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Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers

Published online by Cambridge University Press:  25 December 2008

M. METZGER ,
A. LYONS  and
P. FIFE
M. METZGER
Affiliation:
Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA
A. LYONS
Affiliation:
Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA
P. FIFE
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
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Abstract

Moderately favourable pressure gradient turbulent boundary layers are investigated within a theoretical framework based on the unintegrated two-dimensional mean momentum equation. The present theory stems from an observed exchange of balance between terms in the mean momentum equation across different regions of the boundary layer. This exchange of balance leads to the identification of distinct physical layers, unambiguously defined by the predominant mean dynamics active in each layer. Scaling domains congruent with the physical layers are obtained from a multi-scale analysis of the mean momentum equation. Scaling behaviours predicted by the present theory are evaluated using direct measurements of all of the terms in the mean momentum balance for the case of a sink-flow pressure gradient generated in a wind tunnel with a long development length. Measurements also captured the evolution of the turbulent boundary layers from a non-equilibrium state near the wind tunnel entrance towards an equilibrium state further downstream. Salient features of the present multi-scale theory were reproduced in all the experimental data. Under equilibrium conditions, a universal function was found to describe the decay of the Reynolds stress profile in the outer region of the boundary layer. Non-equilibrium effects appeared to be manifest primarily in the outer region, whereas differences in the inner region were attributed solely to Reynolds number effects.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 617 , 25 December 2008 , pp. 107 - 140
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Balint, J.-L., Wallace, J. & Vukoslavcevic, P. 1991 The velocity and vorticity vector fields of a turbulent boundary layer. Part 2. Statistical properties. J. Fluid Mech. 228, 53–86.Google Scholar
Barenblatt, G., Chorin, A. & Prostokishin, V. 1997 Scaling laws for fully developed turbulent flow in pipes. Appl. Mech. Rev. 50, 413–429.CrossRefGoogle Scholar
Bradshaw, P. 1967 The turbulence structure of equilibrium boundary layers. J. Fluid Mech. 29, 625–645.CrossRefGoogle Scholar
Castillo, L. & George, W. 2001 Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39, 41–47.CrossRefGoogle Scholar
Castillo, L. & Wang, X. 2004 Similarity analysis for nonequilibrium turbulent boundary layers. J. Fluids Engng 126, 827–834.CrossRefGoogle Scholar
Clauser, F. H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aero. Sci. 21, 91–108.CrossRefGoogle Scholar
Clauser, F. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 1–51.CrossRefGoogle Scholar
Coles, D. 1957 Remarks on the equilibrium turbulent boundary layer. J. Aero. Sci. 24, 495–506.CrossRefGoogle Scholar
DeGraaff, D. & Eaton, J. 2000 Reynolds-number scaling of the flat plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.CrossRefGoogle Scholar
Durbin, P. & Belcher, S. 1992 Scaling of adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 238, 699–722.CrossRefGoogle Scholar
Erm, L. P. & Joubert, P. N. 1991 Low-Reynolds-number turbulent boundary layers. J. Fluid Mech. 230, 1–44.CrossRefGoogle Scholar
Falkner, V. & Skan, S. 1931 Solutions of boundary layer equations. Phil. Mag. 12, 865–896.CrossRefGoogle Scholar
Fife, P., Klewicki, J., McMurtry, P. & Wei, T. 2005 a Multiscaling in the presence of indeterminacy: wall-induced turbulence. Multiscale Mod. Sim. 4, 936–959.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
George, W. & Castillo, L. 1997 Zero-pressure gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729.CrossRefGoogle Scholar
Hafez, S. 1991 The structure of accelerated turbulent boundary layers. PhD thesis. University of Melbourne, Australia.Google Scholar
Jones, W. & Launder, B. 1972 Some properties of sink-flow turbulent boundary layers. J. Fluid Mech. 56, 337–351.CrossRefGoogle Scholar
