A quantitative assessment of viscous asymmetric vortex pair interactions

Abstract

The interactions of two like-signed vortices in viscous fluid are investigated using two-dimensional numerical simulations performed across a range of vortex strength ratios, $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}\leqslant 1$, corresponding to vortices of circulation, $\unicode[STIX]{x1D6E4}_{i}$, with differing initial size and/or peak vorticity. In all cases, the vortices evolve by viscous diffusion before undergoing a primary convective interaction, which ultimately results in a single vortex. The post-interaction vortex is quantitatively evaluated in terms of an enhancement factor, $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D6E4}_{end}/\unicode[STIX]{x1D6E4}_{2,start}$, which compares its circulation, $\unicode[STIX]{x1D6E4}_{end}$, to that of the stronger starting vortex, $\unicode[STIX]{x1D6E4}_{2,start}$. Results are effectively characterized by a mutuality parameter, $MP\equiv (S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}$, where the ratio of induced strain rate, $S$, to peak vorticity, $\unicode[STIX]{x1D714}$, for each vortex, $(S/\unicode[STIX]{x1D714})_{i}$, is found to have a critical value, $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$, above which core detrainment occurs. If $MP$ is sufficiently close to unity, both vortices detrain and a two-way mutual entrainment process leads to $\unicode[STIX]{x1D700}>1$, i.e. merger. In asymmetric interactions and mergers, generally one vortex dominates; the weak/no/strong vortex winner regimes correspond to $MP<,=,>1$, respectively. As $MP$ deviates from unity, $\unicode[STIX]{x1D700}$ decreases until a critical value, $MP_{cr}$ is reached, beyond which there is only a one-way interaction; one vortex detrains and is destroyed by the other, which dominates and survives. There is no entrainment and $\unicode[STIX]{x1D700}\sim 1$, i.e. only a straining out occurs. Although $(S/\unicode[STIX]{x1D714})_{cr}$ appears to be independent of Reynolds number, $MP_{cr}$ shows a dependence. Comparisons are made with available experimental data from Meunier (2001, PhD thesis, Université de Provence-Aix-Marseille I).