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
$\varepsilon$
.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics,Applied Mathematics