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The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation

Published online by Cambridge University Press:  01 December 2009

RODERICK JOHNSTONE ,
GARY N. COLEMAN  and
PHILIPPE R. SPALART
RODERICK JOHNSTONE*
Affiliation:
School of Engineering Sciences, University of Southampton, SO17 1BJ, UK
GARY N. COLEMAN
Affiliation:
School of Engineering Sciences, University of Southampton, SO17 1BJ, UK
PHILIPPE R. SPALART
Affiliation:
Boeing Commercial Airplanes, PO Box 3707, Seattle, WA 98124, USA
*
†Email address for correspondence: R.Johnstone@soton.ac.uk
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Abstract

Wall-bounded turbulence in pressure gradients is studied using direct numerical simulation (DNS) of a Couette–Poiseuille flow. The motivation is to include adverse pressure gradients, to complement the favourable ones present in the well-studied Poiseuille flow, and the central question is how the scaling laws react to a gradient in the total shear stress or equivalently to a pressure gradient. In the case considered here, the ratio of local stress to wall stress, namely τ+, ranges from roughly 2/3 to 3/2 in the ‘wall region’. By this we mean the layer believed not to be influenced by the opposite wall and therefore open to simple, universal behaviour. The normalized pressure gradients p+ ≡ dτ+/dy+ at the two walls are −0.00057 and +0.0037. The outcome is in broad agreement with the findings of Galbraith, Sjolander & Head (Aeronaut. Quart. vol. 27, 1977, pp. 229–242) relating to boundary layers (based on measured profiles): the logarithmic velocity profile is much more resilient than two other, equally plausible assumptions, namely universality of the mixing length ℓ = κy and that of the eddy viscosity νt = uτκy. In pressure gradients, with τ+ ≠ 1, these three come into conflict, and our primary purpose is to compare them. We consider that the Kármán constant κ is unique but allow a range from 0.38 to 0.41, consistent with the current debates. It makes a minor difference in the interpretation. This finding of resilience appears new as a DNS result and is free of the experimental uncertainty over skin friction. It is not as distinct in the (rather strong) adverse gradient as it is in the favourable one; for instance the velocity U+ at y+ = 50 is lower by 3% on the adverse gradient side. A plausible cause is that the wall shear stress is small and somewhat overwhelmed by the stress and kinetic energy in the bulk of the flow. The potential of a correction to the ‘law of the wall’ based purely on p+ is examined, with mixed results. We view the preference for the log law as somewhat counter-intuitive in that the scaling law is non-local but also as becoming established and as highly relevant to turbulence modelling.

JFM classification

Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 163 - 175
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Blackwell, B. F., Kays, W. M. & Moffat, R. J. 1972 The Turbulent Boundary Layer on a Porous Plate: An Experimental Study of the Heat Transfer Behaviour with Adverse Pressure Gradients. HMT-16, Thermosciences Division, Mechanical Engineering Department, Standford University.Google Scholar
Bradshaw, P. & Huang, G. P. 1995 The law of the wall in turbulent flow. Proc. R. Soc. Lond. A 451, 165–188.Google Scholar
Chauhan, K., Nagib, H. M. & Monkewitz, P. A. 2007 On the composite logarithmic profile in zero pressure gradient turbulent boundary layers. In 45th AIAA Aerospace Sciences and Meeting and Exhibit, 8–11 January 2007, Reno, Nevada. Also Paper No. 2007-0532. AIAA.Google Scholar
El Telbany, M. M. M. & Reynolds, A. J. 1980 Velocity distributions in plane turbulent channel flows. J. Fluid Mech. 100, 1–29.Google Scholar
Galbraith, R. A. McD. & Head, M. R. 1975 Eddy viscosity and mixing length from measured boundary layer developments. Aeronaut. Quart. 26, 133–154.Google Scholar
Galbraith, R. A. McD., Sjolander, S. & Head, M. R. 1977 Mixing length in the wall region of turbulent boundary layers. Aeronaut. Quart. 27, 229–242.Google Scholar
Head, M. R. & Galbraith, R. A. McD. 1975 Eddy viscosity and entrainment in equilibrium boundary layers. Aeronaut. Quart. 27, 229–242.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re τ = 2003. Phys. Fluids 18, 011702.Google Scholar
Johnson, D. A. & Coakley, T. J. 1990 Improvements to a nonequilibrium algebraic turbulence model. AIAA. J. 28 (11), 2000–2003.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in a fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Lee, M. J. & Kim, J. 1991 The structure of turbulence in simulated plane Couette flow. In Eighth Symposium on Turbulent Shear Flows, Technical University of Munich, Munich, Germany.Google Scholar
Nagano, Y., Tagawa, M. & Tsuji, T. 1992 Effects of adverse pressure gradient on mean flows and turbulence statistics in a boundary layer. In Turbulent Shear Flows 8, (ed. Durst, F., Launder, B. E. & Friedrich, R.). Springer.Google Scholar
Spalart, P. R. 1986 Numerical study of sink-flow boundary layers. J. Fluid Mech. 172, 307–328.CrossRefGoogle Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to R θ = 1410. J. Fluid Mech. 187, 61–98.Google Scholar
Spalart, P. R., Coleman, G. N. & Johnstone, R. 2009 Retraction: direct numerical simulation of the Ekman layer. Phys. Fluids. To appear.Google Scholar
Spalart, P. R. & Watmuff, J. H. 1993 Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371.CrossRefGoogle Scholar
Thielbar, W. H., Kays, W. M. & Moffat, R. J. 1969 The Turbulent Boundary Layer: Experimental Heat Transfer with Blowing, Suction, and Favorable Pressure Gradient. HMT-5, Thermosciences Division, Mechanical Engineering Department, Stanford University.Google Scholar
Volino, R. J. & Simon, T. W. 1997 Velocity and temperature profiles in turbulent boundary layers experiencing streamwise pressure gradients. J. Heat Transfer 119, 433–439.Google 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