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An instability in supersonic boundary-layer flow over a compression ramp

Published online by Cambridge University Press:  26 April 2006

K. W. Cassel
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA
A. I. Ruban
Affiliation:
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Moscow Region, Russia
J. D. A. Walker
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA

Abstract

Separation of a supersonic boundary layer (or equivalently a hypersonic boundary layer in a region of weak global interaction) near a compression ramp is considered for moderate wall temperatures. For small ramp angles, the flow in the vicinity of the ramp is described by the classical supersonic triple-deck structure governing a local viscous-inviscid interaction. The boundary layer is known to exhibit recirculating flow near the corner once the ramp angle exceeds a certain critical value. Here it is shown that above a second and larger critical ramp angle, the boundary-layer flow develops an instability. The instability appears to be associated with the occurrence of inflection points in the streamwise velocity profiles within the recirculation region and develops as a wave packet which remains stationary near the corner and grows in amplitude with time.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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References

Ackeret J., Feldmann F. & Rott N. 1947 Investigations of compression shocks and boundary layers in gases moving at high speed. Translated in Natl Adv. Comm. Aeronaut. Tech. Memo. No. 1113 ETH Zurich No. 10, 1946.
Adamson T. C. & Messiter A. F. 1980 Analysis of two-dimensional interactions between shock waves and boundary layers. Ann. Rev. Fluid Mech. 12, 103138.Google Scholar
Anderson J. D. 1989 Hypersonic and High Temperature Gas Dynamics. McGraw Hill.
Brown S. N., Cheng H. K. & Lee C. J. 1990 Inviscid-viscous interaction on triple-deck scales in a hypersonic flow with strong wall cooling. J. Fluid Mech. 220, 309337.Google Scholar
Brown S. N., Stewartson K. & Williams P. G. 1975 Hypersonic self-induced separation. Phys. Fluids 18, 633639.Google Scholar
Burggraf O. R. 1975 Asymptotic theory of separation and reattachment of a laminar boundary layer on a compression ramp. AGARD-CP-168, Paper No. 10.
Burggraf O. R., Rizzetta D., Werle M. J. & Vatsa V. N. 1979 Effect of Reynolds number on laminar separation of a supersonic stream. AIAA J. 17, 336343.Google Scholar
Cassel K. W. 1993 The effect of interaction on boundary-layer separation and breakdown. PhD thesis Lehigh University, Bethlehem, Pennsylvania.
Cassel K. W., Ruban A. I. & Walker J. D. A. 1995 The influence of wall cooling on hypersonic boundary-layer separation and instability. J. Fluid Mech. (submitted).Google Scholar
Cassel K. W., Smith F. T. & Walker J. D. A. 1994 The onset of instability in unsteady boundary-layer separation. J. Fluid Mech. (submitted).Google Scholar
Chapman D. R., Kuehn D. M. & Larson H. K. 1957 Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition. NACA Tech. Note 3869, pp. 419460.
Cheng H. K. 1993 Perspectives on hypersonic viscous flow research. Ann. Rev. Fluid Mech. 25, 455484.Google Scholar
Drazin P. G. & Reid W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Duck P. W. 1985 Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465498.Google Scholar
Gaster M. 1982 Estimates of the errors incurred in various asymptotic representations of wave packets J. Fluid Mech. 121, 365377.Google Scholar
Gaster M. & Grant I. 1975 An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253269.Google Scholar
Jiang F. 1991 Asymptotic evaluation of three-dimensional wave packets in parallel flows. J. Fluid Mech. 226, 573590.Google Scholar
Kazakov V. A. 1985 Strongly implicit alternately-triangular method for solving problems of asymptotic boundary-layer theory. USSR Comput. Maths Math. Phys. 25, 6873.Google Scholar
Kerimbekov R. M., Ruban A. I. & Walker J. D. A. 1994 Hypersonic boundary-layer separation on a cold wall. J. Fluid Mech. 274, 163195.Google Scholar
