1. The Vortex Ring Problem

This introductory chapter offers the physical and mathematical bases for the description of laminar vortex rings. We start by presenting the incompressible Navier–Stokes equations of motion and the vorticity equation in three dimensions. We use the cylindrical coordinate system which best fits the geometry of the problem. We subsequently simplify the vorticity equation for axisymmetric flows, with or without swirl. We then characterise the structure of vortex rings without swirl by presenting the space distribution of vorticity and the Stokes stream function in both laboratory and vortex frames of reference. The concepts of vortex bubble, core and inner core are introduced to describe the geometry of vortex rings. Circulation, hydrodynamic impulse and energy of the vortex ring are defined as main integral characteristics. We finally derive the Helmholtz–Lamb formula used to calculate the translational velocity of the vortex ring. This chapter is self-contained. However, Appendix A, containing all the details of the derivation of equations in cylindrical coordinates, could be a valuable companion for a reader who is not familiar with the form of equations in this particular coordinate system.