, and thick Hill's-like vortex rings, i.e. , Hill's spherical vortex having a mean core radius of precisely . For mean core radii in between, one must rely on numerical methods. Norbury (1973) found numerically the resulting steady vortex ring of given mean core radius, and this for a set of 14 mean core radii ranging from 0.1 to 1.35. The resulting streamlines defining the core of the ring were tabulated, as well as the translational speed. In addition, the circulation, the hydrodynamic impulse and the kinetic energy of such steady vortex rings were computed and presented in non-dimensional form. Instabilities A kind of azimuthal radiant-symmetric structure was observed by Maxworthy when the vortex ring traveled around a critical velocity, which is between the turbulence and laminar states. Later Huang and Chan reported that if the initial state of the vortex ring is not perfectly circular, another kind of instability would occur. An elliptical vortex ring undergoes an oscillation in which it is first stretched in the vertical direction and squeezed in the horizontal direction, then passes through an intermediate state where it is circular, then is deformed in the opposite way (stretched in the horizontal direction and squeezed in the vertical) before reversing the process and returning to the original state. See also Air vortex cannon Bubble ring – underwater vortex ring Mushroom cloud Toroidal moment Vortex ring gun Vortex ring toy References External links YouTube video of Vortex ring cannon Fluid dynamics lecture covering vortices An animation of a vortex ring Giant vortex ring generator Toy Box Physics: Vortices, Air Cannons, and Mushroom Clouds Thesis on vortex ring formation and interactions Vortex half-ring in a pool, Dianna Cowern (Physics Girl), YouTube More experiments with vortex rings in a pool, Dianna Cowern (Physics Girl), YouTube Aerodynamics Aviation risks Helicopter aerodynamics Vortices