Kozlova I. G. & Mikhailov V. V. 1970 On strong viscous interaction on delta and swept wings. Izv. Akad. Nauk SSSR, Mech. Zhid. i Gaza, No. 6, 9499; see also Fluid Dyn. 5, 982–986.Google Scholar
Lewis J. E., Kubota T. & Lees L. 1968 Experimental investigation of supersonic laminar, two-dimensional boundary-layer separation in a compression corner with and without cooling. AIAA J. 6, 714.Google Scholar
Liepmann H. W. 1946 The interaction between boundary layers and shock waves in transonic flow. J. Aeronaut. Sci. 13, 623637.Google Scholar
Lighthill M. J. 1953 On boundary layers and upstream influence. II. Supersonic flows without separation. Proc. R. SOC. Lond. A 217, 478507.Google Scholar
Messiter A. F. 1979 Boundary layer separation, Proc. 8th US Natl Congr. Appl. Mech., pp. 157179. Western Periodicals, North Hollywood, California.
Messiter A. F. 1983 Boundary-layer interaction theory. Trans. ASME E: J. Appl. Mech. 50, 11041113.Google Scholar
Mikhailov V. V., Neiland V. YA. & Sychev V. V. 1971 The theory of viscous hypersonic flow, Ann. Rev. Fluid Mech. 3. 371396.Google Scholar
Neiland, V. Ya 1969 On the theory of laminar boundary-layer separation in supersonic flow. Izv. Akad. Nauk SSSR, Mech. Zhid. i Gaza, No. 4, 5357; see also Fluid Dyn. 4, 33–35.Google Scholar
Neiland, V. Ya 1970 Upstream propogation of perturbations in a hypersonic flow interacting with a boundary layer. Izv. Akad. Nauk SSSR, Mech. Zhid. i Gaza No. 4, 40–49; see also Fl. Dyn. 5, 559566.Google Scholar
Neiland, V. Ya 1973 Peculiarities of boundary-layer separation on a cooled body and its interaction with a hypersonic flow. Izv. Akad. Nauk SSSR, Mech. Zhid. i Gaza No. 6, 99–109; see also Fluid Dyn. 8, 931–939.Google Scholar
Neiland, V. Ya 1974 Asymptotic problems of the viscous supersonic flow theory. TsAGI Trans., No. 1529.
Neiland, V. Ya 1981 Asymptotic theory for separation and interaction of a boundary layer with supersonic gas flow. Adv. Mech. 4, No. 2, 362.Google Scholar
Peridier V. J., Smith F. T. & Walker J. D. A. 1991 Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re → ∞. J. Fluid Mech. 232, 99132.Google Scholar
Rizzetta D. P., Burggraf O. R. & Jenson R. 1978 Triple-deck solutions for viscous supersonic and hypersonic flow past corners. J. Fluid Mech. 89, 535552.Google Scholar
Ruban A. I. 1978 Numerical solution of the local asymptotic problem of the unsteady separation of a laminar boundary layer in a supersonic flow. USSR Comput. Maths Math. Phys. 18, 175187.Google Scholar
Ruban A. I. 1990 Propagation of wave packets in the boundary layer on a curved surface. Izv. Akad. Nauk SSSR, Mech. Zhid. i Gaza No. 2, 59–68; see also Fluid Dyn. 25, 213–221.Google Scholar
Ryzhov O. S. 1990 The formation of ordered vortex structures from unstable oscillations in the boundary layer USSR Comput. Maths. Math. Phys. 30, 146154.Google Scholar
Ryzhov O. S. & Terent'Ev E. D. 1986 On the transition regime characterizing the start of a vibrator in subsonic boundary layer on a flat plate. Appl. Math. Mech. 50, 974986.Google Scholar
Smith F. T. 1982 On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28 207281.Google Scholar
Smith F. T. 1988 A reversed-flow singularity in interacting boundary layers. Proc. R. SOC. Lond. A 420, 2152.Google Scholar
Smith F. T. & Khorrami A. F. 1991 The interactive breakdown in supersonic ramp flow. J. Fluid Mech. 224, 197215.Google Scholar
Smith F. T., Sykes R. I. & Brighton P. W. M. 1977 A two-dimensional boundary layer encountering a three-dimensional hump. J. Fluid Mech. 83, 163176.Google Scholar
Stewartson K. 1964 Boundary Layers in Compressible Flow. Oxford University Press.
Stewartson K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145239.Google Scholar
Stewartson K. 1981 D'Alembert's Paradox. SIAM Rev. 23, 308343.Google Scholar
Stewartson K. & Williams P. G. 1969 Self-induced separation. Proc. R. SOC. Lond. A 312, 181206.Google Scholar
Terent'Ev E. D. 1987 On the formation of wave packet in the boundary layer on a flat plate. Appl. Math. Mech. 51, 814819.Google Scholar
Turry O. R. & Cowley S. J. 1986 On the stability and the numerical solution of the unsteady interactive boundary-layer separation. J. Fluid Mech. 168, 431456.Google Scholar