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The vibrating ribbon problem revisited

Published online by Cambridge University Press:  26 April 2006

David E. Ashpis  and
Eli Reshotko
David E. Ashpis
Affiliation:
NASA Lewis Research Center, Cleveland, OH 44135. USA
Eli Reshotko
Affiliation:
Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106, USA
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Abstract

A revised formal solution of the vibrating ribbon problem of hydrodynamic stability is presented. The initial formulation of Gaster (1965) is modified by application of the Briggs method and a careful treatment of the complex double Fourier transform inversions. Expressions are obtained in a natural way for the discrete spectrum as well as for the four branches of the continuous spectra. These correspond to discrete and branch-cut singularities in the complex wavenumber plane. The solutions from the continuous spectra decay both upstream and downstream of the ribbon, with the decay in the upstream direction being much more rapid than that in the downstream direction. Comments and clarification of related prior work are made.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 213 , April 1990 , pp. 531 - 547
Copyright
© 1990 Cambridge University Press

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References

Aldoss, T. K. 1982 Initial value study of effects of distributed roughness on boundary layer transition. Ph.D. dissertation, Case Western Reserve University
Ashpis, D. & Reshotko, E. 1985 The vibrating ribbon problem – revisited. Bull. Am. Phys. Soc. 30, 1708.Google Scholar
Ashpis, D. & Reshotko, E. 1986 On the application of Fourier transforms to the linear stability analysis of boundary layers. Case Western Reserve University Rep. FTAS/TR-86-187. (Also Ashpis, D. E., Ph.D. dissertation, Case Western Reserve University. 1986).Google Scholar
Bers, A. 1983 Space-time evolution of plasma instabilities – absolute and convective. In Basic Plasma Physics I (ed. A. A. Galeev & R. N. Sudan), pp. 451–517, . North-Holland.
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.
Case, K. M. 1960 Stability of inviscid plane Couette flow. Phys. Fluids 3, 143–158.Google Scholar
Case, K. M. 1961 Hydrodynamic stability and the inviscid limit. J. Fluid Mech. 10, 420–429.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.Google Scholar
Gaster, M. & Jordinson, R. 1975 On the eigenvalues of the Orr–Sommerfeld equation. J. Fluid Mech. 72, 121–133.Google Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.Google Scholar
Gustavsson, L. H. 1979 Initial-value problem for boundary layer flows. Phys. Fluids 22, 1602–1605.Google Scholar
Huerre, P. 1987 Spatio-temporal instabilities in closed and open flows. In Proc. Intl Workshop on Instabilities and Non-Equilibrium Structures, Valparaiso, Chile, 16–21 December 1985 (ed. E. Tirapegui & D. Villanroel), pp. 141–177. Reidel.
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Hultgren, L. S. & Aggarwal, A. K. 1987 Absolute instability of the Gaussian wake profile. Phys. Fluids 30, 3383–3387.Google Scholar
Koch, W. 1986 Direct resonances in Orr–Sommerfeld problems. Acta Mech. 58, 11–29.Google Scholar
Leib, S. J. & Goldstein, M. E. 1986 The generation of capillary instabilities in a liquid jet. J. Fluid Mech. 168, 479–500.Google Scholar
Lin, S. P. & Lian, Z. W. 1989 Absolute instability of a liquid jet in a gas. Phys. Fluids A 1, 490–493.Google Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.Google Scholar
Miklavčič, M. 1983 Eigenvalues of the Orr–Sommerfeld equation in an unbounded domain. Arch. Rat. Mech. Anal. 83, 221–228.Google Scholar
Miklavčič, M. & Williams, M. 1982 Stability of mean flows over an infinite flat plate. Arch. Rat. Mech. Anal. 80, 57–69.Google Scholar
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers. Phys. Fluids 31, 999–1006.Google Scholar
Monkewitz, P. A. & Sohn, K. D. 1986 Absolute instability in hot jets and their control. AIAA Paper 86–1882.Google Scholar
Murdock, J. W. & Stewartson, K. 1977 Spectra of the Orr–Sommerfeld equation. Phys. Fluids 20, 1404–1411.Google Scholar
Pierrehumbert, R. T. 1986 Spatially amplifying modes of the Charney baroclinic instability problem. J. Fluid Mech. 170, 293–317.Google Scholar
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.Google Scholar
Salwen, H., Kelly, K. A. & Grosch, C. E. 1980 Completeness of spatial eigenfunctions for the boundary layer. Bull. Am. Phys. Soc. 25, 1085.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary layer oscillations and transition on a flat plate. J. Aero. Sci. 14, 69–76. (Also NACA Rep. 909, 1948 .)Google Scholar
Tam, C. K. W. 1971 Directional acoustic radiation from a supersonic jet generated by shear layer instability. J. Fluid Mech. 46, 757–768.Google Scholar
Tam, C. K. W. 1978 Excitation of instability waves in a two-dimensional shear layer by sound. J. Fluid Mech. 89, 357–371.Google Scholar
Tsuge, S. & Rogler, H. L. 1983 The two-dimensional, viscous boundary-value problem for fluctuations in boundary layers. AIAA Paper 83–0044.Google Scholar
Tumin, A. M. & Fedorov, A. V. 1983 Spatial growth of disturbances in a compressible boundary layer. Prikl. Mat. Tech. Fiz. 4, 110–118 (transl. J. Appl. Mech. Tech. Phys., 548–554).Google Scholar
Tumin, A. M. & Fedorov, A. V. 1984 Excitation of instability waves by a localized vibrator in a boundary layer. Prikl. Mat. Tech. Fiz. 6, 65–72 (transl. NASA TM-77873, 1985).Google Scholar
Yang, X. & Zebib, A. 1989 Absolute and convective instability of a cylinder wake. Phys. Fluids A 1, 689–696.Google Scholar