Numerical verification of the similarity laws for the formation of laminar vortex rings

Abstract

From analytical investigations it is well known that the roll-up of an inviscid plane vortex sheet which separates at the edge of a body is a self-similar process which can be described by scaling laws. Unlike plane vortices, ring vortices have a curved rotational axis. For this special vortex type experimental investigations as well as calculations in the literature suggest that the scaling laws are only partially valid. The main goal of this work is to clarify how far these similarity or scaling laws are also valid for the formation of viscid laminar vortex rings. Therefore, the formation process of laminar vortex rings was investigated numerically using a CFD (computational-fluid-dynamics) code. The calculations refer to an experimental setup for which detailed experimental data are available in the literature. In this setup, laminar ring vortices are generated by ejecting water from a circular tube into a quiescent environment by means of a piston. First, a case based on a constant piston velocity was investigated. Comparing calculated and measured data yields a very good agreement. Further calculations were made when forcing the velocity of the piston by three different time-dependent functions. The results of these calculations show that the formation laws for inviscid plane vortices are also valid for the formation process of viscid ring vortices. This applies to the normalized axial and radial position of the vortex centre as well as the normalized diameter of the vortex spiral. However, the similarity laws are valid only if the process is considered in a special frame of reference which moves in conjunction with the front of the jet and if the starting time of the formation process with respect to the starting time of the ejection is taken into account. Additionally, the formation of a ring vortex, which occurs during the start-up process of a free jet flow, was calculated. The results confirm a dependence for the motion of the jet front, which is known from analytical considerations and allows some interesting features to be identified.