Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-14T00:30:21.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristic scales of Townsend’s wall-attached eddies

Published online by Cambridge University Press:  16 April 2019

Adrián Lozano-Durán Open the ORCID record for Adrián Lozano-Durán [Opens in a new window]  and
Hyunji Jane Bae Open the ORCID record for Hyunji Jane Bae [Opens in a new window]
Adrián Lozano-Durán*
Affiliation:
Center for Turbulence Research, Stanford University, CA 94305, USA
Hyunji Jane Bae
Affiliation:
Center for Turbulence Research, Stanford University, CA 94305, USA Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: adrianld@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Townsend (The Structure of Turbulent Shear Flow, 1976, Cambridge University Press) proposed a structural model for the logarithmic layer (log layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multiscale population of eddies attached to the wall. In the attached-eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all length scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.

JFM classification

Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 868 , 10 June 2019 , pp. 698 - 725
Copyright
© 2019 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

Abderrahaman-Elena, N. & García-Mayoral, R. 2017 Analysis of anisotropically permeable surfaces for turbulent drag reduction. Phys. Rev. Fluids 2, 114609.Google Scholar
Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.Google Scholar
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.Google Scholar
Afzal, N. 1976 Millikan’s argument at moderately large Reynolds number. Phys. Fluids 19 (4), 600–602.Google Scholar
Afzal, N. & Yajnik, K. 1973 Analysis of turbulent pipe and channel flows at moderately large Reynolds number. J. Fluid Mech. 61 (1), 23–31.Google Scholar
Agostini, L. & Leschziner, M. 2017 Spectral analysis of near-wall turbulence in channel flow at Re 𝜏 = 4200 with emphasis on the attached-eddy hypothesis. Phys. Rev. Fluids 2, 014603.Google Scholar
del Álamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15, L41–L44.Google Scholar
del Álamo, J. C. & 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. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.Google Scholar
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27 (10), 105103.Google Scholar
Bae, H. J., Lozano-Durán, A., Bose, S. T. & Moin, P. 2018a Dynamic wall model for the slip boundary condition in large-eddy simulation. J. Fluid Mech. 859, 400–432.Google Scholar
Bae, H. J., Lozano-Durán, A., Bose, S. T. & Moin, P. 2018b Turbulence intensities in large-eddy simulation of wall-bounded flows. Phys. Rev. Fluids 3, 014610.Google Scholar
Bailey, S. C. C., Hultmark, M., Smits, A. J. & Schultz, M. P. 2008 Azimuthal structure of turbulence in high Reynolds number pipe flow. J. Fluid Mech. 615, 121–138.Google Scholar
Bakken, O. M., Krogstad, P. Å., Ashrafian, A. & Andersson, H. I. 2005 Reynolds number effects in the outer layer of the turbulent flow in a channel with rough walls. Phys. Fluids 17 (6), 065101.Google Scholar
Balakumar, B. J. & Adrian, R. J. 2007 Large- and very-large-scale motions in channel and boundary-layer flow. Phil. Trans. R. Soc. Lond. A 365, 665–681.Google Scholar
Bullock, K. J., Cooper, R. E. & Abernathy, F. H. 1978 Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow. J. Fluid Mech. 88, 585–608.Google Scholar
Buschmann, M. H. & Gad-el Hak, M. 2003 Generalized logarithmic law and its consequences. AIAA J. 41 (1), 40–48.Google Scholar
Chandran, D., Baidya, R., Monty, J. P. & Marusic, I. 2017 Two-dimensional energy spectra in high-Reynolds-number turbulent boundary layers. J. Fluid Mech. 826, R1.Google Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110.Google Scholar
Chung, D., Monty, J. P. & Ooi, A. 2014 An idealised assessment of Townsend’s outer-layer similarity hypothesis for wall turbulence. J. Fluid Mech. 742, R3.Google Scholar
Coles, D. E. & Hirst, E. 1969 Computation of turbulent boundary layers. In 1968 AFOSR-IFP-Stanford Conference, Belvoir Defense Technical Information Center.Google Scholar
