Energy dissipation in spiral vortex layers wrapped around a straight vortex tube

Abstract

Energy dissipation in spiral layers of high azimuthal vorticity around a straight vortex tube is investigated analytically. Asymptotic expressions of local and total viscous dissipation are obtained for the spiral vortex layers. When a vortex tube, which aligns with a uniform shear flow of a shear rate S, starts with a vortex filament of circulation Γ at an initial instant t=0, it wraps and stretches background vorticity lines around itself to form double spiral vortex layers of intense dissipation. The contribution of the spiral layers to total dissipation per unit axial length is evaluated to be 1.29πν2S2(Γ∕2πν)4∕3t at large vortex Reynolds numbers Γ∕ν⪢1, ν being the kinematic viscosity of fluid. There exists the critical time after which the contribution of the spirals to the total dissipation dominates that of the tube. If the tube is tilted at a small angle α in the direction of the uniform shear vorticity, the spirals around the tube are cross axially sheared into different shapes depending on the sign of α, which leads to local reduction (or enhancement) of the energy dissipation in the spirals at α>0 (or <0). The primary effect of the cross-axial shear on the total dissipation is shown to be −14ανS3Γln(Γ∕2πν)t2 at St∣α∣⪡1 for Γ∕ν⪢1. The contribution to turbulent energy dissipation from spiral structures around a tubular vortex at a large-Reynolds-number limit is also discussed based upon recently reported direct numerical simulations and the present analysis.