Scale-similarity Model Research Articles

Modeling the Subgrid-Scale Scalar Variance: A Priori Tests and Application to Supersaturation in Cloud Turbulence

Abstract The subgrid-scale (SGS) scalar variance represents the “unmixedness” of the unresolved small scales in large-eddy simulations (LES) of turbulent flows. Supersaturation variance can play an important role in the activation, growth, and evaporation of cloud droplets in a turbulent environment, and therefore efforts are being made to include SGS supersaturation fluctuations in microphysics models. We present results from a priori tests of SGS scalar variance models using data collected in turbulent Rayleigh–Bénard convection in the Michigan Tech Pi chamber for Rayleigh numbers Ra ∼ 108–109. Data from an array of 10 thermistors were spatially filtered and used to calculate the true SGS scalar variance, a scale-similarity model, and a gradient model for dimensionless filter widths of h/Δ = 25, 14.3, and 10 (where h is the height of the chamber and Δ is the spatial filter width). The gradient model was found to have fairly low correlations (ρ ∼ 0.2), with the most probable values departing significantly from the one-to-one line in joint probability density functions (JPDFs). However, the scale-similarity model was found to have good behavior in JPDFs and was highly correlated (ρ ∼ 0.8) with the true SGS variance. Results of the a priori tests were robust across the parameter space considered, with little dependence on Ra and h/Δ. The similarity model, which only requires an additional test filtering operation, is therefore a promising approach for modeling the SGS scalar variance in LES of cloud turbulence and other related flows.

Revisiting the modelling framework for the unresolved scalar variance

The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.

A comprehensive study of large eddy simulation models in capturing the dynamics of gas-particle turbulent suspensions

The particle phase attenuates the fluid fluctuations with an increase in volume fraction, and a sudden collapse in the turbulence is observed at a particular particle volume fraction, called critical particle volume loading (CPVL) [P. Muramulla et al. J. Fluid Mech. 889, A28 (2020)]. The present study reports the capability of two different classes of large eddy simulation (LES), viz. anisotropic and eddy viscosity-based, models to capture the turbulence modulation and the sudden disruption of the fluid fluctuations in the particle-laden vertical channel flows. The simulations are performed at two bulk Reynolds numbers of 3300 and 5600 based on the channel width and the bulk averaged fluid velocity. Our study on different LES models shows that approximate deconvolution (ADM) and scale similarity (SS) models accurately predict the critical loading for the Reynolds number of 3300. However, these models predict the critical loading qualitatively only for the Reynolds number of 5600 in the sense that they fail to predict the discontinuity as shown by the direct numerical simulation (DNS) study. The coherent structure model (CSM) predicts the critical loading with an 80% accuracy at both Reynolds numbers. The energy spectral density, production, and particle-induced dissipation spectra are plotted to analyze the distribution across wavenumbers. For all the LES models, a decrease in more than one order of magnitude is observed in the energy spectrum density at the critical loading compared to the unladen flow. The energy density decreases more in the channel center than in the near-wall region for the same particle volume loading. The mean component of particle-induced dissipation is almost two orders of magnitude larger than the particle dissipation spectra of fluctuating energy. The magnitude of streamwise and spanwise dissipation spectra of fluctuating components is higher in the near-wall region than the channel center. However, the magnitude of wall-normal dissipation spectra is higher in the channel center than near the wall region.

Particle subgrid stress models for large Stokes numbers in particle-laden turbulence

