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A note on an algebraic instability of inviscid parallel shear flows

Published online by Cambridge University Press:  19 April 2006

M. T. Landahl
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. and Department of Mechanics, The Royal Institute of Technology, Stockholm, Sweden

Abstract

It is shown that all parallel inviscid shear flows of constant density are unstable to a wide class of initial infinitesimal three-dimensional disturbances in the sense that, according to linear theory, the kinetic energy of the disturbance will grow at least as fast as linearly in time. This can occur even when the disturbance velocities are bounded, because the streamwise length of the disturbed region grows linearly with time. This finding may have implications for the observed tendency of turbulent shear flows to develop a longitudinal streaky structure.

Type
Research Article
Copyright
© 1980 Cambridge University Press

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References

Arnol'd, V. I. 1965 Sov. Math. 6, 773.
Case, K. M. 1960 Phys. Fluids 3, 143.
Chimonas, G. 1979 J. Fluid Mech. 90, 1.
Drazin, P. G. & Howard, L. N. 1966 Adv. Appl. Math. 9, 1.
Eliassen, A., Høiland, J. & Riis, E. 1953 Inst. for Weather & Climate Res. Publ. no. 1, Oslo.
Ellingsen, T. & Palm, E. 1975 Phys. Fluids 18, 487.
Fjørtoft, R. 1950 Geophys. Publ. 17, no. 1, Oslo.
Gustavsson, L. H. 1978 Dept. of Mech. Roy. Inst. Tech. Rep. TRITA-MEK-78-02, Stockholm.
Heisenberg, W. 1924 Ann. Phys. Leipzig 24, 577.
Howard, L. N. 1961 J. Fluid Mech. 10, 509.
Landahl, M. T. 1977 Phys. Fluids Suppl. 20, 55.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Orr, W. M. F. 1907 Proc. Roy. Irish Acad. A 17, 124.
Rayleigh, Lord 1880 Proc. Lond. Math. Soc. 9, 57.
Stuart, J. T. 1965 N.P.L. Aero Rep. 1147; AGARD Rep. 514.
Titschmarsh, E. C. 1939 The theory of Functions, 2d edn. Oxford University Press.
Wilke, L. H. 1967 J. Math. & Phys. 46, 151.