Time tracking and interaction of energy-eddies at different scales

We study the energy cascade through coherent structures obtained in time-resolved simulations of incompressible, statistically steady isotropic turbulence. The structures are defined as geometrically connected regions of the flow with high kinetic energy. We compute the latter by band-pass filtering the velocity field around a scale $r$. We analyse the dynamics of structures extracted with different $r$, which are a proxy for eddies containing energy at those $r$. We find that the size of these ``energy-eddies'' scales with $r$, while their lifetime scales with the local eddy-turnover $r^{2/3}\epsilon^{-1/3}$, where $\epsilon$ is the energy dissipation averaged over all space and time. Furthermore, a statistical analysis over the lives of the eddies shows a slight predominance of the splitting over the merging process. When we isolate the eddies which do not interact with other eddies of the same scale, we observe a parent-child dependence by which, on average, structures are born at scale $r$ during the decaying part of the life of a structure at scale $r'>r$. The energy-eddy at $r'$ lives in the same region of space as that at $r$. Finally, we investigate how interactions between eddies at the same scale are echoed across other scales.