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An asymptotic theory of incompressible turbulent boundarylayer flow over a small hump

Published online by Cambridge University Press:  19 April 2006

R. I. Sykes
R. I. Sykes
Affiliation:
Meteorological Office, Bracknell, Berkshire
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Abstract

A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 101 , Issue 3 , 11 December 1980 , pp. 647 - 670
Copyright
© 1980 Cambridge University Press

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References

Adamson, T. C. & Feo, A. 1975 Interaction between a shock wave and a turbulent boundary layer in transonic flow. SIAM J. of Appl. Math. 29, 121–145.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion of a fluid in turbulent motion. Quart. J. Mech. Appl. Math. 7, 83–103.Google Scholar
Bradshaw, P., Ferriss, D. H. & Atwell, N. P. 1967 Calculation of boundary-layer development, using the turbulent energy equation. J. Fluid Mech. 28, 593–616.Google Scholar
Britter, R. E., Hunt, J. C. R. & Richards, K. J. 1980 Air flow over a two-dimensional hill: studies of velocity speed-up, roughness effects and turbulence. Quart. J. Roy. Met. Soc. (to appear).Google Scholar
Coles, D. 1956 The law of the wake in a turbulent boundary layer. J. Fluid Mech. 1, 191–226.Google Scholar
Counihan, J., Hunt, J. C. R. & Jackson, P. S. 1974 Wakes behind two-dimensional surface obstacles in turbulent boundary layers. J. Fluid Mech. 64, 529–563.Google Scholar
Deaves, D. M. 1975 Wind over hills: a numerical approach. J. Ind. Aerodynamics 1, 371–391.Google Scholar
Donaldson, C. duP. 1971 A progress report on an attempt to construct an invariant model of model of turbulent shear flows. Proc. AGARD Conf. on Turbulent Shear Flows, London, paper B-1.Google Scholar
Hanjalic, K. & Launder, B. E. 1972 A Reynolds stress model of turbulence and its application to thin shear flows. J. Fluid Mech. 52, 609–638.Google Scholar
Hinze, J. O. 1959 Turbulence. McGraw-Hill.
Hunt, J. C. R. 1971 A theory for the laminar wake of a two-dimensional body in a boundary layer. J. Fluid Mech. 49, 159–178.Google Scholar
Irwin, H. P. A. H. & Smith, P. A. 1975 Prediction of the effect of streamline curvature on turbulence. Phys. Fluids 18, 624–630.Google Scholar
Jackson, P. S. & Hunt, J. C. R. 1975 Turbulent wind flow over a low hill. Quart. J. Roy. Met. Soc. 101, 929–955.Google Scholar
Knight, D. 1977 Turbulent flow over a wavy boundary. Boundary Layer Met. 11, 205–222.Google Scholar
Launder, B. E., Reece, G. J. & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537–566.Google Scholar
Mason, P. J. & Sykes, R. I. 1979a Separation effects in Ekman-layer flow over ridges. Quart. J. Roy. Met. Soc. 105, 129–146.Google Scholar
Mason, P. J. & Sykes, R. I. 1979b Flow over an isolated hill of moderate slope. Quart. J. Roy. Met. Soc. 105, 383–395.Google Scholar
Mellor, G. L. 1972 The large Reynolds number asymptotic theory of turbulent boundary layers. Int. J. Engng Sci. 10, 851–873.Google Scholar
Melnik, R. E. & Chow, R. 1975 Asymptotic theory of two-dimensional trailing edge flows. Grunman Res. Dept. Rep. RE-510.Google Scholar
Melnik, R. E. & Grossman, B. 1974 Analysis of the interaction between a weak normal shock wave with a turbulent boundary layer. A.I.A.A. Paper 74–598.Google Scholar
Rotta, J. C. 1951 Statistische Theorie nichthomogener Turbulenz. Z. Phys. 129, 547–572.Google Scholar
Saffman, P. G. 1970 A model for inhomogeneous turbulent flow. Proc. Roy. Soc. A 317, 417–433.Google Scholar
Smith, F. T. 1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803–824.Google Scholar
Smith, F. T., Sykes, R. I. & Brighton, P. W. M. 1977 A two-dimensional boundary layer encountering a three-dimensional hump. J. Fluid Mech. 83, 163–176.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Math. 14, 145–239.Google Scholar
Sykes, R. I. 1978 Stratification effects in boundary-layer flow over hills. Proc. Roy. Soc. A 361, 225–243.Google Scholar
Taylor, P. A. 1977 Numerical studies of neutrally stratified planetary boundary layer flow above gentle topography. I. two-dimensional cases. Boundary Layer Met. 12, 37–60.Google Scholar
Taylor, P. A. & Gent, P. R. 1974 A model of atmospheric boundary layer flow above an isolated two-dimensional ‘hill’; an example of flow above gentle topography. Boundary Layer Met. 7, 349–362.Google Scholar
Taylor, P. A., Gent, P. R. & Keen, J. M. 1976 Some numerical solutions for turbulent boundary-layer flow above fixed, rough, wavy surfaces. Geophys. J. R. Astronomical Soc. 44, 177–201.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
Yajnik, K. 1970 Asymptotic theory of turbulent shear flows. J. Fluid Mech. 42, 411–427.Google Scholar