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We derive new mixed-integer quadratic, quadratically constrained, and second-order cone programming models of distribution system reconfiguration, which are to date the first formulations of the AC problem that have convex continuous relaxations. Each model can be reliably and efficiently solved to optimality using standard commercial software. In the course of deriving each model, we obtain original quadratically constrained and second-order cone approximations to power flow in radial networks.

A dead fish is propelled upstream when its flexible body resonates with oncoming vortices formed in the wake of a bluff cylinder, despite being well outside the suction region of the cylinder. Within this passive propulsion mode, the body of the fish extracts sufficient energy from the oncoming vortices to develop thrust to overcome its own drag. In a similar turbulent wake and at roughly the same distance behind a bluff cylinder, a passively mounted high-aspect-ratio foil is also shown to propel itself upstream employing a similar flow energy extraction mechanism. In this case, mechanical energy is extracted from the flow at the same time that thrust is produced. These results prove experimentally that, under proper conditions, a body can follow at a distance or even catch up to another upstream body without expending any energy of its own. This observation is also significant in the development of low-drag energy harvesting devices, and in the energetics of fish dwelling in flowing water and swimming behind wake-forming obstacles.

Abstract-Significant progress has been made in understanding some of the basic mechanisms of force production and flow manipulation in oscillating foils for underwater use. Biomimetic observations, however, show that there is a lot more to be learned, since many of the functions and details of fish fins remain unexplored.This review focuses primarily on experimental studies on some of the, at least partially understood, mechanisms, which include 1) the formation of streets of vortices around and behind two-and three-dimensional propulsive oscillating foils; 2) the formation of vortical structures around and behind two-and three-dimensional foils used for maneuvering, hovering, or fast-starting; 3) the formation of leading-edge vortices in flapping foils, under steady flapping or transient conditions; 4) the interaction of foils with oncoming, externally generated vorticity; multiple foils, or foils operating near a body or wall.Index Terms-Biomimetics, fish swimming, flapping foil propulsion. I. OVERVIEW OF LITERATURE BIOMIMETIC studies and observations from fish and cetaceans have provided a wealth of information on the kinematics, i.e., how these animals employ their flapping tails and several fins to produce propulsive and maneuvering forces (see reviews in [123] The fluid mechanics and force mechanics of foils have been investigated with the goal of understanding the principles of this different paradigm of propulsion and maneuvering, so as to apply it to enhance existing technology. The tails of some of the fastest swimming animals closely resemble high aspect ratio foils. As a result, flapping foils have been studied extensively using theoretical and numerical techniques [72] II. BASIC PARAMETERS AND DEFINITIONSIn a foil with maximum chord length and maximum span , moving at steady speed , and at an angle of attack , the parameters of relevance are (a) the geometric shape (rectangular, delta-shaped, etc.); (b) the aspect ratio (AR), defined to be equal to the ratio of an average span over an average chord; (c) the angle of attack ; and (d) the Reynolds number , where is the kinematic viscosity of the fluid.

Forces are measured at both ends of rigid cylinders with span 60 cm, performing transverse oscillations within an oncoming stream of water, at Reynolds number Re ≈ 3800. Forced harmonic motions and free vibrations of uniform and tapered cylinders are studied. To study free motions, a novel force-feedback control system has been developed, consisting of: (a) a force transducer, which measures forces on a section of a cylinder moving forward at constant speed; (b) a computer using the measured force signal to drive in real time a numerical simulation of an equivalent mass-dashpot-spring system; (c) a servomotor and linear table which impose, also in real time, the numerically calculated motion on the cylinder section. The apparatus allows very low equivalent system damping and strict control of the parametric values and structure of the equivalent system. Calculation of the cross-correlation coefficient between forces at the two ends of the uniform cylinder reveals five distinct regimes as a function of the nominal reduced velocity V rn : two regimes, for low and high values of V rn , and far away from the value of V rS corresponding to the Strouhal frequency, show small correlation; two regimes immediately adjacent to, but excluding, V rS show strong correlation, close to 1; surprisingly, there is a regime containing the Strouhal frequency, within which correlation is low. Free vibrations with a 40:1 tapered cylinder show that the regime of low correlation, containing the Strouhal frequency, stretches to higher reduced velocities, while lock-in starts at lower reduced velocities.When comparing the amplitude and phase of the lift coefficient measured for free and then for forced vibrations, we obtain close agreement, both for tapered and uniform cylinders. When comparing the cross-correlation coefficient, however, we find that it is much higher in the forced oscillations, especially for the uniform cylinder. Hence, although the force magnitude and phase may be replicated well in forced vibrations, the correlation data suggest that differences exist between free and forced vibration cases.