The turbulence cascade in the near wake of a square prism

<jats:p>We present a study of the turbulence cascade on the centreline of an inhomogeneous and anisotropic near-field turbulent wake generated by a square prism at a Reynolds number of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017003901_inline1"/><jats:tex-math>$Re=3900$</jats:tex-math></jats:alternatives></jats:inline-formula>using the Kármán–Howarth–Monin–Hill equation. This is the fully generalised scale-by-scale energy balance which, unlike the Kármán–Howarth equation, does not require homogeneity or isotropy assumptions. Our data are obtained from a direct numerical simulation and therefore enable us to access all of the processes involved in this energy balance. A significant range of length scales exists where the orientation-averaged nonlinear interscale transfer rate is approximately constant and negative, indicating a forward turbulence cascade on average. This average cascade consists of coexisting forward and inverse cascade behaviours in different scale-space orientations. With increasing distance from the prism but within the near field of the wake, the orientation-averaged nonlinear interscale transfer rate tends to be approximately equal to minus the turbulence dissipation rate even though all of the inhomogeneity-related energy processes in the scale-by-scale energy balance are significant, if not equally important. We also find well-defined near<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017003901_inline2"/><jats:tex-math>$-5/3$</jats:tex-math></jats:alternatives></jats:inline-formula>energy spectra in the streamwise direction, in particular at a centreline position where the inverse cascade behaviour occurs for streamwise oriented length scales.</jats:p>