Abstract
Despite dedicated effort for many decades, statistical description of highly technologically important wall turbulence remains a great challenge. Current models are unfortunately incomplete, or empirical, or qualitative. After a review of the existing theories of wall turbulence, we present a new framework, called the structure ensemble dynamics (SED), which aims at integrating the turbulence dynamics into a quantitative description of the mean flow. The SED theory naturally evolves from a statistical physics understanding of non-equilibrium open systems, such as fluid turbulence, for which mean quantities are intimately coupled with the fluctuation dynamics. Starting from the ensemble-averaged Navier–Stokes (EANS) equations, the theory postulates the existence of a finite number of statistical states yielding a multi-layer picture for wall turbulence. Then, it uses order functions (ratios of terms in the mean momentum as well as energy equations) to characterize the states and transitions between states. Application of the SED analysis to an incompressible channel flow and a compressible turbulent boundary layer shows that the order functions successfully reveal the multi-layer structure for wall-bounded turbulence, which arises as a quantitative extension of the traditional view in terms of sub-layer, buffer layer, log layer and wake. Furthermore, an idea of using a set of hyperbolic functions for modeling transitions between layers is proposed for a quantitative model of order functions across the entire flow domain. We conclude that the SED provides a theoretical framework for expressing the yet-unknown effects of fluctuation structures on the mean quantities, and offers new methods to analyze experimental and simulation data. Combined with asymptotic analysis, it also offers a way to evaluate convergence of simulations. The SED approach successfully describes the dynamics at both momentum and energy levels, in contrast with all prevalent approaches describing the mean velocity profile only. Moreover, the SED theoretical framework is general, independent of the flow system to study, while the actual functional form of the order functions may vary from flow to flow. We assert that as the knowledge of order functions is accumulated and as more flows are analyzed, new principles (such as hierarchy, symmetry, group invariance, etc.) governing the role of turbulent structures in the mean flow properties will be clarified and a viable theory of turbulence might emerge.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kolmogorov A.N.: Equations of turbulent motion of an incompressible viscous fluid. Izv. Acad. Sci. USSR Phys. 6(1–2), 56–58 (1942)
Robinson S.K.: Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639 (1991)
Moin P., Mahesh K.: Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech. 30, 539–578 (1998)
Schoppa W., Hussain F.: Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108 (2002)
Prandtl L.: Zur turbulenten Strömung in Rohren und laengs Platten. Ergeb. Aerodyn. Versuch Geottingen IV. Lieferung 4, 18–29 (1932)
Karman, T.: Mechanische Ahnlichkeit und Turbulenz. Nachr Ges Wiss Gottingen, Math Phys Klasse 58C76. 2 (1930)
Clauser F.H.: The turbulent boundary layer. Adv. Appl. Mech. 4, 2C51.2 (1956)
George W.K., Castillo L.: Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729 (1997)
Jones M.B., Nickels T.B., Marusic I.: On the asymptotic similarity of the zero-pressure-gradient turbulent boundary layer. J. Fluid. Mech. 616, 195–203 (2008)
Zagarola M.V., Smits A.J.: Scaling of the mean velocity profile for tuebulent pipe flow. Phys. Rev. Lett. 78, 2 (1997)
George, W.K.: Recent advancements toward the understanding of turbulent boundary layers. In: Proceedings of the Fourth AIAA Theoretical Fluid Mechanics Meeting, Toronto, Canada, 6–9 June 2005, AIAA Paper 2005–4669
Millikan C.B.: A critical discussion of turbulent flows in channelsand circular tubes. In: Den Hartog, J.P., Peters, H. (eds) Proceedings of the Fifth International Congress on Applied Mechanics, pp. 386–392. Wiley, New York (1938)
Nickels T.B.: Inner scaling for wall-bounded flows subject to large pressure gradients. J. Fluid Mech. 521, 217–239 (2004)
