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Amplitude equations for wave packets in slightly inhomogeneous unstable flows

Published online by Cambridge University Press:  26 April 2006

T. F. Balsa
T. F. Balsa
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
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Abstract

We derive several versions of the (complex) amplitude equation of an inviscid wave packet travelling on a slightly inhomogeneous (and possibly unsteady and viscous) unstable base flow. This is done with complete generality, without any reference to the dimensions of physical and propagation spaces, by using the usual high-frequency ansatz. The final results are extremely simple: volume integrals of a complex wave action density are conserved subject to an appropriate flux and a source term. The latter is expressible in a remarkably concise way in terms of the gradient of the base flow acceleration and vanishes when the base flow is inviscid. The simplicity of our results hinges on a transformation of the dependent variables and on a suitable decomposition of these in cross- and propagation spaces. Our results are also discussed with the help of three different Lagrangian densities and their associated kinematic wave theories which are based on a basic identity due to Hayes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 433 - 455
Copyright
© 1989 Cambridge University Press

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References

Andrews, D. G. & McIntyre, M. E., 1978a An exact theory of nonlinear waves on a Lagrangianmean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E., 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664.Google Scholar
Bernstein, L., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M., 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. A 244, 17–40.Google Scholar
Betchov, R. & Criminale, W. O., 1967 Stability of Parallel Flows. Academic.
Betchov, R. & Szewczyk, A., 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6, 1391–1396.Google Scholar
Bretherton, F. P.: 1968 Propagation in slowly varying waveguides. Proc. R. Soc. Lond. A 302, 555–576.Google Scholar
Chin, W. C.: 1980 Effect of dissipation on dispersion of slowly varying wavetrains. AIAA J. 18, 149–158.Google Scholar
Crighton, D. G. & Gaster, M., 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability. Cambridge University Press.
Gaster, M.: 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347, 271–289.Google Scholar
Gaster, M.: 1981 On transition to turbulence in boundary layers. In Transition and Turbulence (ed. R. Meyer), pp. 95–112. Academic.
Gaster, M. & Grant, I., 1975 An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253–269.Google Scholar
Gaster, M., Kit, E. & Wygnanski, I., 1985 Large-scale structures in a forced turbulent mixing layer. J. Fluid Mech. 150, 23–29.Google Scholar
Hayes, W. D.: 1968 Energy invariant for geometric acoustics in a moving medium. Phys. Fluids 11, 1654–1656.Google Scholar
Hayes, W. D.: 1970 Conservation of action and modal wave action. Proc. R. Soc. Lond. A 320, 187–208.Google Scholar
Itoh, N.: 1980 Linear stability theory for wave packet disturbances in parallel and nearly parallel flows. In Proc. IUTAM Symp. on Laminar-Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 86–95. Springer.
Itoh, N.: 1981 Secondary instability of laminar flows. Proc. R. Soc. Lond. A 375, 565–578.Google Scholar
Landahl, M. T.: 1972 Wave mechanics of breakdown. J. Fluid Mech. 56, 775–802.Google Scholar
Landahl, M. T.: 1982 The application of kinematic wave theory to wave trains and packets with small dissipation. Phys. Fluids 25, 1512–1516.Google Scholar
Lewis, R. M.: 1965 Asymptotic theory of wave-propagation. Arch. Rat. Mech. Anal. 20, 191–250.Google Scholar
Luke, J. C.: 1966 A peturbation method for nonlinear dispersive wave problems. Proc. R. Soc. Lond. A 292, 403–412.Google Scholar
Nayfeh, A. H.: 1980 Three-dimensional stability of growing boundary layers. In Proc. IUTAM Symp. on Laminar-Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 201–217. Springer.
Petesen, R. A. & Samet, M., 1988 On the preferred mode of jet instability. J. Fluid Mech. 194, 153–173.Google Scholar
Russell, J. M.: 1986 Amplitude propagation in slowly varying trains of shear-flow instability waves. J. Fluid Mech. 170, 21–51.Google Scholar
Seliger, R. L. & Whitham, G. B., 1968 Variational principles in continuum mechanics. Proc. R. Soc. Lond. A 305, 1–25.Google Scholar
Stewartson, K.: 1974 Some aspects of nonlinear stability theory. Fluid Dyn. Trans., Polish Acad. Sci. 7, 101–128.Google Scholar
Whitham, G. B.: 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283.Google Scholar
Whitham, G. B.: 1974 Linear and Nonlinear Waves. Wiley.
Wygnanski, I., Champagne, F. & Marasli, B., 1986 On the large-scale structures in twodimensional, small-deficit, turbulent wakes. J. Fluid Mech. 168, 31–71.Google Scholar