Skip to main content

Quantifying wall turbulence via a symmetry approach: a Lie group theory

  • Zhen-Su She (a1), Xi Chen (a1) (a2) and Fazle Hussain (a1) (a2)

First-principle-based prediction of mean-flow quantities of wall-bounded turbulent flows (channel, pipe and turbulent boundary layer (TBL)) is of great importance from both physics and engineering standpoints. Here we present a symmetry-based approach which yields analytical expressions for the mean-velocity profile (MVP) from a Lie-group analysis. After verifying the dilatation-group invariance of the Reynolds averaged Navier–Stokes (RANS) equation in the presence of a wall, we depart from previous Lie-group studies of wall turbulence by selecting a stress length function as a similarity variable. We argue that this stress length function characterizes the symmetry property of wall flows having a simple dilatation-invariant form. Three kinds of (local) invariant forms of the length function are postulated, a combination of which yields a multi-layer formula giving its distribution in the entire flow region normal to the wall and hence also the MVP, using the mean-momentum equation. In particular, based on this multi-layer formula, we obtain analytical expressions for the (universal) wall function and separate wake functions for pipe and channel, which are validated by data from direct numerical simulations (DNS). In conclusion, an analytical expression for the entire MVP of wall turbulence, beyond the log law or power law, is developed in this paper and the theory can be used to describe the mean turbulent kinetic-energy distribution, as well as a variety of boundary conditions such as pressure gradient, wall roughness, buoyancy, etc. where the dilatation-group invariance is valid in the wall-normal direction.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Quantifying wall turbulence via a symmetry approach: a Lie group theory
      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Quantifying wall turbulence via a symmetry approach: a Lie group theory
      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Quantifying wall turbulence via a symmetry approach: a Lie group theory
      Available formats
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence:
Hide All
Alfredsson P. H., Imayamaa S., Lingwood R. J., Orlu R. & Segalini A. 2013 Turbulent boundary layers over flat plates and rotating disks – the legacy of von karman: a stockholm perspective. Eur. J. Mech. (B/Fluids) 40, 1729.
Avsarkisov V., Oberlack M. & Hoyas S. 2014 New scaling laws for turbulent poiseuille flow with wall transpiration. J. Fluid Mech. 746, 99122.
Barenblatt G. I. 1993 Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis. J. Fluid Mech. 248, 513520.
Barenblatt G. I. 1996 Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.
Barenblatt G. I. & Chorin A. J. 2004 A mathematical model for the scaling of turbulence. Proc. Natl Acad. Sci. USA 101 (42), 1502315026.
Batchelor G. K. 1951 Pressure fluctuations in isotropic turbulence. Math. Proc. Camb. Phil. Soc. 47, 359374.
Bluman & Kumei 1989 Symmetries and Differential Equations. Springer.
Cantwell B. J. 2002 Introduction to Symmetry Analysis. Cambridge University Press.
Chen X. & Hussain F. 2017 Similarity transformation for equilibrium boundary layers, including effects of blowing and suction. Phys. Rev. Fluids 2, 034605.
Chen X., Hussain F. & She Z. S. 2016a Predictions of canonical wall-bounded turbulent flows via a modified k–𝜔 equation. J. Turbul. doi:10.1080/14685248.2016.1243244.
Chen X., Hussain F. & She Z. S. 2016b Bulk flow scaling for turbulent channel and pipe flows. Europhys. Lett. 115, 34001.
Chen X. & She Z. S. 2016 Analytic prediction for planar turbulent boundary layers. Science China Physics, Mech. Astron. 59 (11), 114711.
Chen X., Wei B. B., Hussain F. & She Z. S. 2015 Anomalous dissipation and kinetic-energy distribution in pipes at very high Reynolds numbers. Phys. Rev. E 93, 011102(R).
Cipra B. 1996 A new theory of turbulence causes a stir among experts. Science 272 (5264), 951.
Coles D. 1956 The law of the wake in the turbulent boundary layer. J. Fluid Mech. 1, 191226.
Davidson P. A., Kaneda Y., Moffatt K. & Sreenivasan K. R. 2011 A Voyage through Turbulence. Cambridge University Press.
Del Alamo J. C. & Jimenez J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205213.
Driest V. 1956 On turbulent flow near a wall. J. Aeronaut. Sci. (Institute of the Aeronautical Sciences) 23 (11), 10071011.
Falkovich G. 2009 Symmetries of the turbulent state. J. Phys. A 42 (12), 123001.
Frisch U. 1995 Turbulence. Cambridge University Press.
Frisch U. & Parisi G. 1985 Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (ed. Ghil M., Benzi R. & Parisi G.), p. 71. North-Holland.
George W. K. 2005 Recent advancements toward the understanding of turbulent boundary layers. In Proceedings of the Fourth AIAA Theoretical Fluid Mechanics Meeting, Toronto, Canada. AIAA Paper 2005-4669.
Gibson J., Halcrow J. & Cvitanovic P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.
Hoyas S. & Jimenez J. 2006 Scaling of the velocity fluctuations in turbulent channels up to re 𝜏 = 2003. Phys. Fluids 18, 011702.
Hultmark M., Vallikivi M., Bailey S. C. C. & Smits A. J. 2012 Turbulent pipe flow at extreme Reynolds numbers. Phys. Rev. Lett. 108, 094501.
Hussain A. K. M. F. & Zaman K. B. M. Q. 1985 An experimental study of organized motions in the turbulent plane mixing layer. J. Fluid Mech. 159, 85104.
