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On the instability of hypersonic flow past a flat plate

Published online by Cambridge University Press:  26 April 2006

Nicholas D. Blackaby ,
Stephen J. Cowley  and
Philip Hall
Nicholas D. Blackaby
Affiliation:
Department of Mathematics, The University, Manchester M13 9PL, UK
Stephen J. Cowley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Philip Hall
Affiliation:
Department of Mathematics, The University, Manchester M13 9PL, UK
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Abstract

The instability of hypersonic boundary-layer flow over a flat plate is considered. The viscosity of the fluid is taken to be governed by Sutherland's formula, which gives a more accurate representation of the temperature dependence of fluid viscosity at hypersonic speeds than Chapman's approximate linear law. A Prandtl number of unity is assumed. Attention is focused on inviscid instability modes of viscous hypersonic boundary layers. One such mode, the ‘vorticity’ mode, is thought to be the fastest growing disturbance at high Mach numbers, M [Gt ] 1; in particular it is believed to have an asymptotically larger growth rate than any viscous instability. As a starting point we investigate the instability of the hypersonic boundary layer which exists far downstream from the leading edge of the plate. In this regime the shock that is attached to the leading edge of the plate plays no role, so that the basic boundary layer is non-interactive. It is shown that the vorticity mode of instability operates on a different lengthscale from that obtained if a Chapman viscosity law is assumed. In particular, we find that the growth rate predicted by a linear viscosity law overestimates the size of the growth rate by O((log M)½). Next, the development of the vorticity mode as the wavenumber decreases is described. It is shown, inter alia, that when the wavenumber is reduced to O(M-3/2) from the O(1) initial, ‘vorticity-mode’ scaling, ‘acoustic’ modes emerge.

