Vortex Dynamics

Abstract

Dynamics of vortices of filament type is introduced for the following three cases: deformation of a vortex filament by self-induction, entanglement of two vortex filaments by self-and mutual inductions and reconnection of two vortex filaments. The first two cases are treated by the use of vorticity equation without the viscous term, while the third one requires a treatment with the effect of viscosity. Key words: Vorticity, Self-Induction, Mutual Induction, Entanglement, Reconnection What is Vortex?The vortex is a swirling motion of fluid, where the angular velocity of the motion is largest in its central region and decreases with the distance from the center. The vortex has been familiar to us since many years, and has been often expressed as artworks such as "Study of water falling into still water" by Leonardo da Vinci and "Panel with red and white apricot flowers" by Korin Ogata (Japanese artist in 18-19 c.). Recently, we often see satellite photos of typhoon, which is a large scale vortex with large angular velocity in the central part, called "eye of a typhoon".In most vortices including the typhoon the fluid velocity u(r ) vanishes at the center, increases linearly with the radius r and then decreases, as is shown in Fig. 1. The angular velocity u(r )/r is nearly uniform in the central region, which corresponds to the eye of a typhoon. On the other hand, the vorticity, which is defined below, vanishes out of this central region. Of course, vortices appearing in the nature receive various disturbances from the environment, and the distributions shown in Fig. 1 are much distorted.The vorticity ω(x) is obtained from a velocity vector u(x) = (u, v, w) by the following differential operation:where ∇ = (∂/∂ x, ∂/∂y, ∂/∂z). The physical meaning of the vorticity is a strength of rotation of a small fluid element around its center. The equation governing the vorticity can be derived from the Navier-Stokes equation (a basic equation governing the dynamics of viscous fluid), and the result is given below (refer some textbooks for its derivation, such as Batchelor (1967) and Imai (1973)).Copyright c Society for Science on Form, Japan.doi:10.5047/forma.2015.s005The second and the third terms in the left-hand side indicate, respectively, convection of the vorticity by the flow field u and increase of vorticity by stretching of vortex by u. The right-hand side indicates the diffusion of the vorticity due to the viscosity. If the fluid is inviscid (non-viscous), there are some conserved quantities in the vortex dynamics. In order to understand the conserved quantities, the Stokes' theorem given below is necessary. Consider a closed smooth surface in the 3D space with its edge denoted by C, then the following equation is satisfied by a vector field u defined in this space:where n is a normal vector on the surface and d is a line element of the edge C. If rot u is replaced by ω, it is rewritten aswhere is called a circulation and indicates a strength of the part of vorticity distribution within the closed curve C, as sho...