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Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

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

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 291 , 25 May 1995 , pp. 57 - 81
Copyright
© 1995 Cambridge University Press

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