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Amplitude propagation in slowly varying trains of shear-flow instability waves

Published online by Cambridge University Press:  21 April 2006

J. M. Russell
J. M. Russell
Affiliation:
School of Aerospace, Mechanical, and Nuclear Engineering, The University of Oklahoma, Norman, OK 73019, USA
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Abstract

The analog of Whitham's law of conservation of wave action density is derived in the case of Rayleigh instability waves. The analysis allows for wave propagation in two space dimensions, non-unidirectionality of the background flow velocity profiles and weak horizontal nonuniformity and unsteadiness of those profiles. The small disturbance equations of motion in the Eulerian flow description are subject to a change of dependent variable in which the new variable represents the pressure-driven part of a disturbance material coordinate function as a function of the Cartesian spatial coordinates and time. Several variational principles expressing the physics of the small disturbance equations of motion are presented in terms of this new variable. A law of conservation of ‘bilinear wave action density’ is derived by a method intermediate between those of Jimenez and Whitham (1976) and Hayes (1970a). The distinction between the observed square amplitude of an amplified wavetrain and the wave action density is discussed. Three types of algebraic focusing are discussed, the first being the far-field ‘caustics’, the second being near-field ‘movable singularities’, and the third being a focusing mechanism due to Landahl (1972) which we here derive under somewhat weaker hypotheses.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 170 , September 1986 , pp. 21 - 51
Copyright
© 1986 Cambridge University Press

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References

Chin W. C.1980 AIAA J. 18, 149.
Drazin, P. G. & Howard L. N.1966 Adv. Appl. Mech. 9, 1.
Eckart C.1963 Phys. Fluids 6, 1037.
Goldstein H.1980 Classical Mechanics. Addison-Wesley.
Hayes W. D.1970a Proc. R. Soc. Lond. A 320, 187.
Hayes W. D.1970b Proc. R. Soc. Lond. A 320, 209.
Hille E.1969 Lectures on Ordinary Differential Equations. Addison-Wesley.
Itoh N.1980 In Proc. IUTAM Symp. on Laminar—Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 86–95. Springer.
Itoh N.1981 Proc. R. Soc. Lond. A 375, 565.
Jimenez, J. & Whitham G. B.1976 Proc. R. Soc. Lond. A 349, 277.
Klebanoff P. S., Tidstrom, K. D. & Sargent L. M.1962 J. Fluid Mech. 12, 1.
Landahl M. T.1972 J. Fluid Mech. 56, 775.
Landahl M. T.1982 Phys. Fluids 25, 1512.
Lighthill M. J.1978 Waves in Fluids. Cambridge University Press.
Nayfeh A. H.1980 In Proc. IUTAM Symp. on Laminar—Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 201–217. Springer.
Seliger, R. L. & Whitham G. B.1968 Proc. R. Soc. Lond. A 305, 1.
Whitham G. B.1974 Linear and Nonlinear Waves. Wiley-Interscience.