Jones, M., Marusic, I. & Perry, A. 2001 Evolution and structure of sink-flow turbulent boundary layers. J. Fluid Mech. 428, 1–27.CrossRefGoogle Scholar
Klewicki, J., Gendrich, C., Foss, J. & Falco, R. 1990 On the sign of the instantaneous spanwise vorticity component in the near-wall region of turbulent boundary layers. Phys. Fluids A 2, 1497–1500.CrossRefGoogle Scholar
Long, R. & Chen, T. 1981 Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59.CrossRefGoogle Scholar
Lyons, A. 2007 Experimental study of the mean momentum balance in sink-flow turbulent boundary layers. Master's thesis, University of Utah, Salt Lake City, UT.Google Scholar
McKeon, B., Li, J., Jiang, W., Morrison, J. & Smits, A. 2004 Further observations on the mean velocity distribution in fully-developed pipe flow. J. Fluid Mech. 501, 135–147.CrossRefGoogle Scholar
Metzger, M. & Klewicki, J. 2003 Development and characterization of a probe to measure scalar transport. Meas. Sci. Technol. 14, 1437–1448.CrossRefGoogle Scholar
Millikan, C. 1939 A critical discussion of turbulent flows in channel and circular tubes. In Proceedings of the Fifth International Congress of Applied Mathematics (ed. Hartog, J. & Peters, H.), pp. 386–392. New York: Wiley.Google 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
Murlis, J., Tsai, H. M. & Bradshaw, P. 1982 The structure of turbulent boundary layers at low reynolds number. J. Fluid Mech. 122, 13–56.CrossRefGoogle Scholar
Narasimha, R. & Sreenivasan, K. 1979 Relaminarization of fluid flows. Adv. Appl. Mech. 19, 221–309.CrossRefGoogle Scholar
Panton, R. 1990 Scaling turbulent wall layers. J. Fluids Engng 112, 425–432.CrossRefGoogle Scholar
Park, S. & Wallace, J. 1993 The influence of instantaneous velocity gradients on turbulence properties measured with multi-sensor hot-wire probes. Exps. Fluids 16, 17–26.CrossRefGoogle Scholar
Perry, A., Marusic, I. & Jones, M. 2002 On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients. J. Fluid Mech. 461, 61–91.CrossRefGoogle Scholar
Perry, A., Marusic, I. & Li, J. 1994 Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis. Phys. Fluids 6, 1024–1035.CrossRefGoogle Scholar
Piquet, J. 2001 Turbulent Flows. Springer-Verlag.Google Scholar
Pope, S. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prandtl, L. 1925 Bericht ber untersuchungen zur ausgebildeten turbulenz. Z. Angew. Math. Mech. 5, 136–139.CrossRefGoogle Scholar
Rotta, J. C. 1962 Turbulent boundary layers in incompressible flow. Prog. Aero. Sci. 2, 1–220.CrossRefGoogle Scholar
Skare, P. E. & Krogstad, P. A. 1994 A turbulent equilibrium boundary layer near separation. J. Fluid Mech. 272, 319–348.CrossRefGoogle Scholar
Spalart, P. 1986 Numerical study of sink-flow boundary layers. J. Fluid Mech. 172, 307–328.CrossRefGoogle Scholar
Spalart, P. 1988 Direct simulation of a turbulent boundary layer up to R θ = 1410. J. Fluid Mech. 187, 61–98.CrossRefGoogle Scholar
Sreenivasan, K. & Sahay, A. 1997 The persistence of viscous effects in the overlap region, and the mean velocity in turbulent pipe and channel flows. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton, R.), pp. 253–272. Computational Mechanics Publications.Google Scholar
Townsend, A. 1956 The Structure of Turbulent Shear Flow, 1st edn.Cambridge University Press.Google Scholar
Townsend, A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97–120.CrossRefGoogle Scholar
Townsend, A. 1976 The Structure of Turbulent Shear Flow, 2nd edn.Cambridge University Press.Google Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 a Properties of the mean momentum balance in boundary layer, pipe, and channel flows. J. Fluid Mech. 522, 303–327.CrossRefGoogle Scholar
Wei, T., McMurtry, P., Klewicki, J. & Fife, P. 2005 b Mesoscaling of Reynolds shear stress in turbulent channel and pipe flows. AIAA J. 43, 2350–2353.CrossRefGoogle Scholar
Wosnik, M., Castillo, L. & George, W. 2000 A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145.CrossRefGoogle Scholar
Yaglom, A. 1979 Similarity laws for constant-pressure and pressure-gradient turbulent wall flows. Annu. Rev. Fluid Mech. 11, 505–540.CrossRefGoogle Scholar
Zhu, Y. & Antonia, R. 1995 Effect of wire separation on X-probe measurements in a turbulent flow. J. Fluid Mech. 287, 199–223.CrossRefGoogle Scholar