A numerical study of some vortex ring phenomena using direct numerical simulation (DNS)

Abstract

Direct numerical simulation (DNS) is used to study some aspects of the dynamics of vortex rings in viscous, incompressible flow at Reynolds numbers (defined as the ratio of the initial circulation to the kinematic viscosity) in the range of 10³ to 10⁴. Firstly, the effect of the particular initial core azimuthal vorticity profile of a vortex ring on its subsequent evolution in unbounded flow is studied. Vortex rings with a wide range of initial core vorticity profiles are shown to relax to a common equilibrium state. Additionally the behaviour of the equilibrium vortex ring at large times is studied. When the slenderness ratio of the vortex rings increases beyond a particular limit, the vortex rings diverge from the common equilibrium state and follow paths determined by the viscosity of the fluid. Secondly, the interaction of a laminar vortex ring with a non-deformable, free-slip surface at an oblique angle of incidence leading to the phenomenon of vortex reconnection is investigated. Specifically the effect of Reynolds number on the dynamics of the reconnection process is studied. The scaling of the reconnection timescale with the Reynolds number is obtained. At high Reynolds numbers the reconnection process leads to a breakdown of the entire vortex ring structure to a turbulent-like flow. This phenomenon is shown to be related to the mechanics of the reconnection process. Finally, the dynamics of vortex rings with swirl in unbounded flow is studied. Two different types of vortex rings with swirl were considered: i) Vortex rings with Gaussian distributions of core azimuthal vorticity and core azimuthal velocity and ii) Steady state solutions of the Euler equations for vortex rings with swirl. Both types of vortex rings develop an elongated axial vortex after initialisation. The existence of a maximum limit for the swirl on a vortex ring is shown above which the vortex rings undergo a rapid de-swirling readjustment. A helical instability occurring in vortex rings due to swirl at high Reynolds numbers is presented. A relation is shown to exist between one of the modes of the helical instability and the geometric parameters of the vortex ring.

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