The effect of streamwise braid vortices on the particle dispersion in a plane mixing layer. I. Equilibrium points and their stability

Abstract

The dynamics of small, heavy, spherical particles are investigated in an analytical model of the stretched counterrotating streamwise braid vortices commonly found in three-dimensionally evolving mixing layers. The flow field consists of two superimposed rows of Stuart vortices of opposite sign, with an additional two-dimensional strain field. The particle dynamics are determined by a balance of inertial, gravitational, and viscous drag forces, i.e., the dimensionless Stokes and Froude numbers, St and Fr, as well as by the dimensionless strain rate, and the Stuart vortex family parameter. Equilibrium points for the particles, as well as their stability criteria, are determined analytically, both in the absence and in the presence of gravity, and for different orientations of the gravity vector. In the absence of gravity, accumulation of low St particles can occur at the center of the braid vortices. An analytical expression for the critical particle diameter, below which accumulation is possible, is derived. The presence of gravity can lead to the emergence of multiple equilibrium points, whose stability properties depend on their locations. For a horizontal mixing layer flow and strong gravity effects, unconditional accumulation can occur midway between the streamwise braid vortices in the upwelling regions. Conditionally stable accumulation regions exist a short horizontal distance away from the centers of the braid vortices. If the gravity vector lies within the plane of the mixing layer, accumulation points exist only for moderate strengths of gravity. Under these circumstances, conditional accumulation is possible near the streamwise vortex centers.