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On the logarithmic mean profile

  • J. KLEWICKI (a1), P. FIFE (a2) and T. WEI (a3)

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.

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Corresponding author

Email address for correspondence: joe.klewicki@unh.edu

References

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Adrian, R., Meinhart, C. & Tomkins, C. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 154.
Afzal, N. 1982 Fully developed turbulent flow in a pipe: an intermediate layer. Ing.-Arch. 52, 355377.
Barenblatt, G. 1996 Scaling, Self-Similarity and Intermediate Asymptotics. Cambridge University Press.
Buschmann, M. & Gad-el-Hak, M. 2007 Recent developments in scaling of wall-bounded flows. Prog. Aerosp. Sci. 42, 419467.
Cantwell, B. 2002 Introduction to Symmetry Analysis. Cambridge University Press.
Eyink, G. 2008 Turbulent flow in pipes and channels as cross-stream “inverse cascades” of vorticity. Phys. Fluids 20, 125101.
Fife, P., Klewicki, J., McMurtry, P. & Wei, T. 2005 a Multiscaling in the presence of indeterminacy: wall-induced turbulence. Multiscale Model. Simul. 4, 936959.
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, 781807.
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, 165189.
Ganapathisubramani, B., Longmire, E. & Marusic, I. 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 3546.
George, W. & Castillo, L. 1997 Zero-pressure gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689729.
Hamman, C., Klewicki, J. & Kirby, M. 2008 On the Lamb vector divergence in Navier–Stokes flows. J. Fluid Mech. 610, 261284.
Hansen, A. 1964 Similarity Analyses of Boundary Value Problems in Engineering. Prentice-Hall.
Hoyas, S. & Jimenez, J. 2006 Scaling of the velocity fluctuations in turbulent channels upto Re τ = 2003. Phys. Fluids 18, 011702.
Izakson, A. 1937 On the formula for the velocity distribution near walls. Tech. Phys. USSR IV, 2, 155162.
von Kármán, T. 1930 Mechanische ahnlichkeit und turbulenz. Nachr. Ges. Wiss. Gottingen, Math.-Phys. Klasse. 58–76.
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. 1532. Aichi Shuppan.
Klewicki, J., Fife, P., Wei, T. & McMurtry, P. 2006 Overview of a methodology for scaling the indeterminate equations of wall-turbulence. AIAA J. 44, 24752484.
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, 823839.
Marusic, I. 2001 On the role of large-scale structures in wall turbulence. Phys. Fluids 13, 735743.
Metzger, M., Adams, P. & Fife, P. 2008 Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers. J. Fluid Mech. 617, 107140.
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. 386392. Wiley.
Monkewitz, P., Chauhan, K. & Nagib, H. 2008 Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys. Fluids 20, 105102.
Moser, R., Kim, J. & Mansour, N. 1999 Direct numerical simulation of turbulent channel flow up to Re τ = 590. Phys. Fluids 11, 943945.
Nagib, H. & Chauhan, K. 2008 Variation of von Karman coefficient in cannonical flows. Phys. Fluids 20, 101518.
Oberlack, M. 2001 A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299328.
Osterlund, J., Johansson, A., Nagib, H. & Hites, M. 2000 A note on the overlap region in turbulent boundary layers. Phys. Fluids 12, 14.
Panton, R. 2005 Review of wall turbulence as described by composite expansions. Appl. Mech. Rev. 58, 136.
Perry, A. & Chong, M. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173217.
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, 361388.
Pope, S. 2000 Turbulent Flows. Cambridge University Press.
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. Springer.
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.
Tennekes, H. & Lumley, J. 1972 A First Course in Turbulence. MIT Press.
Townsend, A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
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, 371398.
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.
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 Scaling heat transfer in fully developed turbulent channel flow. Intl J. Heat Mass Transfer 48, 52845296.
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, 23502353.
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally-zero-pressure gradient flat-plate boundary layer. J. Fluid Mech. 660, 541.
Zagarola, M. & Smits, A. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 3379.
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On the logarithmic mean profile

  • J. KLEWICKI (a1), P. FIFE (a2) and T. WEI (a3)

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