For the Stokes number based on the Kolmogorov time scale ${St}_K$ up to $O(10^{2})$ , the particle subgrid stress (particle stress) in the volume-average framework is studied by comparing the fluid residual stress, the particle Smagorinsky model and the particle scale-similarity model. To obtain the numerical database of the particle-laden turbulence, two-way coupling direct numerical simulation is carried out with isotropic and anisotropic forcing conditions. As the particle stress is related to the local flow structure, which is not reflected by ${St}_K$ , a new Stokes number ${St}_R$ is introduced to extract the effect of the intensity of the fluid velocity fluctuation in the averaging volume. The degrees of agreement of the principal axes (eigenvectors) of the particle stress models to those of the fully resolved particle stress are regarded as functions of ${St}_R$ regardless of the averaging volume size. The fluid residual stress model shows the highest degree of agreement over a small ${St}_R$ range for both of the forcing cases, and similar predominance is also observed for the correlation coefficient reflecting the magnitude of the particle stress. The effects of ${St}_R$ , ${St}_K$ , the averaging volume size and the Reynolds number on the model coefficients are investigated based on the intensities of the deviatoric and isotropic parts of the fully resolved particle stress and its models. The Stokes number ${St}_R$ is found to be an essential factor to determine the model coefficients, as each effect is extracted reasonably by fixing ${St}_R$ .

Subgrid modeling of the filtered equation of state with application to real-fluid turbulent mixing at supercritical pressures

Simulation of turbulent mixing and combustion at supercritical pressures requires the use of a real-fluid equation of state (EOS) to represent the nonideal, nonlinear thermodynamic behavior of fluids under these conditions. The simplified representation of the filtered EOS in the large eddy simulation methodology introduces inconsistencies in the computed filtered thermodynamic state. This study investigates these inconsistencies and novel subgrid modeling approaches to address these issues, using high-resolution direct numerical simulation of a transcritical mixing layer. Errors incurred by not accounting for subgrid effects in the EOS are quantified, and fundamental insights are drawn regarding the nature of these effects. Then, different modeling approaches are proposed and investigated to obtain a more accurate representation of the filtered EOS. The evaluation of the filtered EOS in terms of the Reynolds-filtered state variables is considered instead of the conventional Favre-filtered variables. A dynamic gradient model is formulated by building upon the ideas of dynamic modeling to render a functional form for the subgrid EOS expressed in terms of the resolved flow gradients. A scale-similarity model formulation for the subgrid EOS is also constructed and examined. Finally, a model for the filtered EOS is derived using a presumed filtered density function that accounts for the effect of subgrid-scale fluctuations. The performance of each model is evaluated using various metrics, and the relative accuracy of each modeling approach is compared and contrasted at different filter sizes.

Evaluation of flamelet/progress variable model for the applications in supersonic combustion using hybrid RANS/LES approach

The Flamelet/Progress Variables (FPV) model coupled with large eddy simulation (LES) has been performed to appraise its applicability and performance for supersonic hydrogen combustion, based on the simulations of strut based scramjet and cavity based combustors. For the two typical schemes involved in the present work, the validation of the numerical solver compiled by in-house code is realized by the comparisons with the experimental measurements including temperature, velocity, and wall pressure. The methods for calculating mixture fraction variance in the framework of LES are also investigated and examined in both cases. It can be observed that the diffusion combustion mode is in the dominant state for the whole reactive zone in both cases, where the FPV model is physically compatible. Through the comparisons of the scale similarity method and the transport equation method for calculating mixture fraction variance, the importance of calculating this sub-grid turbulent scalar flux is emphasized. The results show that the mixture fraction variance calculated by the scale similarity model is locally high, but its diffusion is insufficient, which makes the local temperature of the combustion flow field overpredicted and concentrated. On the contrary, the transport equation of the mixture fraction variance can overcome this problem and improve the combustion prediction accuracy, thus, it is more recommended if the computing resources are guaranteed. Qualitatively, the Flamelet/Progress Variables (FPV) model with the large eddy simulation approach can accurately capture the structure and characteristics of the supersonic combustion flow field.

