  We use DNS to study inter-scale and inter-space energy exchanges in the
near-field of a turbulent wake of a square prism in terms of the KHMH equation
written for a triple decomposition of the velocity field accounting for the
quasi-periodic vortex shedding. Orientation-averaged terms of the KHMH are
computed on the plane of the mean flow and on the geometric centreline. We
consider locations between $2$ and $8$ times the width $d$ of the prism. The
mean flow produces kinetic energy which feeds the vortex shedding coherent
structures. In turn, these structures transfer energy to the stochastic
fluctuations over all length-scales $r$ from the Taylor length $\lambda$ to $d$
and dominate spatial turbulent transport of two-point stochastic turbulent
fluctuations. The orientation-averaged non-linear inter-scale transfer rate
$\Pi^{a}$ which was found to be approximately independent of $r$ by Alves
Portela et. al. (2017) in the range $\lambda\le r \le 0.3d$ at a distance
$x_{1}=2d$ from the square prism requires an inter-scale transfer contribution
of coherent structures for this approximate constancy. However, the
near-constancy of $\Pi^a$ at $x_1=8d$ which was also found by Alves Portela et.
al. (2017) is mostly due to stochastic fluctuations. Even so, the proximity of
$-\Pi^a$ to the turbulence dissipation rate $\varepsilon$ in the range
$\lambda\le r\le d$ at $x_1=8d$ requires contributions of the coherent
structures. Spatial inhomogeneity also makes a direct and distinct contribution
to $\Pi^a$, and the constancy of $-\Pi^a/\varepsilon$ 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,
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.
