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On hypersonic self-induced separation, hydraulic jumps and boundary layers with algebraic growth

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  26 February 2010

J. Gajjar  and
F. T. Smith
J. Gajjar
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2BZ
F. T. Smith
Affiliation:
Department of Mathematics, Imperial College, London, SW7 2BZ
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Summary

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 ( = 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

MSC classification

Secondary: 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
Type
Research Article
Information
Mathematika , Volume 30 , Issue 1 , June 1983 , pp. 77 - 93
Copyright
Copyright © University College London 1983

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References

Abramowitz, M. and Stegun, I. A.. Handbook of Math. Functions (Nat. Bur. of Standards, US Dept. of Commerce, 1964).Google Scholar
Brown, S. N. and Williams, P. G.. J. Inst. Math. Applies., 16 (1975), 175–192.CrossRefGoogle Scholar
Brown, S. N., Stewartson, K. and Williams, P. G.. Proc. Roy. Soc. Edinburgh, 74A (1974/1975), 21.Google Scholar
Brown, S. N., Stewartson, K. and Williams, P. G.. Phys. Fluids, 18 (1975), 633–639.CrossRefGoogle Scholar
Chester, W.. J.Fluid Meek, 24 (1966), 367.CrossRefGoogle Scholar
Craik, A. D. D., Latham, R. C., Fawkes, M. J. and Gribbon, P. W. F.. J. Fluid Meek, 112 (1981), 347–362.CrossRefGoogle Scholar
Dijkstra, D.. Proc. 6th Int. Conf. Num. Meth. Fluid. Dyn. (Tbilisi, U.S.S.R., 1978).Google Scholar
Gajjar, J.. Ph.D. Thesis (University of London, 1983) (in preparation).Google Scholar
Henderson, F. M.. Open channel flow (Macmillan, 1966).Google Scholar
Jones, C. W. and Watson, E. J.. Ch. V of Laminar Boundary Layers, ed. Rosenhead, L. (Oxford University Press, 1963).Google Scholar
Kurihara, M.. Res. Inst. for Fluid Eng., Dyusyu Imp. Univ., Japan, 3 (1946), 11.Google Scholar
Lamb, H.. Hydrodynamics, 6th ed. (Cambridge University Press, 1932).Google Scholar
Rajaratnam, N.. Advs. in Hydrosci., 4 (1967), 197–280.CrossRefGoogle Scholar
Reyhner, T. A. and Lotz, I. FlÜgge. Int. J. Nonl. Meek, 3 (1968), 173.Google Scholar
Rizzetta, D. P., Burggraf, D. R. and Jenson, R.. J. Fluid Meek, 89 (1978), 535–552.Google Scholar
Smith, F. T.. J. Fluid Meek, 78 (1976), 709–736.CrossRefGoogle Scholar
Smith, F. T.. J. Fluid Meek, 99 (1980), 185–224.Google Scholar
Smith, F. T.. United Technologies Res. Center, E. Hartford, U.S.A., Report (1983) in preparation. Also, submitted to Proc. Roy. Soc. A.Google Scholar
Stewartson, K.. Advs. in Appl. Mechs., 14 (1974), 145.Google Scholar
Stewartson, K. and Williams, P. G.. Mathematika, 20 (1973), 98–208.CrossRefGoogle Scholar
Tani, K.. J. Phys. Soc. Japan, 4 (1948), 212.CrossRefGoogle Scholar
Watson, E. J.. J. Fluid Meek, 20 (1964), 481–499.Google Scholar