Dynamical systems model of entrainment due to coherent structures

Abstract

The lifetime of a turbulent shear flow is inversely related to the amount of surrounding fluid it entrains. A dynamical systems model of such a flow, dominated by coherent structures, was recently proposed [ R. Govindarajan Phys. Rev. Lett. 88 134503 (2002)], where an axisymmetric heated jet was modelled by a pair of leap-frogging vortex rings. In the present paper we study the equivalent two-dimensional flow, namely, two pairs of corotating point vortices, of opposite sign. We prove that the system is nonintegrable for any finite separation between the pairs, and show that entrainment is a natural consequence of chaotic advection. The effect of core diffusion due to viscosity is studied, and it is shown that the entrainment rate is reduced with increasing viscosity, by a reduction in leap-frogging frequency.