A computational study of three-dimensional vortex–cylinder interaction is reported for the case where the nominal orientation of the cylinder axis is normal to the vortex axis. The computations are performed using a new tetrahedral vorticity element method for incompressible viscous fluids, in which vorticity is interpolated using a tetrahedral mesh that is refit to the Lagrangian computational points at each timestep. Fast computation of the Biot-Savart integral for velocity is performed using a box-point multipole acceleration method for distant tetrahedra and Gaussian quadratures for nearby tetrahedra. A moving least-square method is used for differentiation, and a flux-based vorticity boundary condition algorithm is employed for satisfaction of the no-slip condition. The velocity induced by the primary vortex is obtained using a filament model and the Navier–Stokes computations focus on development of boundary-layer separation from the cylinder and the form and dynamics of the ejected secondary vorticity structure. As the secondary vorticity is drawn outward by the vortex-induced flow and wraps around the vortex, it has a substantial effect both on the essentially inviscid flow field external to the boundary layer and on the cylinder surface pressure field. Cases are examined with background free-stream velocity oriented in the positive and negative directions along the cylinder axis, with free-stream velocity normal to the cylinder axis, and with no free-stream velocity. Computations with no free-stream velocity and those with free-stream velocity tangent to the cylinder axis exhibit similar secondary vorticity structures, consisting of a vortex loop (or hairpin) that wraps around the primary vortex and is attached to the cylinder boundary layer at two points. Computations with free-stream velocity oriented normal to the cylinder axis exhibit secondary vorticity structure of a markedly different character, in which the secondary eddy remains close to the cylinder boundary and has a quasi-two-dimensional form for an extended time period.