A note on velocity, vorticity and helicity of inviscid fluid elements

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Abstract
A brief description is given, with a new geometrical derivation, of the changes in velocity, vorticity and helicity of fluid elements and fluid volumes in inviscid flow. When a compact material volume Vb moves with a velocity νb in a flow which at infinity has a velocity U∞ and uniform vorticity Ω, it is shown that in general there is a net change δHE in the integral of helicity HE in the external region VE outside the volume, i.e. HE = ∫νEu·Ωdν changes by δHE, where u and Ω are the velocity and vorticity fields. When the vorticity at infinity is weak (i.e. , and CM is the added mass coefficient. So for a sphere moving along the axis of a pure rotating flow, δHE = −½νb (νb·Ω), which is negative. Larger values of the local helicity (u·Ω) are generated by the flow around a volume when (νb−U∞) Λ Ω ≠ 0, but for symmetric volumes there is no net contribution to δHE if (νb−U∞) Λ Ω = 0. These results are used to develop some new physical concepts about helicity in turbulent flow, in particular concerning the helicity associated with eddy motions in rotating flows and the relative speed Eb of the boundary defining a region of turbulent flow moving into an adjacent region of weak or non-existent turbulence.

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