Skip to main content Accessibility help
×
Home

Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number

  • Myoungkyu Lee (a1) and Robert D. Moser (a1) (a2)

Abstract

The transport equations for the variances of the velocity components are investigated using data from direct numerical simulations of incompressible channel flows at friction Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}$ ) up to $Re_{\unicode[STIX]{x1D70F}}=5200$ . Each term in the transport equation has been spectrally decomposed to expose the contribution of turbulence at different length scales to the processes governing the flow of energy in the wall-normal direction, in scale and among components. The outer-layer turbulence is dominated by very large-scale streamwise elongated modes, which are consistent with the very large-scale motions (VLSM) that have been observed by many others. The presence of these VLSMs drives many of the characteristics of the turbulent energy flows. Away from the wall, production occurs primarily in these large-scale streamwise-elongated modes in the streamwise velocity, but dissipation occurs nearly isotropically in both velocity components and scale. For this to happen, the energy is transferred from the streamwise-elongated modes to modes with a range of orientations through nonlinear interactions, and then transferred to other velocity components. This allows energy to be transferred more-or-less isotropically from these large scales to the small scales at which dissipation occurs. The VLSMs also transfer energy to the wall region, resulting in a modulation of the autonomous near-wall dynamics and the observed Reynolds number dependence of the near-wall velocity variances. The near-wall energy flows are more complex, but are consistent with the well-known autonomous near-wall dynamics that gives rise to streaks and streamwise vortices. Through the overlap region between outer- and inner-layer turbulence, there is a self-similar structure to the energy flows. The VLSM production occurs at spanwise scales that grow with $y$ . There is transport of energy away from the wall over a range of scales that grows with $y$ . Moreover, there is transfer of energy to small dissipative scales which grows like $y^{1/4}$ , as expected from Kolmogorov scaling. Finally, the small-scale near-wall processes characterised by wavelengths less than 1000 wall units are largely Reynolds number independent, while the larger-scale outer-layer processes are strongly Reynolds number dependent. The interaction between them appears to be relatively simple.

Copyright

Corresponding author

Email address for correspondence: rmoser@ices.utexas.edu

Footnotes

Hide All

Present address: Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94550, USA.

