Analysis of Steady Vortex Rings Using Contour Dynamics Method for Fluid Velocity

A water-jet can be used as one of the propulsion systems for ships and marine life. When a jet is injected to obtain thrust, a vortex ring is formed at a nozzle and then propagated downstream (Krueger et al., 2008). Furthermore, a vortex ring is generated due to volcanic eruption or nuclear explosion (Akhmetov, 2009). The flow of a vortex ring is formulated with the Helmholtz vorticity equation in inviscid and incompressible fluids (Batchelor, 1967). A steady vortex ring was first reported by Helmholtz (1867) who examined a vortex ring of an small circular cross section, while a spherical vortex was first analyzed by Hill (1894). Norbury (1973) analyzed a vortex ring in a steady state for general circumstances, which is referred to as the Norbury–Fraenkel family (N-F family) of vortex rings. A dynamic analysis is required for analyzing the instability due to the disturbance or interaction between vortex rings. A contour dynamics (CD) method for fluid velocity is used for analyzing the complex evolution of the contour of a vortex core. The CD method is a two-dimensional or axisymmetric flow analysis method due to the isolated vorticity in an inviscid, incompressible, and irrotational flow field (Pullin, 1992; Smith et al., 2018). The CD method can drastically reduce the burden of computations because the computation is performed in the form of line integrals on the boundary contour of the vorticity region. The fluid velocity on the contour is calculated using the CD method and then applied with time integrals to estimate the dynamic changes in the shape of the vortex core. Zabusky et al. (1979) introduced the CD method in dynamic analysis of two-dimensional vortex patches. Various examples of dynamic analysis for threedimensional axisymmetric vortex rings are provided in the study by Shariff et al. (1989). In this study, the CD method was applied to the analysis of the N-F family of vortex rings which are flows in steady state. Choi (2020) combined the CD method for a stream function (Shariff et al., 1989) and the direct shape-calculation method, and thus obtained results that were superior that those reported by Norbury (1973) wherein surface integrals and Fourier analysis were used. As a follow-up study to Choi (2020), in this study, we analyzed the N-F family of vortex rings using the CD method for fluid velocity examined in studies by Shariff et al. (1989) and Shariff et al. (2008). A stream function has been mostly used for analyzing a vortex ring in a steady state (Batchelor, 1967; Fraenkel, 1970; Fraenkel, 1972; Norbury 1973). In this study, we examined whether the CD method for fluid velocity, which is used in dynamic analysis, can also be applied to the analysis of a vortex ring in Journal of Ocean Engineering and Technology [ARTICLE IN PRESS] https://doi.org/10.26748/KSOE.2021.070 pISSN 1225-0767 eISSN 2287-6715