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The structure of a three-dimensional turbulent boundary layer

  • A. T. Degani (a1), F. T. Smith (a2) and J. D. A. Walker (a1)


The three-dimensional turbulent boundary layer is shown to have a self-consistent two-layer asymptotic structure in the limit of large Reynolds number. In a streamline coordinate system, the streamwise velocity distribution is similar to that in two-dimensional flows, having a defect-function form in the outer layer which is adjusted to zero at the wall through an inner wall layer. An asymptotic expansion accurate to two orders is required for the cross-stream velocity which is shown to exhibit a logarithmic form in the overlap region. The inner wall-layer flow is collateral to leading order but the influence of the pressure gradient, at large but finite Reynolds numbers, is not negligible and can cause substantial skewing of the velocity profile near the wall. Conditions under which the boundary layer achieves self-similarity and the governing set of ordinary differential equations for the outer layer are derived. The calculated solution of these equations is matched asymptotically to an inner wall-layer solution and the composite profiles so formed describe the flow throughout the entire boundary layer. The effects of Reynolds number and cross-stream pressure gradient on the cross-stream velocity profile are discussed and it is shown that the location of the maximum cross-stream velocity is within the overlap region.



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Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover.
Anderson, S. D. & Eaton, J. K. 1989 Reynolds stress development in pressure-driven three-dimensional turbulent boundary layers. J. Fluid Mech. 202, 263294.
Baldwin, B. S. & Lomax, H. 1978 Thin layer approximation and algebraic model for separated turbulent flows. AIAA Paper 78-257, 16th Aerosciences Meeting, Huntsville, Alabama, January 16–18.
Barnwell, R. W., Wahls, R. A. & Dejarnette, F. E. 1988 A defect stream function, law of the wall/wake method for turbulent boundary layers. AIAA Paper 88-0127, 26th Aerospace Sciences Meeting, Reno, Nevada.
Bradshaw, P. & Pontikos, N. S. 1985 Measurements in the turbulent boundary layer on an ‘infinite’ swept wing. J. Fluid Mech. 159, 105130.
Cebeci, T. & Smith, A. M. O. 1974 Analysis of Turbulent Boundary Layers. Academic.
Clauser, F. H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21, 91108.
Clauser, F. H. 1956 The turbulent boundary layer. Adv. Appl. Mech. 4, 151.
Degani, A. T. 1991 The three-dimensional turbulent boundary layer–theory and application. Ph.D. thesis, Lehigh University.
Degani, A. T., Smith, F. T. & Walker, J. D. A. 1992 The three-dimensional turbulent boundary layer near a plane of symmetry. J. Fluid Mech. 234, 329360.
Degani, A. T. & Walker, J. D. A. 1991 Computation of three-dimensional turbulent boundary layers using the embedded-function method. AIAA Paper 92-0440, 30th Aerospace Sciences Meeting, Reno, Nevada, January 6–9.
Fendell, F. E. 1972 Singular perturbation and turbulent shear flow near walls. J. Astro. Sci. 20, 129165.
Fernholz, H. H. & Vagt, J.-D. 1981 Turbulence measurements in an adverse-pressure-gradient three-dimensional turbulent boundary layer along a circular cylinder. J. Fluid Mech. 111, 233269.
Goldberg, U. & Reshotko, E. 1984 Scaling and modeling of three-dimensional pressure-driven turbulent boundary layers. AIAA J. 22, 914920.
Hornung, H. G. & Joubert, P. N. 1963 The mean velocity profile in three-dimensional turbulent boundary layers. J. Fluid Mech. 15, 368384.
Johnston, J. P. 1960 On three-dimensional turbulent boundary layers generated by secondary flow. Trans. ASME D: J. Basic Engng 82, 233246.
Mellor, G. L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Intl J. Engng Sci. 10, 851873.
Mellor, G. L. & Gibson, D. M. 1966 Equilibrium turbulent boundary layers. J. Fluid Mech. 24, 225253.
Nash, J. F. & Patel, V. C. 1972 Three-Dimensional Turbulent Boundary Layers. Atlanta: SBC Technical Book.
Pierce, F. J., McAllister, J. E. & Tennant, M. H. 1983 A review of near-wall similarity models in three-dimensional turbulent boundary layers. Trans. ASME I: J. Fluids Engng 105, 251262.
Pierce, F. J. & Zimmerman, B. B. 1973 Wall shear stress inference from two- and three-dimensional turbulent boundary layer velocity profiles. Trans. ASME I: J. Fluids Engng 95, 6167.
Prahlad, T. S. 1973 Mean velocity profiles in three-dimensional incompressible turbulent boundary layers. AIAA J. 11, 359365.
Rubesin, M. R. & Viegas, J. R. 1985 A critical examination of the use of wall functions as boundary conditions in aerodynamic calculations. Third Symp. on Numerical and Physical Aspects of Aerodynamic Flows, California State University, Long Beach, January 21–24.
Smith, C. R., Walker, J. D. A., Haidari, A. H. & Sobrun, U. 1991 On the dynamics of near-wall turbulence. Phil. Trans. R. Soc. Lond. A 336, 131175.
Van den Berg, B., Elsenaar, A., Lindhout, J. P. F. & Wesseling, P. 1975 Measurements in an incompressible three-dimensional turbulent boundary layer, under infinite swept-wing conditions, and comparison with theory. J. Fluid Mech. 70, 127147.
Walker, J. D. A., Abbott, D. E., Scharnhorst, R. K. & Weigand, G. G. 1989 A wall-layer model for the velocity profile in turbulent flows. AIAA J. 27, 140149.
Walker, J. D. A., Ece, M. C. & Werle, M. J. 1991 An embedded function approach for turbulent flow prediction. AIAA J. 29, 18101818.
Walker, J. D. A. & Stewartson, K. 1974 Separation and the Taylor column for a hemisphere. J. Fluid Mech. 66, 767789.
Wie, Y.-S. & DeJarnette, F. R. 1988 Numerical investigation of three-dimensional separation using the boundary-layer equations. AIAA Paper 88-0617, 26th Aerospace Sciences Meeting, Reno, Nevada.
Yajnik, K. S. 1970 Asymptotic theory of turbulent shear flows. J. Fluid Mech. 42, 411427.
Yuhas, L. J. & Walker, J. D. A. 1982 An optimization technique for the development of two-dimensional steady turbulent boundary layer models. Rep. FM-1. Dept. Mech. Eng. and Mech., Lehigh Univ.; also AFOSR-TR-0417.
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The structure of a three-dimensional turbulent boundary layer

  • A. T. Degani (a1), F. T. Smith (a2) and J. D. A. Walker (a1)


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