On the generation and flux of enstrophy in isotropic turbulence

Abstract

We investigate the manner in which enstrophy is distributed across the different scales in three-dimensional, isotropic turbulence. Unlike inviscid invariants, such as energy, enstrophy can be generated in a turbulent flow, and so we propose that the effects of inertia be divided into two distinct processes: the generation of enstrophy above a given scale r, G ^L (r), and the flux of enstrophy across scale r to smaller scales, F(r). Explicit expressions for G ^L (r) and F(r) are obtained in terms of two-point triple correlations. These are used to show that, in the inertial sub-range: (i) there is a flux of enstrophy of orderF(r)∼ ϵ/r ², shadowing the flux of energy; (ii) the rate of generation of enstrophy is governed by G ^L (r) = 5⟨ ω²(Δ u _///Δ u _// r)⟩, which admits a simple physical interpretation in terms of vortex stretching at scale r; and (iii) there is a near perfect balance between the generation of enstrophy above scale r and the flux across scale r to smaller scales, F(r) ≈ G ^L (r). (Here ϵ is the energy dissipation rate, η is the Kolmogorov scale, x′ = x + r, ω = 1½2(ω+ω′), Δ u = u′− u and Δ u _// is the component of Δ u parallel to r.) This balance between the flux and generation of enstrophy in the inertial range suggests that the physical processes responsible for the generation and flux of enstrophy are one and the same. This is consistent with Helmholtz's laws. That is, as enstrophy is increased at a particular scale by the inviscid stretching of vortices (i.e. blobs, sheets or tubes of vorticity), so these vortices contract.

Keywords:

• homogeneous turbulence
• enstrophy flux
• enstrophy generation

Acknowledgement

This work was partly supported by a JSPS grant-in-aid for Japan–UK Bilateral Joint Projects.