The evolution of an elliptic vortex ring

Abstract

The evolution of a vortex ring in an ideal fluid under self-induction from a flat and elliptic configuration is followed numerically using the cut-off approximation (Crow 1970) for the velocity at the vortex. Calculations are presented for four different axes ratios of the initial ellipse. A particular choice is made for the core size and vorticity distribution in the core of the vortex ring. When the initial axes ratio is close to 1, the vortex ring oscillates periodically. The periodicity is lost as more eccentric cases are considered. For initial axes ratio 0·2, the calculations suggest a break-up of the ring through the core at one portion of the ring touching that at another, initially distant, portion of the ring.Results from quantitative experiments, conducted at moderate Reynolds number with the vortex rings produced by puffing air through elliptic orifices, are compared with the calculations. The agreement is fairly good and it is found that a vortex ring produced from an orifice of axes ratio 0·2 breaks up into two smaller rings. The relevance of the results to the vortex trail of an aircraft is discussed.

Vortex generation by deep-water breaking waves

“…It is natural to then consider the simple model proposed by Helmholtz (1858, see also Tait 1867), and quantified by Taylor (1953) (see also Saffman 1992, § 6.4), in which a half vortex ring is generated by impulsively forcing a thin disc, half submerged, through a perfect fluid with no flow separation. Breaking waves may have an asymmetry between the horizontal and vertical scales, so that a generalization of Taylor's (1953) work to elliptical vortex rings, by Dhanak & De Bernardinis (1981), is more applicable to our study. Elliptical vortex rings can be generated by slightly modifying the thought experiment of Helmholtz (1858) to consider the flow generated by impulsively forcing a thin elliptical disc, submerged along its major axis to the depth of its semi-minor axis, through a fluid.…”

“…This problem was considered by Dhanak & De Bernardinis (1981) in their work on elliptical vortex rings. Now, Dhanak & De Bernardinis (1981) find that the velocity potential at the surface of the disc is 18) where ± refers to the front and rear of the disc, respectively; e is the eccentricity and E (e) is the complete elliptic integral of the second kind:…”

“…The circulation around a contour starting at (x, y, z) = (0 + , 0, 0) and ending at (0 − , 0, 0) is given by Dhanak & De Bernardinis (1981) …”

“…Dhanak & De Bernardinis (1981) found that elliptical vortex rings will oscillate in shape as they propagate, and for cases where the vortex ring has large eccentricity, can potentially break-up into multiple rings. The time scale of these oscillations is on the order of 4πA 2 /Γ .…”

“…The fluid impulse P (Batchelor 1967, § 7.2), associated with the flow of the forced elliptical disc presented in § 2.2 is given by Dhanak & De Bernardinis (1981) …”