WRAP, TILT AND STRETCH OF VORTICITY LINES AROUND A STRONG THIN STRAIGHT VORTEX TUBE IN A SIMPLE SHEAR-FLOW

The mechanism of wrap, tilt and stretch of vorticity lines around a st rong thin straight vortex tube of circulation Gamma starting with a vo rtex filament in a simple shear flow (U = SX2 (X) over cap(1)$, S bein g a shear rate)is investigated analytically. An asymptotic expression for the vorticity held is obtained at large Reynolds number Gamma/nu m uch greater than 1, nu being the kinematic viscosity of fluid, and dur ing the initial time S t much less than 1 of evolution as well as S t much less than (Gamma/nu)(1/2). Th, vortex tube, which is inclined fro m the streamwise (X-1) direction both in;he vertical (X-2) and spanwis e (X-3) 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 swir ling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuth al vorticity increases up to O ((Gamma/nu)S-1/3) at distance r = O((Ga mma/nu)(1/3)(nu t)(1/2)) from the vortex tube. The spirals induce axia l flows of the same spiral shape with alternate sign in adjacent spira ls which in turn tilt the simple shear vorticity toward the axial dire ction. As a result, the vorticity lines wind helically around the vort ex tube accompanied by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S(2)t, t he direction of which is opposite to that of the vortex tube at r = O ((Gamma/nu)(1/2)(nu t)(1/2)) where the vorticity magnitude is stronges t. In the near region r much less than (Gamma/nu)(1/3)(nu t)(1/2), On other band, a viscous cancellation takes place in tightly wrapped vort icity of alternate sign, which leads to the disappearance of the vorti city 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 di rection of the simple shear vorticity (a cyclonic vortex) is intensifi ed, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this effect a ttains a maximum (or minimum) value of +/-S-2/3(3/2) when the vortex t ube is oriented in the direction of (X) over cap(1)$(X) over cap(2)$+/ -(X) over cap(3)$. The present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes ob served in various kinds of turbulence such as helical winding of vorti city lines around a vortex tube, the dominance of cyclonic vortex tube s,the appearance of opposite-signed vorticity around streamwise vortic es and zig-zag arrangement of streamwise vortices in homogeneous isotr opic turbulence, homogeneous shear turbulence and near-wall turbulence .