Davidson, P. A. & Krogstad, P.-A. 2009 A simple model for the streamwise fluctuations in the log-law region of a boundary layer. Phys. Fluids 21 (5), 055105.Google Scholar
Davidson, P. A., Nickels, T. B. & Krogstad, P.-A. 2006 The logarithmic structure function law in wall-layer turbulence. J. Fluid Mech. 550, 51–60.Google Scholar
De Graaff, D. B. & Eaton, J. K. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.Google Scholar
Dong, S., Lozano-Durán, A., Sekimoto, A. & Jiménez, J. 2017 Coherent structures in statistically stationary homogeneous shear turbulence. J. Fluid Mech. 816, 167–208.Google Scholar
Flores, O. & Jiménez, J. 2006 Effect of wall-boundary disturbances on turbulent channel flows. J. Fluid Mech. 566, 357–376.Google Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22 (7), 071704.Google Scholar
Flores, O., Jiménez, J. & 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
Guala, M., Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–542.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18 (1), 011702.Google Scholar
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20 (10), 101511.Google Scholar
Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2012 Turbulent pipe flow at extreme Reynolds numbers. Phys. Rev. Lett. 108, 094501.Google Scholar
Hunt, J. C. R. & Morrison, J. F. 2000 Eddy structure in turbulent boundary layers. Eur. J. Mech. (B/Fluids) 19, 673–694.Google Scholar
Hwang, Y. & Bengana, Y. 2016 Self-sustaining process of minimal attached eddies in turbulent channel flow. J. Fluid Mech. 795, 708–738.Google Scholar
Hwang, Y. & Cossu, C. 2010 Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105, 044505.Google Scholar
Jelly, T. O., Jung, S. Y. & Zaki, T. A. 2014 Turbulence and skin friction modification in channel flow with streamwise-aligned superhydrophobic surface texture. Phys. Fluids 26 (9), 095102.Google Scholar
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36 (1), 173–196.Google Scholar
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.Google Scholar
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.Google Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Jiménez, J. & Moser, R. D. 2007 What are we learning from simulating wall turbulence? Phil. Trans. R. Soc. Lond. A 365 (1852), 715–732.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.Google Scholar
von Kármán, T. 1930 Mechanische Ahnlichkeit und Turbulenz. In Proceedings Third International Congr. Applied Mechanics, Stockholm, pp. 85–105.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Kim, K. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Re 𝜏 ≈ 5200. J. Fluid Mech. 774, 395–415.Google Scholar
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.Google Scholar
Lozano-Durán, A. & Bae, H. J. 2016 Turbulent channel with slip boundaries as a benchmark for subgrid-scale models in LES. In Annual Research Briefs, Center for Turbulence Research, Stanford University, pp. 97–103.Google Scholar
Lozano-Durán, A. & Bae, H. J. 2019 Characteristic scales of Townsend’s wall attached eddies. In Annual Research Briefs, Center for Turbulence Research, Stanford University, pp. 183–196.Google Scholar
Lozano-Durán, A. & Borrell, G. 2016 Algorithm 964: an efficient algorithm to compute the genus of discrete surfaces and applications to turbulent flows. ACM Trans. Math. Softw. 42 (4), 34:1–34:19.Google 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.Google Scholar
Lozano-Durán, A., Hack, M. J. P. & Moin, P. 2018 Modeling boundary-layer transition in direct and large-eddy simulations using parabolized stability equations. Phys. Rev. Fluids 3, 023901.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014a Effect of the computational domain on direct simulations of turbulent channels up to Re 𝜏 = 4200. Phys. Fluids 26 (1), 011702.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014b Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.Google Scholar
Luchini, P. 2017 Universality of the turbulent velocity profile. Phys. Rev. Lett. 118, 224501.Google Scholar
Martell, M. B., Perot, J. B. & Rothstein, J. P. 2009 Direct numerical simulations of turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 620, 31–41.Google Scholar
