Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: Quasi-periodic structures

Abstract

A plane wake or jet supports sinuous and varicose instability modes. The nonlinear interaction between them following their linear development was described previously by Leib and Goldstein [“Nonlinear interaction between the sinuous and varicose instability modes in a plane wake,” Phys. Fluids A 1, 513–521 (1989)] using the strongly nonlinear non-equilibrium critical-layer approach in the case of the Bickley jet for which the frequencies of the sinuous and varicose modes have an integer ratio of 2. This paper develops the theory for general profiles where the frequencies of the sinuous and varicose modes are non-commensurable. The disturbance is quasi-periodic in time and space and must be expressed as a function of two phase variables. Using matched asymptotic expansions simultaneously with the multi-scale method, we derived a set of coupled evolution equations governing the development of the amplitudes and critical-layer vorticities of these modes. The evolution system is solved for the base-flow profiles mimicking those in experiments. The sinuous mode suppresses the varicose mode but also causes the latter to saturate in a highly oscillatory manner. The varicose mode inhibits the sinuous mode initially. However, in the later stage, it lends the sinuous mode a significantly higher saturating amplitude. For a wide range of initial modal compositions and Reynolds numbers, the ratio of the varicose mode amplitude to that of the sinuous mode eventually tends to an almost constant value in the range of 0.4–0.6, in line with the experimental measurement. Due to the self and mutual interactions, the vorticities roll up to form vortices, which are non-symmetric in the transverse direction and quasi-periodic in the streamwise direction as well as in time. With such an increased complexity, the vortices resemble those observed in experiments. The nonlinear interactions of the sinuous and varicose modes in the critical layer generate all harmonics in the main layer, as a result of which the perturbation is non-periodic and may even appear “random-like.”

Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field

“…In particular, we see that the shearwise length scale of the perturbations is smaller, and dominated by the mode for the streamfunction, and the mode for the flux function. The flow contains convergent and divergent regions, consistent with varicose modes in other systems (Mattingly & Criminale 1972; Drazin & Reid 1981; Mikhaylov & Wu 2020). While these modes appear (to our knowledge) to be unnoticed in the literature, they are not the focus of this paper, and will be discussed in greater detail in future work.…”

“…Another class of unstable modes that also emerges is what we call the ‘varicose’ KH modes hereafter, where the term varicose here is used by analogy with varicose modes in, for example, shear instabilities in jets (Mattingly & Criminale 1972; Drazin & Reid 1981; Mikhaylov & Wu 2020), or the instabilities present in the streaky flows generated as part of the self-sustaining process in wall-bounded shear flows (Waleffe 1995, 1997). While the varicose KH modes also have , they can be distinguished from the sinuous KH modes because they have no -averaged shearwise flow or field, and thus .…”