6 - Creation of vorticity

Summary

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 = U_y 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).