3-DIMENSIONAL DISTURBANCES TO A MIXING LAYER IN THE NONLINEAR CRITICAL-LAYER REGIME

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 k(z) much less than the streamwise one k(x)) in a mixing layer upsilon(x) = 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 similar to x(2/3) is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of k(z) 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 k(z) = O(k(x)) such a transition cannot be realized in the framework of weakly nonlinear theory.