Subgrid-scale Stress Tensor Research Articles

Subgrid-scale model considering the inverse energy cascade using an artificial neural network

For the closure of the subgrid-scale (SGS) stress tensor, an artificial neural network (ANN)-based SGS model that takes account of the inverse energy cascade in isotropic turbulence is developed. The data required for training this ANN-based SGS model are provided by direct numerical simulation of isotropic turbulence with an inverse energy cascade. Two input features, the root mean square of the rate-of-strain tensor and the product of the eigenvalues of the rate-of-strain tensor, are employed to characterize the inverse energy cascade. An a priori test reveals that the ANN-based model adequately predicts the SGS stress tensor in the backward energy transfer process, and the predictive capability of the gradient model is found to be slightly poorer than that of the ANN-based model, while that of the Smagorinsky model is not satisfactory. In comparison with the gradient model, the ANN-based model even predicts a few backward energy transfer events in the stage of excessive energy dissipation. In addition, the off-diagonal component of the SGS stress tensor, rather than the diagonal component, may be intimately associated with the inverse energy cascade. The ANN-based SGS model presented here is expected to provide inspiration for future investigations of the construction of SGS models that take account of the inverse energy cascade.

Autonomous large-eddy simulations of turbulence using eddy viscosity derived from the subgrid-scale similarity stress tensor

A previously developed method for large-eddy simulations (LES), based on spectral eddy-viscosity models, is generalised to the physical space representation. The method estimates the subgrid-scale (SGS) energy transfer using a similarity-type model expression for the SGS tensor obtained using Gaussian filtering of velocity fields advanced in the simulations. Following steps for the spectral space representation, the SGS transfer in the physical space is used to obtain a spatially varying eddy viscosity at each time step in LES. The computed eddy viscosity is employed to model the SGS stress tensor in the familiar Boussinesq form for use in LES. The method is tested in LES of isotropic turbulence at high Reynolds numbers where the inertial range dynamics is expected and for lower Reynolds number decaying turbulence under conditions of the classical Comte-Bellot and Corrsin experiments. In both cases the agreement with reference data is very good and the SGS transfer computed for the proposed eddy-viscosity model is highly correlated with the transfer computed for the similarity-type stress tensor.

A priori test of Large-Eddy Simulation models for the Sub-Grid Scale turbulent stress tensor in perfect and transcritical compressible real gas Homogeneous Isotropic Turbulence

In this study, the focus lies on the modeling of the subgrid-scale stress tensor in the context of real gas flows. By conducting tests on perfect and dense gas configurations using six different models, the research findings highlight the superiority of gradient models over eddy-viscosity models in capturing the structural behavior of the stress tensor. However, the models’ performance experiences a significant decline when considering the intricate thermodynamics of real gases. To address this challenge, dynamic models that take into account the characteristics of the flow are also tested, resulting in some improvements in the outcomes. Despite these advancements, the achieved results still fall short of being entirely satisfactory. In particular, the models fail to accurately predict the occurrence of large values of the subgrid-scale stress tensor in the real gas flow. The complex thermodynamic nature of real gases therefore significantly influences the subgrid-scale stress tensor and the outcomes of this study underscore the urgent need for continued research to advance our understanding and modeling capabilities of subgrid-scale terms in real gas flows, paving the way for statistically accurate Large-Eddy Simulations of industrial real gas flows using moderately refined grids.

Neural-network-based mixed subgrid-scale model for turbulent flow

An artificial neural-network-based subgrid-scale (SGS) model, which is capable of predicting turbulent flows at untrained Reynolds numbers and on untrained grid resolution is developed. Providing the grid-scale strain-rate tensor alone as an input leads the model to predict a SGS stress tensor that aligns with the strain-rate tensor, and the model performs similarly to the dynamic Smagorinsky model. On the other hand, providing the resolved stress tensor as an input in addition to the strain-rate tensor is found to significantly improve the prediction of the SGS stress and dissipation, and thereby the accuracy and stability of the solution. In an attempt to apply the neural-network-based model trained for turbulent flows with a limited range of the Reynolds number and grid resolution to turbulent flows at untrained conditions on untrained grid resolution, special attention is given to the normalisation of the input and output tensors. It is found that the successful generalization of the model to turbulence for various untrained conditions and resolution is possible if distributions of the normalised inputs and outputs of the neural network remain unchanged as the Reynolds number and grid resolution vary. In a posteriori tests of the forced and the decaying homogeneous isotropic turbulence and turbulent channel flows, the developed neural-network model is found to predict turbulence statistics more accurately, maintain the numerical stability without ad hoc stabilisation such as clipping of the excessive backscatter, and to be computationally more efficient than the algebraic dynamic SGS models.

