Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-04-26T17:59:53.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 2. Integrated energy and $A_{1}$

Published online by Cambridge University Press:  12 November 2019

Woutijn J. Baars Open the ORCID record for Woutijn J. Baars [Opens in a new window]  and
Ivan Marusic Open the ORCID record for Ivan Marusic [Opens in a new window]
Woutijn J. Baars*
Affiliation:
Department of Engineering, Aarhus University, 8000 Aarhus C, Denmark
Ivan Marusic
Affiliation:
Department of Mechanical Engineering, The University of Melbourne, VIC 3010, Australia
*
†Email address for correspondence: baars@eng.au.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

Scalings of the streamwise velocity energy spectra in turbulent boundary layers were considered in Part 1 (Baars & Marusic, J. Fluid Mech., vol. 882, 2020, A25). A spectral decomposition analysis provided a means to separate out attached and non-attached eddy contributions and was used to generate three spectral sub-components, one of which is a close representation of the spectral signature induced by self-similar, wall-attached turbulence. Since sub-components of the streamwise turbulence intensity $\overline{u^{2}}$ follow from an integration of the velocity energy spectra, we here focus on the scaling of the former. Wall-normal profiles and Reynolds-number trends of the three individual, additive sub-components of the streamwise turbulence intensity are examined. Based on universal trends across all Reynolds numbers considered, some evidence is given for a Townsend–Perry constant of $A_{1}=0.98$, which would describe the wall-normal logarithmic decay of the turbulence intensity per Townsend’s attached-eddy hypothesis. It is also demonstrated how this constant can be consistent with the Reynolds-number increase of the streamwise turbulence intensity in the near-wall region.

JFM classification

Turbulent Flows: Turbulent boundary layers Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , A26
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.)

