WEAKLY NONLINEAR-THEORY OF THE ALTERNATION OF MODES IN A CIRCULAR SHEAR-FLOW

Experimental investigations on the origin and evolution of structure in circular shear flows, made under different conditions by different groups of authors, reveal a number of common regularities. (i) When the difference DELTAOMEGA between the angular velocities of the centre and the periphery is smaller than a certain critical value (DELTAOMEGA)C, the flow is axisymmetric. (ii) When DELTAOMEGA = (DELTAOMEGA)C, a pattern appears consisting of m(C) vortices. (iii) With a subsequent adiabatic growth of DELTAOMEGA (at a certain (DELTAOMEGA)mC+ > (DELTAOMEGA)C), transition to a pattern with m(C) - 1 vortices occurs, but a pattern with m(C) + 1 vortices never arises (although in terms of linear theory the modes m(C) - 1 and m(C) + 1 are, equivalent). Subsequent growth of DELTAOMEGA leads to the transition (m(C) - 1) - (m(C) - 2), etc. (iv) As DELTAOMEGA decreases, a cascade of inverse transitions of the form m - 1 --> m up to m = mc occurs. and the transition m - 1 --> m proceeds at a smaller value of DELTAOMEGA compared with the transition m --> m - 1, i.e. hysteresis occurs. This paper offers a weakly nonlinear theory which makes it possible to describe the change of the order of symmetry of the wave pattern (number of vortices) with a change of DELTAOMEGA and to ascertain conditions under which the above regularities occur. Some particular examples of the calculation of several models of shear flows are given, and it is shown that direct transitions (m --> m - 1) can be described in terms of a weakly nonlinear theory only for flows with a sufficiently large curvature of the shear layer, i.e. when D = L/R = O(1), where L is the width of the shear layer, and R is its radius, and at not too large m (m(C) = 4,5). If D much less than 1, a description of direct transitions requires a strongly nonlinear theory and is beyond the scope of this paper. Inverse transitions (m - 1 --> m, m less-than-or-equal-to m(C)) admit a weakly nonlinear treatment at any D.