On asymmetric vortex pair interactions in shear

Abstract

This study examines the two-dimensional interaction of two unequal co-rotating viscous vortices in uniform background shear. Numerical simulations are performed for vortex pairs having various circulation ratios $\varLambda _0 = \varGamma _{1,0}/\varGamma _{2,0} = (\omega _{1,0}/\omega _{2,0})(a^2_{1,0}/a^2_{2,0}) \leqslant 1$ , corresponding to different initial characteristic radii $a_{i,0}$ and peak vorticities $\omega _{i,0}$ of each vortex $i=1,2$ , in shears of various strengths $\zeta _0 = \omega _S/\omega _{2,0}$ , where $\omega _S$ is the constant vorticity of the shear. Two primary flow regimes are observed: separations ( $\zeta _0 < \zeta _{sep} < 0$ ), in which the vortices move apart continuously, and henditions ( $\zeta _0 > \zeta _{sep}$ ), in which the interaction results in a single vortex (where $\zeta _{sep}$ is the adverse shear strength beyond which separation occurs). Vortex motion and values of $\zeta _{sep}(\varLambda _0)$ are well-predicted by a point-vortex model for unequal vortices. In vortex-dominated henditions, shear varies the peak–peak distance $b$ , and vortex deformation. The main convective interaction begins when core detrainment of one vortex is established, and proceeds similarly to the no-shear ( $\zeta _0 = 0$ ) case: merger occurs if the second vortex also detrains, engendering mutual entrainment; otherwise straining out occurs. Detrainment requires persistence of straining of both sufficient magnitude, as indicated by relative straining above a consistent critical value, $(S/\omega )_i > (S/\omega )_{cr}$ , where $S$ is the strain rate magnitude at the vorticity peak, and conducive direction. Hendition outcomes are assessed in terms of an enhancement factor $\varepsilon \equiv \varGamma _{end}/\varGamma _{2,start}$ . Although $\varepsilon$ generally varies with $\zeta _0$ , $(a^2_{1,0} /a^2_{2,0} )$ and $(\omega _{1,0}/\omega _{2,0})$ in a complicated manner, this variation is well-characterized by the pair's starting enstrophy ratio, $Z_2/Z_1$ . Within a transition region between merger and straining out (approximately $1.65 < Z_2/Z_1 < 1.9$ ), shear of either sense may increase $\varepsilon$ .