Schooling, an archetype of collective behavior, emerges from the interactions of fish responding to visual and other informative cues mediated by their aqueous environment. In this context, a fundamental and largely unexplored question concerns the role of hydrodynamics. Here, we investigate schooling by modeling swimmers as vortex dipoles whose interactions are governed by the Biot-Savart law. When we enhance these dipoles with behavioral rules from classical agent based models we find that they do not lead robustly to schooling due to flow mediated interactions. In turn, we present dipole swimmers equipped with adaptive decision-making that learn, through a reinforcement learning algorithm, to adjust their gaits in response to non-linearly varying hydrodynamic loads. The dipoles maintain their relative position within a formation by adapting their strength and school in a variety of prescribed geometrical arrangements. Furthermore, we identify schooling patterns that minimize the individual and the collective swimming effort, through an evolutionary optimization. The present work suggests that the adaptive response of individual swimmers to flow-mediated interactions is critical in fish schooling.Comment: 18 pages, 12 figure

The notion of a finite dipole is introduced as a pair of equal and opposite strength point vortices (i.e. a vortex dipole) separated by a finite distance. Equations of motion for N finite dipoles interacting in an unbounded inviscid fluid are derived from the modified interaction of 2N independent vortices subject to the constraint that the inter-vortex spacing of each constrained dipole, , remains constant. In the absence of all other dipoles and background flow, a single dipole moves in a straight line along the perpendicular bisector of the line segment joining the two point vortices comprising the dipole, with a self-induced velocity inversely proportional to . When more than one dipole is present, the velocity of the dipole centre is the sum of the self-induced velocity and the average of the induced velocities on each vortex comprising the pair due to all the other dipoles. Each dipole orients in the direction of shear gradient based on the difference in velocities on each of the two vortices in the pair. Several numerical experiments are shown to illustrate the interactions between two and three dipoles in abreast and tandem configurations. We also show that equilibria (multi-poles) can form as a result of the interactions, and we study the stability of polygonal equilibria, showing that the N = 3 case is linearly stable, whereas the N > 3 case is linearly unstable.

We present simulations of two interacting moving cylinders immersed in a two-dimensional incompressible, viscous flow. Simulations are performed by coupling a wavelet-adapted, remeshed vortex method with the Brinkman penalization and projection approach. This method is validated on benchmark problems and applied to simulations of a master-slave pair of cylinders. The master cylinder's motion is imposed and the slave cylinder is let free to respond to the flow. We study the relative role of viscous and inertia effects in the cylinders interactions and identify related sharp transitions in the response of the slave. The observed differences in the behavior of cylinders with respect to corresponding potential flow simulations are discussed. In addition, it is observed that in certain situations the finite size of the slave cylinders enhances the transport so that the cylinders are advected more effectively than passive tracers placed, respectively, at the same starting position.

The interaction of two arbitrary bodies immersed in a two-dimensional inviscid fluid is investigated. Given the linear and angular velocities of the bodies, the solution of the potential flow problem with zero circulation around both bodies is reduced to the determination of a suitable Laurent series in a conformally mapped domain that satisfies the boundary conditions. The potential flow solution is then used to determine the force and moment acting on each body by using generalized Blasius formulas. The current formulation is applied to two examples. First, the case of two rigid circular cylinders interacting in an unbounded domain is investigated. The forces on two cylinders with prescribed motion ͑forced-forced͒ is determined and compared to previous results for validation purposes. We then study the response of a single "free" cylinder due to the prescribed motion of the other cylinder ͑forced-free͒. This forced-free situation is used to justify the hydrodynamic benefits of drafting in aquatic locomotion. In the case of two neutrally buoyant circular cylinders, the aft cylinder is capable of attaining a substantial propulsive force that is the same order of magnitude of its inertial forces. Additionally, the coupled interaction of two cylinders given an arbitrary initial condition ͑free-free͒ is studied to show the differences of perfect collisions with and without the presence of an inviscid fluid. For a certain range of collision parameters, the fluid acts to deflect the cylinder paths just enough before the collision to drastically affect the long time trajectories of the bodies. In the second example, the flapping of two plates is explored. It is seen that the interactions between each plate can cause a net force and torque at certain instants in time, but for idealized sinusoidal motions in irrotational potential flow, there is no net force and torque acting at the system center.

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