Two-point enstrophy budget and energy cascade in turbulence

According to the Kolmogorov's four-fifths law, the prominent feature of high Reynolds number flows is the energy transfer from large to small scales which is described by a single scalar quantity, the average dissipation rate. Kolmogorov's groundbreaking intuition was reducing the complex problem of turbulence to its essential features, by assuming homogeneity and isotropy. However, actual turbulent flows have a much richer physics, involving, beyond energy transfer, anisotropic production and inhomogeneous spatial fluxes. The energy injection/release associated with these phenomena gives rise to a split cascade where energy flows simultaneously both to small and large scales. The split in forward and reverse cascade is particularly relevant in wall turbulence where it challenges turbulence closures and theories. To shed light on these processes, we consider the exact evolution equations for the second-order moment of the two-point velocity and vorticity increments. Preliminary results on the flow settings of a temporally evolving boundary layer are presented and are shown to unveil interesting physical mechanisms at the basis of the split cascade phenomenon.