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.

Dissipation in unsteady turbulence

“…We reproduce a simple derivation of the non-equilibrium correction here. The same results were obtained by [23,24] using similarity arguments, using Kovaznay's closure in [25] and using more sophisticated closures in [22,26].We start from the evolution equation for the kinetic energy spectrum at high Reynolds numbers at scales where both production and dissipation mechanisms can be ne-arXiv:1610.04054v1 [physics.flu-dyn]…”

Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling

“…If we carefully visit the regions of coherent structures, it can be found that none of these assumptions is satisfied: i) the Reynolds number may be not high (usually Re λ < 1000), which does not lead to obvious inertial range [27]; ii) the flow is usually inhomogeneous; iii) the local isotropy is not usually satisfied even in small scales; iv) the flow is in non-equilibrium. Existing studies have focused on and (partly) solved the first three problems [27][28][29][30][31][32][33][34], however, there is still few attempts on the fourth problem, i.e., the non-equilibrium fact, in the domain of SGS modeling [35,7]. In this case, in the following we would like to introduce our previous attempt of considering the non-equilibrium in a simplest situation, i.e., the homogeneous isotropic turbulence (HIT).…”

“…However, LES simulations for capturing largescale coherent structures (e.g., separation, transition and boundary layer) do not usually yield satisfactory results, while some of them are even worse than the Reynolds-averaged Navier-Stokes (RANS) results [4]. This bad performance also happens in the nonequilibrium regions, even if the flow is free from boundaries [5][6][7][8][9]. Various attempts have been performed to repair this problem.…”

“…It is not appropriate to assume that this part can yield a short-time forward non-linear energy transfer flux, which is represented by the SGS dissipation in LES (see Refs. [40,7,8] for detailed discussions on the imbalance between the non-linear energy transfer and the viscous dissipation). Hence, a natural correction is to assume null dissipation for the time-reversed part.…”

Flow resistance in natural, turbulent channel flows: The need for a fluvial fluid mechanics

“…This will then reflect the topology of the flow under consideration, permitting phase lags as a consequence of local variations in dissipation. Perhaps more importantly for representing complex processes, it will, as a consequence, permit reverse cascades of energy whereby subfilter scale energy generation can sustain large-scale oscillation, which is more realistic than the absolutely dissipative Smagorinsky approach [Horiuti and Tamaki, 2013]. Such a model may also be implemented within the dynamic framework [Ghosal et al, 1995] and, hence, the triple filtering approach of Porte-Agel and coworkers [Porte-Agel et al, 2000;Bou-Zeid et al, 2005] described above, providing significant greater flexibility in the evaluation of the modeling coefficients.…”