Impulse extremization in vortex formation for application in low speed maneuvering of underwater vehicles

Compact zero-mass pulsatile jet actuators are proposed for low speed maneuvering and station keeping of small underwater vehicles. The flow field of such jets are initially dominated by vortex ring formation. To this end, impulse extremization in vortex ring formation is considered. It is shown that a pinched-off vortex ring characterizes the extremum impulse accumulated by the leading vortex ring in a vortex ring formation process. An appropriate scaling for vortex ring impulse is found and the limiting values of the non-dimensionalized impulses are established. An estimate for the non-dimensional impulses is provided by equating their values from the slug model with their values from a vortex in the Norbury family of vortices. For a vortex ring generator with constant kinetic energy and circulation generation rate, the pinched-off vortex ring has a maximum impulse I-nd(E) 11 normalized by the circulation and energy. On the other hand, for a vortex ring generator with constant rate of circulation generation at a constant translational velocity, a pinched-off vortex ring produces a minimum impulse I-nd(Gamma) approximate to 0.12 normalized by the circulation and translational velocity. Direct numerical simulations of vortex ring formation and vortex ring pinch-off process are performed and the estimated values of the non-dimensionalized impulses are confirmed. A formula for calculating the translational velocity of a vortex ring, generated through an impulsive ejection of fluid from an orifice, is presented. Variation of the translational velocity as a function of the stroke ratio and the non-dimensional mean core radius is calculated. Application of such vortex generators as zero-mass pulsatile jets for low speed maneuvering of underwater vehicles is also demonstrated. The actuators could be used in two ways: (i) to improve the low speed maneuvering and hovering capabilities of traditional propeller driven underwater vehicles, (ii) and as a synthetic jet for flow control at higher cruising speeds. A model for calculating the rotation rate of the underwater vehicle is also proposed and verified.