In Chap. 3, an asymptotic theory was developed for the motion of slender vortex filaments at high Reynolds number. The theory is based on the assumptions that the characteristic radius of curvature of the filament centerline is much larger than the core size, and that the vortex core remains well-separated from solid boundaries. Under these assumptions, the asymptotic theory provides (1) a leading-order prediction of the filament centerline velocity and (2) evolution equations for the leading-order vorticity and axial velocity structures. The centerline velocity is given as the sum of a nonlocal velocity expressed as a convolution over the centerline with nonsingular kernel, and a local contribution that is a function of the local curvature, and of the instantaneous core structure. Meanwhile, the local core structure evolves according to stretching of the filament centerline and vortex diffusion phenomena. Thus, the filament motion is generally closely coupled with the evolution of its vortex core.
Ting, L., Klein, R., & Knio, O. M. (2007). Numerical simulation of slender vortex filaments. In Applied Mathematical Sciences (Switzerland) (Vol. 161, pp. 227–283). Springer. https://doi.org/10.1007/978-3-540-68582-1_5