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The interactive breakdown in supersonic ramp flow

Published online by Cambridge University Press:  26 April 2006

F. T. Smith  and
A. Farid Khorrami
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
A. Farid Khorrami
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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Abstract

The separating flow induced on a ramp under a supersonic main stream is discussed, for high Reynolds numbers, according to the interactive-boundary-layer approach. There are two principal motivations for the study, apart from the recent general upsurge of interest in compressible boundary-layer separation and stall. The first is the need for more progress to be made in the numerical determination of strongly reversed flows than has proved possible in computations hitherto. This is tackled by means of a new computational scheme based in effect on a global Newton iteration procedure but coupled with linearized shooting, second-order accurate windward differencing and linear multi-sweeping. The specific case addressed is the triple-deck version for steady laminar two-dimensional motion, although the present scheme, like the theory described below, also has broader application, for example to subsonic, hypersonic and/or unsteady interactive flows. The second motivation is to compare closely with the recent theoretical prediction (Smith 1988a) of a local breakdown or stall occurring in any interactive boundary-layer solution at a finite value of the controlling parameter, a say, within the reversed-flow region; the breakdown produces a large adverse pressure gradient and minimum negative surface shear locally. The first quantitative comparisons are made between the theory and computational results, derived in this work at values of a, here the scaled ramp angle, greater than those obtainable before, but with a fixed outer boundary. The agreement, while not complete, seems to prove fairly affirmative overall and tends to support the suggestion (in Smith 1988a) that, contrary to most earlier expectations, in general there is a finite upper limit on the extent to which the interacting boundary-layer approach can be taken on its own. A similar conclusion holds for unsteady interactive boundary layers concerning a finite-time breakdown (Smith 19886) and boundary-layer transition, and in the present context the local nonlinear breakdown provides an explanation for the severe computational difficulties encountered previously as well as for a form of airfoil stall.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 197 - 215
Copyright
© 1991 Cambridge University Press

