Vorticity Stretching and Energy Dissipation around a Straight Vortex Tube in a Uniform Shear Flow

The interaction of a straight diffusing vortex tube with a background uniform shear flow is investigated analytically. The vortex tube is aligned with the axial uniform shear flow βye_z and simultaneously the tube undergoes the cross-axial weak shear αye_x. In the limit of |α|/β≪1,the asymptotic solution to the Navier-Stokes equation is obtained to examine energy dissipation as well as axial and cross-axial vorticity components around the vortex tube. Vortex lines of the axial shear are wrapped and stretched by the tube to intensify the azimuthal vorticity. This vorticity stretching and intensification are found to be enhanced (or reduced) when the vorticity of the cross-axial shear is anti-parallel (or parallel) to that of the tube, which leads to the enhancement (or reduction) of the total energy dissipation around the vortex tube. It is shown that at the initial time of evolution βt<0.623(Г/2πν)^<1/3> the energy dissipation associated with the tube itself is dominant, while at a later time βt>0.623(Г/2πν)^<1/3> the dissipation of the wrapped vortex layers dominates that of the tube, where Γ is the circulation of the tube and ν is the kinematic viscosity.