Formation of a vortex sheet

In perfect barotropic fluid, acted upon by conservative forces with a single-valued potential, the first Helmholtz law (§1.5) says that it is not possible to endow a fluid particle with vorticity, and Kelvin's circulation theorem (§1.6) shows that the circulation around a material circuit is zero if initially zero. The question arises whether vorticity can be created without violating these theorems, and without invoking viscosity, non-conservative forces or baroclinic effects. There is no a priori reason why they are not important in subsequent motion if present initially, and so one wishes to know if vorticity can be created without appeal to these effects.

Klein [1910] addressed this question with his Kaffeelöffel experiment. (See also Betz [1950].) The conclusion is that the Helmholtz and Kelvin theorems preclude the generation of piece-wise continuous vorticity, but do not prevent the formation of vortex sheets or the generation of circulation. Consider Klein's experiment. A two-dimensional plate of width 2a is set in motion through a perfect incompressible fluid with velocity U normal to the plate. We introduce the complex potential w(z) = φ+iψ, z = x+iy. The boundary conditions are ψ = Uy on x = 0, |y| < a (the axes are taken to coincide instantaneously with the plate with the y-axis along the plate and the x-axis in the direction of motion), and w ∼ 0 as z → ∞ (circulation at infinity is not allowed).

Vortex jumps

It has been supposed so far that the velocity and vorticity fields are continuous and have continuous derivatives. In a real viscous fluid, these assumptions are generally regarded as appropriate, and the velocity field is assumed analytic everywhere, except possibly at an initial instant o, when the speed of boundaries changes in a non-analytic manner. However, if we go to the the limit of vanishingly small viscosity and consider ideal fluids, to which the Helmholtz laws apply, the mathematics allows non-analytic behaviour and we cannot assert on physical grounds that only continuous fields should be considered. Velocity or vorticity fields with singularities or discontinuities are indeed of considerable importance. The singularities are, of course, not arbitrary and must be consistent with integral forms of the Euler equations or equivalently with the conservation of mass, momentum and energy. In particular, the dynamical constraint that pressure is continuous across a surface of discontinuity must be satisfied unless there are also singularities in the external force fields. We suppose in this chapter that the density is uniform and put it equal to unity unless explicitly stated otherwise. Also, external forces are supposed conservative unless non-conservative forces are explicitly introduced.

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 Göttingen school in the first half of this century – has received continuing impetus from problems arising in physics, engineering and mathematics. As aptly remarked by Küchemann [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 (Poincaré [1893]).