Article purchase

Temporarily unavailable

References

Abe, H., Kawamura, H. & Choi, H. 2004 Very large-scale structures and their effects on the wall shear-stress fluctuations in a turbulent channel flow up to Re 𝜏 = 640. Trans. ASME J. Fluids Engng 126, 835–843.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
Baars, W. J., Hutchins, N. & Marusic, I. 2016 Spectral stochastic estimation of high-Reynolds-number wall-bounded turbulence for a refined inner-outer interaction model. Phys. Rev. Fluids 1, 054406.Google Scholar
Baars, W. J., Hutchins, N. & Marusic, I. 2017a Reynolds number trend of hierarchies and scale interactions in turbulent boundary layers. Phil. Trans. R. Soc. A 375, 20160077.Google Scholar
Baars, W. J., Hutchins, N. & Marusic, I. 2017b Self-similarity of wall-attached turbulence in boundary layers. J. Fluid Mech. 823, R2.Google Scholar
Baars, W. J. & Marusic, I. 2020 Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra. J. Fluid Mech. 882, A25.Google Scholar
Baidya, R., Philip, J., Hutchins, N., Monty, J. P. & Marusic, I. 2017 Distance-from-the-wall scaling of turbulent motions in wall-bounded flows. Phys. Fluids 29, 020712.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
Chandran, D., Baidya, R., Monty, J. P. & Marusic, I. 2017 Two-dimensional energy spectra in a high Reynolds number turbulent boundary layer. J. Fluid Mech. 826, R1.Google Scholar
Chen, X., Hussain, F. & She, Z.-S. 2018 Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses. J. Fluid Mech. 850, 401–438.Google Scholar
Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474–504.Google Scholar
Cossu, C. & Hwang, Y. 2017 Self-sustaining processes at all scales in wall-bounded turbulent shear flows. Phil. Trans. R. Soc. A 375, 20160088.Google Scholar
Davidson, P. A. & Krogstad, P.-Å. 2009 A simple model for the streamwise fluctuations in the log-law region of a boundary layer. Phys. Fluids 21, 055105.Google Scholar
de Giovanetti, M., Hwang, Y. & Choi, H. 2016 Skin-friction generation by attached eddies in turbulent channel flow. J. Fluid Mech. 808, 511–538.Google Scholar
de Giovanetti, M., Sung, H. J. & Hwang, Y. 2017 Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions. J. Fluid Mech. 832, 483–513.Google Scholar
DeGraaff, D. B. & 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. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.Google 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.Google Scholar
Hu, R. & Zheng, X. 2018 Energy contributions by inner and outer motions in turbulent channel flows. Phys. Rev. Fluids 3, 084607.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 (9), 094501.Google Scholar
Hutchins, N., Chauhan, K., Marusic, I. & Klewicki, J. 2012 Towards reconciling the large-scale structure of turbulent boundary layers in the atmosphere and laboratory. Boundary-Layer Meteorol. 145, 273–306.Google Scholar
Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. A 365, 647–664.Google Scholar
Hutchins, N., Nickels, T. B., Marusic, I. & Chong, M. S. 2009 Hot-wire spatial resolution issues in wall-bounded turbulence. J. Fluid Mech. 635, 103–136.Google Scholar
Hwang, J. & Sung, H. J. 2018 Wall-attached structures of velocity fluctuations in a turbulent boundary layer. J. Fluid Mech. 856, 958–983.Google Scholar
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.Google Scholar
Hwang, Y. & Cossu, C. 2010a Amplification of coherent streaks in the turbulent Couette flow: an inputoutput analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.Google Scholar
Hwang, Y. & Cossu, C. 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.Google Scholar
Hwang, Y., Willis, A. P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for Re 𝜏 up to 1000. J. Fluid Mech. 802, R1.Google Scholar
Kerhervé, F., Roux, S. & Mathis, R. 2017 Combining time-resolved multi-point and spatially-resolved measurements for the recovering of very-large-scale motions in high Reynolds number turbulent boundar layer. Exp. Therm. Fluid Sci. 82, 102–115.Google Scholar
Klewicki, J., Fife, P. & Wei, T. 2009 On the logarithmic mean profile. J. Fluid Mech. 638, 73–93.Google Scholar
Klewicki, J. C. 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. Trans. ASME J. Fluids Engng 132, 094001.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.Google Scholar
Kwon, Y. S., Hutchins, N. & Monty, J. P. 2016 On the use of the Reynolds decomposition in the intermittent region of turbulent boundary layers. J. Fluid Mech. 794, 5–16.Google Scholar
Laval, J.-P., Vassilicos, J. C., Foucaut, J.-M. & Stanislas, M. 2017 Comparison of turbulence profiles in high-Reynolds-number turbulent boundary layers and validation of a predictive model. J. Fluid Mech. 814, R2.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
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
Marusic, I., Baars, W. J. & Hutchins, N. 2017 Scaling of the streamwise turbulence intensity in the context of inner-outer interactions in wall-turbulence. Phys. Rev. Fluids 2, 100502.Google Scholar
Marusic, I., Chauhan, K. A., Kulandaivelu, V. & Hutchins, N. 2015 Evolution of zero-pressure-gradient boundary layers from different tripping conditions. J. Fluid Mech. 783, 379–411.Google Scholar
