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Barotropic instability of the Bickley jet

Published online by Cambridge University Press:  26 April 2006

S. A. Maslowe
S. A. Maslowe
Affiliation:
Department of Mathematics, McGill University, Montreal, P.Q. H3A 2K6, Canada
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Abstract

The linear stability of the zonal shear flow $\overline{u} = - \sec h^2 y$ is investigated in the framework of the beta-plane approximation. This retrograde jet is known to be more unstable than its eastward-propagating counterpart and has some surprising characteristics. First, this is a rare example of a flow in which barotropically unstable modes occur that do not have a critical point. Secondly, singular neutral modes exist in which the critical point occurs at the centre of the jet, where $\overline{u}^{\prime}_{\rm c}=0 $. It is shown in this paper that such singular modes form part of the stability boundary both for the varicose mode and also for the radiating sinuous mode.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 417 - 426
Copyright
© 1991 Cambridge University Press

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References

Ahmadi, A. R. & Widnall, S. E. 1985 Unsteady lifting-line theory as a singular perturbation problem. J. Fluid Mech. 153, 59–81.Google Scholar
Ashley, H. & Landhal, L. 1965 Aerodynamics of Wings and Bodies, pp. 88–98. Addison-Wesley.
Cheng, H. K. 1976 On lifting-line theory in unsteady aerodynamics. USCAE. Dept. of Aero. Engng Rep. 133.Google Scholar
Cheng, H. K. 1978 Lifting-line theory of oblique wings. AIAA J. 16, 1211–1213.Google Scholar
Cheng, H. K. & Murillo, L. E. 1984 Lunate-tail swimming propulsion as a problem of curved lifting-line in unsteady flow. J. Fluid Mech. 143, 327–350.Google Scholar
Guermond, J.-L. 1987 A new systematic formula for the asymptotic expansion of singular integrals. Z. Angew. Math. Phys. 38, 717–729.Google Scholar
Guermond, J.-L. 1988 Une nouvelle approche des développements asymptotiques d'intégrales. CR Acad. Sci. Paris I 307, 881–886.Google Scholar
Guermond, J.-L. 1990 A generalized lifting-line theory for curved and swept wings. J. Fluid Mech. 211, 497–513 (and Corrigendum 229 (1991), 695).Google Scholar
Guiraud, J. P. & Slama, G. 1981 Sur la théorie asymptotique de la ligne portante en régime incompressible oscillatoire. La Rech. Aérospat. 1, 1–6.Google Scholar
Hadamard, J. 1932 Lectures on Cauchy's Problem in Linear Differential Equation, pp. 133–153. Dover.
Hess, J. L. 1972 Calculation of potential flow about arbitrary three-dimensional lifting bodies. Douglas Aircraft Co. Rep. MDC J5679–01.Google Scholar
James, E. C. 1975 Lifting-line theory for an unsteady wing as a singular perturbation problem. J. Fluid Mech. 70, 753–771.Google Scholar
Kida, T. & Miyai, Y. 1978 An alternative treatment of lifting-line theory as a perturbation problem. Z. Angew. Math. Phys. 29, 591–607.Google Scholar
Lavoine, J. 1959 Calcul Symbolique, Distributions et Pseudo-fonctions, pp. 15–21. Paris: CNRS.
Lavoine, J. 1963 Transformation de Fourier des Pseudo-fonctions, pp. 13–45. Paris: CNRS.
Prandtl, L. 1921 Application of modern hydrodynamics to aeronautics. NACA Rep. 116.Google Scholar
Sclavounos, P. D. 1987 An unsteady lifting-line theory. J. Engng Maths 21, 201–226.Google Scholar
Sears, W. R. 1941 Some aspect of non-stationary airfoil theory and its practical application. J. Aero. Sci. 8, 104–108.Google Scholar
Sellier, A. 1990 Une théorie unifiée de la ligne portante. Thesis, University Paris VI.
Theodorsen, T. 1935 General theory of aerodynamics instability and the mechanism of flutter. NACA TR 496.Google Scholar
Van Dyke, M. D. 1964 Lifting-line theory as a singular perturbation problem. Appl. Math. Mech. 28, 90–101.Google Scholar
Van Holten, T. 1976 Some notes on unsteady lifting-line theory. J. Fluid Mech. 77, 561–579.Google Scholar