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 = SX 2 X 1 , S being a shear rate) is investigated analytically. An asymptotic expression for the vorticity field is obtained at a large Reynolds number Γ /ν 1, ν being the kinematic viscosity of fluid, and during the initial time S t 1 of evolution as well as S t (Γ /ν) 1/2 . The vortex tube, which is inclined from the streamwise (X 1 ) direction both in the vertical (X 2 ) and spanwise (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 swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity increases1/2 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 S 2 t, the direction of which is opposite to that of the vortexwhere the vorticity magnitude is strongest. In the near region r (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 ±S 2 /3 3/2 when the vortex tube is oriented in the direction of X 1 + X 2 ∓ X 3 . 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 oppositesigned vorticity around streamwise vortices and a zig-zag arrangement of streamwise 116 G. Kawahara, S. Kida, M. Tanaka and S. Yanase vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.

Vortex Dynamics in Turbulence

“…This scenario includes both the generalized Burgers' vortex and the stretched-spiral vortex. A further special case has been analyzed by Kawahara et al (1997). They consider the diffusion of a strong line vortex embedded in a simple background shear flow, and they analyze the distortion of the vortex lines of this shear by the line vortex as it rotates and diffuses.…”

“…There is clear potential for constructing a cascade mechanism. No turbulence calculations with this model are reported by Kawahara et al (1997).…”

A classification method for vortex sheet and tube structures in turbulent flows

“…We consider that the deviation from the linear profile was generated because Burgers' vortex layer is susceptible to small disturbances, however, it may imply the existence of some deviations from the pure Burgers' vortex layer model. 13,27,28 It is shown that the sheetlike structures in which the amplitude of strain is large are decomposed into two groups with different scalings between Ϯ and Ϯ . Because Ϯ and Ϯ consist of first-and second-order products, respectively, of the velocity gradient tensor, an estimation of flatness and curvature of the object can be accomplished by combining these two eigenvalues.…”

A Survey on a Class of Exact Solutions of the Navier–Stokes Equations and a Model for Turbulence

“…Some comments of related problems are in order. The present study is related with of works of [PA84,M85,N,C89,S90,KKTY97] which investigate motion of diffusive vortex in a simple shear. The difference is that in the present case the axial velocity u z (r, θ) decays in x-y plane, rather than a simple shear e.g.…”