Monkewitz P.A., Chauhan K.A., Nagib H.M.: Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19, 115101–115112 (2007)
Wosnik M., Castillo L., Geroge W.K.: A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145 (2000)
Buschmanna M.H., Gad-el-Hak M.: Recent developments in scaling of wall-bounded flows. Prog. Aerosp. Sci. 42, 419–467 (2007)
Barenblatt G.I.: Scaling laws for fully developed turbulent shear flows. Part 1,Basic hypotheses and analysis. J. Fluid Mech. 248, 513–520 (1993)
Barenblatt G.I., Chorin A.J., Prostokishin V.M.: Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech. 410, 263–283 (2000)
Sreenivansan R.: The importance of higher-order effects in the Barenblatt–Chorin theory of wall-bounded fully developed turbulent shear flows. Phys. Fluids 10, 1037C9 (1998)
Oberlack M.: Symmetries, invariance and scaling-Laws in inhomogeneous turbulent shear flows. Flow Turbulence Combus. 62, 111–135 (1999)
Oberlack M.: A unified approach for symmetries in plane parallel turbulent shear flows. Flow Turbulence Combus. 427, 299–328 (2001)
L’vov V.S., Procaccia I., Rudenco O.: Universal model of finite Reynolds number turbulent flow in channels and pipes. Phys. Rev. Let. 100(1–4), 050504 (2008)
Zagarola M.V., Perry A.E., Smits A.J.: Laws or power laws: The scaling in the overlap region. Phys. Fluids 9, 2094 (1997)
George W.K.: Is there a universal log-law for turbulent wall-bounded flow?. Phil. Trans. Roy. Soc. A 365, 789–806 (2007)
Nagib H.M., Chauhan K.A.: Variations of von Kármán coefficient in canonical flows. Phys. Fluids 20, 101518 (2008)
Monkewitz P.A., Chauhan K.A., Nagib H.M.: of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys. Fluids 20, 105102 (2008)
Oberlack M., Rosteck A.: New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discret. Contin. Dyn. Syst. Ser. S 3(3), 451–471 (2010)
Marusic I., McKeon B.J., Monkewitz P.A. et al.: Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys. Fluids 22, 065103 (2010)
She Z.S., Leveque E.: Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336 (1994)
She Z.S., Zhang Z.X.: Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems. Acta Mech. Sin. 25, 279–294 (2009)
Hoyas, S., Jimenez, J.: Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003. Phys. Fluids 18, 011702 (2006). http://torroja.dmt.upm.es/ftp/channels/
Gao H., Fu D.-X., Ma Y.-W. et al.: Direct numerical simulation of supersonic turbulent boundary layer flow. Chin. Phys. Lett. 22, 1709 (2005)
Iwamoto, K., Suzuki, Y., Kasagi, N.: Database of fully developed channel flow. THTLAB Internal Report No. ILR-0201 (2002). http://www.thtlab.t.u-tokyo.ac.jp
Batchelor, G.K.: Pressure fluctuations in isotropic turbulence. In: Proceedings of Cambridge Philosophical Society 47, 359 (1951)
Pope S.B.: Turbulent Flows. Cambridge Univ. Press, Cambridge (2000)
She Z.S., Hu N., Wu Y.: Structural ensemble dynamics based closure model for wall-bounded turbulent flow. Acta. Mech. Sin. 25, 731–736 (2009)
Bradshaw P., Ferriss D.H., Atwell N.P.: Calculation of boundary layer development using the turbulent energy equation. J. Fluid Mech. 28, 593–616 (1967)
Wilcox D.C.: Turbulence Modeling for CFD. DCW Industries, Philadelphia (2006)
Moser R.D., Kim J., Mansour N.N.: Direct numerical simulation of turbulent channel flow up to Re = 590. Phys. Fluids 11, 943–945 (1999)
Acknowledgments
Special thanks go to X.L. Li who supply us with his compressible TBL simulation code and some data, and Y.S. Zhang for analyzing TBL data. The authors have benefitted from discussions with a number of colleagues, including S.Y. Chen, C.B. Lee, J. Chen, W.T. Bi, J.J. Tao, Y.P. Shi, and students N. Hu, J. Pei, Y.Z. Wang, Z.X. Zhang, J.H. Xie.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The project was supported by the National Natural Science Foundation of China (90716008) and the National Basic Research Program of China (2009CB724100).
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
She, ZS., Chen, X., Wu, Y. et al. New perspective in statistical modeling of wall-bounded turbulence. Acta Mech Sin 26, 847–861 (2010). https://doi.org/10.1007/s10409-010-0391-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-010-0391-y