| Abstract: | The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer. |
