We use Direct Numerical Simulation (DNS) data to study inter-scale and inter-space energy exchanges in the near-field of a turbulent wake of a square prism in terms of a Kármán-Howarth-Monin-Hill (KHMH) equation written for a triple decomposition of the velocity field which takes into account the presence of quasi-periodic vortex shedding
coherent structures. Concentrating attention on the plane of the mean flow and on the geometric centreline, we calculate orientation-averages of every term in the KHMH equation. The near-field considered here ranges between 2 and 8 times the width d of the square prism and is very inhomogeneous and out of equilibrium so that non-stationarity
and inhomogeneity contributions to the KHMH balance are dominant. The mean flow produces kinetic energy which feeds the vortex shedding coherent structures. In turn, these coherent structures transfer their energy to the stochastic turbulent fluctuations over all length-scales r from the Taylor length to d and dominate spatial turbulent transport of small-scale two-point stochastic turbulent fluctuations. The orientation averaged non-linear inter-scale transfer rate a which was found to be approximately independent of r by Alves Portela et al. (2017) in the range 6 r 6 0:3d at a distance x1 = 2d from the square prism requires an inter-scale transfer contribution of coherent structures for this approximate constancy. However, the near-constancy of a in the range 6 r 6 d at x1 = 8d which was also found by Alves Portela et al. (2017) is mostly
attributable to stochastic fluctuations. Even so, the proximity of 􀀀 a to the turbulence dissipation rate " in the range 6 r 6 d at x1 = 8d does require inter-scale transfer contributions of the coherent structures. Spatial inhomogeneity also makes a direct and distinct contribution to a, and the constancy of 􀀀 a=" close to 1 would not have been possible without it either in this near-field flow. Finally, the pressure-velocity term is also an important contributor to the KHMH balance in this near-field, particularly at scales r larger than about 0:4d, and appears to correlate with the purely stochastic non-linear inter-scale transfer rate when the orientation average is lifted.