A model of unsteady line-vortex dynamics in free-surface gravity-wave flow

Kurzfassung

Meandering is the prototype of the unsteady dynamics of line vortices observed in experiments and has never been examined subject to free-surface gravity-wave flow explicitly before. With this study, we pursue the objective to make progress in developing a theoretical vortex-meandering model from first principles. As in recently proposed models, we start our modelling from the experimentally measurable vortex-centre integral, which we consider as the definition of vortex meandering. We then derive an equation of motion for the vortex centre assuming three-dimensional vorticity dynamics, which has not been reported in the literature previously. Considering meandering to correspond to the lateral displacement of the vortex as a whole, we demonstrate theoretically that practically all terms in this equation of motion mutually cancel and that the problem cannot be closed with the experimentally available two-dimensional data alone. As suggested by our analysis, we therefore assume vortex meandering to be associated with an infinity of viscously damped displacement waves excited by the free stream turbulence and propagating along the core. The resulting power spectrum closely resembles that of an Ornstein–Uhlenbeck process, as proposed recently. We compare all aspects and assumptions of our model development with an experiment of a single line vortex conducted in a free-surface recirculating water channel. The effect of the free-surface gravity wave on the vortex-meandering dynamics has not been examined before, but is found to have a non-negligible contribution of the same order as the free stream turbulence in the facility. Overall, we confirm our model assumptions and find remarkably good principal agreement between the model characteristics and the experiment at the 5 % level of significance. Despite this overall good agreement, there remains the considerable deficiency that the model-predicted vortex-response time scale is approximately two orders different from the experimental one. While the reason for this inconsistency is not clear yet, the results are encouraging and hopefully help solving the problem of vortex meandering eventually.