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On a class of compressible laminar boundary-layer flows and the solution behaviour near separation

Published online by Cambridge University Press:  20 April 2006

Antonios Liakopoulos  and
Chen-Chi Hsu
Antonios Liakopoulos
Affiliation:
Department of Engineering Sciences, University of Florida, Gainesville, Florida 32611
Chen-Chi Hsu
Affiliation:
Department of Engineering Sciences, University of Florida, Gainesville, Florida 32611
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Abstract

A class of compressible laminar boundary-layer flows subject to adverse pressure gradients of different magnitude is studied using a finite-element–differential method in which the assumed solutions are represented by classical cubic spline functions. The numerical integration process for the reduced initial-value problem has been carried out directly to at least one integration step upstream of the separation point, and very accurate numerical results have been obtained for a large number of integration steps extremely close to separation. The skin-friction and heat-transfer coefficients for nearly zero-heat-transfer, cooled-wall and heated-wall cases, computed under the assumption of constant Prandtl number Pr = 1.0 as well as Pr = 0.72, have clearly exhibited the same distinctive behaviour near separation. It is deduced that Buckmaster's series expansions for the solution near separation, derived on the assumptions of cooled wall and Pr = 1.0, are valid for all the cases considered. By matching the numerical results with Buckmaster's expansions, accurate distributions of skin friction and heat transfer have been obtained up to the separation point. Moreover, the importance of Prandtl number on the solution is evidenced from the numerical results presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 149 , December 1984 , pp. 339 - 353
Copyright
© 1984 Cambridge University Press

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References

Buckmaster, J. 1970 The behaviour of a laminar compressible boundary layer on a cold wall near a point of zero skin friction. J. Fluid Mech. 44, 237.Google Scholar
Curle, N. B. 1958 The steady compressible laminar boundary layer, with arbitrary pressure gradient and uniform wall temperature. Proc. R. Soc. Lond. A 249, 206.Google Scholar
Curle, N. 1978 Development and separation of a compressible laminar boundary layer under the action of a very sharp adverse pressure gradient. J. Fluid Mech. 84, 385.Google Scholar
Davies, T. & Walker, G. 1977 On solutions of the compressible laminar boundary layer equations and their behaviour near separation. J. Fluid Mech. 80, 279.Google Scholar
Gear, C. W. 1969 The automatic integration of stiff ordinary differential equations. In Information Processing (ed. A. J. H. Morrel), p. 187. North-Holland.
Goldstein, S. 1948 On laminar boundary layer flow near a position of separation. Q. J. Mech. Appl. Maths 1, 43.Google Scholar
Hsu, C. C. 1976 The use of splines for the solution of the boundary-layer equations. Tech. Rep. AFFDL-TR-75-158.Google Scholar
Hsu, C. C. 1980 A finite element—differential method for a class of boundary-layer flows. In Proc. 3rd Intl Conf. on Finite Elements in Flow Problems, Alberta, Canada.
Hsu, C. C. & Chang, T. H. 1982 On a numerical solution of incompressible turbulent boundary layer flow. In Finite Element Flow Analysis (ed. T. Kawai), p. 219. University of Tokyo Press.
Hsu, C. C. & Liakopoulos, A. 1982 Nonsimilar solutions of compressible laminar boundary layer flows by a semi-discretization method. In Finite Element Flow Analysis (ed. T. Kawai), p. 395. University of Tokyo Press.
Hunt, R. & Wilks, G. 1980 On the behaviour of the laminar boundary-layer equations of mixed convection near a point of zero skin friction. J. Fluid Mech. 101, 377.Google Scholar
Kaplun, S. 1967 In Fluid Mechanics and Singular Perturbations (ed. P. A. Lagestrom, L. N. Howard & C. S. Liu). Academic.
Liakopoulos, A. 1982 A finite element—differential method for compressible turbulent boundary-layer flows. Ph.D. dissertion, University of Florida.
Liakopoulos, A. 1984 Computation of high speed turbulent boundary-layer flows using the k turbulence model. Intl J. Numer. Meth. Fluids (to appear).Google Scholar
Merkin, J. H. 1969 The effect of buoyancy forces on the boundary-layer flow over a semi-infinite vertical flat plate in a uniform free stream. J. Fluid Mech. 35, 439.Google Scholar
Poots, G. 1960 A solution of the compressible laminar boundary layer equation with heat transfer and adverse pressure gradient. Q. J. Mech. Appl. Maths 13, 57.Google Scholar
Stewartson, K. 1958 On Goldstein's theory of laminar separation. Q. J. Mech. Appl. Maths 13, 57.Google Scholar
Stewartson, K. 1962 The behaviour of a laminar compressible boundary layer near a point of zero skin-friction. J. Fluid Mech. 12, 117.Google Scholar
Werle, M. J. & Senechal, G. D. 1973 A numerical study of separating supersonic laminar boundary layers. Trans. ASME E: J. Appl. Mech. 40, 679.Google Scholar