Random jellyfish: energetics and diffusion in a sea of inertial swimmers

We address the energetics and diffusion of particles amidst a random distribution of swimmers in a viscous, inertial fluid. At intermediate Reynolds number, the main mechanism of mixing is the induced net particle displacement (drift). Several experiments have examined this drift for small jellyfish, which produce vortex rings that trap and transport a fair amount of fluid. Inviscid theory implies infinite particle displacements for the trapped fluid, so the effect of viscosity must be included to understand the damping of real vortex motion. We use a model viscous vortex ring to compute particle displacement and other moments. Fluid entrainment at the tail end of a growing vortex 'envelope' is found to play an important role in the total fluid transport and drift. Newer vortices produced by other jellyfish in the bloom overwhelm older vortices, limiting drift and dictating the effective particle diffusion. Our results are robust in the sense that any self-propulsion by isolated impulses produces the same long-time and far-field velocity fields, which determine how the density of swimmers scales with the effective diffusivity.