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51 (1), 49–74.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.Google Scholar
Mellor, G. L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Intl J. Engng Sci. 10 (10), 851–873.Google Scholar
Meneveau, C. & Marusic, I. 2013 Generalized logarithmic law for high-order moments in turbulent boundary layers. J. Fluid Mech. 719, R1.Google Scholar
Millikan, C. B. 1938 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings 5th International Congr. Applied Mechanics, New York (ed. Hartog, J. P. D. & Peters, H.), pp. 386–392. Wiley.Google Scholar
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), L55–L58.Google Scholar
Mizuno, Y. & Jiménez, J. 2011 Mean velocity and length-scales in the overlap region of wall-bounded turbulent flows. Phys. Fluids 23 (8), 085112.Google Scholar
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without walls. J. Fluid Mech. 723, 429–455.Google 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.Google Scholar
Moisy, F. & Jiménez, J. 2004 Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 513, 111–133.Google Scholar
Monty, J. P., Stewart, J. A., Williams, R. C. & Chong, M. S. 2007 Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156.Google 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.Google Scholar
Morrison, W. R. B. & Kronauer, R. E. 1969 Structural similarity for fully developed turbulence in smooth tubes. J. Fluid Mech. 39 (1), 117–141.Google Scholar
Nikuradse, J. 1933 Laws of flow in rough pipes. VDI Forschungsheft 361.Google Scholar
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328.Google Scholar
Orlandi, P. 2000 Fluid Flow Phenomena: A Numerical Toolkit. Springer.Google Scholar
Park, H., Park, H. & Kim, J. 2013 A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids 25, 110815.Google Scholar
Perry, A. E. & Abell, C. J. 1975 Scaling laws for pipe-flow turbulence. J. Fluid Mech. 67, 257–271.Google Scholar
Perry, A. E. & Abell, C. J. 1977 Asymptotic similarity of turbulence structures in smooth- and rough-walled pipes. J. Fluid Mech. 79, 785–799.Google Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.Google Scholar
Phillips, W. R. C. 1987 The wall region of a turbulent boundary layer. Phys. Fluids 30 (8), 2354–2361.Google Scholar
Prandtl, L. 1925 Bericht über die Entstehung der Turbulenz. Z. Angew. Math. Mech. 5, 136–139.Google Scholar
Raupach, M. R., Antonia, R. A. & Rajagopalan, S. 1991 Rough-wall turbulent boundary layers. Appl. Mech. Rev. 44 (1), 1–25.Google Scholar
Rosenfeld, A. & Kak, A. C. 1982 Digital Picture Processing, 2nd edn. vols. 1 and 2. Academic Press.Google Scholar
Rotta, J. 1962 Turbulent boundary layers in incompressible flow. Prog. Aerosp. Sci. 2 (1), 1–95.Google Scholar
Seo, J., García-Mayoral, R. & Mani, A. 2015 Pressure fluctuations and interfacial robustness in turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 783, 448–473.Google Scholar
Seo, J. & Mani, A. 2016 On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces. Phys. Fluids 28 (2), 025110.Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 26 (10), 105109.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43 (1), 353–375.Google Scholar
Tomkins, C. D. & Adrian, R. J. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 37–74.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. Cambridge University Press.Google Scholar
Tuerke, F. & Jiménez, J. 2013 Simulations of turbulent channels with prescribed velocity profiles. J. Fluid Mech. 723, 587–603.Google Scholar
Vallikivi, M., Ganapathisubramani, B. & Smits, A. J. 2015 Spectral scaling in boundary layers and pipes at very high Reynolds numbers. J. Fluid Mech. 771, 303–326.Google Scholar
Wosnik, M., Castillo, L. & George, W. K. 2000 A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145.Google Scholar
Wray, A. A.1990 Minimal-storage time advancement schemes for spectral methods. NASA Tech. Rep. MS 202 A-1. Ames Research Center.Google Scholar
Zagarola, M. V. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.Google Scholar