Vortex ring
A vortex ring, also called a toroidal vortex, is a torus-shaped vortex in a fluid; that is, a region where the fluid mostly spins around an imaginary axis line that forms a closed loop. The dominant flow in a vortex ring is said to be toroidal, more precisely poloidal.
Vortex rings are plentiful in turbulent flows of liquids and gases, but are rarely noticed unless the motion of the fluid is revealed by suspended particles—as in the smoke rings which are often produced intentionally or accidentally by smokers. Fiery vortex rings are also a commonly produced trick by fire eaters. Visible vortex rings can also be formed by the firing of certain artillery, in mushroom clouds, in microbursts, and rarely in volcanic eruptions.
A vortex ring usually tends to move in a direction that is perpendicular to the plane of the ring and such that the inner edge of the ring moves faster forward than the outer edge. Within a stationary body of fluid, a vortex ring can travel for relatively long distance, carrying the spinning fluid with it.
Structure
In a typical vortex ring, the fluid particles move in roughly circular paths around an imaginary circle that is perpendicular to those paths. As in any vortex, the velocity of the fluid is roughly constant except near the core, so that the angular velocity increases towards the core, and most of the vorticity is concentrated near it.Unlike a sea wave, whose motion is only apparent, a moving vortex ring actually carries the spinning fluid along. Just as a rotating wheel lessens friction between a car and the ground, the poloidal flow of the vortex lessens the friction between the core and the surrounding stationary fluid, allowing it to travel a long distance with relatively little loss of mass and kinetic energy, and little change in size or shape. Thus, a vortex ring can carry mass much further and with less dispersion than a jet of fluid. That explains, for instance, why a smoke ring keeps traveling long after any extra smoke blown out with it has stopped and dispersed. These properties of vortex rings are exploited in the vortex ring gun for riot control, and vortex ring toys such as the air vortex cannons.
Formation
Formation process
The formation of vortex rings has fascinated the scientific community for more than a century, starting with William Barton Rogers who made sounding observations of the formation process of air vortex rings in air, air rings in liquids, and liquid rings in liquids. In particular, William Barton Rogers made use of the simple experimental method of letting a drop of liquid fall on a free liquid surface; a falling colored drop of liquid, such as milk or dyed water, will inevitably form a vortex ring at the interface due to the surface tension.A method proposed by G. I. Taylor to generate a vortex ring is to impulsively start a disk from rest. The flow separates to form a cylindrical vortex sheet and by artificially dissolving the disk, one is left with an isolated vortex ring. This is the case when someone is stirring their cup of coffee with a spoon and observing the propagation of a half-vortex in the cup.
In a laboratory, vortex rings are formed by impulsively discharging fluid through a sharp-edged nozzle or orifice. The impulsive motion of the piston/cylinder system is either triggered by an electric actuator or by a pressurized vessel connected to a control valve. For a nozzle geometry, and at first approximation, the exhaust speed is uniform and equal to the piston speed. This is referred as a parallel starting jet. It is possible to have a conical nozzle in which the streamlines at the exhaust are directed toward the centerline. This is referred as a converging starting jet. The orifice geometry which consists in an orifice plate covering the straight tube exhaust, can be considered as an infinitely converging nozzle but the vortex formation differs considerably from the converging nozzle, principally due to the absence of boundary layer in the thickness of the orifice plate throughout the formation process. The fast moving fluid is therefore discharged into a quiescent fluid. The shear imposed at the interface between the two fluids slows down the outer layer of the fluid relatively to the centerline fluid. In order to satisfy the Kutta condition, the flow is forced to detach, curl and roll-up in the form of a vortex sheet. Later, the vortex sheet detaches from the feeding jet and propagates freely downstream due to its self-induced kinematics. This is the process commonly observed when a smoker forms smoke rings from their mouth, and how vortex ring toys work.
Secondary effects are likely to modify the formation process of vortex rings. Firstly, at the very first instants, the velocity profile at the exhaust exhibits extrema near the edge causing a large vorticity flux into the vortex ring. Secondly, as the ring grows in size at the edge of the exhaust, negative vorticity is generated on the outer wall of the generator which considerably reduces the circulation accumulated by the primary ring. Thirdly, as the boundary layer inside the pipe, or nozzle, thickens, the velocity profile approaches the one of a Poiseuille flow and the centerline velocity at the exhaust is measured to be larger than the prescribed piston speed. Last but not least, in the event the piston-generated vortex ring is pushed through the exhaust, it may interact or even merge with the primary vortex, hence modifying its characteristic, such as circulation, and potentially forcing the transition of the vortex ring to turbulence.
Vortex ring structures are easily observable in nature. For instance, a mushroom cloud formed by a nuclear explosion or volcanic eruption, has a vortex ring-like structure. Vortex rings are also seen in many different biological flows; blood is discharged into the left ventricle of the human heart in the form of a vortex ring and jellyfishes or squids were shown to propel themselves in water by periodically discharging vortex rings in the surrounding. Finally, for more industrial applications, the synthetic jet which consists in periodically-formed vortex rings, was proved to be an appealing technology for flow control, heat and mass transfer and thrust generation
Vortex formation number
Prior to Gharib et al., few studies had focused on the formation of vortex rings generated with long stroke-to-diameter ratios, where is the length of the column of fluid discharged through the exhaust and is the diameter of the exhaust. For short stroke ratios, only one isolated vortex ring is generated and no fluid is left behind in the formation process. For long stroke ratios, however, the vortex ring is followed by some energetic fluid, referred as the trailing jet. On top of showing experimental evidence of the phenomenon, an explanation of the phenomenon was provided in terms of energy maximisation invoking a variational principle first reported by Kelvin and later proven by Benjamin, or Friedman & Turkington. Ultimately, Gharib et al. observed the transition between these two states to occur at a non-dimensional time, or equivalently a stroke ratio, of about 4. The robustness of this number with respect to initial and boundary conditions suggested the quantity to be a universal constant and was thus named formation number.The phenomenon of 'pinch-off', or detachment, from the feeding starting jet is observed in a wide range of flows observed in nature. For instance, it was shown that biological systems such as the human heart or swimming and flying animals generate vortex rings with a stroke-to-diameter ratio close to the formation number of about 4, hence giving ground to the existence of an optimal vortex ring formation process in terms of propulsion, thrust generation and mass transport. In particular, the squid lolliguncula brevis was shown to propel itself by periodically emitting vortex rings at a stroke-ratio close to 4. Moreover, in another study by Gharib et al, the formation number was used as an indicator to monitor the health of the human heart and identify patients with dilated cardiomyopathy.