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Helicity generation and vorticity dynamics in helically symmetric flow

Published online by Cambridge University Press:  26 April 2006

Masanori Takaoka
Masanori Takaoka
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan Present address: Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan.
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Abstract

Helicity generation and vorticity dynamics in a helically symmetric flow are studied numerically by solving the Navier–Stokes equations in an unbounded domain. The helical symmetry reduces the problem to one in two dimensions, which makes it practicable to use an analytical expression for vortex surfaces. Furthermore, the field still retains three-dimensional aspects, such as helicity and vortex stretching. To every vortex surface there corresponds an inviscid invariant of helicity.

Our initial conditions are chosen as two cases of twisted elliptical tubes of high vorticity. The first case has elliptical vortex surfaces, which is a helically symmetric version of the initial condition employed by Aref & Zawadzki (1991), but the second case has axisymmetric vortex surfaces. The total helicity inside every vortex surface is zero in both the initial fields. It is found that vortex stretching plays an important role in the time evolution of the first case, but not of the second case.

We examine the relation of the vorticity dynamics to the helicity generation by using the representation of the vortex lines and the vortex surfaces rather than the equi-vorticity surfaces. This leads to new concepts for the mechanisms of formation of the spiral vortex structures observed for the two cases. The detailed investigation of the helicity generation is done by examining the distribution of the helicity on each vortex surface and the Fourier spectrum of helicity. The processes of helicity generation due to the vortex stretching are different for each initial condition. The viscosity dependence of ‘inviscid’ invariants shows that with smaller viscosity, only in the first case is more helicity generated. This is because in the first case where there is vortex stretching the contact zones of the adjacent vortex layers are elongated and the local vorticity is intensified to an extent limited by viscosity. Thus with smaller viscosity a more intense vortex is reconnected to generate more helicity. As expected, in both cases more of the energy is preserved with smaller viscosity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 319 , 25 July 1996 , pp. 125 - 149
Copyright
© 1996 Cambridge University Press

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