An efficient model for subgrid-scale velocity enrichment for large-eddy simulations of turbulent flows

In some applications of large-eddy simulation (LES), in addition to providing a closure model for the subgrid-scale stress tensor, it is necessary to also provide means to approximate the subgrid-scale velocity field. In this work, we derive a new model for the subgrid-scale velocity that can be used in such LES applications. The model consists in solving a linearized form of the momentum equation for the subgrid-scale velocity using a truncated Fourier-series approach. Solving within a structured grid of statistically homogeneous sub-domains enables the treatment of inhomogeneous problems. It is shown that the generated subgrid-scale velocity emulates key properties of turbulent flows, such as the right kinetic energy spectrum, realistic strain–rotation relations, and intermittency. The model is also shown to predict the correct inhomogeneous and anisotropic velocity statistics in unbounded flows. The computational costs of the model are still of the same order as the costs of the LES.

A data-driven dynamic nonlocal subgrid-scale model for turbulent flows

We developed a novel autonomously dynamic nonlocal turbulence model for the large and very large eddy simulation (LES, VLES) of homogeneous isotropic turbulent flows. The model is based on a generalized (integer-to-noninteger)-order Laplacian of the filtered velocity field, and a novel dynamic model has been formulated to avoid the need for tuning the model constant. Three data-driven approaches were introduced for the determination of the fractional-order to have a model that is totally free of any tuning parameter. Our analysis includes both the a priori and the a posteriori tests. In the former test, using a high-fidelity and well-resolved dataset from direct numerical simulations (DNSs), we computed the correlation coefficients for the stress components of the subgrid-scale (SGS) stress tensor and the one we get directly from the DNS results. Moreover, we compared the probability density function of the ensemble-averaged SGS forces for different filter sizes. In the latter, we employed our new model along with other conventional models including the static and dynamic Smagorinsky models into our pseudo-spectral solver and tested the final predicted quantities. The results of the newly developed model exhibit an expressive agreement with the ground-truth DNS results in all components of the SGS stress and forces. Also, the model exhibits promising results in the VLES region as well as the LES region, which could be remarkably important for cost-efficient nonlocal turbulence modeling, e.g., in meteorological and environmental applications.

Subgrid-scale models of isotropic turbulence need not produce energy backscatter

This investigation questions the importance of inverse interscale energy fluxes, the so-called energy backscatter, for the modelling of the energy cascade in large-eddy simulations (LES) of turbulent flows. The invariance of the filtered Navier–Stokes equations to divergence-preserving transformations of the subgrid-scale (SGS) stress tensor is exploited here to explore alternative representations of the local SGS energy fluxes. Numerical optimisation procedures are applied to the SGS stress tensor – obtained by filtering isotropic turbulence flow fields – to find alternative stresses that satisfy the filtered Navier–Stokes equations, but that produce negligible backscatter. These alternative SGS stresses show that backscatter represents not a flux of energy from the subgrid to the resolved scales, but conservative spatial fluxes, and that it need not be modelled to reproduce the local energetic exchange between the resolved and the subgrid scales in LES. From the perspective of statistical mechanics, it is argued that this is a consequence of the strong statistical irreversibility of inertial-range dynamics, which precludes inverse energy cascades even in a local sense. These findings show that the energy cascade is strongly unidirectional locally, and that it can be modelled as an irreversible sink of energy, justifying the extended use of purely dissipative SGS models in LES.

Dynamic subgrid-scale LES model for turbulent non-Newtonian flows: A priori and a posteriori analyses of Burgers turbulence

Large Eddy Simulation (LES) of turbulent non-Newtonian flows involves two additional closures, namely the Non-Newtonian SubGrid-Scale (NNSGS) stress tensor and filtered viscosity. Here, dynamic closures are proposed for NNSGS, eliminating the need for model calibration. In addition, for the primary evaluation of LES closures, two canonical case studies are designed by the extension of Newtonian forced Burgers turbulence to include power-law viscosity rheology. The characteristics of the proposed non-Newtonian turbulence are studied carefully using direct numerical simulation. For instance, it is revealed that the shear-thinning effect intensifies the small-scale motions and elevates the energy spectrum function at high wave-numbers. The subsequent a priori and a posteriori studies manifest that the NNSGS modeling is important for the present test cases. The omission of this term results in the under-prediction of kinetic energy due to the substantial backscatter effect of NNSGS. The proposed dynamic Smagorinsky based closure is recommended, based on the correlation coefficient measure in the a priori tests and the energy spectrum function in the a posteriori tests, for LES of turbulence at high shear-thinning rheology. The use of the present NNSGS dynamic model improved the predictions, however, further effort to achieve more accurate closures for NNSGS and especially the filtered viscosity is still necessary in future studies.

