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

Published online by Cambridge University Press:  21 January 2020

Michael Heisel Open the ORCID record for Michael Heisel [Opens in a new window] ,
Charitha M. de Silva Open the ORCID record for Charitha M. de Silva [Opens in a new window] ,
Nicholas Hutchins Open the ORCID record for Nicholas Hutchins [Opens in a new window] ,
Ivan Marusic Open the ORCID record for Ivan Marusic [Opens in a new window]  and
Michele Guala Open the ORCID record for Michele Guala [Opens in a new window]
Michael Heisel*
Affiliation:
St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55414, USA Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Charitha M. de Silva
Affiliation:
School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney2052, Australia
Nicholas Hutchins
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Victoria3010, Australia
Ivan Marusic
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Victoria3010, Australia
Michele Guala
Affiliation:
St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55414, USA Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: heise070@umn.edu
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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.

JFM classification

Turbulent Flows: Turbulent boundary layers Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Atmospheric flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , R1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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, 1–54.CrossRefGoogle Scholar
Baars, W. J., Hutchins, N. & Marusic, I. 2017 Self-similarity of wall-attached turbulence in boundary layers. J. Fluid Mech. 823, R2.CrossRefGoogle Scholar
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, 609–633.CrossRefGoogle Scholar
Brutsaert, W. 2013 Evaporation into the Atmosphere: Theory, History and Applications, vol. 1. Springer Science & Business Media.Google Scholar
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, 119–151.CrossRefGoogle Scholar
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, 1037–1071.CrossRefGoogle ScholarPubMed
Eisma, J., Westerweel, G., Ooms, G. & Elsinga, G. E. 2015 Interfaces and internal layers in a turbulent boundary layer. Phys. Fluids 27 (5), 055103.CrossRefGoogle Scholar
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, 31–64.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
George, W. K. 2007 Is there a universal log law for turbulent wall-bounded flows? Phil. Trans. R. Soc. Lond. A 365 (1852), 789–806.CrossRefGoogle Scholar
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, 704–747.CrossRefGoogle Scholar
Hwang, J. & Sung, H. J. 2018 Wall-attached structures of velocity fluctuations in a turbulent boundary layer. J. Fluid Mech. 856, 958–983.CrossRefGoogle Scholar
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.CrossRefGoogle Scholar
Ishihara, T., Kaneda, Y. & Hunt, J. C. R. 2013 Thin shear layers in high Reynolds number turbulence – DNS results. Flow Turbul. Combust. 91 (4), 895–929.CrossRefGoogle Scholar
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36 (1), 173–196.CrossRefGoogle Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.CrossRefGoogle Scholar
von Kármán, T. 1930 Mechanische Ähnlichkeit und Turbulenz. Gött. Nachr. 5, 58–76.Google Scholar
Klewicki, J. C. 2013 A description of turbulent wall-flow vorticity consistent with mean dynamics. J. Fluid Mech. 737, 176–204.CrossRefGoogle Scholar
Klewicki, J. C., Fife, P. & Wei, T. 2009 On the logarithmic mean profile. J. Fluid Mech. 638, 73–93.CrossRefGoogle Scholar
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, 554–590.CrossRefGoogle 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
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.CrossRefGoogle ScholarPubMed
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74.CrossRefGoogle Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
McKeon, B. 2019 Self-similar hierarchies and attached eddies. Phys. Rev. Fluids 4 (8), 082601(R).CrossRefGoogle Scholar
Meinhart, C. D. & Adrian, R. J. 1995 On the existence of uniform momentum zones in a turbulent boundary layer. Phys. Fluids 7 (4), 694–696.CrossRefGoogle Scholar
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), 163–187.Google Scholar
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, 319–338.CrossRefGoogle Scholar
Nemes, A., Jacono, D. L., Blackburn, H. M. & Sheridan, J. 2015 Mutual inductance of two helical vortices. J. Fluid Mech. 774, 298–310.CrossRefGoogle Scholar
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Verhandlg. III. Intern. Math. Kong, Heidelberg, pp. 484–491. B. G. Teubner.Google Scholar
Prandtl, L. 1925 Bericht über Untersuchungen zur Ausgebildeten Turbulenz. Z. Angew. Math. Mech. 5, 136–139.CrossRefGoogle Scholar
Prandtl, L. 1932 Zur turbulenten Strömung in Röhren und längs Platten. Ergebn. Aerodyn. Versuchsanst 4, 18–29.Google Scholar
Prandtl, L. & Schlichting, H. 1934 Das Widerstandsgesetz rauher Platten. Werft-Redeerei-Hafen 15, 1–4.Google Scholar
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, 307–346.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 1999 Boundary-Layer Theory, 8th edn. Springer.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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
de Silva, C. M., Hutchins, N. & Marusic, I. 2016 Uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 786, 309–331.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
de Silva, C. M., Philip, J., Hutchins, N. & Marusic, I. 2017 Interfaces of uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 820, 451–478.CrossRefGoogle Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43 (1), 353–375.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers.CrossRefGoogle Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, vol. 2. Cambridge University Press.Google Scholar
Tsinober, A. 2001 An Informal Introduction to Turbulence. Springer.Google Scholar
Zagarola, M. V. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.CrossRefGoogle Scholar