Self-similar roll-up of a vortex sheet driven by a shear flow: Hyperbolic double spiral

In this paper, we consider the roll-up of an infinite vortex sheet and investigate its self-similar behavior. We address the question of whether the unsteady double spiral produced by the curvature singularity in finite time exhibits self-similar behavior. We find a self-similar solution of the double-spiral vortex sheet, which in fact, is a hyperbolic spiral. The radius of the spiral asymptotically grows with time and is proportional to the inverse of the angle from the spiral center. The curvature singularity plays the role of triggering spiral formation, but the source of vorticity for forming the spiral is the initial vorticity of the sheet. We show analytically that the self-similar solution satisfies the Birkhoff-Rott equation asymptotically. Numerical validation is also given by applying the blob-regularization model to the vortex sheet with a periodic perturbation. We examine various asymptotic relations among primitive variables for the spiral turns and find agreement of numerical results of the inner turns of the vortex sheet with the analytic solution. Our study clarifies contrasting results on the existence of the self-similar double-spiral of a large structure in the previous studies. Our solution also suggests the possibility of bifurcation of the self-similar solution of the double-spiral as the sheet strength varies.