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

TL;DR

This study examines the energy transfer mechanisms in non-equilibrium grid-generated turbulence, revealing how production, transport, and dissipation scale and influence the energy cascade differently from equilibrium turbulence.

Contribution

It provides detailed measurements of scale-by-scale energy transfer in non-equilibrium turbulence and compares it with equilibrium conditions, highlighting the effects of non-equilibrium scaling on the energy cascade.

Findings

Production and transport are large-scale phenomena not affecting small-scale energy transfer.

The peak energy transfer scales as $(ar{u^2})^{3/2}/ ext{ell}$ in both regimes.

In non-equilibrium turbulence, an imbalance exists between energy transfer and dissipation at small scales.

Abstract

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 $m\approx 1 \approx n$ ($Re_M$ 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.