Mixed modeling for large-eddy simulation: The single-layer and two-layer minimum-dissipation-Bardina models

Predicting the behavior of turbulent flows using large-eddy simulation requires a modeling of the subgrid-scale stress tensor. This tensor can be approximated using mixed models, which combine the dissipative nature of functional models with the capability of structural models to approximate out-of-equilibrium effects. We propose a mathematical basis to mix (functional) eddy-viscosity models with the (structural) Bardina model. By taking an anisotropic minimum-dissipation (AMD) model for the eddy viscosity, we obtain the (single-layer) AMD–Bardina model. In order to also obtain a physics-conforming model for wall-bounded flows, we further develop this mixed model into a two-layer approach: the near-wall region is parameterized with the AMD–Bardina model, whereas the outer region is computed with the Bardina model. The single-layer and two-layer AMD–Bardina models are tested in turbulent channel flows at various Reynolds numbers, and improved predictions are obtained when the mixed models are applied in comparison to the computations with the AMD and Bardina models alone. The results obtained with the two-layer AMD–Bardina model are particularly remarkable: both first- and second-order statistics are extremely well predicted and even the inflection of the mean velocity in the channel center is captured. Hence, a very promising model is obtained for large-eddy simulations of wall-bounded turbulent flows at moderate and high Reynolds numbers.

Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network

An artificial neural network (ANN) is used to establish the relation between the resolved-scale flow field and the subgrid-scale (SGS) stress tensor, to develop a new SGS model for large-eddy simulation (LES) of isotropic turbulent flows. The data required for training and testing of the ANN are provided by performing filtering operations on the flow fields from direct numerical simulations (DNSs) of isotropic turbulent flows. We use the velocity gradient tensor together with filter width as input features and the SGS stress tensor as the output labels for training the ANN. In the a priori test of the trained ANN model, the SGS stress tensors obtained from the ANN model and the DNS data are compared by computing the correlation coefficient and the relative error of the energy transfer rate. The correlation coefficients are mostly larger than 0.9, and the ANN model can accurately predict the energy transfer rate at different Reynolds numbers and filter widths, showing significant improvement over the conventional models, for example the gradient model, the Smagorinsky model and its dynamic version. A real LES using the trained ANN model is performed as the a posteriori validation. The energy spectrum computed by the improved ANN model is compared with several SGS models. The Lagrangian statistics of fluid particle pairs obtained from the improved ANN model almost approach those from the filtered DNS, better than the results from the Smagorinsky model and dynamic Smagorinsky model.

Verification and validation of a variable–density solver for fire safety applications

An iterative variable–density solver is developed and implemented in Code_Saturne, a finite-volume code. The method of manufactured solutions (MMS) is used to investigate the convergence and the order of accuracy of the algorithm. A well-documented large-scale helium buoyant plume is simulated by large-eddy simulation (LES) to validate the code against experimental data. MMS verification confirms the second-order accuracy of the code. LES of helium plumes are performed by using the dynamic Smagorinsky model for subgrid-scale (SGS) stress tensor and five dynamic SGS scalar flux models including, the classical dynamic eddy diffusivity model (DEDM), two models based on the generalized gradient diffusion assumption, and two other mixed models based on the Taylor series expansion. Predictions obtained with the DEDM provide on the whole a good agreement with the experimental data although the mixing rate at vicinity of the plume centerline is underestimated. The other SGS scalar flux models reduce these discrepancies, especially close to the plume source. In addition, helium mass fraction and its fluctuation are more sensitive to grid refinement than SGS scalar closures which suggest that the small-scale buoyancy effects are not fully captured by the present LES. The validated LES code provides state-of-the-art predictions as compared to other numerical studies and can be applied to fire safety applications in the future.

Subgrid-scale stress parameterization for anisotropic turbomachinery flow as deduced from stereoscopic particle image velocimetry measurements

The present study searches for parametric subgrid-scale models for the turbomachinery flow based on single-plane stereoscopic particle image velocimetry measurements. A group of deformation tensors are used as the predictors to parameterize the subgrid-scale stress tensor as the response. Missing terms in the rotation rate tensor are completed with a novel approach which is based on geometrical relations between deformation and subgrid-scale tensors. The optimal relations for parametric subgrid-scale models are shaped after minimizing the square of error between modeled and real SGS stresses. The most superior and inferior one-predictor models are based on strain rate and rotation rate tensors, respectively. The two-predictor models show the most efficient prediction of subgrid-scale stress where the correlation coefficient is up to 43% larger than that of the classical Smagorinsky model. The three- to five-predictor models are inefficient since they give minor enhancement of correlation coefficient with a considerably larger computational cost.

