal layers in a turbulent boundary layer. Phys. Fluids 27 (5), 055103.
Elsinga, G. E., Ishihara, T., Goudar, M. V., da Silva, C. B. & Hunt, J. C. R. 2017 The scaling of straining motions in homogeneous isotropic turbulence. J. Fluid Mech. 829, 3164.
Flack, K. A., Schultz, M. P. & Connelly, J. S. 2007 Examination of a critical roughness height for outer layer similarity. Phys. Fluids 19 (9), 095104.
George, W. K. 2007 Is there a universal log law for turbulent wall-bounded flows? Phil. Trans. R. Soc. Lond. A 365 (1852), 789806.
Heisel, M., Dasari, T., Liu, Y., Hong, J., Coletti, F. & Guala, M. 2018 The spatial structure of the logarithmic region in very-high-Reynolds-number rough wall turbulent boundary layers. J. Fluid Mech. 857, 704747.
Hwang, J. & Sung, H. J. 2018 Wall-attached structures of velocity fluctuations in a turbulent boundary layer. J. Fluid Mech. 856, 958983.
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254289.
Ishihara, T., Kaneda, Y. & Hunt, J. C. R. 2013 Thin shear layers in high Reynolds number turbulence – DNS results. Flow Turbul. Combust. 91 (4), 895929.
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36 (1), 173196.
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.
von Kármán, T. 1930 Mechanische Ähnlichkeit und Turbulenz. Gött. Nachr. 5, 5876.
Klewicki, J. C. 2013 A description of turbulent wall-flow vorticity consistent with mean dynamics. J. Fluid Mech. 737, 176204.
Klewicki, J. C., Fife, P. & Wei, T. 2009 On the logarithmic mean profile. J. Fluid Mech. 638, 7393.
Laskari, A., de Kat, R., Hearst, R. J. & Ganapathisubramani, B. 2018 Time evolution of uniform momentum zones in a turbulent boundary layer. J. Fluid Mech. 842, 554590.
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100130.
L’vov, V. S., Procaccia, I. & Rudenko, O. 2008 Universal model of finite Reynolds number turbulent flow in channels and pipes. Phys. Rev. Lett. 100, 054504.
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 4974.
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.
McKeon, B. 2019 Self-similar hierarchies and attached eddies. Phys. Rev. Fluids 4 (8), 082601(R).
Meinhart, C. D. & Adrian, R. J. 1995 On the existence of uniform momentum zones in a turbulent boundary layer. Phys. Fluids 7 (4), 694696.
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the surface layer of the atmosphere. Tr. Akad. Nauk. SSSR Geophiz. Inst. 24 (151), 163187.
Morris, S. C., Stolpa, S. R., Slaboch, R. E. & Klewicki, J. C. 2007 Near-surface particle image velocimetry measurements in a transitionally rough-wall atmospheric boundary layer. J. Fluid Mech. 580, 319338.
Nemes, A., Jacono, D. L., Blackburn, H. M. & Sheridan, J. 2015 Mutual inductance of two helical vortices. J. Fluid Mech. 774, 298310.
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Verhandlg. III. Intern. Math. Kong, Heidelberg, pp. 484491. B. G. Teubner.
Prandtl, L. 1925 Bericht über Untersuchungen zur Ausgebildeten Turbulenz. Z. Angew. Math. Mech. 5, 136139.
Prandtl, L. 1932 Zur turbulenten Strömung in Röhren und längs Platten. Ergebn. Aerodyn. Versuchsanst 4, 1829.
Prandtl, L. & Schlichting, H. 1934 Das Widerstandsgesetz rauher Platten. Werft-Redeerei-Hafen 15, 14.
Priyadarshana, P. J. A., Klewicki, J. C., Treat, S. & Foss, J. F. 2007 Statistical structure of turbulent-boundary-layer velocity–vorticity products at high and low Reynolds numbers. J. Fluid Mech. 570, 307346.
Saxton-Fox, T. & McKeon, B. J. 2017 Coherent structures, uniform momentum zones and the streamwise energy spectrum in wall-bounded turbulent flows. J. Fluid Mech. 826, R6.
Schlichting, H. & Gersten, K. 1999 Boundary-Layer Theory, 8th edn. Springer.
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.
de Silva, C. M., Gnanamanickam, E. P., Atkinson, C., Buchmann, N. A., Hutchins, N., Soria, J. & Marusic, I. 2014 High spatial range velocity measurements in a high Reynolds number turbulent boundary layer. Phys. Fluids 26, 025117.
de Silva, C. M., Hutchins, N. & Marusic, I. 2016 Uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 786, 309331.
de Silva, C. M., Philip, J., Chauhan, K., Menevau, C. & Marusic, I. 2013 Multiscale geometry and scaling of the turbulent-nonturbulent interface in high Reynolds number boundary layers. Phys. Rev. Lett. 111, 044501.
de Silva, C. M., Philip, J., Hutchins, N. & Marusic, I. 2017 Interfaces of uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 820, 451478.
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43 (1), 353375.
Squire, D. T., Morrill-Winter, C., Hutchins, N., Marusic, I., Schultz, M. P. & Klewicki, J. C. 2016 Smooth- and rough-wall boundary layer structure from high spatial range particle image velocimetry. Phys. Rev. Fluids 1 (6), 064402.
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, vol. 2. Cambridge University Press.
Tsinober, A. 2001 An Informal Introduction to Turbulence. Springer.
Zagarola, M. V. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 3379.
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On the mixing length eddies and logarithmic mean velocity profile in wall turbulence

  • Michael Heisel (a1) (a2), Charitha M. de Silva (a3), Nicholas Hutchins (a4), Ivan Marusic (a4) and Michele Guala (a1) (a2)...

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