Footnotes

References

Hide All
Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Ingenieur-Archiv 52, 355377.
Ahn, J., Lee, J. & Sung, H. J. 2017 Contribution of large-scale motions to the Reynolds shear stress in turbulent pipe flows. Intl J. Heat Fluid Flow 66, 209216.
Ahn, J., Lee, J. H., Lee, J., Kang, J.-h. & Sung, H. J. 2015 Direct numerical simulation of a 30R long turbulent pipe flow at Re 𝜏 = 3008. Phys. Fluids 27, 065110.
Ahn, J. & Sung, H. J. 2017 Relationship between streamwise and azimuthal length scales in a turbulent pipe flow. Phys. Fluids 29 (10), 105112.
del Álamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41L44.
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.
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, 329358.
Antonia, R. A., Djenidi, L. & Spalart, P. R. 1994 Anisotropy of the dissipation tensor in a turbulent boundary layer. Phys. Fluids 6 (7), 24752479.
Antonia, R. A., Kim, J. & Browne, L. W. B. 1991 Some characteristics of small-scale turbulence in a turbulent duct flow. J. Fluid Mech. 233, 369388.
Aulery, F., Dupuy, D., Toutant, A., Bataille, F. & Zhou, Y. 2017 Spectral analysis of turbulence in anisothermal channel flows. Comput. Fluids 151, 115131.
Baltzer, J. R., Adrian, R. J. & Wu, X. 2013 Structural organization of large and very large scales in turbulent pipe flow simulation. J. Fluid Mech. 720, 236279.
Bolotnov, I. A., Lahey, R. T. Jr, Drew, D. A., Jansen, K. E. & Oberai, A. A. 2010 Spectral analysis of turbulence based on the DNS of a channel flow. Comput. Fluids 39 (4), 640655.
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.
Chin, C., Philip, J., Klewicki, J., Ooi, A. & Marusic, I. 2014 Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows. J. Fluid Mech. 757, 747769.
Cimarelli, A., De Angelis, E. & Casciola, C. M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436451.
Cimarelli, A., De Angelis, E., Jiménez, J. & Casciola, C. M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417436.
Cimarelli, A., De Angelis, E., Schlatter, P., Brethouwer, G., Talamelli, A. & Casciola, C. M. 2015 Sources and fluxes of scale energy in the overlap layer of wall turbulence. J. Fluid Mech. 771, 407423.
DeGraaff, D. B. & Eaton, J. K. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319346.
Domaradzki, J. A., Liu, W., Härtel, C. & Kleiser, L. 1994 Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6 (4), 15831599.
Fernholz, H. H. & Finley, P. J. 1996 The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data. Prog. Aerosp. Sci. 32, 245311.
Ganapathisubramani, B., Hutchins, N., Monty, J. P., Chung, D. & Marusic, I. 2012 Amplitude and frequency modulation in wall turbulence. J. Fluid Mech. 712, 6191.
Guala, M., Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521542.
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511.
Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2012 Turbulent pipe flow at extreme Reynolds numbers. Phys. Rev. Lett. 108, 094501.
Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2013 Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow. J. Fluid Mech. 728, 376395.
Hutchins, N. & Marusic, I. 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 128.
Hutchins, N. & Marusic, I. 2007b Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365, 647664.
Hutchins, N., Nickels, T. B., Marusic, I. & Chong, M. S. 2009 Hot-wire spatial resolution issues in wall-bounded turbulence. J. Fluid Mech. 635, 103136.
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254289.
Jeong, J., Hussain, F., Schoppa, W. & Kim, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185214.
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 2745.
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215236.
Jiménez, J. & Moser, R. D. 2007 What are we learning from simulating wall turbulence? Phil. Trans. R. Soc. Lond. A 365, 715732.
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335359.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kraheberger, S., Hoyas, S. & Oberlack, M. 2018 DNS of a turbulent Couette flow at constant wall transpiration up to Re 𝜏 = 1000. J. Fluid Mech. 835, 421443.
Kunkel, G. J. & Marusic, I. 2006 Study of the near-wall-turbulent region of the high-Reynolds-number boundary layer using an atmospheric flow. J. Fluid Mech. 548, 375402.
Lee, M., Malaya, N. & Moser, R. D. 2013 Petascale direct numerical simulation of turbulent channel flow on up to 786K cores. In The International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 111. ACM Press.
Lee, M. & Moser, R. D. 2015a Direct numerical simulation of turbulent channel flow up to Re 𝜏 = 5200. J. Fluid Mech. 774, 395415.
Lee, M. & Moser, R. D. 2015b Spectral analysis on Reynolds stress transport equation in high Re wall-bounded turbulence. In Ninth International Symposium on Turbulence and Shear Flow Phenomena, art. 4A-3.
Lee, M., Ulerich, R., Malaya, N. & Moser, R. D. 2014 Experiences from leadership computing in simulations of turbulent fluid flows. Comput. Sci. Engng 16 (5), 2431.
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561583.
Luchini, P. 2018 Structure and interpolation of the turbulent velocity profile in parallel flow. Eur. J. Mech. (B/Fluids) 71, 1534.
Lumley, J. L. 1964 Spectral energy budget in wall turbulence. Phys. Fluids 7 (2), 190196.
Lumley, J. L. 1975 Pressure–strain correlation. Phys. Fluids 18 (6), 750.
Lumley, J. L. 1979 Computational modeling of turbulent flows. In Advances in Applied Mechanics, pp. 123176. Elsevier.
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 1544.
Marusic, I., Mathis, R. & Hutchins, N. 2010a High Reynolds number effects in wall turbulence. Intl J. Heat Fluid Flow 31 (3), 418428.
Marusic, I., Mathis, R. & Hutchins, N. 2010b Predictive model for wall-bounded turbulent flow. Science 329, 193196.
Marusic, I., McKeon, B. J., Monkewitz, P. A., Nagib, H. M., Smits, A. J. & Sreenivasan, K. R. 2010c Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22, 065103.
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311337.
Millikan, C. B. 1938 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings of the Fifth International Congress for Applied Mechanics, pp. 386392.
Mizuno, Y. 2015 Spectra of turbulent energy transport in channel flows. In 15th European Turbulence Conference.
Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171187.
Mizuno, Y. & Jiménez, J. 2011 Mean velocity and length-scales in the overlap region of wall-bounded turbulent flows. Phys. Fluids 23, 085112.
Monkewitz, P. A. 2017 Revisiting the quest for a universal log-law and the role of pressure gradient in ‘canonical’ wall-bounded turbulent flows. Phys. Rev. Fluids 2 (9), 094602.
Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431442.
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, 99131.
Nagib, H. M. & Chauhan, K. A. 2008 Variations of von Kármán coefficient in canonical flows. Phys. Fluids 20, 101518.
Perot, B. & Moin, P. 1995 Shear-free turbulent boundary layers. Part 1. Physical insights into near-wall turbulence. J. Fluid Mech. 295, 199227.
Perry, A. E., Henbest, S. & Chong, M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163199.
Perry, A. E. & Marusic, I. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis. J. Fluid Mech. 298, 361388.
Richter, D. H. 2015 Turbulence modification by inertial particles and its influence on the spectral energy budget in planar Couette flow. Phys. Fluids 27 (6), 063304.
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.
de Silva, C. M., Marusic, I., Woodcock, J. D. & Meneveau, C. 2015 Scaling of second- and higher-order structure functions in turbulent boundary layers. J. Fluid Mech. 769, 654686.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. The MIT Press.
Tomkins, C. D. & Adrian, R. J. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 3774.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
Vallikivi, M., Ganapathisubramani, B. & Smits, A. J. 2015 Spectral scaling in boundary layers and pipes at very high Reynolds numbers. J. Fluid Mech. 771, 303326.
Woodcock, J. D. & Marusic, I. 2015 The statistical behaviour of attached eddies. Phys. Fluids 27, 015104.
Wu, X., Baltzer, J. R. & Adrian, R. J. 2012 Direct numerical simulation of a 30r long turbulent pipe flow at r = 685: large- and very large-scale motions. J. Fluid Mech. 698, 235281.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number

  • Myoungkyu Lee (a1) and Robert D. Moser (a1) (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed