Rotating equilibria of vortex sheets

Abstract

We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.

Finite rotating and translating vortex sheets

“…It is an interesting open question as to whether an analogous uniqueness result could be established for the translating equilibrium discussed in § 2.2. The search for new equilibrium configurations involving vortex sheets remains an active area of research (O'Neil, 2018b,a;Protas & Sakajo, 2020).…”

“…Interestingly, as proved in Lopes Filho et al (2003Filho et al ( , 2007, while the rotating equilibrium can be interpreted as a weak solution of the 2D Euler equation, the translating equilibrium cannot. The rotating equilibrium was recently generalized to configurations involving multiple straight segments with one endpoint at the centre of rotation and the other at a vertex of a regular polygon by Protas & Sakajo (2020). By allowing for the presence of point vortices in the far field O'Neil (2018b,a) were able to find more general equilibria involving multiple vortex sheets, including curved ones, in both rotating and translating frames of reference.…”

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

“…) which is a rotating solution with angular velocity [2]. Protas-Sakajo [31] generalized this solution and proved the existence of several others made out of segments rotating about a common center of rotation with endpoints at the vertices of a regular polygon by solving a Riemann-Hilbert problem, even finding some of them analytically.…”

Finite rotating and translating vortex sheets

“…(2007), while the rotating equilibrium can be interpreted as a weak solution of the two-dimensional Euler equation, the translating equilibrium cannot. The rotating equilibrium was recently generalized to configurations involving multiple straight segments with one endpoint at the centre of rotation and the other at a vertex of a regular polygon by Protas & Sakajo (2020). By allowing for the presence of point vortices in the far field, O'Neil (2018 a , b ) was able to find more general equilibria involving multiple vortex sheets, including curved ones, in both rotating and translating frames of reference.…”