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Nonlinear spatial evolution of helical disturbances to an axial jet

Published online by Cambridge University Press:  26 April 2006

S. M. Churilov  and
I. G. Shukhman
S. M. Churilov
Affiliation:
Institute of Solar-Terrestrial Physics, Irkutsk, 664033, PO Box 4026, Russia
I. G. Shukhman
Affiliation:
Institute of Solar-Terrestrial Physics, Irkutsk, 664033, PO Box 4026, Russia
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Abstract

We investigate the weakly nonlinear spatial evolution of helical disturbances of an axisymmetrical jet which are the analogue of three-dimensional disturbances, such as a single oblique wave (the wave vector is directed at an angle to the main flow velocity) in plane-parallel flows. It is shown that when a supercriticality is large enough, the perturbation amplitude A grows in the streamwise direction (along z) explosively: A ∼ (z0 — z)−5/2, though more slowly than in the case of essentially three-dimensional disturbances in the form of a pair of oblique waves (A ∼ (z0 — z)−3; Goldstein & Choi 1989). The nonlinearity needed for such a growth, is due equally to the cylindricity of shear layer and to the spatial character of the evolution (in the temporal problem the ‘evolution’ contribution is absent). At a smaller supercriticality, the evolution equation has a non-local (integral in z) nonlinearity, unusual for the regime of a viscous critical layer. Scenarios of disturbance development for different levels of supercriticality are studied, with proper account taken of viscous broadening of the flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 281 , 25 December 1994 , pp. 371 - 402
Copyright
© 1994 Cambridge University Press

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References

Churilov, S. M. & Shukhman, I. G. 1987 The nonlinear development of disturbances in a zonal shear flow. Geophys. Astrophys. Fluid Dyn. 38, 145–175.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1988 Nonlinear stability of stratified shear flow in the regime with an unsteady critical layer. J. Fluid Mech. 194, 187–216.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1993 Critical layer and nonlinear evolution of disturbances in weakly supercritical shear flows. Preprint ISTP No 4-93, Irkutsk. Also Izv. RAN. Fiz. Atmos. Okeana (in press), and XVIIIth Intl Congr. on Theor. and Appl. Mech. Haifa, Israel, 1992, Abstracts, pp. 39–40.
Cohen, J. & Wygnanski, I. 1987a The evolution of instabilities in the axisymmetric jet. Part 1. The linear growth of disturbances near the nozzle. J. Fluid Mech. 176, 191–219.Google Scholar
Cohen, J. & Wygnanski, I. 1987b The evolution of instabilities in the axisymmetric jet. Part 2. The flow resulting from the interaction between two waves. J. Fluid Mech. 176, 221–235.Google Scholar
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120.Google Scholar
Goldstein, M. E. & Hultgren, L. S. 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295–330 (referred to herein as GH).Google Scholar
Goldstein, M. E. & Leib, S. J. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–161.Google Scholar
Huerre, P. & Scott, J. F. 1980 Effect of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Land. A 371, 509–524.Google Scholar
Maslowe, S. A. 1973 Finite-amplitude Kelvin–Helmholtz billows. Boundary-Layer Met. 5, 43–52.Google Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521–536.Google Scholar
Reutov, V. P. 1982 An unsteady critical layer and nonlinear stage of instability in a flat Poiseuille flow. Zh. Prikl. Mekh. Tekhn. Fiz. 4, 43–54. (in Russian).Google Scholar
Shukhman, I. G. 1989 Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid. J. Fluid Mech. 200, 425–450. Also, Preprint SibIZMIR No. 30–87, Irkutsk, 1987 (in Russian).Google Scholar
Shukhman, I. G. 1991 Nonlinear evolution of spiral density waves generated by the instability of shear layer in a rotating compressible fluid. J. Fluid Mech. 233, 587–612.Google Scholar
Smith, F. T. & Blennerhassett, P. 1992 Nonlinear interaction of oblique three-dimensional Tollmien-Schlichting waves and longitudinal vortices, in channel flows and boundary layers. Proc. R. Soc. Land. A 436, 585–602.Google Scholar
Wu, X. 1993 Nonlinear temporal–spatial modulation of near planar Rayleigh waves in shear flows: formation of streamwise vortices. J. Fluid Mech. 256, 685–719.Google Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–721.Google Scholar