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Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

Published online by Cambridge University Press:  19 March 2014

Sergio Pirozzoli
Sergio Pirozzoli*
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma ‘La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
*
†Email address for correspondence: sergio.pirozzoli@uniroma1.it
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Abstract

We reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

JFM classification

Turbulent Flows: Turbulent boundary layers Turbulent Flows: Turbulent Flows Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014 , pp. 378 - 397
Copyright
© 2014 Cambridge University Press 

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References

del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.Google Scholar
Bernardini, M. & Pirozzoli, S. 2009 A general strategy for the optimization of Runge–Kutta schemes for wave propagation phenomena. J. Comput. Phys. 228, 4182–4199.Google Scholar
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to $Re_{\tau }=4000$. J. Fluid Mech. 742, 171–191.CrossRefGoogle Scholar
Bernardini, M., Pirozzoli, S., Quadrio, M. & Orlandi, P. 2013 Turbulent channel flow simulations in convecting reference frames. J. Comput. Phys. 232, 1–6.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Cess, R. D.1958 A survey of the literature on heat transfer in turbulent tube flow. Res. Rep. 8-0529-R24. Westinghouse Research.Google Scholar
Clauser, F. H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 1–51.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.Google Scholar
Dean, R. B. 1978 Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow. Trans. ASME: J. Fluids Engng 100, 215–223.Google Scholar
DeGraaff, D. B. & Eaton, J. K. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Durand, W. F. 1935 Aerodynamic Theory. Springer.Google Scholar
Flores, O. & Jimenez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.Google Scholar
George, W. K. & Castillo, L. 1997 Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of velocity fluctuations in turbulent channels up to ${R}e_{\tau } = 2003$. Phys. Fluids 18, 011702.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J. 2004 Further observations on the mean velocity in fully-developed pipe flow. J. Fluid Mech. 501, 135–147.Google Scholar
McKeon, B. J., Zagarola, M. V. & Smits, A. J. 2005 A new friction factor relationship for fully developed pipe flow. J. Fluid Mech. 538, 429–443.CrossRefGoogle Scholar
Millikan, C. B. 1938 A critical discussion of turbulent flows in channel and circular tubes. In Proc. of the Fifth International Congress of Applied Mechanics (ed. Den Hartog, J. P. & Peters, H.), pp. 386–392. Wiley.Google Scholar
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.Google Scholar
Monty, J. P., Stewart, J. A., Williams, R. C. & Chong, M. S. 2007 Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156.Google Scholar
Musker, A. J. 1979 Explicit expression for the smooth wall velocity distribution in a turbulent boundary layer. AIAA J. 17, 655–657.Google Scholar
Nagib, H. M. & Chauhan, K. A. 2008 Variations of von Kármán coefficient in canonical flows. Phys. Fluids 20, 101518.CrossRefGoogle Scholar
Nagib, H. M. & Chauhan, K. A. 2009 Criteria for assessing experiments in zero pressure gradient boundary layers. Fluid Dyn. Res. 41, 021404.Google Scholar
Nagib, H. M., Chauhan, K. A. & Monkewitz, P. A. 2007 Approach to an asymptotic state of zero pressure gradient turbulent boundary layers. Phil. Trans. R. Soc. Lond. A 365, 755–770.Google Scholar
Nagib, H., Christophorou, C., Rüedi, J. D., Monkewitz, P. A. & Österlund, J. M. 2004 Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress? AIAA Paper 2004-2392.Google Scholar
Orlandi, P. 2000 Fluid Flow Phenomena: A Numerical Toolkit. Kluwer.Google Scholar
Österlund, J. M., Johansson, A. V., Nagib, H. M. & Hites, M. H. 2000 A note on the overlap region in turbulent boundary layers. Phys. Fluids 12, 1–4.Google Scholar
Pirozzoli, S. 2012 On the size of the energy-containing eddies in the outer turbulent wall layer. J. Fluid Mech. 702, 521–532.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2011 Large-scale organization and inner–outer layer interactions in turbulent Couette–Poiseuille flows. J. Fluid Mech. 680, 534–563.Google Scholar
Prandtl, L. 1925 Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. Angew. Math. Mech. 5, 136–139.Google Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow, with application to Malkus theory. J. Fluid Mech. 27, 253–272.Google Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.Google Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. 8th edn. Springer.Google Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–226.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schultz, M. P. & Flack, K. A. 2013 Reynolds-number scaling of turbulent channel flow. Phys. Fluids 25, 025104.Google Scholar
Simens, M. P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for a turbulent boundary layers. J. Comput. Phys. 228, 4218–4231.Google Scholar
Smith, M. W. & Smits, A. J. 1995 Visualization of the structure of supersonic turbulent boundary layers. Exp. Fluids 18, 288–302.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. 2nd edn. Cambridge University Press.Google Scholar
Trefethen, A. E., Trefethen, L. N. & Schmid, P. J. 1999 Spectra and pseudospectra for pipe Poiseuille flow. Comput. Meth. Appl. Mech. Engng 175, 413–420.Google Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flow. J. Fluid Mech. 573, 303–327.CrossRefGoogle Scholar
Wosnik, M., Castillo, L. & George, W. K. 2000 A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2008 A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112.Google Scholar
Zagarola, M. V. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.Google Scholar
Zanoun, E.-S., Nagib, H. & Durst, F. 2009 Refined $C_f$ relation for turbulent channels and consequences for high-Re experiments. Fluid Dyn. Res. 41, 021405.Google Scholar