The passive advection of tracer panicles is considered in open two-dimensional incompressible flows with chaotic time dependence. As illustrative examples we investigate flows produced by chaotically moving ideal point vortices. The advection problem can be seen as a chaotic scattering process in a chaotically driven Hamiltonian system. Studying the motion of tracer ensembles, we present numerical evidence for the existence of a bounded chaotic set containing infinitely many aperiodic trajectories never leaving the mixing region of the flow. These ensembles converge to filamental patterns which, however, do not follow self-similar scaling. Nevertheless, they possess a fractal dimension after averaging over several finite-time realizations of the flow. We propose random maps as simple models of the phenomenon.