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

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 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 δH_E in the integral of helicity H_E in the external region V_Eoutside the volume, i.e. H_E = ∫_νEu·Ωdν changes by δH_E, where u and Ω are the velocity and vorticity fields. When the vorticity at infinity is weak (i.e. $|{\boldmath \Omega}|V^{\frac{1}{3}}_{\rm b}\ll |U_{\infty} - v_{\rm b}|)$ and when Ω is parallel to v_b and U_∞, the change in the external helicity integral, δH_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), δH_E = -((ν_b−U_∞)·Ω) ν_bC_H, where $C_H = \frac{1}{3}(1+C_M)$, and C_M is the added mass coefficient. So for a sphere moving along the axis of a pure rotating flow, δH_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 δH_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.