Assessment of turbulence models using DNS data of compressible plane free shear layer flow

The present paper uses the detailed flow data produced by direct numerical simulation (DNS) of a three-dimensional, spatially developing plane free shear layer to assess several commonly used turbulence models in compressible flows. The free shear layer is generated by two parallel streams separated by a splitter plate, with a naturally developing inflow condition. The DNS is conducted using a high-order discontinuous spectral element method (DSEM) for various convective Mach numbers. The DNS results are employed to provide insights into turbulence modelling. The analyses show that with the knowledge of the Reynolds velocity fluctuations and averages, the considered strong Reynolds analogy models can accurately predict temperature fluctuations and Favre velocity averages, while the extended strong Reynolds analogy models can correctly estimate the Favre velocity fluctuations and the Favre shear stress. The pressure–dilatation correlation and dilatational dissipation models overestimate the corresponding DNS results, especially with high compressibility. The pressure–strain correlation models perform excellently for most pressure–strain correlation components, while the compressibility modification model gives poor predictions. The results of an a priori test for subgrid-scale (SGS) models are also reported. The scale similarity and gradient models, which are non-eddy viscosity models, can accurately reproduce SGS stresses in terms of structure and magnitude. The dynamic Smagorinsky model, an eddy viscosity model but based on the scale similarity concept, shows acceptable correlation coefficients between the DNS and modelled SGS stresses. Finally, the Smagorinsky model, a purely dissipative model, yields low correlation coefficients and unacceptable accumulated errors.

Large eddy simulation of non-Newtonian viscous fluids with low grid dependency using an anisotropy-resolving subgrid-scale model

Large eddy simulation (LES) is expected to be an efficient and accurate numerical method for predicting the behavior of non-Newtonian fluids. However, it has been reported that the LES results obtained using eddy viscosity subgrid-scale (SGS) models have a high dependency on the grid resolution for wall turbulence. In order to resolve this problem, in the present study, a mixed SGS model combining an isotropic eddy-viscosity model and a scale-similarity model, which is proposed by Inagaki and Abe (2017), is applied to non-Newtonian fluids obeying the power law. The model performance is tested in plane channel flows and pipe flows for the power-law index n of 0.5–1.15, using a wide range of grid resolution. The results demonstrate that, whereas conventional eddy viscosity SGS models significantly depend on the grid resolution, the present SGS model successfully prevents such grid dependency and yields accurate results even with relatively coarse grids for both pseudoplastic and dilatant fluids, where the modeled SGS shear stress consistently compensates the damped grid-scale (GS) shear stress. When decreasing the value of n (higher shear-thinning property), it has been recognized that the anisotropy of the turbulence is augmented. Since the present SGS model can express the anisotropy of the turbulence stress, the results from n=1 down to n=0.69 are in fairly good agreement with the DNS data, regardless of the grid resolution used, predicting the reduction of wall-friction drag, quantitatively. Upon further decreasing the value of n to 0.5, most eddy viscosity models do not maintain a turbulent state, whereas the present model maintains turbulence and provides results that are in reasonable agreement with the DNS data. The validity of the present model is also confirmed for high-Reynolds-number flows up to Reτ=750, where the grid independency is retained. These results demonstrate that the present model can remarkably reduce the computational cost of LES for non-Newtonian fluids.

Direct numerical simulation of supercritical oxy-methane mixing layers with CO2 substituted counterparts

Direct numerical simulations of temporally developing, three-dimensional, CH4/CO2, CH4/O2, and CO2/O2 mixing layers are conducted at a supercritical pressure of 300 atm. To effectively model the supercritical regime, the employed formulation includes the compressible form of the governing equations, the cubic Peng–Robinson equation of state and a generalized formulation for heat and mass flux vectors derived from non-equilibrium thermodynamics and fluctuation theory. A linear inviscid stability analysis is also performed for each case, to determine its most unstable wavelength. Flow visualizations reveal the presence of high density gradient magnitude regions for all three mixing layers, with conditional averages indicating increased presence of heavier fluid species within these regions. No significant departures are observed from perfect gas behavior, with compressibility factors very close to unity for all three mixing cases. Applicability of presumed probability density function methods is examined for the three supercritical mixing layers. An a priori analysis is also conducted to investigate various simplifying assumptions employed in modeling various subgrid scale (SGS) flux models. Two additional terms are identified in the large eddy simulation equations, the gradient of SGS contribution of pressure in the momentum equation and the gradient of SGS contribution of heat flux in energy equation, whose magnitudes are similar and comparable with their respective resolved terms. The performance of the scale similarity model to represent these additional terms is investigated. Finally, the performance of Smagorinsky, gradient, and scale similarity models is also investigated.

