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Singularities in solutions of the three-dimensional laminar-boundary-layer equations

Published online by Cambridge University Press:  20 April 2006

James C. Williams
James C. Williams
Affiliation:
Department of Aerospace Engineering, Auburn University, Alabama
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Abstract

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.

A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 160 , November 1985 , pp. 257 - 279
Copyright
© 1985 Cambridge University Press

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References

Cebeci, T., Khatteb, A. K. & Stewartson, K. 1981 Three-dimensional laminar boundary layers and the ok of accessibility. J. Fluid Mech. 107, 57–87.Google Scholar
Goldstein, S. 1948 On laminary boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1, 43–69.Google Scholar
Maskell, E. C. 1955 Flow separation in three dimensions. R. Aircraft Establ. Rep. AER 2565.
Moore, F. K. 1958 On the separation of the unsteady laminar boundary layer. In Boundary Layer Research (ed. H. G. Görtler), pp. 296–310. Springer.
Nanbu, K. 1977 Unsteady Falkner–Skan flow. Z. angew. Math. Phys. 22, 1167–1172.Google Scholar
Nash, J. F. & Patel, V. C. 1972 Three-dimensional turbulent boundary layers. SBC Technical Books, Atlanta, Georgia.
Raetz, G. S. 1957 A method of calculating three-dimensional laminar layers of steady compressible flows. Northrop Aircraft, Inc., rep. NAI-58–73.
Rott, M. 1956 Unsteady viscous flow in the vicinity of a stagnation point. Q. Appl. Maths 13, 444–51.Google Scholar
Sears, W. R. 1956 Some recent developments in airfoil theory. J. Aero. Sci. 23, 490–499.Google Scholar
Smith, S. H. 1967 The impulsive motion of a wedge in a viscous fluid. Z. angew. Math. Phys. 18, 508–522.Google Scholar
Stewartson, K. 1951 On the impulsive motion of a flate plate in a viscous fluid. Q. J. Mech. Appl. Maths 4, 182–198.Google Scholar
Telionis, D. P. & Costis, C. E. 1983 Three-dimensional laminar separation. Dept Engng Sci. Mech. Virginia Poly. Inst. and State Univ. Rep. DTNSRDC-ASED-CR-04-83.
Telionis, D. P., Tsahalis, D. T. & Werle, M. J. 1973 Numerical investigation of unsteady boundary layer separation. Phys. Fluids 16, 968–73.Google Scholar
Wang, J. C. T. & Shen, F. 1978 Unsteady boundary layers with flow reversal and the associated heat transfer problem. AIAA J. 16, 1025–1029.Google Scholar
Williams, J. C. 1975 Semi-similar solutions to the three-dimensional laminar boundary layer equations. Appl. Sci. Res. 31, 161–86.Google Scholar
Williams, J. C. 1981 Unsteady development of the boundary layer in the vicinity of a rear stagnation point. In Numerical and Physical Aspects of Aerodynamic Flows (ed T. Cebeci), pp. 347–364. Springer.
Williams, J. C. 1982 Flow development in the vicinity of the sharp trailing edge of bodies impulsively set into motion. J. Fluid Mech. 115, 27–37.Google Scholar
Williams, J. C. & Johnson, W. D. 1974 Semi-similar solutions to unsteady boundary layer flows including separation. AIAA J. 12, 1388–93.Google Scholar
Williams, J. C. & Johnson, W. D. 1975 New solutions to the unsteady laminar boundary layer equations including the approach to separation. In Unsteady Aerodynamics – Proc. Symp. at University of Arizona, 18–20 March 1975 (ed. R. B. Kinney).
Williams, J. C. & Rhyne, T. B. 1980 Boundary layer development on a wedge impulsively set into motion. SIAM J. Appl. Maths 38, 215–224.Google Scholar
Williams, J. C. & Stewartson, K. 1982 Flow development in the vicinity of the sharp trailing edge of bodies impulsively set into motion. Part 2. J. Fluid Mech. 131, 177–194.Google Scholar