Nonequilibrium state in energy spectra and transfer with implications for topological transitions and SGS modeling

This paper reviews the recent progress in studies on the nonequilibrium statistics of turbulence. The structure of the energy spectrum in the inertial subrange is studied using direct numerical simulation (DNS) data for turbulence in a periodic box at high Reynolds numbers. A perturbation expansion for the energy spectrum about a base Kolmogorov k(-5/3) steady state yields additional -7/3 and -9/3 power components that are induced by the fluctuation of the dissipation rate epsilon and represents a nonequilibrium state. The nonequilibrium component is extracted by applying a conditional sampling on d epsilon/dt to the DNS data, and it is shown that the deviation from the base -5/3 spectrum fits the -7/3and -9/3 power slopes. The temporal development of the spectrum is divided into two regimes, phases 1 and 2. The large amount of energy contained in the low-wavenumber range in Phase 1 is cascaded to the small scales in Phase 2. This energy transfer is accomplished by the reversal in the sign of the -7/3 power component. Correlation of the appearance of the nonequilibrium spectrum and the transition in the mode of the configuration of the stretched spiral vortex is discussed. Occurrence of transition is identified using the helicity. Subgrid-scale (SGS) modeling in LES that accounts for the effect of unsteadiness and nonequilibrium state is considered by employing the transport equation for the SGS energy (one-equation model). Perturbation expansion about the Kolmogorov -5/3 energy spectrum which constitutes a base equilibrium state in the inertial subrange yields -7/3 spectrum as in the unfiltered case. These spectra are extracted in the DNS data, and their roles in the generation of the energy cascade are revealed. The SGS energy spectrum which governs the one-equation model is sought in a perturbative manner. Besides the base -5/3 spectrum assumed in the Smagorinsky model, -7/3 power component is derived, which is induced by temporal variations of SGS energy. The nonequilibrium Smagorinsky model in which estimate of the SGS energy based on the -7/3 spectrum is added to the Smagorinsky model is proposed. Assessment of the nonequilibrium Smagorinsky model in forced homogeneous isotropic turbulence showed that the performance of the Smagorinsky and one-equation models for prediction of temporal variations of turbulence energy is not satisfactory, but improvement is achieved in the new model.