Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T01:37:30.099Z Has data issue: false hasContentIssue false
  • English
  • Français

Secondary instability in rotating-disk flow

Published online by Cambridge University Press:  26 April 2006

S. Balachandar ,
C. L. Streett  and
M. R. Malik
S. Balachandar
Affiliation:
High Technology Corporation, PO Box 7262, Hampton, VA 23666. USA Present address: Department of Theoretical and Applied Mechanics. University of Illinois, Urbana.IL 61801, USA.
C. L. Streett
Affiliation:
NASA Langley Research Center, Hampton, VA 23665. USA
M. R. Malik
Affiliation:
High Technology Corporation, PO Box 7262, Hampton, VA 23666. USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Primary instability of the three-dimensional boundary layer on a rotating disk introduces periodic modulation of the mean flow in the form of stationary crossflow vortices. Here we study the stability of this modulated mean flow with respect to secondary disturbances. These secondary disturbances are found to have quite large growth rates compared to primary disturbances. Both fundamental and subharmonic resonance cases are considered and their corresponding results indicate that the growth rate and the frequency of the secondary instability are insensitive to the exact nature of the resonance condition. The threshold primary stationary crossflow vortex amplitude for secondary instability found in this three-dimensional incompressible boundary layer is significantly larger than that for a two-dimensional boundary layer which is subjected to Tollmien–Schlichting instability. The secondary instability results in a pair of travelling counter-rotating vortices, tilted up and oriented at an angle to the primary stationary crossflow vortices. The computed velocity signals and flow visualization, evaluated based on this secondary disturbance structure, are compared with experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 323 - 347
Copyright
© 1992 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

Balakumar, P. & Malik, M. R. 1990 Travelling disturbances in rotating disk flow. Theo. Comput. Fluid Dyn. 2, 125–137.Google Scholar
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.Google Scholar
Dallmann, U. & Bieler, H. 1987 Analysis and simplified prediction of primary instability of three-dimensional boundary-layer flows. AIAA-87-1337.
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576.Google Scholar
Faller, A. J. & Kaylor, R. E. 1966 Investigations of stability and transition in rotating boundary layers. In Dynamics of Fluids and Plasmas (ed. S. I. Pai). Academic.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A248, 155–199.Google Scholar
Hall, P. 1986 An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disk. Proc. R. Soc. Lond. A406, 93–106.Google Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.Google Scholar
Herbert, T. 1984 Analysis of the sub-harmonic route to transition in boundary layers. AIAA-84-0009.
Herbert, T. 1985 Three-dimensional phenomenon in the transitional flat plate boundary layers. AIAA-85-0489.
Herbert, T. 1988 Secondary instability in boundary layers. Ann. Rev. Fluid. Mech. 20, 487–526.Google Scholar
Herbert, T., Bertolotti, F. P. & Santos, G. R. 1985 Floquet analysis of secondary instability in shear flows. In Proc. ICASE/NASA Workshop on Stability of Time-dependent and Spatially Varying Flows, Aug. 19-20, Hampton, VA (ed. D. L. Dwoyer & M. Y. Hussaini). Springer.
Kachanov, Y. S. & Levchenko, V. Y. 1984 The resonant interaction of disturbances at laminar—turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.Google Scholar
Kármán, The. Von 1940 Uber Laminare und Turbulente Reibung. Z. Angew. Math. Mech. 20, 241–253.Google Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1–34.Google Scholar
Kleiser, L. & Laurien, E. 1985 Three-dimensional numerical simulation of laminar—turbulent transition and its control by periodic disturbances. In Proc. IUTAM Symp. Novosibirsk, 1984 (ed. V. V. Kozlov). Springer.
Kobayashi, R., Kohama, Y. & Takamadate, Ch. 1980 Spiral vortices in boundary layer transition regime on a rotating disk. Acta Mechanica 35, 71–82.Google Scholar
Kohama, Y. 1984 Study on boundary layer transition of a rotating disk. Acta Mechanica 50, 193–199.Google Scholar
Kohama, Y. 1987 Cross-flow instability in rotating disk boundary layer. AIAA-87-1340.
Kohama, Y., Saric, W. S. & Hoos, J. A. 1991 A high-frequency, secondary instability of crossflow vortices that leads to transition In Proc. R. Aeronaut. Soc. Conf. on Boundary-Layer Transition and Control. Cambridge University.
Mack, L. M. 1984 Boundary-layer linear stability theory In AGARD Special Course on Stability and Transition, March 26–30 Rhode-st-Genese, Belgium.
Mack, L. M. 1985 The wave pattern produced by a point source on a rotating disk. AIAA-85-0490.
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating disk flow. J. Fluid Mech. 164, 275–287.Google Scholar
Malik, M. R., Wilkinson, S. P. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA-81-4225.
Nishioka, M., Asai, M. & Iida, S. 1981 Wall phenomenon in the final stage of transition to turbulence. In Transition and Turbulence (ed. R. E. Meyer), pp. 113–126. Academic.
Poll, D. I. A. 1985 Some observations of the transition process on the windward face of a long yawed cylinder. J. Fluid Mech. 150, 329–356.Google Scholar
Singer, B. A., Meyer, F. & Kleiser, L. 1989 Nonlinear developments of crossflow vortices. In Proc. ICASE/NASA Workshop on Stability and Transition, Hampton, VA (ed. M. Y. Hussaini & R. G. Voigt). Springer.
Streett, C. L & Hussaini, M. Y. 1991 A numerical simulation of the appearance of chaos in finite-length Taylor—Couette flow. Intl. J. Appl. Numer. Maths Special Issue on Transition to Turbulence 7, 41–72.Google Scholar
Wilkinson, S. P. & Malik, M. R. 1983 Stability experiments in the flow over a rotating disk. AIAA-83-1760