A generalized vortex ring model

Abstract

A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness is given by the relation = a t b , where a is a positive number and 1/4 6 b 6 1/2. In the case in which a = √ 2ν, where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν * is equal to . This generalization is performed both in the case of a fixed vortex ring radius R 0 and increasing vortex ring radius. In the latter case, the so-called second Saffman's formula is modified. In the case of fixed R 0 , the predicted vorticity distribution for short times shows a close agreement with a Gaussian form for all b and compares favourably with available experimental data. The time evolution of the location of the region of maximal vorticity and the region in which the velocity of the fluid in the frame of reference moving with the vortex ring centroid is equal to zero is analysed. It is noted that the locations of both regions depend upon b, the latter region being always further away from the vortex axis than the first one. It is shown that the axial velocities of the fluid in the first region are always greater than the axial velocities in the second region. Both velocities depend strongly upon b. Although the radial component of velocity in both of these regions is equal to zero, the location of both of these regions changes with time. This leads to the introduction of an effective radial velocity component; the latter case depends upon b. The predictions of the model are compared with the results of experimental measurements of vortex ring parameters reported in the literature.

Modelling of confined vortex rings

“…For example, variations in energy over time were accurately captured, while circulation decay and the evolution of the translational velocity were predicted fairly well, which enables us to recommend the model for practical engineering applications. The fit with DNS data could be further improved if a power different to 1/2 was considered in expression (4.2) was considered; it was earlier shown that the Kaplanski-Rudi model for viscous vortex ring could be generalised for power-laws with exponents between 0.25 and 0.5 (see Kaplanski et al (2009)). The main aim of this paper, however, is not to find the best fit with DNS data but to assess the general capabilities of the model to reproduce realistic confined vortex rings.…”

A model for confined vortex rings with elliptical-core vorticity distribution

“…The generalisation of Eq (2.7) for turbulent vortex rings was discussed by Kaplanski et al (2009). The Reynolds number of the flow is introduced as Re = Γ 0 /ν and the parameter ε = R 0 /R w < 1 quantifies the confinement of the vortex ring.…”

Reynolds-number effect on vortex ring evolution in a viscous fluid

“…12 In the case b = 1/4 some of the predictions of Eqs. (1)-(6) are similar to the turbulent vortex ring models, assuming that the time-dependent effective viscosity ν * (introduced instead of ν) is equal to L L .…”