Vortex sheet

Abstract: An axi-symmetric and swirling vortex sheet is investigated as the simplest flow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stability of a single sheet but instability for other configurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and growth rates (or frequencies) can be explicitly expressed. Subsequent nonlinear evolution is numerically simulated using a vortex method. The numerical results for an axi-symmetric swirling sheet with a vortex line along the axis of symmetry show detachment of a vortex ring from the sheet into the outer fluid, and collapse of the sheet onto the vortex line at some points. Vortex cancellation, which in the presence of viscosity would likely lead to vortex line reconnection, seems to occur in both of these phenomena. The evolution of two co-axial, axi-symmetric, swirling vortex sheets is similar.

Abstract: A new family of solutions to the steady Euler equations corresponding to spatially periodic states of a free shear layer is reported. This family bifurcates from a parallel shear layer of finite thickness and uniform vorticity, and extends continuously to a shear layer consisting of a row of concentrated pointlike vortices. The energetic properties of the family are considered, and it is concluded that a vortex in a row of uniform vortices produced by periodic roll-up of a vortex sheet must have a major axis of length approximately 50% or more of the distance between vortex centres; it is also concluded that vortex amalgamation events tend to reduce vortex size relative to spacing. The geometric and energetic properties of the solutions confirm the mathematical basis of the tearing mechanism of shear-layer growth first proposed in an approximate theory of Moore & Saffman (1975).

Abstract: An exact calculation is given of the acoustic radiation from a time dependent flow coupled to an inhomogeneous solid surface. Specifically, the flow consists of a vortex sheet leaving a semi-infinite plate and undergoing a two-dimensional spatial Kelvin-Helmholtz instability. In the absence of the plate, such an instability mode of the vortex sheet generates no sound. In the presence of a rigid plate, it is found that the intensity-directivity law is $I\sim U^{4}$sin$^{2}\frac{1}{2}\theta $, with $\theta $ measured from the downstream direction. If the plate is compliant and fluid loading effects high, the radiation is weaker, with $I\sim U^{5}$sin$^{2}\theta $. These results agree completely with those predicted from general theories of the scattering of the near-field of point quadrupoles by large wedge-shaped surfaces (Ffowcs Williams & Hall 1970; Crighton & Leppington 1970, 1971). Imposition of the 'rectified' Kutta condition of Orszag & Crow (1970) does not modify the sound field. Application of the 'full' Kutta condition, that the sheet leaves the plate at zero gradient, results in an enormous increase in the radiation, with $I\sim U^{2}$cosec$^{2}\frac{1}{2}\theta $.