The three-dimensional instabilities and destruction of the viscous Hill’s vortex

• Published in: Theoretical and computational fluid dynamics 2021-06, Vol.35 (3), p.363-379
• Main Author: Orlandi, P.
• Format: Article
• Language: English
• Subjects: Amplitude; Amplitudes; Analysis; Classical and Continuum Physics; Computational fluid dynamics; Computational Science and Engineering; Direct numerical simulation; Disturbances; Engineering; Engineering Fluid Dynamics; Euler-Lagrange equation; Evolution; Fluid flow; High Reynolds number; Mathematical analysis; Navier-Stokes equations; Numerical analysis; Original Article; Reynolds number; Shape; Stability; Stability analysis; Surface stability; Three dimensional flow; Vortices
• ISSN: 0935-4964; 1432-2250
• DOI: 10.1007/s00162-021-00563-1

Hill’s vortex is a three-dimensional vortex structure form-preserving solution of the Euler equations (Hill in Philos Trans R Soc Lond A 185:213–245, 1894). For small amplitude axisymmetric disturbances on the external surface, the linear stability analysis by Moffat and Moore (J Fluid Mech 87:749–760, 1978) predicted the formation of a tail. Successive linear and nonlinear investigations confirmed this fact and in addition observed that the shape of the tail was linked to number of small amplitude azimuthal disturbances of the surface. In this paper, the Navier–Stokes equations are solved, at high Reynolds number, by imposing large amplitude axisymmetric and three-dimensional disturbances on the surface of the vortex. The axisymmetric disturbances are convected in the rear side, are dumped and form an axisymmetric wave increasing at the same rate as that in the linear stability analysis. The azimuthal disturbances produce a hierarchy of structures inside the vortex, and in a short-time evolution, the shape of the vortex is maintained. For a long-time evolution, direct numerical simulations show that Hill’s vortex for azimuthal disturbances loses its original form for the formation of a wide range of energy containing scales characteristic of three-dimensional flows. Although a true turbulent state has not been reached, the DNS of this simple vortex structure shows the passage from a vortex dominated to a turbulent state.