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

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 V b moves with a velocity ν b 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 δ H E in the integral of helicity H E in the external region V E outside the volume, i.e. H E = ∫ ν E u ·Ωdν changes by δ H E , where u and Ω are the velocity and vorticity fields. When the vorticity at infinity is weak (i.e. | Ω|V1/3b |U∞ - vb|) and when Ω is parallel to v b and U ∞ , the change in the external helicity integral, δ H E is proportional to the dipole strength of ν b . For the case of volumes with reflectional symmetry about an axis parallel to their direction of motion (e.g. axisymmetric volumes), δ H E = -((ν b − U ∞ )·Ω) ν b C H , where CH = 1/3(1+CM) , and C M is the added mass coefficient. So for a sphere moving along the axis of a pure rotating flow, δ H E = −½ν b (ν b ·Ω), which is negative. Larger values of the local helicity ( u ·Ω) are generated by the flow around a volume when (ν b − U ∞ ) Λ Ω ≠ 0, but for symmetric volumes there is no net contribution to δ H E if (ν b − 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 E b of the boundary defining a region of turbulent flow moving into an adjacent region of weak or non-existent turbulence.