An EDQNM study of the dissipation rate in isotropic non-equilibrium turbulence

Abstract

The energy cascade from large to small scales is a robust feature of three-dimensional turbulence. In statistically steady turbulence, the average dissipation is in equilibrium with the energy injected in the system. A global quantity measuring the deviations from such a flux equilibrium is the normalised dissipation rate $C_{ϵ}$, corresponding to the viscous dissipation, normalised by quantities associated with the largest scales of the system. Recent investigations have pointed out how this normalised dissipation rate varies in unsteady flows. We focus on two test-cases to assess non-equilibrium in isotropic turbulence. These cases are, respectively, turbulence in the presence of a large-scale periodic forcing and turbulence with reversed initial conditions. We show, using the Eddy-Damped Quasi-Normal Markovian closure, that for turbulence in the presence of periodic forcing, a $C_{ϵ}\sim R_{\lambda }^{-15/14}$ scaling is reproduced ($R_{\lambda }$ indicating the Taylor-scale Reynolds number) when the forcing-frequency is adjusted to be of the order of the inverse of the integral time-scale. It is shown that the spectrum can be decomposed into an equilibrium spectrum, governed by Kolmogorov scaling in the inertial range, and a perturbation spectrum, proportional to $k^{-7/3}$, k being the wavenumber. For reversed turbulence, a novel procedure is introduced to prescribe initial conditions for the nonlinear transfer. Subsequently a clear transient with a scaling $C_{ϵ}\sim R_{\lambda }^{-2}$ is observed in the dynamics.

Keywords:

• Isotropic turbulence
• EDQNM
• non-equilibrium

Acknowledgements

The book ‘turbulence in fluids’ by Marcel Lesieur [39] has played an essential role in the understanding of turbulence of one of the authors. It has accompanied WB during his carreer and the pages of chapter VII on EDQNM have all detached from their bindings. The code used in this work to integrate the EDQNM equations is practically the same as the one Marcel Lesieur and his colleagues used since the 1970s.

LF is supported by the Science Center for Gas Turbine Project (Project No. P2022-C-III-001-001). For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

Disclosure statement

No potential conflict of interest was reported by the author(s).