Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T12:48:30.891Z Has data issue: false hasContentIssue false
  • English
  • Français

On the first-mode instability in subsonic, supersonic or hypersonic boundary layers

Published online by Cambridge University Press:  21 April 2006

F. T. Smith
F. T. Smith
Affiliation:
Mathematics Department, University College, Gower Street, London, WC1E 6BT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The work, addressing subsonic, supersonic and hypersonic boundary-layer instability, is motivated by the need for more understanding of compressible transition at high global Reynolds numbers Re. In the supersonic case, the so-called ‘first modes’ of instability found/suggested by previous Orr-Sommerfeld computations can be identified as triple-deck oblique ones, directed outside the local wave-Mach-cone directions. Less oblique instability modes inside are not of Orr-Sommerfeld form since they are substantially affected by non-parallel flow effects. The maximum linear growth rates are determined for a range of supersonic and subsonic free-stream Mach numbers M∞, and comparisons are made with previous computations, showing fairly good agreement at moderate Mach numbers. A second mound of unstable wavenumbers and frequencies is also evident. In addition, the nonlinear version is set up and emphasized (with attention drawn to a recent paper by the author (1988a) showing the possibility of nonlinear break-up), and certain extremes are examined including those of transonic and hypersonic boundary layers. In the hypersonic limit a new regime is found (for many conditions, including the insulated plate), namely $M_{\infty \sim Re^{\frac{1}{10}}$, in which non-parallel-flow effects enter to control the main disturbances, and it is concluded that the restriction $M_{\infty \ll Re^{\frac{1}{10}}$ applies to the Orr-Sommerfeld approach. This is a very severe restriction in practice.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 198 , January 1989 , pp. 127 - 153
Copyright
1989 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brotherton-Ratcliffe, R. V. & Smith, F. T.1987 Complete breakdown of an unsteady interactive boundary layer (over a surface distortion or in a liquid layer). Mathematika 34, 86–100.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability. Cambridge University Press.
Duck, P. W.: 1987 Presentation at Euromech Colloquium. Exeter, UK.
Kendall, J. M.: 1975 Wind-tunnel experiments relating to supersonic and hypersonic boundary layer transition, AIAA J. 13, 290–299.Google Scholar
Laufer, J. & Vrebalovich, T., 1960 Stability and transition of a laminar boundary layer on an insulated flat plate. J. Fluid Mech. 9, 257–299.Google Scholar
Lysenko, V. I. & Maslov, A. A., 1984 The effect of cooling on supersonic boundary-layer stability. J. Fluid Mech. 147, 39–52.Google Scholar
Mack, L. M.: 1975 Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13, 278–289.Google Scholar
Mack, L. M.: 1984 Boundary-layer linear stability theory. AGARD Rep. 709.Google Scholar
Mack, L. M.: 1986 In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini). Springer.
Malik, M. R.: 1982 COSAL - a black-box compressible stability analysis code for transition prediction in three-dimensional boundary layers. NASA CR-165925.Google Scholar
Malik, M. R.: 1987 Prediction and control of transition in hypersonic boundary layers. AIAA paper no. 87–1414.Google Scholar
Maslov, A. A.: 1987 Presentation at Euromech Colloquium, Exeter, UK.
Miles, J. W.: 1960 The hydrodynamic stability of a thin film of liquid in uniform shearing motion. J. Fluid Mech. 8, 593–610.Google Scholar
Neiland, V. Ya., 1970 Upstream propagation of disturbances in hypersonic boundary layer interactions. Izv. Akad. Nauk, SSSR, Mekh. Zhid.i Gaza, 4, 40–49.Google Scholar
Pate, S. R. & Schueler, C. J., 1969 Radiated aerodynamic noise effects on boundary-layer transition in supersonic and hypersonic wind tunnels: AIAA J. 7, 450–457.
Ryzhov, O. S.: 1987 Presentation at Euromech Colloquium, Exeter, UK.
Smith, F. T.: 1979a On the nonparallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T.: 1979b Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 366, 91–109; 368, 573–589.Google Scholar
Smith, F. T.: 1985 Two-dimensional disturbance travel, growth and spreading in boundary layers. UTRC Rep. 85–36 (and see J. Fluid Mech.169, 1986, 353–377).Google Scholar
Smith, F. T.: 1986 The strong nonlinear growth of three-dimensional disturbances in boundary layers. Utd. Tech. Res. Cent., E. Hartford, Conn., Rep. UT 86-10.Google Scholar
Smith, F. T.: 1988a Finite-time break-up can occur in any unsteady interactive boundary layer. Mathematika (submitted).Google Scholar
Smith, F. T.: 1988b A reversed-flow singularity in interacting boundary layers. Proc. R. Soc. Lond. A (in press).Google Scholar
Smith, F. T. & Stewart, P. A., 1987 The resonant-triad nonlinear interaction in boundary-layer transition. J. Fluid Mech. 179, 227–252.Google Scholar
Stewartson, K.: 1964 Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.
Stewartson, K.: 1974 Multi-structured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.Google Scholar
Stewartson, K. & Williams, P. G., 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar