Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-07T09:52:08.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear resonant instability in boundary layers

Published online by Cambridge University Press:  29 March 2006

Alex D. D. Craik
Alex D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, St Andrews, Fife, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

An investigation is made of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer. The triads considered are those comprising a two-dimensional wave and two oblique waves propagating at equal and opposite angles to the flow direction and such that all three waves have the same phase velocity in the downstream direction. For such a resonant triad remarkably powerful wave interations take place, which may cause a continuous and rapid transfer of energy from the primary shear flow to the disturbance. It appears that the oblique waves can grow particularly rapidly and it is suggested that such preferential growth may be responsible for the rapid development of three-dimensionality in unstable boundary layers. The non-linear energy transfer primarily takes place in the vicinity of the critical layer where the downstream propagation velocity of the waves equals the velocity of the primary flow.

The theoretical analysis is initially carried out for a general primary velocity profile; then, in order to demonstrate the essential features of the results, precise interaction equations are derived for a particular profile consisting of a layer of constant shear bounded by a uniform flow. Some exact solutions of the general interaction equations are presented, one of which has the property that the wave amplitudes become indefinitely large at a finite time. The possible relevance of the present theoretical model to the experiments of Klebanoff, Tidstrom & Sargent (1962) is examined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 50 , Issue 2 , 29 November 1971 , pp. 393 - 413
Copyright
© 1971 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

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Washington DC: National Bureau of Standards.
Benney, D. J. 1961 A non-linear theory for oscillations in a parallel flow. J. Fluid Mech. 10, 209.Google Scholar
Benney, D. J. 1964 Finite amplitude effects in an unstable laminar boundary layer. Phys. Fluids, 7, 319.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids, 4, 656.Google Scholar
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Craik, A. D. D. 1968 Resonant gravity-wave interactions in a shear flow. J. Fluid Mech. 34, 531.Google Scholar
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1971 A non-linear instability burst in plane parallel flow. Submitted to J. Fluid Mech.Google Scholar
Ince, E. L. 1956 Ordinary Differential Equations. Dover.
Kaplan, R. E. 1964 The stability of laminar incompressible boundary layers in the presence of compliant boundaries. M.I.T. Aero-Elastic and Structures Res. Lab. ASRL-TR 116-1.
Kelly, R. E. 1968 On the resoilant interaction of neutral disturbances in inviscid shear flows. J. Fluid Mech. 31, 789.Google Scholar
Klebanoff, P. S. & Tidstrom, K. D. 1959 Evolution of amplified waves leading to transition in a boundary layer with zero pressure gradient. N.A.S.A. Tech. Note no. D-195.Google Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1.Google Scholar
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Norair Rep. NOR-59-383, Hawthorne, California.Google Scholar
Raetz, G. S. 1964 Current status of resonance theory of transition. Norair Rep. NOR-64-111, Hawthorne, California.Google Scholar
Reid, W. H. 1965 The stability of parallel flows. In Basic Developments in Fluid Mechanics, vol. I (ed. M. Holt). Academic.
Ross, J. A., Barnes, F. H., Burns, J. G. & Ross, M. A. S. 1970 The flat plate boundary layer, Part 3. Comparison of theory with experiment. J. Fluid Mech. 43, 819.Google Scholar
Shen, S. F. 1954 Calculated amplified oscillations in plane Poiseuille and Blasius flows. J. Aero. Sci. 21, 62.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. Roy. Soc. A 309, 551.Google Scholar
Tani, I. 1969 Boundary-layer transition. In Annual Review of Fluid Mechanics. vol. I, pp. 169–196. (ed. W. R. Sears & M. Van Dyke). Annual Reviews Inc.
Stuart, J. T. 1962a On three-dimensional non-linear effects in the stability of parallel flows. Adv. Aero. Sci. 3-4, 121.
Stuart, J. T. 1962b Non-linear effects in hydrodynamic stability. Proc. 10th Internat. Congr. Appl. Mech. (Stresa 1960), pp. 63–97. Elsevier.
Tietjens, O. 1925 Beiträge zur Entstehung der Turbulenz Z. angew. Math. Mech. 5, 200.Google Scholar
Watson, J. 1960 Three-dimensional disturbances in flow between parallel plates. Proc. Roy. Soc. A 254, 562.Google Scholar
Wazzan, A. R., Okamura, T. & Smith, A. M. O. 1966 Spatial stability study of some Falkner-Skan similarity profiles. Proc. Fifth U.S. Nat. Cong. Appl. Mech. p. 836. ASME University of Minnesota.