Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T22:26:26.507Z Has data issue: false hasContentIssue false
  • English
  • Français

The mean logarithm emerges with self-similar energy balance

Published online by Cambridge University Press:  01 October 2020

Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]  and
Myoungkyu Lee Open the ORCID record for Myoungkyu Lee [Opens in a new window]
Yongyun Hwang*
Affiliation:
Department of Aeronautics, Imperial College London, South KensingtonSW7 2AZ, UK
Myoungkyu Lee
Affiliation:
Combustion Research Facility, Sandia National Laboratories, Livermore, CA94550, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The attached eddy hypothesis of Townsend (The Structure of Turbulent Shear Flow, 1956, Cambridge University Press) states that the logarithmic mean velocity admits self-similar energy-containing eddies which scale with the distance from the wall. Over the past decade, there has been a significant amount of evidence supporting the hypothesis, placing it to be the central platform for the statistical description of the general organisation of coherent structures in wall-bounded turbulent shear flows. Nevertheless, the most fundamental question, namely why the hypothesis has to be true, has remained unanswered over many decades. Under the assumption that the integral length scale is proportional to the distance from the wall $y$, in the present study we analytically demonstrate that the mean velocity is a logarithmic function of $y$ if and only if the energy balance at the integral length scale is self-similar with respect to $y$, providing a theoretical basis for the attached eddy hypothesis. The analysis is subsequently verified with the data from a direct numerical simulation of incompressible channel flow at the friction Reynolds number $Re_\tau \simeq 5200$ (Lee & Moser, J. Fluid Mech., vol. 774, 2015, pp. 395–415).

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , R6
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355–377.CrossRefGoogle Scholar
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.CrossRefGoogle Scholar
Baars, W. J. & Marusic, I. 2020 a Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra. J. Fluid Mech. 882, A25.CrossRefGoogle Scholar
Baars, W. J. & Marusic, I. 2020 b Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 2. Integrated energy and $a_1$. J. Fluid Mech. 882, A26.CrossRefGoogle Scholar
Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474–504.CrossRefGoogle Scholar
Doohan, P., Willis, A. P. & Hwang, Y. 2019 Shear stress-driven flow: the state space of near-wall turbulence as $Re_\tau \rightarrow \infty$. J. Fluid Mech. 874, 606–638.CrossRefGoogle Scholar
Eckhardt, B. & Zammert, S. 2018 Small scale exact coherent structures at large Reynolds numbers in plane Couette flow. Nonlinearity 31, R66–R77.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. 723, 264–288.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2011 Self-sustained processes in the logarithmic layer of turbulent channel flows. Phys. Fluids 23, 061702.CrossRefGoogle Scholar
Hwang, Y. & Eckhardt, B. 2020 Attached eddy model revisited using a minimal quasilinear approximation. J. Fluid Mech. 894, A23.CrossRefGoogle Scholar
von Kármán, T. 1931 Mechanical similitude and turbulence. NACA TM-611.Google Scholar
Kawata, T. & Alfredsson, P. H. 2018 Inverse interscale transport of the Reynolds shear stress in plane Couette turbulence. Phys. Rev. Lett. 120, 244501.CrossRefGoogle ScholarPubMed
Klewicki, J. C. 2013 Self-similar mean dynamics in turbulent wall flows. J. Fluid Mech. 718, 596–621.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 209–303.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_\tau \approx 5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Lee, M. & Moser, R. D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.CrossRefGoogle Scholar
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. J. 2017 The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1.CrossRefGoogle Scholar
Mizuno, Y. & Jiménez, J. 2011 Mean velocity and length-scales in the overlap region of wallbounded turbulent flows. Phys. Fluids 16, 085112.CrossRefGoogle Scholar
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without wall. J. Fluid Mech. 723, 429–455.CrossRefGoogle Scholar
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.CrossRefGoogle Scholar
Morrison, J. F., McKeon, B. J., Jiang, W. & Smits, A. J. 2004 Scaling of the streamwise velocity component in turbulent pipe flow. J. Fluid Mech. 508, 99–131.CrossRefGoogle Scholar
Panton, R. L. 2007 Composite asymptotic expansions and scaling wall turbulence. Phil. Trans. R. Soc. A 365 (1852), 733–54.CrossRefGoogle ScholarPubMed
Perry, A. E. & Abel, J. C. 1975 Scaling laws for pipe-flow turbulence. J. Fluid Mech. 67, 257–271.CrossRefGoogle Scholar
Perry, A. E. & Abel, J. C. 1977 Asymptotic similarity of turbulence structures in smooth- and rough-walled pipes. J. Fluid Mech. 79, 785–799.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.CrossRefGoogle Scholar
Prandtl, L. 1925 Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. Angew. Math. Mech. 5, 136–139.CrossRefGoogle Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta ^{+} \approx 2000$. Phys. Fluids 25, 105102.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1967 A First Course in Turbulence. MIT Press.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow, 1st edn. Cambridge University Press.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Yang, Q., Willis, A. P. & Hwang, Y. 2019 Exact coherent states of attached eddies in channel flow. J. Fluid Mech. 862, 1029–1059.CrossRefGoogle Scholar