Stability of confined vortex sheets
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O R I G I NA L A RT I C L E
Bartosz Protas
Stability of confined vortex sheets
Received: 4 January 2020 / Accepted: 21 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose a simple model for the evolution of an inviscid vortex sheet in a potential flow in a channel with parallel walls. This model is obtained by augmenting the Birkhoff–Rott equation with a potential field representing the effect of the solid boundaries. Analysis of the stability of equilibria corresponding to flat sheets demonstrates that in this new model the growth rates of the unstable modes remain unchanged as compared to the case with no confinement. Thus, in the presence of solid boundaries the equilibrium solution of the Birkhoff–Rott equation retains its extreme form of instability with the growth rates of the unstable modes increasing in proportion to their wavenumbers. This linear stability analysis is complemented with numerical computations performed for the nonlinear problem which show that confinement tends to accelerate the growth of instabilities in the nonlinear regime. Keywords Vortex sheets · Stability analysis · Channel flows 1 Introduction Shear layers play an important role in fluid mechanics as they appear in many flows of industrial and geophysical significance when boundary layers separate from solid objects. A key property of shear layers is that under typical conditions they are unstable and undergo the Kelvin–Helmholtz instability as a result of which the vorticity from the shear layer rolls up into big vortices. When occurring recurrently, this phenomenon can in turn give rise to a turbulent cascade. In this investigation, we are interested in a simple inviscid model of the Kelvin–Helmholtz instability occurring in a channel with solid walls. In the context of more realistic flows, this problem was studied using an approach based on the theory of viscous potential flows in [7] where the effect of various problem parameters on the growth rates of unstable modes was analyzed. Analogous questions relevant for the stability of confined jets in circular geometries in the presence of heat and mass transfer were considered in [3]. Inviscid vortex sheets, represented as one-dimensional (1D) curves across which the tangential velocity component exhibits a discontinuity and evolving under their own induction in a potential flow, have been frequently invoked as a mathematical abstraction of actual viscous shear layers [16]. In unbounded domains, they admit an elegant description in terms of the Birkhoff–Rott equation. This singular integro-differential equation has a number of interesting properties—in particular, its equilibrium solution representing a flat (undeformed) vortex sheet is highly unstable to small-wavelength perturbations, a fact that underlies the illposedness of the Birkhoff–Rott model [13]. As a result, computational studies involving the Birkhoff–Rott equation typically require some regularization in order to track its long-time evolution, usually in the form of the wel
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