On the simultaneous cascades of energy, helicity, and enstrophy in incompressible homogeneous turbulence

We characterise the incompressible turbulence cascade in terms of the concurrent inter-scale and inter-space exchanges of the scale-by-scale energy, helicity and enstrophy. The governing equations for the scale-by-scale helicity and enstrophy are derived in a similar fashion to that of the second order structure function obtained by Hill (2002). We apply these equations to both forced periodic turbulence and a von Kármán flow, focusing on scales in the dissipative range r = 2.5η, the near-dissipative range r = 0.5λ and the onset inertial range r = λ (where η and λ are the Kolmogorov and Taylor length scales, respectively). The well-known random sweeping effect in the energy cascade between unsteady and non-linear transport is found to extend across all quantities considered. Additional mechanisms within the individual cascades are identified. Across cascades, the helicity cascade was found to be decorrelated from the others at all scales, however a connection between the energy cascade and helicity itself is identified kinematically. Further, the large-scale sweeping motions captured by the two-point velocity sum Σu i = u i + u i are shown to be a key element in the cascade of energy