Inviscid two dimensional vortex dynamics and a soliton expansion of the sinh-Poisson equation

Abstract

The dynamics of inviscid, steady, two dimensional flows is examined for the case of a hyperbolic sine functional relation between the vorticity and the stream function. The 2-soliton solution of the sinh-Poisson equation with complex wavenumbers will reproduce the Mallier-Maslowe pattern, a row of counter-rotating vortices. A special 4-soliton solution is derived and the corresponding flow configuration is studied. By choosing special wavenumbers complex flows bounded by two rigid walls can result. A conjecture regarding the number of recirculation regions and the wavenumber of the soliton expansion is offered. The validity of the new solution is verified independently by direct differentiation with a computer algebra software. The circulation and the vorticity of these novel flow patterns are finite and are expressed in terms of well defined integrals. The questions of the linear stability and the nonlinear evolution of a finite amplitude disturbance of these steady vortices are left for future studies.