Marusic, I. & Kunkel, G. J. 2003 Streamwise turbulence intensity formulation for flat-plate boundary layers. Phys. Fluids 15 (8), 2461–2464.Google Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010a Predictive model for wall-bounded turbulent flow. Science 329 (5988), 193–196.Google Scholar
Marusic, I., McKeon, B. J., Monkewitz, P. A., Nagib, H. M., Smits, A. J. & Sreenivasan, K. R. 2010b Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22, 065103.Google Scholar
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 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
Marusic, I., Uddin, A. K. M. & Perry, A. E. 1997 Similarity law for the streamwise turbulence intensity in zero-pressure-gradient turbulent boundary layers. Phys. Fluids 9 (12), 3718–3726.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2011 A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows. J. Fluid Mech. 681, 537–566.Google Scholar
Metzger, M. M., Klewicki, J. C., Bradshaw, K. L. & Sadr, R. 2001 Scaling the near-wall axial turbulent stress in the zero pressure gradient boundary layer. Phys. Fluids 13 (6), 1819–1821.Google Scholar
Monkewitz, P. A. & Nagib, H. M. 2015 Large-Reynolds-number asymptotics of the streamwise normal stress in zero-pressure-gradient turbulent boundary layers. J. Fluid Mech. 783, 474–503.Google Scholar
Monkewitz, P. A., Nagib, H. M. & Boulanger, W. 2017 Comparing the three possible scalings of stream-wise normal stress in turbulent boundary layers. In 10th International Symposium on Turbulence and Shear Flow Phenomena.Google Scholar
Morrill-Winter, C., Philip, J. & Klewicki, J. 2017 An invariant representation of mean inertia: theoretical basis for a log law in turbulent boundary layers. J. Fluid Mech. 813, 594–617.Google Scholar
Morrison, J. F., Jiang, W., McKeon, B. J. & Smits, A. J. 2002 Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow. Phys. Rev. Lett. 88 (21), 214501.Google Scholar
Nickels, T. B., Marusic, I., Hafez, S. & Chong, M. S. 2005 Evidence of the k 1 -1 law in a high-Reynolds-number turbulent boundary layer. Phys. Rev. Lett. 95, 074501.Google Scholar
Örlü, R., Fiorini, T., Segalini, A., Bellani, G., Talamelli, A. & Alfredsson, P. H. 2017 Reynolds stress scaling in pipe flow turbulence—first results from CICLoPE. Phil. Trans. R. Soc. A 375, 20160187.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. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.Google Scholar
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199.Google Scholar
Perry, A. E. & Li, J. D. 1990 Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers. J. Fluid Mech. 218, 405–438.Google Scholar
Rosenberg, B. J., Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2013 Turbulence spectra in smooth- and rough-wall pipe flow at extreme Reynolds numbers. J. Fluid Mech. 731, 46–63.Google Scholar
Samie, M.2017 Sub-miniature hot-wire anemometry for high Reynolds number turbulent flows. PhD thesis, The University of Melbourne, Melbourne, Australia.Google Scholar
Samie, M., Marusic, I., Hutchins, N., Fu, M. K., Fan, Y., Hultmark, M. & Smits, A. J. 2018 Fully-resolved measurements of turbulent boundary layer flows up to Re 𝜏 = 20000. J. Fluid Mech. 851, 391–415.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.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.Google Scholar
Solak, I. & Laval, J.-P. 2018 Large-scale motions from a direct numerical simulation of a turbulent boundary layer. Phys. Rev. E 98, 033101.Google Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to Re 𝜃 = 1410. J. Fluid Mech. 187, 61–98.Google Scholar
Sreenivasan, K. R. & Sahay, A. 1997 The persistence of viscous effects in the overlap region and the mean velocity in turbulent pipe and channel flows. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton, R.), pp. 253–272. Computational Mechanics Publications.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.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
Vassilicos, J. C., Laval, J.-P., Foucaut, J.-M. & Stanislas, M. 2015 The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow. J. Fluid Mech. 774, 324–341.Google Scholar
Vincenti, P., Klewicki, J. C., Morrill-Winter, C., White, C. M. & Wosnik, M. 2013 Streamwise velocity statistics in turbulent boundary layers that spatially develop to high Reynolds number. Exp. Fluids 54 (12), 1–13.Google Scholar
Wang, W., Pan, C. & Wang, J. 2018 Quasi-bivariate variational mode decomposition as a tool of scale analysis in wall-bounded turbulence. Exp. Fluids 59 (1), 1–18.Google Scholar
Wei, T., Fife, P., Klewicki, J. C. & McMurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.Google Scholar
Willert, C. E., Soria, J., Stanislas, M., Klinner, J., Amili, O., Eisfelder, M., Cuvier, C., Bellani, G., Fiorini, T. & Talamelli, A. 2017 Near-wall statistics of a turbulent pipe flow at shear Reynolds numbers up to 40 000. J. Fluid Mech. 826, R5.Google Scholar
Winkel, E. S., Cutbirth, J. M., Ceccio, S. L., Perlin, M. & Dowling, D. R. 2012 Turbulence profiles from a smooth flat-plate turbulent boundary layer at high Reynolds number. Exp. Therm. Fluid Sci. 40, 140–149.Google Scholar
Yamamoto, Y. & Tsuji, Y. 2018 Numerical evidence of logarithmic regions in channel flow at Re 𝜏 = 8000. Phys. Rev. Fluids 3, 012602.Google Scholar