Iwamoto K., Suzuki Y. & Kasagi N.2002 Database of fully developed channel flow. Tech. Rep. ILR-0201, see
Kadanoff L. P. 2009 More is the same; phase transitions and mean field theories. J. Stat. Phys. 137 (5–6), 777797.
Kelbin O., Cheviakov A. F. & Oberlack M. 2013 New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows. J. Fluid Mech. 721, 340366.
Klewicki J. 2013 Self-similar mean dynamics in turbulent wall flows. J. Fluid Mech. 718, 596621.
Klewicki J., Chin C., Blackburn H., Ooi A. & Marusic I. 2012 Emergence of the four layer dynamical regime in turbulent pipe flo. Phys. Fluids 24, 045107.
Kolmogorov A. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Dokl. Akad. Nauk SSSR 30, 301305.
Lindgren B., Osterlund J. M. & Johansson A. V. 2004 Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers. J. Fluid Mech. 502, 127152.
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 (5), 054504.
Marati N., Davoudi J., Casciola C. M. & Eckhardt B. 2006 Mean profiles for a passive scalar in wall-bounded flows from symmetry analysis. J. Turbul. 7, N61.
Marusic I., McKeon B. J., Monkewitz P. A., Nagib H. M., Smits A. J. & Sreenivasan K. R. 2010 Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22 (6), 065103.
Marusic I., Monty J. P., Hultmark M. & Smits A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.
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, 135147.
Millikan C. B. 1938 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings 5th International Congress on Applied Mechanics, Cambridge, MA.
Monkewitz P. A., Chauhan K. A. & Nagib H. M. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19, 115101.
Monty J. P.2005 Developments in smooth wall turbulent duct flows. PhD Thesis, University of Melbourne.
Nagib H. M. & Chauhan K. A. 2008 Variations of von karman coefficient in canonical flows. Phys. Fluids 20, 101518.
Nickels T. B. 2004 Inner scaling for wall-bounded flows subject to large pressure gradients. J. Fluid Mech. 521, 217239.
Oberlack M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299328.
Oberlack M. & Rosteck A. 2010 New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. J. Discrete Continuous Dyn. Syst. S 3, 451471.
Olver P. J. 1995 Equivalence, Invariants and Symmetry. Cambridge University Press.
Panton R. L. 2007 Composite asymptotic expansions and scaling wall turbulence. Phil. Trans. R. Soc. Lond. A 365, 733754.
Perry A. E. & Chong M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173217.
Perry A. E., Henbest S. M. & Chong M. S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163199.
Pope S. B. 2000 Turbulent Flows. Cambridge University Press.
Prandtl L. 1925 Bericht uber die entstehung der turbulenz. Z. Angew. Math. Mech 5, 136.
Reynolds W. C. & Hussain A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54, 263288.
Schlatter P., Li Q., Brethouwer G., Johansson A. V. & Henningson D. S. 2010 Simulations of spatially evolving turbulent boundary layers up to Re𝜃 = 4300. Intl J. Heat Fluid Flow 31 (3), 251261.
Schoppa W. & Hussain F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.
Segalini A., Orlu R. & Alfredsson P. H. 2013 Uncertainty analysis of the von karman constant. Exp. Fluids 54, 1460.
She Z. S., Chen X., Wu Y. & Hussain F. 2010 New perspective in statistical modeling of wall-bounded turbulence. Acta Mechanica Sin. 26 (6), 847861.
She Z. S. & Leveque E. 1994 Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72 (3), 336.
She Z. S., Wu Y., Chen X. & Hussain F. 2012 A multi-state description of roughness effects in turbulent pipe flow. New J. Phys. 14 (9), 093054.
She Z. S. & Zhang Z. X. 2009 Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems. Acta Mechanica Sin. 25 (3), 279294.
Smits A. J. & Marusic I. 2013 Wall-bounded turbulence. Phys. Today 66 (9), 25.
Smits A. J., McKeon B. J. & Marusic I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43 (1), 353375.
Townsend A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
Van Dyke M. 1964 Perturbation Methods in Fluid Mechanics, vol. 964. Academic.
Von Karman T. 1930 Mechanische ähnlichkeit und turbulenz, nachr. ges. wiss. göttingen, math.-phys. kl.(1930) 58–76. In Proc. 3. Int. Cong. Appl. Mech., pp. 85105.
Wei T., Fife P., Klewicki J. & Mcmurtry P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303327.
White F. M. 2006 Viscous Fluid Flow. McGraw-Hill.
Wilcox D. C. 2006 Turbulence Modeling for CFD. DCW Industries La Canada.
Wu B., Bi W. T., Hussain F. & She Z. S. 2017 On the invariant mean velocity profile for compressible turbulent boundary layers. J. Turbul. 18, 186202.
Wu X. H. & Moin P. 2008 A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81112.
Wu Y., Chen X., She Z. S. & Hussain F. 2012 Incorporating boundary constraints to predict mean velocities in turbulent channel flow. Science China Phys., Mech. Astron. 55 (9), 1691.
Wu Y., Chen X., She Z. S. & Hussain F. 2013 On the karman constant in turbulent channel flow. Phys. Scr. 2013 (T155), 014009.
Zagarola M. V. & Smits A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 3379.
Zhang Y. S., Bi W. T., Hussain F., Li X. L. & She Z. S. 2012 Mach-number-invariant mean-velocity profile of compressible turbulent boundary layers. Phys. Rev. Lett. 109, 054502.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 16
Total number of PDF views: 389 *
Loading metrics...

Abstract views

Total abstract views: 290 *
Loading metrics...

* Views captured on Cambridge Core between 22nd August 2017 - 15th December 2017. This data will be updated every 24 hours.