Eigensensitivity analysis of subgrid-scale stresses in large-eddy simulation of a turbulent axisymmetric jet

The study of complex turbulent flows by means of large-eddy simulation approaches has become increasingly popular in many scientific and engineering applications. The underlying filtering operation of the approach enables to significantly reduce the spatial and temporal resolution requirements by means of representing only large-scale motions. However, the small-scale stresses and their effects on the resolved flow field are not negligible, and therefore require additional modeling. As a consequence, the assumptions made in the closure formulations become potential sources of model-form uncertainty that can impact the quantities of interest. The objective of this work, thus, is to perform a model-form sensitivity analysis in large-eddy simulations of an axisymmetric turbulent jet following an eigenspace-based strategy recently proposed. The approach relies on introducing perturbations to the decomposed subgrid-scale stress tensor within a range of physically plausible values. These correspond to discrepancy in magnitude (trace), anisotropy (eigenvalues) and orientation (eigenvectors) of the normalized, small-scale stresses with respect to a given tensor state, such that propagation of their effects can be assessed. The generality of the framework with respect to the six degrees of freedom of the small-scale stress tensor makes it also suitable for its application within data-driven techniques for improved subgrid-scale modeling.

A priori study of the subgrid energy transfers for small-scale dynamo in kinematic and saturation regimes

The statistical properties of the subgrid energy transfers of homogeneous small-scale dynamo are investigated during the kinematic, nonlinear, and statistically saturated stages. We carry out an a priori analysis of data obtained from an ensemble of direct numerical simulations on 5123 grid points and at unity magnetic Prandtl number. In order to provide guidance for subgrid-scale (SGS) modelling of different types of energy transfer that occur in magnetohydrodynamic dynamos, we consider the SGS stress tensors originating from inertial dynamics, Lorentz force, and the magnetic induction separately. We find that all SGS energy transfers display some degree of intermittency as quantified by the scale-dependence of their respective probability density functions. Concerning the inertial dynamics, a depletion of intermittency occurs in the presence of a saturated dynamo.

Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation

Data-driven machine learning algorithms, random forests and artificial neural network (ANN), are used to establish the subgrid-scale (SGS) model for large-eddy simulation. A total of 30 flow variables are examined as the potential input features. A priori tests indicate that the ANN algorithm provides a better solution for this regression problem. The relative importance of the input variables is evaluated by the two algorithms. It reveals that the gradient of filtered velocity and the second derivative of filtered velocity account for a vast majority of the importance. Besides, a pattern is found for the dependence of each component of the SGS stress tensor on the input features. Accordingly, a new uniform ANN model is proposed to provide closure for all the components of the SGS stress, and a correlation coefficient over 0.7 is reached. The proposed new model is tested by large-eddy simulation of isotropic turbulence. By examining the energy budget and the dissipative properties, the ANN model shows good agreement with direct numerical simulation and it provides better predictions than the Smagorinsky model and the dynamic Smagorinsky model. The current research suggests that data-driven algorithms are effective approaches to help us discover knowledge from large amounts of data.

Assessment of mean and fluctuating velocity and temperature of CuO/water nanofluid in a horizontal channel: Large eddy simulation

A comprehensive investigation applying the large eddy simulation approach to turbulent forced convection of CuO/water nanofluid flowing through a horizontal channel is carried out. Dealing with the sub-grid scale stress tensor and heat flux vector, the wall-adopting local eddy-viscosity model is employed. The periodic boundary condition is imposed to the streamwise and spanwise directions, while the no-slip and constant heat flux are applied to the walls. The results indicate that adding nanoparticles into the base fluid increases the dimensionless mean velocity and fluctuations of velocity and temperature. This increment is more evident for turbulent Reynolds stress and turbulent heat flux in the streamwise direction than the other directions. Therefore, higher energy is transferred between nanofluid layers which results in a higher amount of heat transfer than the pure water. It is also observed that the nanoparticles enhance the turbulence energy at all frequencies, and the decay in the fluctuations occurs at the higher wavenumbers.

