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Weakly nonlinear theory of the alternation of modes in a circular shear flow

  • S. M. Churilov (a1) and I. G. Shukhman (a1)


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 ΔΩ between the angular velocities of the centre and the periphery is smaller than a certain critical value (ΔΩ)c, the flow is axisymmetric. (ii) When ΔΩ = (ΔΩ)c, a pattern appears consisting of mc vortices. (iii) With a subsequent adiabatic growth of ΔΩ (at a certain (ΔΩ)mc+ > (ΔΩ)c), transition to a pattern with mc − 1 vortices occurs, but a pattern with mc + 1 vortices never arises (although in terms of linear theory the modes mc − 1 and mc + 1 are equivalent). Subsequent growth of ΔΩ leads to the transition (mc − 1) → (mc − 2), etc. (iv) As ΔΩ 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 ΔΩ compared with the transition mm − 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 ΔΩ 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 (mm −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 DL/R = O(1), where L is the width of the shear layer, and R is its radius, and at not too large m (mc = 4,5). If D [Lt ] 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 [les ] mc) admit a weakly nonlinear treatment at any D.



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Weakly nonlinear theory of the alternation of modes in a circular shear flow

  • S. M. Churilov (a1) and I. G. Shukhman (a1)


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