Finally, the inviscid instability of the boundary layer near the leading-edge interaction zone is discussed. Particular attention is focused on the strong-interaction zone which occurs sufficiently close to the leading edge. We find that the vorticity mode in this regime is again unstable. The fastest growing mode is centred in the adjustment layer at the edge of the boundary layer where the temperature changes from its large, O(M2). value in the viscous boundary layer, to its O(1) free-stream value. The existence of the shock indirectly, but significantly, influences the instability problem by modifying the basic flow structure in this layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 369 - 416
Copyright
© 1993 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.
Balsa, T. F. & Goldstein, M. E. 1990 On the instabilities of supersonic mixing layers: a high Mach number asymptotic theory. J. Fluid Mech. 216, 585– 611.Google Scholar
Blackaby, N. D. 1991 On viscous, inviscid and centrifugal instability mechanisms in compressible boundary layers, including non-linear vortex/wave interaction and the effects of large Mach number on transition. Ph.D. thesis, University of London.
Brown, S. N., Khorrami, A. F., Neish, A. & Smith, F. T. 1991 On hypersonic boundary-layer interactions and transition. Phil. Trans. R. Soc. Lond. A. 335, 139– 152.Google Scholar
Brown, S. N. & Stewartson, K. 1975 A non-uniqueness of the hypersonic boundary layer. Q. J. Mech. Appl. Maths XXVIII, 75– 90.Google Scholar
Bush, W. B. 1966 Hypersonic strong similarity solutions for flow past a flat plate. J. Fluid Mech. 25, 51– 64.Google Scholar
Bush, W. B. & Cross, A. K. 1967 Hypersonic weak interaction similarity solutions for flow past a flat plate. J. Fluid Mech. 29, 349– 359.Google Scholar
Cowley, S. J. & Hall, P. 1990 On the instability of flow past a wedge. J. Fluid Mech. 214, 17– 43 (referred to herein as CH.)Google Scholar
Dunn, D. W. & Lin, C. C. 1955 On the stability of the laminar boundary layer in a compressible fluid. J. Aero. Sci. 22, 455– 477.Google Scholar
Fischer, M. C. & Weinstein, L. M. 1972 Cone transitional boundary-layer structure at Me = 14. AIAA J. 10, 699– 701.Google Scholar
Freeman, N. C. & Lam, S. H. 1959 On the Mach number independence principle for a hypersonic boundary layer. Princeton University Rep. 471.
Fu, Y., Hall, P. & Blackaby, N. D. 1990 On the Gortler instability in hypersonic flows: Sutherland law fluids and real gas effects. ICASE Rep. 90-85; and to appear in Phil. Trans. R. Soc. Lond. A.
Gajjar, J. S. B. & Cole, J. W. 1989 Upper branch stability of compressible boundary layer flows. Theor. Comput. Fluid Dyn. 1, 105– 123.Google Scholar
Goldstein, M. E. & Wundrow, D. W. 1990 Spatial evolution of nonlinear acoustic mode-instabilities on hypersonic boundary layers. J. Fluid Mech. 219, 585– 607.Google Scholar
Grubin, S. E. & Trigub, V. N. 1993a The asymptotic theory of a hypersonic boundary layer stability. J. Fluid Mech. 246, 361– 380.Google Scholar
Grubin, S. E. & Trigub, V. N. 1993b The long-wave limit in the asymptotic theory of a hypersonic boundary layer stability. J. Fluid Mech. 246, 381– 395.Google Scholar
Hall, P. & Fu, Y. 1989 Görtler vortices at hypersonic speeds. Theor. Comput. Fluid Mech. 1, 125– 134.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641.Google Scholar
Heisenberg, W. 1924 Über Stabilität und Turbulenz von Flussigkeitsströmen. Ann. Phys. Lpz. (4) 74, 577– 627.Google Scholar
Jackson, T. L. & Grosch, C. E. 1989 Inviscid spatial instability of a compressible mixing layer. J. Fluid Mech. 208, 609– 638.Google Scholar
Jackson, T. L. & Grosch, C. E. 1991 Inviscid spatial instability of a compressible mixing layer. Part 3. J. Fluid Mech. 224, 159– 175.Google Scholar
Lee, R. S. & Cheng, H. K. 1969 On the outer edge problem of a hypersonic boundary layer. J. Fluid Mech. 38, 161– 179.Google Scholar
Lees, L. 1953 On the boundary layer equations in hypersonic flow and their approximate solutions. J. Aero. Sci. 30, 143– 145.Google Scholar
Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA Tech. Note 1115.Google Scholar
Lin, C. C. 1945a On the stability of two-dimensional parallel flows, Part I. Q. Appl. Maths. 3, 117– 142.Google Scholar
Lin, C. C. 1945b On the stability of two-dimensional parallel flows, Part II. Q. Appl. Maths 3, 218– 234.Google Scholar
Lin, C. C. 1945c On the stability of two-dimensional parallel flows, Part III. Q. Appl. Maths 3, 277– 301.Google Scholar
Luniev, V. V. 1959 On the similarity of hypersonic viscous flows around slender bodies. Prikl. Mat. Mech. 23, 193– 197.Google Scholar
Mack, L. M. 1969 Boundary-layer stability theory. Document 900-277. Rev. A. Jet Propulsion Laboratory, Pasadena, CA.
Mack, L. M. 1984 Boundary-layer linear stability theory. In Special Course on Stability and Transition of Laminar Flow. AGARD Rep. 709.
Mack, L. M. 1987 Review of linear compressible stability theory. In Stability of Time Dependent and Spatially Varying Flows, (ed. D. L. Dwoyer & M. Y. Hussaini). Springer.
Malmuth, N. D. 1991 Inviscid stability of hypersonic strong interaction flow over a flat plate. J. Fluid Mech. (submitted).Google Scholar
Papageorgiou, D. T. 1990 Linear instability of the supersonic wake behind a flat plate aligned with a uniform stream. Theor. Comput. Fluid Dyn. 1. 327– 348.Google Scholar
Papageorgiou, D. T. 1991 The stability of two-dimensional wakes and shear-layers at high Mach numbers. Phys. Fluids A 3, 793– 802.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311– 350.Google Scholar
Reshotko, E. & Khan, M. M. S. 1979 Stability of the laminar boundary layer on a blunted plate in supersonic flow. In IUTAM Symp. Stuttgart, Germany on Laminar-Turbulent Transition. Springer.
Ryzhov, O. S. 1984 Stability and separation of viscous flows. In IUTAM Symp. Novosibirsk, USSR on Laminar-Turbulent Transition. Springer.
Seddougui, S. O. & Bassom, A. P. 1991 Nonlinear instability of hypersonic flow past a wedge. Theor. Comput. Fluid Dyn. (submitted).Google Scholar
Seddougui, S. O., Bowles, R. I. & Smith, F. T. 1991 Surface-cooling effects on compressible boundary layer instability. Eur. J. Mech. B/ Fluids 10, 117– 145.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91– 109.Google Scholar
Smith, F. T. 1989 On the first-mode instability in subsonic, supersonic or hypersonic boundary layers. J. Fluid Mech. 198, 127– 154.Google Scholar
Smith, F. T, & Brown, S. N. 1990 The inviscid instability of a Blasius boundary layer at large values of the Mach number. J. Fluid Mech. 219, 499– 518 (referred to herein as SB).Google Scholar
Stewartson, K. 1955 On the motion of a flat plate at high speed in a viscous compressible fluid – II. Steady motion. J. Aero. Sci. 22, 303– 309.Google Scholar
Stewartson, K. 1964 Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.
Tollmien, W. 1929 Über die Entstehung der Turblenz. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, 21– 44.Google Scholar
Wundrow, D. W. 1992 Nonlinear spatial evolution of inviscid instabilities on hypersonic boundary layers. J. Fluid Mech. (in press).Google Scholar