The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation
The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation
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 ?+, 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+ ? d\tau+/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 \ell=\?appa y and that of the eddy viscosity \nu_t =u_\tau \kappa y. In pressure gradients, with \tau+ \not= 1, these three come into conflict, and our primary purpose is to compare them. We consider that the K´arm´an constant \kappa 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.
turbulent boundary layers, near-wall similarity, direct numerical simulation
Johnstone, Roderick
8ac02aa2-776b-4f80-b44d-1a5cf8682f21
Coleman, Gary N.
ea3639b9-c533-40d7-9edc-3c61246b06e0
Spalart, Philippe R.
8b92da5d-561c-4c7d-be1f-97f27de7c2a3
1 December 2009
Johnstone, Roderick
8ac02aa2-776b-4f80-b44d-1a5cf8682f21
Coleman, Gary N.
ea3639b9-c533-40d7-9edc-3c61246b06e0
Spalart, Philippe R.
8b92da5d-561c-4c7d-be1f-97f27de7c2a3
Johnstone, Roderick, Coleman, Gary N. and Spalart, Philippe R.
(2009)
The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation.
Journal of Fluid Mechanics.
(doi:10.1017/S0022112009992333).
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 ?+, 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+ ? d\tau+/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 \ell=\?appa y and that of the eddy viscosity \nu_t =u_\tau \kappa y. In pressure gradients, with \tau+ \not= 1, these three come into conflict, and our primary purpose is to compare them. We consider that the K´arm´an constant \kappa 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.
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Published date: 1 December 2009
Keywords:
turbulent boundary layers, near-wall similarity, direct numerical simulation
Organisations:
Aerodynamics & Flight Mechanics
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Local EPrints ID: 71722
URI: http://eprints.soton.ac.uk/id/eprint/71722
ISSN: 0022-1120
PURE UUID: 43f8adc3-2795-437c-a7d1-816c5594027d
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Date deposited: 21 Dec 2009
Last modified: 13 Mar 2024 20:40
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Author:
Roderick Johnstone
Author:
Gary N. Coleman
Author:
Philippe R. Spalart
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