A-posteriori LES assessment of subgrid-scale closures for bounded passive scalars

Scale similarity or gradient models represent attractive, functionally simple expressions for large eddy simulation (LES) subgridscale (SGS) models, showing excellent behaviour in a-priori LES studies for small to moderate filter sizes. However, when applied a-posteriori to real LES calculations, they frequently suffer from numerically unstable behavior. A recent regularization approach revealed promising results for both wall-bounded and free turbulent flows. One attractive feature of this modelling strategy is that it can potentially be applied in many different contexts, such as momentum transport in single or two phase flows, different fluids such as non-Newtonian fluids, etc. The aim of the present study is to test the application of this regularisation technique to turbulent transport of bounded passive scalars. The performance of the model together with suitable discretization strategies will be assessed for turbulent plane free jet simulations. LES results will be benchmarked against DNS calculations of the same configuration obtained with the same code. The new model exhibits good results for momentum and scalar transport, outperforming standard eddy-viscosity models particularly with respect to the prediction of second order moments. Further, they provide good stability and are easy to implement. While the influence of the SGS model for scalar transport is not negligible, it is shown that an appropriate momentum SGS-closure is likely to be more important.

Role of various scale-similarity models in stabilized mixed subgrid-scale model

We investigate the physical role of various scale-similarity models in the stabilized mixed model [K. Abe, “An improved anisotropy-resolving subgrid-scale model with the aid of a scale-similarity modeling concept,” Int. J. Heat Fluid Flow 39, 42 (2013); M. Inagaki and K. Abe, “An improved anisotropy-resolving subgrid-scale model for flows in laminar–turbulent transition region,” Int. J. Heat Fluid Flow 64, 137 (2017)] and evaluate their performance in turbulent channel flows. Among various models in the present study, the original model combined with the scale-similarity model for the subgrid-scale (SGS)-Reynolds term yields the best prediction for the anisotropy of the grid-scale (GS) velocity fluctuations and the SGS stress, even in coarse grid resolutions. Moreover, it successfully predicts large intensities of the spectra close to the cut-off scale in accordance with the filtered direct numerical simulation, whereas other models predict a rapid decay of the spectra in the low-wavelength region. To investigate the behavior of the models close to the cut-off scale, we analyze the budget equation for the GS Reynolds stress spectrum. The result shows that the scale-similarity model for the SGS-Reynolds term plays a role in the enhancement of the wall-normal velocity fluctuation close to the cut-off scale. Thereby, it activates turbulence close to the cut-off scale, leading to a reproduction of the proper streak structures observed in wall-bounded turbulent flows. The reproduction of velocity fluctuations close to the cut-off scale and turbulent structures is a key element for further development of SGS models.

2D turbulence closures for the barotropic jet instability simulation

AbstractIn the present work the possibility of turbulence closure applying to improve barotropic jet instability simulation at coarse grid resolutions is considered. This problem is analogous to situations occurring in eddy-permitting ocean models when Rossby radius of deformation is partly resolved on a computational grid. We show that the instability is slowed down at coarse resolutions. As follows from the spectral analysis of linearized equations, the slowdown is caused by the small-scale normal modes damping arising due to numerical approximation errors and nonzero eddy viscosity. In order to accelerate instability growth, stochastic and deterministic kinetic energy backscatter (KEBs) parameterizations and scale-similarity model were applied. Their utilization led to increase of the growth rates of normal modes and thus improve characteristic time and spatial structure of the instability.

