Nonequilibrium energy spectrum in the subgrid-scale one-equation model in large-eddy simulation

Abstract

The subgrid-scale (SGS) modeling in large-eddy simulation (LES) which accounts for the effect of unsteadiness and nonequilibrium state in the SGS is considered. Unsteadiness is incorporated by considering the spectral evolution in the forced homogeneous isotropic turbulence using the transport equation for the SGS energy. As for the unfiltered spectrum, perturbative expansion of the Kovasnay spectral model about the Kolmogorov −5/3 energy spectrum which constitutes a base equilibrium state in the inertial subrange, yields the extra components with −7/3 and −9/3 powers. It is shown that these spectra are actually extracted in the direct numerical simulation (DNS) data and these components govern the unsteady energy transfer. As for the SGS real-space representation of the spectral model, we consider the SGS one-equation model. The perturbation expansion is applied to the one-equation model by setting the base SGS energy as the standard Smagorinsky model, which assumes the equilibrium state in the SGS and its spectral counterpart is the Kolmogorov −5/3 spectrum. The solution yields the terms whose spectral counterparts are the components with −7/3 and −9/3 powers. These additional terms are induced by temporal variations of the base SGS energy. In the temporal variations of the grid-scale energy, SGS energy, SGS production term, and SGS dissipation which are obtained by applying the filter to the DNS data, it is shown that these quantities lag in time in this order. This time-lag is not realized in the standard Smagorinsky model and the one-equation model because the SGS dissipation is defined so that it instantaneously adjusts to the SGS energy. In the one-equation model, the direction of the energy cascade in the initial period is opposite to that obtained in the DNS data. To retrieve correct time-lag and direction of energy transfer, we relax this instantaneous adjustment and propose the nonequilibrium Smagorinsky model. In this nonequilibrium model, the SGS energy incurred by the −7/3 spectrum is added to the base Smagorinsky energy. Assessment in actual LES shows that the time-lag predicted using the standard Smagorinsky and the one-equation models is inaccurate, whereas good agreement with the DNS data is achieved in the nonequilibrium Smagorinsky model. Extraction of the grid-scale nonequilibrium energy spectrum yields the −7/3 and −9/3 components in addition to the base −5/3 spectrum. In the nonequilibrium Smagorinsky model, continuation of the grid-scale spectra into the SGS is established for the −5/3 and −7/3 components. As a result, the unsteady energy transfer is more accurately predicted, whereas the standard Smagorinsky model does not have the SGS counterpart for the −7/3 component. Feasibility of employing the eddy-viscosity approximation to account for the transfer in the period in which −9/3 spectrum prevails is discussed.