The energy cascade in grid-generated non-equilibrium decaying turbulence

We investigate non-equilibrium turbulence where the non-dimensionalised dissipation coefficient $C_{varepsilon}$ scales as $C_{varepsilon} sim Re_{M}^{m}/Re_{ell}^{n}$ with $mapprox 1 approx n$ ($Re_M$ and $Re_{ell}$ are global/inlet and local Reynolds numbers respectively) by measuring the downstream evolution of the scale-by-scale energy transfer, dissipation, advection, production and transport in the lee of a square-mesh grid and compare with a region of equilibrium turbulence (i.e. where $C_{varepsilon}approx mathrm{constant}$). These are the main terms of the inhomogeneous, anisotropic version of the von K{a}rm{a}n-Howarth-Monin equation. It is shown in the grid-generated turbulence studied here that, even in the presence of non-negligible turbulence production and transport, production and transport are large-scale phenomena that do not contribute to the scale-by-scale balance for scales smaller than about a third of the integral-length scale, $ell$, and therefore do not affect the energy transfer to the small-scales. In both the non-equilibrium and the equilibrium decay regions, the peak of the scale-by-scale energy transfer scales as $(overline{u^2})^{3/2}/ell$ ($overline{u^2}$ is the variance of the longitudinal fluctuating velocity). In the non-equilibrium case this scaling implies an imbalance between the energy transfer to the small scales and the dissipation. This imbalance is reflected on the small-scale advection which becomes larger in proportion to the maximum energy transfer as the turbulence decays whereas it stays proportionally constant in the further downstream equilibrium region where $C_{varepsilon} approx mathrm{constant}$ even though $Re_{ell}$ is lower.