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On the logarithmic mean profile

Published online by Cambridge University Press:  23 September 2009

J. KLEWICKI ,
P. FIFE  and
T. WEI
J. KLEWICKI*
Affiliation:
Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA
P. FIFE
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
T. WEI
Affiliation:
Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 16802, USA
*
†Email address for correspondence: joe.klewicki@unh.edu
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Abstract

Elements of the first-principles-based theory of Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fife et al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936; J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged Navier–Stokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Kármán coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 638 , 10 November 2009 , pp. 73 - 93
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Adrian, R., Meinhart, C. & Tomkins, C. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.CrossRefGoogle Scholar
Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355–377.CrossRefGoogle Scholar
Barenblatt, G. 1996 Scaling, Self-Similarity and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Buschmann, M. & Gad-el-Hak, M. 2007 Recent developments in scaling of wall-bounded flows. Prog. Aerosp. Sci. 42, 419–467.CrossRefGoogle Scholar
Cantwell, B. 2002 Introduction to Symmetry Analysis. Cambridge University Press.Google Scholar
Eyink, G. 2008 Turbulent flow in pipes and channels as cross-stream “inverse cascades” of vorticity. Phys. Fluids 20, 125101.CrossRefGoogle Scholar
Fife, P., Klewicki, J., McMurtry, P. & Wei, T. 2005 a Multiscaling in the presence of indeterminacy: wall-induced turbulence. Multiscale Model. Simul. 4, 936–959.CrossRefGoogle Scholar
Fife, P., Klewicki, J. & Wei, T. 2009 Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. J. Discrete Continuous Dyn. Syst. 24, 781–807.CrossRefGoogle Scholar
Fife, P., Wei, T., Klewicki, J. & McMurtry, P. 2005 b Stress gradient balance layers and scale hierarchies in wall bounded turbulent flows. J. Fluid Mech. 532, 165–189.CrossRefGoogle Scholar
Ganapathisubramani, B., Longmire, E. & Marusic, I. 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46.CrossRefGoogle Scholar
George, W. & Castillo, L. 1997 Zero-pressure gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729.CrossRefGoogle Scholar
Hamman, C., Klewicki, J. & Kirby, M. 2008 On the Lamb vector divergence in Navier–Stokes flows. J. Fluid Mech. 610, 261–284.CrossRefGoogle Scholar
Hansen, A. 1964 Similarity Analyses of Boundary Value Problems in Engineering. Prentice-Hall.Google Scholar
Hoyas, S. & Jimenez, J. 2006 Scaling of the velocity fluctuations in turbulent channels upto Re τ = 2003. Phys. Fluids 18, 011702.CrossRefGoogle Scholar
Izakson, A. 1937 On the formula for the velocity distribution near walls. Tech. Phys. USSR IV, 2, 155–162.Google Scholar
von Kármán, T. 1930 Mechanische ahnlichkeit und turbulenz. Nachr. Ges. Wiss. Gottingen, Math.-Phys. Klasse. 58–76.Google Scholar
Kawamura, H., Abe, H. & Shingai, K. 2000 DNS of turbulence and heat transport in a channel flow with different Reynolds and Prandtl numbers and boundary conditions. In Turbulence Heat and Mass Transfer 3 (Proceedings of the Third Intl Symp. on Turbulence Heat and Mass Transfer), pp. 15–32. Aichi Shuppan.Google Scholar
Klewicki, J., Fife, P., Wei, T. & McMurtry, P. 2006 Overview of a methodology for scaling the indeterminate equations of wall-turbulence. AIAA J. 44, 2475–2484.CrossRefGoogle Scholar
Klewicki, J., Fife, P., Wei, T. & McMurtry, P. 2007 A physical model of the turbulent boundary layer consonant with mean momentum balance structure. Phil. Trans. R. Soc. A 365, 823–839.CrossRefGoogle ScholarPubMed
Marusic, I. 2001 On the role of large-scale structures in wall turbulence. Phys. Fluids 13, 735–743.CrossRefGoogle Scholar
Metzger, M., Adams, P. & Fife, P. 2008 Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers. J. Fluid Mech. 617, 107–140.CrossRefGoogle Scholar
Millikan, C. B. 1939 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings of Fifth International Congress of Applied Mechanics, pp. 386–392. Wiley.Google Scholar
Monkewitz, P., Chauhan, K. & Nagib, H. 2008 Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys. Fluids 20, 105102.CrossRefGoogle Scholar
Moser, R., Kim, J. & Mansour, N. 1999 Direct numerical simulation of turbulent channel flow up to Re τ = 590. Phys. Fluids 11, 943–945.CrossRefGoogle Scholar
Nagib, H. & Chauhan, K. 2008 Variation of von Karman coefficient in cannonical flows. Phys. Fluids 20, 101518.CrossRefGoogle Scholar
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328.CrossRefGoogle Scholar
Osterlund, J., Johansson, A., Nagib, H. & Hites, M. 2000 A note on the overlap region in turbulent boundary layers. Phys. Fluids 12, 1–4.CrossRefGoogle Scholar
Panton, R. 2005 Review of wall turbulence as described by composite expansions. Appl. Mech. Rev. 58, 1–36.CrossRefGoogle Scholar
Perry, A. & Chong, M. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A. & 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, 361–388.CrossRefGoogle Scholar
Pope, S. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. Springer.CrossRefGoogle Scholar
Spalart, P., Coleman, G. & Johnstone, R. 2008 Direct numerical simulation of the Eckman layer: a step in Reynolds number, and cautious support for a log law with shifted origin. Phys. Fluids 20, 101507.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Wei, T., Fife, P., & Klewicki, J. 2007 On scaling the mean momentum balance and its solutions in turbulent Couette–Poiseuille flow. J. Fluid Mech. 573, 371–398.CrossRefGoogle Scholar
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, 303–327.CrossRefGoogle Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 Scaling heat transfer in fully developed turbulent channel flow. Intl J. Heat Mass Transfer 48, 5284–5296.CrossRefGoogle Scholar
Wei, T., McMurtry, P., Klewicki, J. & Fife, P. 2005 Meso scaling of the Reynolds shear stress in turbulent channel and pipe flows. AIAA J. 43, 2350–2353.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally-zero-pressure gradient flat-plate boundary layer. J. Fluid Mech. 660, 5–41.CrossRefGoogle Scholar
Zagarola, M. & Smits, A. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.CrossRefGoogle Scholar