Counter-rotating vortex patches in shear: a model of the effect of wind shear on aircraft trailing vortices

Abstract

The shape and stability of two–dimensional, uniform patches of vorticity in incompressible inviscid fluid are calculated. The object of the study is to investigate the effect of wind shear on trailing vortices. The wind shear is modelled by simple shear with velocities in the plane perpendicular to the trailing vorticity. The first portion of this study models the trailing vortices as a pair of uniform straight counter–rotating line vortices which are essentially point vortices. They move under their self–induced velocities and the external shear. It is shown that the equations for the positions of the vortices can be solved exactly, and they move on parabolic trajectories with the horizontal and vertical separations kept constant. The trajectories are weakly unstable: infinitesimal disturbances grow algebraically in time. But, apart from the effect of wind shear on the horizontal displacement and an asymmetry of the streamlines, wind shear has no significant effect on the motion of the vortex pair under the point vortex approximation. The situation is different when the finite size of the trailingvortex cores is taken into account. Following a method originally suggested in a personal communication from Jimenez, Schwarz functions are used to find the shape of steady flows and their stability to infinitesimal two–dimensional disturbances. It is found that the vortex patches may be deformed significantly, or very slightly, depending upon the sense of rotation of the trailing vortex relative to the sense of rotation in the external simple shear. When co–rotating, the patch remains circular to a good approximation, as would a single co–rotating patch in a uniform shear, a case which can be solved exactly. But when counter–rotating, the patch may be significantly deformed into a long, thin shape, perpendicular to the line joining the centroids of the patches. The configuration is unstable to small disturbances but the eigenfunction corresponding to a deformation of the co–rotating vortex patch is small compared with that for the counter–rotating patch. This type of asymmetric behaviour is seen in the observations and suggests that the asymmetric properties are due to wind shear, although other explanations may be significant. It is shown that the heavy numerical calculations which are required by the Schwarz function approach for two or more patches can be bypassed by use of an approximation called the elliptic patch model, which gives equations of Hamiltonian form for the position of the vortex centroid and the orientation of the elliptical patch which models the vortices. The Moore–Saffman dispersion relation for steady states can be shown to be a consequence of the model.

Additional details