Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow

Abstract

The mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation Γ starting with a vortex filament in a simple shear flow (U=SX₂Xˆ₁, S being a shear rate) is investigated analytically. An asymptotic expression for the vorticity field is obtained at a large Reynolds number Γ/ν[Gt ]1, ν being the kinematic viscosity of fluid, and during the initial time St[Lt ]1 of evolution as well as St[Lt ](Γ/ν)^1/2. The vortex tube, which is inclined from the streamwise (X₁) direction both in the vertical (X₂) and spanwise (X₃) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. The simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity increases up to O((Γ/ν)^1/3S) at distance r=O((Γ/ν)^1/3 (νt)^1/2) from the vortex tube. The spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. As a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S²t, the direction of which is opposite to that of the vortex tube at r=O((Γ/ν)^1/2 (νt)^1/2) where the vorticity magnitude is strongest. In the near region r[Lt ](Γ/ν)^1/3 (νt)^1/2, on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. Only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. As a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this effect attains a maximum (or minimum) value of ±S²/3^3/2 when the vortex tube is oriented in the direction of Xˆ₁+Xˆ₂∓ Xˆ₃. The present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.