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References

Bodonyi, R. J. & Smith, F. T., 1985 On short-scale inviscid instabilities in the flow past surface-mounted obstacles and other nonparallel motions. Aeron. J. June/July, 205–212.Google Scholar
Bodonyi, R. J. & Smith, F. T., 1986 Shock-wave laminar boundary layer interaction in supercritical transonic flow. Computers Fluids 14, 97–108.Google Scholar
Bowles, R. I.: 1990 Applications of nonlinear viscous-inviscid interactions in liquid layer flows and transonic boundary layer transition. Ph.D. thesis, University of London.
Brown, S. N., Khorrami, A. F., Neish, A. & Smith, F. T., 1991 Hypersonic boundary-layer interactions and transition. Phil. Trans. R. Soc. Lond. (to appear).Google Scholar
Carter, J. E.: 1972 NASA Tech. Rep. TR-R-385.
Conlisk, A. T. The pressure field in intense vortex-boundary layer interaction. Presented at Reno, Nevada. January 1989. AlAA Paper 89–0293.
Davis, R. T.: 1984 AlAA Paper 84–1614 (presented at Snowrnass, Colorado, June 1984).
Davis, R. T. & Werle, M. J., 1982 Progress on interacting boundary layer computations at high Reynolds numbers. In Proc. 1st Symp. Num. Phys. Aspects Aerodyn. Flows, Long Beach, CA (ed. T. Cebeci), pp. 187–210. Springer.
Duck, P. W.: 1985 Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465–498.Google Scholar
Duck, P. W. & Burggraf, O. R., 1982 Spectral computation of triple-deck flows. In Proc. 1st Symp. Num. Phys. Aspects Aerodyn. Flows, Long Reach, CA (ed. T. Cebeci), pp. 145–158. Springer.
Elliott, J. W. & Smith, F. T., 1986 Separated supersonic flow past a trailing edge at incidence. Computers Fluids 14, 109–116.Google Scholar
Gajjar, J.: 1983 On some viscous interaction problems in incompressible fluid flows. Ph.D. thesis, University of London.
Gittler, P. H. & Kluwick, A., 1987a Nonuniqueness of triple-deck solutions for axisymmetric supersonic flow with separation. In Boundary-layer Separation (ed. S. N. Brown & F. T. Smith). Springer.
Gittler, P. H. & Kluwick, A., 1987b Triple-deck solutions for supersonic flow past flared cylinders. J. Fluid Mech. 179, 469–487.Google Scholar
Gittler, P. H. & Kluwick, A., 1989 Interacting laminar boundary layers in quasi-two-dimensional flow. Fluid Dyn. Res. 5, 29–47.Google Scholar
Hoyle, J. M.: 1991 Ph.D. thesis, University of London (in preparation).
Hoyle, J. M., Smith, F. T. & Walker, J. D. A. 1990 On sublayer eruption and vortex formation. Proc. IMACS Conf., Boulder, CO., June 1990 (to appear, 1990, Computer Phys Commun.).
Khorrami, A. F.: 1991 Hypersonic aerodynamics on flat plates and thin airfoils. D. Phil, thesis, University of Oxford (submitted).
Lighthill, M. J.: 1953 On boundary layers and upstream influence II. Supersonic flows without separation. Proc. R. Soc. Lond. A 217, 478–507.Google Scholar
Messiter, A. F.: 1983 Boundary-layer interaction theory. Trans. ASME E: J. Appl. Mech. 30, 1104–1113.Google Scholar
Napolitano, M., Werle, M. J. & Davis, R. T., 1978 Numerical solutions of the triple-dick equations for supersonic and subsonic flow past a hump. Rep. AFL-78–6–42. University of Cincinnati; see also AlAA J. 17, no. 7, 1979.
Peridier, V. J. (1989) Ph.D. thesis, Lehigh University
Peridier, V. J., Walker, J. D. A. & Smith, F. T. 1988 Methods for the calculation of unsteady separation. AIAA Paper 88–0604, presented at Reno, Nevada, January 1988.Google Scholar
Peridier, V. J., Walker, J. D. A. & Smith, F. T. 1990 Vortex-induced boundary-layer separation. Parts I and II. J. Fluid Mech. (submitted). (1990); also presentations by Walker and Smith at Workshop on Unsteady Separation, Ohio State University, January 1990.Google Scholar
Rizzbtta, D. P., Burggraf, O. R. & Jenson, R., 1978 Triple-deck solutions for viscous supersonic and hypersonic flow past corners. J. Fluid Mech. 89, 535–552.Google Scholar
Ruban, A. I.: 1978 J. Vychisl. Mat. Fiz. 18, 1253–1265.
Smith, F. T.: 1977 The laminar separation of an incompressible fluid streaming past a smooth surface. Proc. R. Soc. Lond. A 356, 433–463.Google Scholar
Smith, F. T.: 1982 On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28, 207–281.Google Scholar
Smith, F. T.: 1985 Two-dimensional disturbance travel, growth and spreading in boundary layers. Utd Tech. Res. Center Rep. 85–36.Google Scholar
Smith, F. T.: 1986a Two-dimensional disturbance travel, growth and spreading in boundary layers. J. Fluid Mech. 169, 353–377.Google Scholar
Smith, F. T.: 1986b Steady and unsteady boundary-layer separation. Ann. Rev. Fluid Mech. 18, 197–220.Google Scholar
Smith, F. T.: 1987 Separating flow: small-scale, large-scale and nonlinear unsteady effects. In Boundary-layer Separation (ed. S. N. Brown & F. T. Smith). Springer.
Smith, F. T.: 1988a A reversed-flow singularity in interacting boundary layers, (based on Utd. Tech. Res. Cent. Rep UT88–12) Proc. R. Soc. Lond. A 420, 21–52.Google Scholar
Smith, F. T.: 1988b Finite-time break-up can occur in any unsteady interactive boundary-layer. Mathematika 35, 256–273.Google Scholar
Smith, F. T. & Merkin, J. H., 1982 Triple-deck solutions for subsonic flow past humps, concave or convex corners and wedged trailing edges. Computers Fluids 10, 7–25.Google Scholar
Smith, F. T. & Stewartson, K., 1973 On slot-injection into a supersonic laminar boundary-layer. Proc. R. Soc. Lond. A 332, 1–22.Google Scholar
Stewaktson, K.: 1974 Multi-structured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.Google Scholar
Stewartson, K.: 1981 D'Alembert's paradox. SIAM REv. 23, 308–343.Google Scholar
Stewartson, K. & Williams, P. G., 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar
Stewartson, K. & Williams, P. G., 1973 Self-induced separation II. MathemMika 20, 98–108.Google Scholar
Sychev, V. V.: 1987 Asymptotic Theory of Separated Flows. Moscow: Nauka.
Veldman, A. E. P.: 1981 New quasi-simultaneous method to calculate interacting boundary layers. AIAA J. 19, 79–85.Google Scholar
Werle, M. J., Burggraf, O. R., Rizzetta, D. P. & Vatsa, V. N. 1979 AIAA J. 17, 336–356.
Williams, B. R.: 1989 Viscous-inviscid interaction schemes for separated flow. Proc. R. Aero. Soc. Mtg, Prediction & Exploitation of Separated Flows, April 1989, London.Google Scholar