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

Abstract

The interaction of a straight diffusing vortex tube with a background uniform shear flow is investigated analytically. The vortex tube is aligned with the longitudinal uniform shear flow βye_z and simultaneously the tube undergoes the transverse weak shear αye_χ. In the limit of |α|/β&Lt;1, the asymptotic solution to the Navier-Stokes equation is obtained to examine energy dissipation as well as axial and cross-flow vorticity components around the vortex tube.Vortex lines of the longitudinal 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 transverse shear is anti-parallel (or parallel) to that of the tube, which leads to the enchancement (or reduction) of the total energy dissipation around the vortex tebu. It is shown that at initial time of evolution βt<0.623(Γ/2πν)^1/3the energy dissipation associated with the tube itself is dominant, while at later time βt>0.623(Γ/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 the kinematic viscosity.