Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T16:19:23.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional stability of an elliptical vortex in a straining field

Published online by Cambridge University Press:  20 April 2006

A. C. Robinson  and
P. G. Saffman
A. C. Robinson
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, California 91125 Present address: Sandia National Laboratories, Albuquerque, New Mexico 87185.
P. G. Saffman
Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, California 91125
Get access Rights & Permissions [Opens in a new window]

Abstract

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 142 , May 1984 , pp. 451 - 466
Copyright
© 1984 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Natl Bur. Stand./Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Blanch, G. 1966 Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, 51–97.Google Scholar
Boyd, J. P. 1978 Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems. Mon. Weather Rev. 106, 1192–1203.Google Scholar
Clemm, D. S. 1969 Algorithm 352: Characteristic values and associated solutions of Mathieu's differential equation. Comm. ACM 12, 399–407.Google Scholar
Garbow, B. S., Boyle, J. M., Dongarra, J. J. & Moler, C. B. 1977 Matrix Eigensystem Routines — EISPACK Guide Extension. Lecture Notes in Computer Science vol. 51. Springer.
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence (ed. Olsen, Goldburg & Rogers), pp. 339–354. Plenum.
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Neu, J. 1983 The dynamics of a columnar vortex in an imposed strain. Math. Sci. Res. Inst. Rep. 022-83, Berkeley, Calif.Google Scholar
Pierrehumbert, R. T. 1980 The structure and stability of large vortices in an inviscid flow. M.I.T. Fluid Dyn. Lab. Rep. 80–1.
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.Google Scholar
Robinson, A. C. & Saffman, P. G. 1982 Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411–427.Google Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.Google Scholar
Sale, A. H. J. 1970 Remark on algorithm 352 [S22]; Characteristic values and associated solutions of Mathieu's differential equation. Comm. ACM 13, 750.Google Scholar
Tsai, C-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.Google Scholar
Ward, R. C. 1981 Balancing the generalized eigenvalue problem. SIAM J. Sci. Stat. Comp. 2, 141–152.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar
Widnall, S. E. & Tsai, C-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287, 273–305.Google Scholar
Wiltse, J. C. & King, M. J. 1958 Values of the Mathieu functions. The Johns Hopkins Univ. Rad. Lab. Tech. Rep. AF-53, Baltimore, Md.