An a priori DNS analysis of scale similarity based combustion models for LES of non-premixed jet flames

In this work, recently developed finite-rate dynamic scale similarity (SS) sub-grid scale (SGS) combustion models have been a priori assessed and compared with the Eddy Dissipation Concept (EDC) and “no model” approaches based on a Direct Numerical Simulation (DNS) database of a temporally evolving non-premixed jet flame. Two different filter widths, one placed in the inertial range and the other in the near dissipation range, have been used. The analyses were carried out in two time instants corresponding to instants of maximum local extinction and re-ignition. Conditional averaged filtered chemical source terms, conditioned on different parameters in the composition space, have been presented. Improvements are observed using the dynamic SS models compared to the two other approaches in the prediction of filtered chemical source terms of individual species while using larger filter widths. However, discrepancies still exists using the dynamic SS model on the turbulent/non-turbulent interfaces of the jet, mainly in the prediction of the oxidizer consumption rate.

New Dynamic Scale Similarity Based Finite-Rate Combustion Models for LES and a priori DNS Assessment in Non-premixed Jet Flames with High Level of Local Extinction

In this work, the performances of two recently developed finite-rate dynamic scale similarity (SS) sub-grid scale (SGS) combustion models (named DB and DC) for non-premixed turbulent combustion are a priori assessed based on three Direct Numerical Simulation (DNS) databases. These numerical experiments feature temporally evolving syngas jet flames with different Reynolds (Re) numbers (2510, 4487 and 9079), experiencing a high level of local extinction. For comparison purposes, the predicting capability of these models is compared with three classical non-dynamic SS models, namely the scale similarity resolved reaction rate model (SSRRRM or A), the scale similarity filtered reaction rate model (SSFRRM or B), and a SS model derived by the “test filtering” approach (C), as well as an existing dynamic version of SSRRRM (DA). Improvements in the prediction of heat release rates using a new dynamic model DC are observed in high Re flame case. By decreasing Re, dynamic procedures produce results roughly similar to their non-dynamic counterparts. In the lowest Re, the dynamic methods lead to higher errors.

A posteriori tests of subgrid-scale models in strongly anisothermal turbulent flows

This paper studies the large-eddy simulation of anisothermal low Mach number turbulent channel flows. We consider the large-eddy simulations of the low Mach number equations in two formulations, the velocity formulation and the Favre formulation. In both formulations, we investigate the subgrid-scale modeling of the two most significant subgrid terms of the filtered low Mach number equations: the momentum convection subgrid term and the density-velocity correlation subgrid term. To this end, the predictions of large-eddy simulations implementing the models are compared to filtered direct numerical simulations. We address several types of subgrid-scale models: functional eddy-viscosity or eddy-diffusivity models, structural models, tensorial models, and dynamic versions of these models. For the momentum convection subgrid term, we recommend the use of the scale-similarity model and the constant-parameter or dynamic tensorial anisotropic minimum-dissipation (AMD) model. For the density-velocity correlation subgrid term, several models are able to improve temperature-related statistics, for instance, the AMD model and the scale-similarity model. More accurate results are obtained with the Favre formulation than with the velocity formulation.

A novel method for expressing anisotropy of subgrid-scale components for thermal and scalar fields

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively increase the accuracy of large eddy simulation (LES). For example, Abe (2013) recently proposed a stabilized mixed model, which can successfully express the anisotropy of the SGS stress, and remarkably improves the predictive performance for wall turbulence at coarse grid resolutions without sacrificing computational stability. In the present work, this approach is extended for thermal and scalar field modeling to express the anisotropy of SGS heat (scalar) flux. A priori tests using direct numerical simulation data demonstrated that the proposed model can accurately predict the anisotropic SGS heat flux in channel flow and its variation with Prandtl number in the range 0.1–2. Its effectiveness is also confirmed by a posteriori tests under flow conditions corresponding to those in the a priori tests.

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

A scale-similarity model of a two-point two-time Lagrangian velocity correlation (LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows (HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313 (2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values (Rλ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity. Moreover, the dispersion velocity V normalized by the Kolmogorov velocity Vη ≡ η/τη in which η and τη are the Kolmogorov space and time scales, respectively, scales with the Reynolds number Rλ as $$V/V_\eta\propto{R_\lambda^{1.39}}$$ obtained from the numerical data.