A dynamic simplified Lund model for large-eddy simulation and its application to a centrifugal pump

PurposeConstruct a new sub-grid scale (SGS) model which can improve the efficiency and maintain comparative accuracy comparing to the existing dynamic cubic non-linear SGS model (DCNM).Design/methodology/approachThe polynomial constitutive relation between the SGS stress tensor and both strain and rotation rate is selected as a basement. Simplification is achieved by eliminating the solid-body rotation term and adopting the assumption proposed by Kosovic. A dynamic procedure is applied to calculate three model coefficients in the new model. The new model (named dynamic simplified Lund model) and DCNM are applied to the rotating channel flow and the internal flow in a centrifugal pump impeller to examine the performance.FindingsThe new model is as accurate as DCNM but decreases 25 per cent computational resources. The ability of capturing rotation effect and reflecting backscatter is verified through cases. In addition, good numerical stability is shown during the calculation.Research limitations/implicationsMore benchmark and engineering cases should be used to get further confidence on the new model.Practical implicationsThe new model is promising in industrial application with the advantage of both accuracy and efficiency. For the flow with large-scale separation or more complicate phenomenon, the model is thought to give accurate flow structure.Originality/valueA new non-linear SGS model is proposed in this paper. The accuracy, numerical stability and efficiency are validated for this model. Therefore, it is promising in the prediction of the flow structure in centrifugal pumps.

Evaluating the modulated gradient model in large eddy simulation of channel flow with OpenFOAM

ABSTRACT Recently, a new family of subgrid-scale (SGS) models, termed as gradient-based models, has been introduced to calculate the SGS stresses in large eddy simulation (LES). In the present work, the modulated gradient model (MGM) was implemented in the OpenFOAM package, and the pimpleFoam solver was improved to be adopted with non-eddy viscosity models. The MGM is a new, nonlinear model that uses the local equilibrium hypothesis to assess the SGS kinetic energy and the velocity gradient tensor to calculate the relative weight of the different components of the SGS stress tensor. To evaluate the accuracy of the MGM along with the modified pimpleFoam solver, a turbulent channel flow was simulated at the three different frictional Reynolds numbers of 180, 395 and 590. Furthermore, the results were compared with direct numerical simulation data, as well as the numerical results obtained by the established SGS models such as the dynamic Smagorinsky model (DSM). A suitable accuracy for the first- and second-order turbulence parameters was reported. Moreover, it was demonstrated that MGM is computationally efficient compared to the DSM in treating channel flow.

Multi-scale properties of large eddy simulations: correlations between resolved-scale velocity-field increments and subgrid-scale quantities

ABSTRACTWe provide analytical and numerical results concerning multi-scale correlations between the resolved velocity field and the subgrid-scale (SGS) stress-tensor in large eddy simulations (LES). Following previous studies for Navier–Stokes equations, we derive the exact hierarchy of LES equations governing the spatio-temporal evolution of velocity structure functions of any order. The aim is to assess the influence of the subgrid model on the inertial range intermittency. We provide a series of predictions, within the multifractal theory, for the scaling of correlation involving the SGS stress and we compare them against numerical results from high-resolution Smagorinsky LES and from a-priori filtered data generated from direct numerical simulations (DNS). We find that LES data generally agree very well with filtered DNS results and with the multifractal prediction for all leading terms in the balance equations. Discrepancies are measured for some of the sub-leading terms involving cross-correlation between resolved velocity increments and the SGS tensor or the SGS energy transfer, suggesting that there must be room to improve the SGS modelisation to further extend the inertial range properties for any fixed LES resolution.

On the Validity of Boussinesq Approximation in Variable Property Turbulent Mixed Convection Channel Flows

ABSTRACTThe present study investigates the effects of fluid thermo-physical property variations on fully developed turbulent mixed convection flow in vertical and horizontal parallel plate channels where walls are kept at different temperatures. The main attention is paid to the validity of the Boussinesq approximation when the temperature differences are enhanced. Reynolds number based on the channel half-width and friction velocity is assumed to be 150. The governing equations are solved utilizing the large eddy simulation and the low Mach number approach on a finite-volume basis. Dealing with the sub-grid scale stress tensor and the heat flux vector, the dynamic Smagorinsky model is used. All the numerical simulations are carried out by developing a solver in OpenFOAM. The present results display that an increase in the walls' temperature difference, applying the Boussinesq approximation and neglecting the dependency of thermal conductivity and dynamic viscosity on temperature, can result in relatively large errors. Deviations of up to 33% and 45% in the prediction of friction coefficient and Nusselt number, respectively, are shown. This is true for both vertical and horizontal configurations.