Vortex Dynamics

Abstract

In the past three decades, the study of vortices and vortex motions - which originated in Helmholtz's great paper of 1858, 'Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen' (translated by Tait [1867]), and continued in the brilliant work of Lord Kelvin and others in the nineteenth century, and Prandtl and his Gottingen school in the first half of this century - has received continuing impetus from problems arising in physics, engineering and mathematics. As aptly remarked by Kiichemann [1965], vortices are the 'sinews and muscles of fluid motions'. Aerodynamic problems of stability, control, delta wing aerodynamics, high lift devices, the jumbo jet wake hazard phenomenon, among other concerns, have led to a myriad of studies. Smith [1986] reviews some of this work. The realisation that many problems involving interfacial motion can be cast in the form of vortex sheet dynamics has stimulated much interest. The discovery (rediscovery?) of coherent structures in turbulence has fostered the hope that the study of vortices will lead to models and an understanding of turbulent flow, thereby solving or at least making less mysterious one of the great unsolved problems of classical physics. Vortex dynamics is a natural paradigm for the field of chaotic motion and modern dynamical system theory. It is perhaps not well known that the father of modern dynamics and chaos wrote a monograph on vorticity (Poincait [1893]). The theory of line vortices and vortex rings is a part of modern macroscopic treatments of liquid helium II, as is described by Donnelly and Roberts [1974]. Even Kelvin's [1867a] vortex theory of matter may one day have some topical interest.