Prediction of Combustion and Heat Release Rates in Non-Premixed Syngas Jet Flames Using Finite-Rate Scale Similarity Based Combustion Models

Generating energy from combustion is prone to pollutant formation. In energy systems working under non-premixed combustion mode, rapid mixing is required to increase the heat release rates. However, local extinction and re-ignition may occur, resulting from strong turbulence–chemistry interaction, especially when rates of mixing exceed combustion rates, causing harmful emissions and flame instability. Since the physical mechanisms for such processes are not well understood, there are not yet combustion models in large eddy simulation (LES) context capable of accurately predicting them. In the present study, finite-rate scale similarity (SS) combustion models were applied to evaluate both heat release and combustion rates. The performance of three SS models was a priori assessed based on the direct numerical simulation of a temporally evolving syngas jet flame experiencing high level of local extinction and re-ignition. The results show that SS models following the Bardina’s “grid filtering” approach (A and B) have lower errors than the model based on the Germano’s “test filtering” approach (C), in terms of mean, root mean square (RMS), and local errors. In mean, both Bardina’s based models capture well the filtered combustion and heat release rates. Locally, Model A captures better major species, while Model B retrieves radicals more accurately.

An inconsistence-free integral-based dynamic one- and two-parameter mixed model

ABSTRACTThe development of the dynamic procedure as well as the deeper understanding of the link between filtering, modelling and numerics, allowed Large Eddy Simulation (LES) to make great progresses during the last years. Among several modelling approaches, the scale-similar-based modelling is based on the observation that the smallest resolved scales are the most active in the interaction with the unresolved ones. Owing to the low dissipation introduced by the scale-similar models (SSMs), the coupling with the eddy-viscosity model is often used in the so-called mixed models. Dynamic version of mixed models is historically based on the application of the test-filtering on the differential form of the filtered momentum equation. Such an approach is used for both the one and the two-coefficients mixed models. The use of the differential form of the filtered equations produces the well-known mathematical inconsistence caused by the need to extract arbitrarily the model functions out of filtering. It is known that, along with the eddy viscosity assumption, the magnitude of the Germano identity error (GIE) is strongly influenced. The mathematical inconsistence in the extraction of the dynamic eddy viscosity coefficient was recently superseded by using the new integral-based formulation. However, owing to the intrinsic limits of the Smagorinsky model, also in those results, the GIE is still remarkable therefore, the present paper presents a new formulation to the integral-based dynamic procedure for both one and two-coefficients mixed models (IDMM). The original contributions of the present paper can be summarised: (1) A theoretical analysis comparing the spectral errors for the differential and integral-based SSM, assessing that the errors are less relevant for the integral form; (2) The implementation of one and two parameters IDMM for the simulation of turbulence in a plane channel flow, assessing the reduction of the GIE and the good behaviour of the statistics that are compared with those of the other LES codes used in the LESinItaly project.

Improvement of the SGS model by using a scale-similarity model based on the analysis of SGS force and SGS energy transfer

The subgrid-scale (SGS) model is improved by using a scale-similarity model based on the analysis of the SGS force and SGS energy transfer around an elliptic Burgers vortex. Abe (2013) proposed an anisotropy-resolving SGS model in which the Bardina term of a scale-similarity model is mixed with an eddy viscosity model under a new concept wherein the Bardina term does not affect the SGS energy transfer although it affects the SGS force. By using the concept, we propose a scale-similarity model with the Clark term. The SGS energy transfer is determined by the scale-similarity term and not by the eddy viscosity term while the SGS force is improved by using the SGS kinetic energy (Abe, 2013). It is observed that the Clark term yields higher spatial correlation with the true distributions of the SGS force and SGS energy transfer around the elliptic vortex when compared to the Bardina term. The SGS model based on the Clark term exhibits good performance for turbulent channel flows with respect to Reτ=180 and 590 even in extremely coarse grid resolutions. Specifically, the SGS model with the SGS kinetic energy fairly improves the mean streamwise velocity profile.