Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T23:02:11.474Z Has data issue: false hasContentIssue false
  • English
  • Français

The drag on an undulating surface induced by the flow of a turbulent boundary layer

Published online by Cambridge University Press:  26 April 2006

S. E. Belcher ,
T. M. J. Newley  and
J. C. R. Hunt
S. E. Belcher
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
T. M. J. Newley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Present address: c/o BP Exploration, Fairbanks, Alaska, USA.
J. C. R. Hunt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate, using theoretical and computational techniques, the processes that lead to the drag force on a rigid surface that has two—dimensional undulations of length L and height H (with H/L [Lt ] 1) caused by the flow of a turbulent boundary layer of thickness h. The recent asymptotic analyses of Sykes (1980) and Hunt, Leibovich & Richards (1988) of the linear changes induced in a turbulent boundary layer that flows over an undulating surface are extended in order to calculate the leading-order contribution to the drag. It is assumed that L is much less than the natural lengthscale h* = hU0/u* over which the boundary layer evolves (u* is the unperturbed friction velocity and U0 a mean velocity scale in the approach flow). At leading order, the perturbation to the drag force caused by the undulations arises from a pressure asymmetry at the surface that is produced by the thickening of the perturbed boundary layer in the lee of the undulation. This we term non-separated sheltering to distinguish it from the mechanism proposed by Jeffreys (1925). Order of magnitude estimates are derived for the other mechanisms that contribute to the drag; the next largest is shown to be smaller than the non-separated sheltering effect by O(u*/U0). The theoretical value of the drag induced by the non-separated sheltering effect is in good agreement with both the values obtained by numerical integration of the nonlinear equations with a second-order-closure model and experiments. Although the analytical solution is developed using the mixing-length model for the Reynolds stresses, this model is used only in the inner region, where the perturbation shear stress has a significant effect on the mean flow. The analytical perturbation shear stresses are approximately equal to the results from a higher-order closure model, except where there is strong acceleration or deceleration. The asymptotic theory and the results obtained using the numerical model show that the perturbations to the Reynolds stresses in the outer region do not directly contribute a significant part of the drag. This explains why several previous analyses and computations that use the mixing-length model inappropriately throughout the flow lead to values of the drag force that are too large by up to 100%.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 249 , April 1993 , pp. 557 - 596
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. R. 1972 A Handbook of Mathematical Functions. Dover.
Belcher, S. E. 1990 Turbulent boundary layer flow over undulating surfaces. PhD thesis, University of Cambridge.
Belcher, S. E. & Hunt, J. C. R. 1993a Turbulent shear flow over slowly moving waves. J. Fluid Mech. 251 (to appear).Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1993b A wall function for rapidly perturbed turbulent boundary layer flows. In preparation.
Bradley, E. F. 1980 An experimental study of the profiles of wind speed, shearing stress and turbulent intensities at the crest of a large hill. Q. J. R. Met. Soc. 106, 101– 124.Google Scholar
Bradshaw, P. 1967 ‘Inactive’ motion and pressure fluctuations in a turbulent boundary layer. J. Fluid Mech. 30, 241– 258.Google Scholar
Bradshaw, P., Ferris, D. H. & Atwell, N. P. 1967 Calculation of boundary layer development using the turbulent kinetic energy equation. J. Fluid Mech. 28, 593– 616.Google Scholar
