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Author(s):
G. Kawahara, S. Kida, M. Tanaka and S. Yanase
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Title:
Wrap, Tilt and Stretch of Vorticity Lines around a Strong Straight Vortex Tube in a Simple Shear Flow
Date of publication:
Oct. 1996
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Key words:
none
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Abstract:
The mechanism of wrap, tilt and stretch of vorticity lines around a strong straight vortex tube with circulation Gamma in a simple shear flow (U = SX_2 hat{X}_1, S being a shear rate) is investigated analytically. An asymptotic expression of the vorticity field is obtained at a large Reynolds number Gamma/nu gg 1, nu being the kinematic viscosity of fluid, and during the initial time St ll 1 of evolution as well as St ll (Gamma/nu)^1/2. The vortex tube, which is inclined from the streamwise (X_1) direction both to 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 two spiral vortex layers of high normal vorticity. The magnitude of the normal vorticity increases up to O ((Gamma/nu)^1/3 S) at distance r = O ((Gamma/nu)^1/3 (nu t)^1/2) apart 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 with 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 vortex tube at gamma = O ((Gamma/nu)^1/2 (nu t)^1/2) where the magnitude is strongest. In the near region gamma is less than or equal to (Gamma/nu)^1/4 (nu t)^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 normal vorticity. 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 to the opposite direction (an anti-cyclonic vortex) is weakened. The growth rate of vorticity due to this effect attains a maximum (or minimum) value of pm S^2 / 3^3/2 when the vortex tube is oriented to the direction of hat{X}_l + hat{X}_2 mp hat{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 helically 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.
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