Existence and Stability of Vortices and Vortex Arrays

The stability to three-dimensional disturbances of three classical steady vortex configurations in an incompressible inviscid fluid is studied in the limit of small vortex cross-sectional area and long axial disturbance wavelength. The configurations examined are the single infinite vortex row, the Karman vortex street of staggered vortices and the symmetric vortex street. It is shown that the single row is most unstable to a two-dimensional disturbance, while the Karman vortex street is most unstable to a three-dimensional disturbance over a significant range of street spacing ratios. The symmetric vortex street is found to be most unstable to three-dimensional or two-dimensional symmetric disturbances depending on the spacing ratio of the street. Short remarks are made concerning the relevance of the calculations to the observed instabilities in free shear layer, wake and boundary layer type flows. The three-dimensional linear stability of a steady rectilinear vortex of elliptical cross-section existing in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted for weak strain persists for finite strain and that the weak strain results continue to be quantitatively valid for finite strain. Parametric dependence of the growth rates of the unstable modes on the strain and the axial disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest. The radially symmetric Burgers' vortex is an example of a solution to the Navier-Stokes equations in which the intensification of vorticity due to vortex stretching is balanced by the diffusion of vorticity through viscosity. We present analytical solutions obtained from a perturbation analysis as well as numerical computations of non-symmetric Burgers' vortices in which the radial flow field in a plane perpendicular to the vorticity is non-symmetric. We also demonstrate the linear stability of the symmetric Burgers' vortex to a restricted class of two-dimensional perturbations.