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The stability of spatially periodic flows

Published online by Cambridge University Press:  20 April 2006

D. N. Beaumont
D. N. Beaumont
Affiliation:
Mathematics Department, University Walk, Bristol, BS8 1TW
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Abstract

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 108 , July 1981 , pp. 461 - 474
Copyright
© 1981 Cambridge University Press

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References

Drazin, P. G. 1977 Proc. Roy. Soc. A 356, 411–432.
Drazin, P. G. & Howard, L. N. 1966 Adv. Appl. Mech. 9, 1–89.
Gill, A. E. 1974 Geophys. Fluid Dyn. 6, 29–47.
Gotoh, K. 1965 J. Phys. Soc. Japan 28, 164–169.
Green, J. S. A. 1974 J. Fluid Mech. 62, 273–287.
Howard, L. N. 1959 J. Math. Phys. 37, 283–298.
Howard, L. N. 1961 J. Fluid Mech. 10, 509–512.
Jordan, D. W. & Smith, P. 1977 Nonlinear Ordinary Differential Equations. Oxford University Press.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lorenz, E. N. 1972 J. Atmos. Sci. 29, 258–269.
Rayleigh, J. W. S. 1945 The Theory of Sound, vol. 2, cha. 21. Dover.
Squire, H. B. 1933 Proc. Roy. Soc. A 142, 621–628.
Synge, J. L. 1938 Hydrodynamic stability. Semi-centenn. Publ. Amer. Math. Soc. 2, 227–269.Google Scholar
Tatsumi, T. & Gotoh, K. 1960 J. Fluid Mech. 7, 433–441.