Skip to main content
    • Aa
    • Aa

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

  • A. C. Robinson (a1) (a2) and P. G. Saffman (a1)

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.

Hide All
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, 5197.
Boyd, J. P. 1978 Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems. Mon. Weather Rev. 106, 11921203.
Clemm, D. S. 1969 Algorithm 352: Characteristic values and associated solutions of Mathieu's differential equation. Comm. ACM 12, 399407.
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, 155168.
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 35173520.
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. 339354. 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, 413425.
Neu, J. 1983 The dynamics of a columnar vortex in an imposed strain. Math. Sci. Res. Inst. Rep. 022-83, Berkeley, Calif.
Pierrehumbert, R. T. 1980 The structure and stability of large vortices in an inviscid flow. M.I.T. Fluid Dyn. Lab. Rep. 801.
Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 5982.
Robinson, A. C. & Saffman, P. G. 1982 Three-dimensional stability of vortex arrays. J. Fluid Mech. 125, 411427.
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625639.
Sale, A. H. J. 1970 Remark on algorithm 352 [S22]; Characteristic values and associated solutions of Mathieu's differential equation. Comm. ACM 13, 750.
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, 721733.
Ward, R. C. 1981 Balancing the generalized eigenvalue problem. SIAM J. Sci. Stat. Comp. 2, 141152.
Widnall, S. E., Bliss, D. B. & Tsai, C-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 3547.
Widnall, S. E. & Tsai, C-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287, 273305.
Wiltse, J. C. & King, M. J. 1958 Values of the Mathieu functions. The Johns Hopkins Univ. Rad. Lab. Tech. Rep. AF-53, Baltimore, Md.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 73 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd October 2017. This data will be updated every 24 hours.