Skip to main content Accessibility help
×
Home

Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid

  • I. G. Shukhman (a1)

Abstract

A model has been constructed for a mixing layer of a rotating fluid with a large Reynolds number which is an analogue of a mixing-layer model for a plane flow widely used in the literature. The angular velocity profile in such a model has the form: \[ \Omega(r) = {\textstyle\frac{1}{2}}(\Omega_1 + \Omega_2)-{\textstyle\frac{1}{2}}(\Omega_1 - \Omega_2)\tan h\left(\frac{1}{D}\ln\frac{r}{R}\right), \] where r is the distance from the rotation axis; and R, Ω1,2, and D are the model's parameters. The model permits a relatively simple analytical study of the stability for two-dimensional disturbances. It is shown that the stability is defined by the ‘shear-width’ parameter D, namely the model is unstable when D < Dcrit = ½. In a weakly supercritical flow (|DDcrit| [Lt ] 1), one mode with azimuthal number m = 2 develops. In this case two vortices are produced in the vicinity of a critical layer (CL), i.e. a radius where the wave's azimuthal velocity Ωp coincides with the rotation velocity Ω(r). A study is made of their nonlinear evolution corresponding to different CL regimes: viscous, nonlinear, and unsteady. It is found that the instability saturates at a low enough level and the equilibrium amplitude depends on the degree of supercriticality ΔD = |DDcrit|, but the character of this dependence is different in different regions of the supercriticality parameter ΔD.

It is shown that, despite the specific form of the velocity profile in the model under consideration, results concerning the critical-layer dynamics have a high degree of universality. In particular, it becomes possible to formulate the criterion that the instability will be saturated at a low level for an arbitrary weakly supercritical flow.

Copyright

References

Hide All
Benney, D. G. & Maslowe, S. A., 1975 Stud. Appl. Maths 54, 181205.
Churilov, S. M.: 1988 Geophys. Astrophys. Fluid Dyn. (submitted).
Churilov, S. M. & Shukhman, I. G., 1987a J. Fluid Mech. 180, 120.
Churilov, S. M. & Shukhman, I. G., 1987b Geophys. Astrophys. Fluid Dyn. 38, 145175.
Churilov, S. M. & Shukhman, I. G., 1987c Proc. R. Soc. Lond. A 409, 351367.
Fried, B. D., Liu, C. S., Sagdeev, R. Z. & Means, R. V., 1970 Bull. Am. Phys. Soc. 15, 1421.
Galeev, A. A. & Sagdeev, R. Z., 1973 Voprosy Teorii Plazmy (Plasma Theory Problems) (ed. M. A. Leontovich), vol. 7, pp. 3145. Moscow: Atomizdat (in Russian; English translation is available).
Haberman, R.: 1972 Stud. Appl. Maths 51, 139161.
Haberman, R.: 1976 SIAM J. Math. Anal. 7, 7081.
Huerre, P.: 1980 Phil. Trans. R. Soc. Lond. A 293, 643675.
Huerre, P.: 1987 Proc. R. Soc. Lond. A 409, 369381.
Huerre, P. & Scott, J. F., 1980 Proc. R. Soc. Lond. A 371, 509524.
Kadomtsev, B. B.: 1976 Collective Phenomena in Plasmas, p. 154. Moscow: Nauka (in Russian).
Michalke, A.: 1964 J. Fluid Mech. 19, 543556.
Onishchenko, I. N., Linetsky, A. R., Matsiborko, N. G., Shapiro, V. D. & Shevchenko, V. I., 1970 Pisma v Zh. Eksp. & Teor. Fiz. (Sov. Phys. JETP Lett.) 12, 407411.
Robinson, J. L.: 1974 J. Fluid Mech. 63, 723752.
Schade, H.: 1964 Phys. Fluids 7, 623628.
Stewartson, K.: 1978 Geophys. Astrophys. Fluid Dyn. 9, 185200.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

Nonlinear stability of a weakly supercritical mixing layer in a rotating fluid

  • I. G. Shukhman (a1)

Metrics

Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.