Britter, R. E., Hunt, J. C. R. & Richards, K. J. 1981 Air flow over a 2-d hill: studies of velocity speed-up, roughness effects and turbulence. Q. J. R. Met. Soc. 107, 91– 110.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech. 61, 625– 706.Google Scholar
Hunt, J. C. R. 1978 A review of the theory of rapidly distorted turbulent flows and its applications. Fluid Dyn. Trans. 9, 121– 152.Google Scholar
Hunt, J. C. R. 1984 Turbulence structure in thermal convection and shear-free boundary layers. J. Fluid Mech. 138, 161– 184.Google Scholar
Hunt, J. C. R., Leibovich, S. & Richards, K. J. 1988 Turbulent shear flows over low hills. Q. J. R. Met. Soc. 114, 1435– 1471 (referred to herein as HLR.)Google Scholar
Hunt, J. C. R., Newley, T. M. J. & Weng, W. S. 1990 Analysis and computation of turbulent boundary layers in varying pressure gradients. In Proc. IM A Conf. on Computational Methods in Aeronautical Fluid Mechanics (ed. P. Stow). Clarendon.
Hunt, J. C. R., Tampieri, F., Weng, W. S. & Carruthers, D. J. 1991 Air flow and turbulence over complex terrain: a colloquium and computational workshop. J. Fluid Mech. 227, 667– 689.Google Scholar
Jacobs, S. J. 1987 An asymptotic theory for the turbulent flow over a progressive wave. J. Fluid Mech. 174, 69– 80.Google Scholar
Jeffreys, H. 1925 On the formation of water waves by wind. Proc. R. Soc. Lond. A 107, 189– 206.Google Scholar
Launder, B. E., Reece, G. T. & Rodi, W. 1975 The development of a Reynolds stress turbulent closure. J. Fluid Mech. 68, 537– 566 (referred to herein as LRR.)Google Scholar
Mason, P. J. 1986a On the parameterisation of orographic drag. In Proc. ECMWF Workshop on Observation, Theory and Modelling of Orographic Effects. ECMWF, Shinfield Park, Reading.
Mason, P. J. 1986a Flow over the summit of an isolated hill. Boundary Layer Met. 37, 385– 405.Google Scholar
Mason, P. J. & King, J. C. 1985 Measurements and predictions of flow and turbulence over an isolated hill of low slope. Q. J. R. Met. Soc. 111, 617– 640.Google Scholar
Maxey, M. R. 1982 Distortion of turbulence in flows with parallel streamlines. J. Fluid Mech. 124, 261– 282.Google Scholar
Newley, T. M. J. 1986 Turbulent air flow over low hills. PhD thesis, University of Cambridge.
Smith, F. T., Brighton, P. M. W., Jackson, P. S. & Hunt, J. C. R. 1981 On boundary layer flow past two-dimensional bodies. J. Fluid Mech. 113, 123– 152.Google Scholar
Stewartson, K. 1974 Multi-structured boundary layers on flat plates and related bodies. Adv. Appl. Maths 14, 145– 239.Google Scholar
Sykes, R. I. 1980 An asymptotic theory of incompressible turbulent flow over a small hump. J. Fluid Mech. 101, 647– 670.Google Scholar
Tampieri, F. 1987 Separation features of boundary layer flow over valleys. Boundary Layer Met. 40, 295– 307.Google Scholar
Taylor, P. A. & Gent, P. R. 1974 A model of atmospheric boundary layer flow above an isolated hill: an example of flow over gentle topography. Boundary Layer Met. 7, 349– 362.Google Scholar
Taylor, P. A., Mason, P. J. & Bradley, E. F. 1987 Boundary layer flow over low hills. Boundary Layer Met. 39, 107– 132.Google Scholar
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97– 120.Google Scholar
Townsend, A. A. 1972 Flow in a deep turbulent boundary layer over a surface distorted by water waves. J. Fluid Mech. 55, 719– 735.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.
Townsend, A. A. 1980 Sheared turbulence and additional distortion. J. Fluid Mech. 98, 171– 191.Google Scholar
Weng, W.-S., Hunt, J. C. R., Carruthers, D. J., Warren, A., Wiggs, G. F. S., Livingstone, I. & Castro, I. 1991 Air flow and sand transport over sand dunes. Acta Mechanica (suppl) 2, 1– 22.Google Scholar
Zeman, O. & Jensen, N. O. 1987 Modification to turbulence characteristics in flow over hills. Q. J. R. Met. Soc. 113, 55– 80.Google Scholar
Zeman, O. & Tennekes, J. L. 1975 A self-contained model for the pressure terms in the turbulent stress equations of the neutral atmospheric boundary layer. J. Atmos. Sci. 32, 1808– 1813.Google Scholar
Zilker, D. P. & Hanratty, T. J. 1979 Influence of the amplitude of a solid wavy boundary on a turbulent flow. Part 2. Separated flows. J. Fluid Mech. 90, 257– 271.Google Scholar