Axial flow in trailing line vortices

Abstract

Axial flow in the core of a laminar steady trailing vortex from the tip of a semi-infinite wing is analyzed assuming small departure of the axial velocity from the free-stream velocity. It is further assumed that the axial pressure gradient is determined by the swirl velocities of an ideal infinite line vortex in which the radial and the associated axial velocity variations are neglected in the equation for the angular momentum. The axial and lateral variations of the axial velocity depend on the strength of the vortex and initial axial velocity distribution which must be specified at some station behind the wing except at the virtual origin of the vortex where a nonintegrable singularity exists. Numerical solutions for the axial velocity are obtained using the axial pressure gradient given by the line vortex and analytical solutions are obtained using an equivalent axial pressure gradient with good agreement between the two sets of axial velocity distributions. Resolution of the previous uncertainties in this field is given which were due to the unrecognized singularity at the virtual origin of the vortex. Using the calculated axial velocity the neglected radial and the associated axial fluxes of angular momentum are determined and the limits of validity of the theory presented here in terms of a suitably defined vortex Reynolds number and a nondimensional distance measured from the virtual origin of the vortex are given.