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

Abstract

We investigate non-equilibrium turbulence where the non-dimensionalised dissipation coefficient Cε scales as Cε∼ReMm/Reℓn with m ≈ 1 ≈ n (ReM and Reℓ 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 where Cε ≈ constant. These are the main terms of the inhomogeneous, anisotropic version of the von Kármán-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, ℓ, and therefore do not affect the energy transfer to the small-scales. In both the non-equilibrium region where Cε∼ReMm/Reℓn and further downstream where Cε ≈ constant, the peak of the scale-by-scale energy transfer scales as (u2¯)3/2/ℓ (u2¯ 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 region where Cε ≈ constant even though Reℓ is lower.