Subgrid Terms Research Articles

Use of Neural Networks for Stable, Accurate and Physically Consistent Parameterization of Subgrid Atmospheric Processes With Good Performance at Reduced Precision

AbstractA promising approach to improve climate‐model simulations is to replace traditional subgrid parameterizations based on simplified physical models by machine learning algorithms that are data‐driven. However, neural networks (NNs) often lead to instabilities and climate drift when coupled to an atmospheric model. Here, we learn an NN parameterization from a high‐resolution atmospheric simulation in an idealized domain by accurately calculating subgrid terms through coarse graining. The NN parameterization has a structure that ensures physical constraints are respected, such as by predicting subgrid fluxes instead of tendencies. The NN parameterization leads to stable simulations that replicate the climate of the high‐resolution simulation with similar accuracy to a successful random‐forest parameterization while needing far less memory. We find that the simulations are stable for different horizontal resolutions and a variety of NN architectures, and that an NN with substantially reduced numerical precision could decrease computational costs without affecting the quality of simulations.

Joint-velocity scalar energy probability density function method for large eddy simulations of compressible flow

The combination of large eddy simulation (LES) and probability density function (PDF) methods is a general framework to model turbulent reactive flows. The coupled approach provides direct closures for the nonlinear subgrid source terms typical of chemically reacting flows. LES-PDF methods have a wide range of applicability and they are started to be used in high-speed flows with strong compressibility effects. However, PDF formulations are more complex in compressible flows, where mechanical and thermodynamic contributions are more coupled. The paper presents a novel PDF framework that uses a full thermodynamic closure (scalar-energy-density-velocity) with the Eulerian Monte Carlo stochastic field approach. The work uses simple closures for the subgrid terms using the advantages of the Eulerian formulation and recasts the stochastic equations in a pseudoconservative form. The resultant formulation is applied to three canonical compressible flows: turbulent shock-tubes, compressible homogeneous turbulence, and a reactive free-moving premixed flame. All cases show large density and pressure fluctuations. The effects of underlying numerical schemes and PDF closures to represent compressible effects are investigated along with the statistical convergence of the method.

Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe

Abstract

Variational Multi-Scale Super-Resolution: A Data-Driven Approach for Reconstruction and Predictive Modeling of Unresolved Physics

The variational multiscale (VMS) formulation formally segregates the evolution of the coarse-scales from the fine-scales. VMS modeling requires the approximation of the impact of the fine scales in terms of the coarse scales. In linear problems, our formulation reduces the problem of learning the sub-scales to learning the projected element Green's function basis coefficients. For the purpose of this approximation, a special neural-network structure - the variational super-resolution N-N (VSRNN) - is proposed. The VSRNN constructs a super-resolved model of the unresolved scales as a sum of the products of individual functions of coarse scales and physics-informed parameters. Combined with a set of locally non-dimensional features obtained by normalizing the input coarse-scale and output sub-scale basis coefficients, the VSRNN provides a general framework for the discovery of closures for both the continuous and the discontinuous Galerkin discretizations. By training this model on a sequence of $L_2-$projected data and using the subscale to compute the continuous Galerkin subgrid terms, and the super-resolved state to compute the discontinuous Galerkin fluxes, we improve the optimality and the accuracy of these methods for the convection-diffusion problem, linear advection and turbulent channel flow. Finally, we demonstrate that - in the investigated examples - the present model allows generalization to out-of-sample initial conditions and Reynolds numbers. Perspectives are provided on data-driven closure modeling, limitations of the present approach, and opportunities for improvement.

Flame Surface Density Transport Statistics for High Pressure Turbulent Premixed Bunsen Flames in the Context of Large Eddy Simulation

ABSTRACT The implications of elevated pressure on the statistical behavior of the flame surface density (FSD) transport statistics together with the behavior of selected established sub-models of the unclosed terms of the FSD transport equation have been analyzed in the context of large eddy simulation. For this purpose, five turbulent premixed Bunsen flames have been considered from an existing database. Four of the Bunsen flames are characterized by different pressure levels and are located on the boundary of the wrinkled and the corrugated flamelet (CF) regimes to allow for the possibility and clear identification of combustion instabilities that are often observed at high pressures. The fifth flame is in the thin reaction zones regime and serves as a reference case for the purpose of comparison. For a given filter width to flame thickness ratio, the terms of the FSD transport equation and their models behave in a qualitatively similar manner for different pressure levels. However, as the flame thickness decreases with increasing pressure, it is unlikely that a high pressure flame will be simulated for the same spatial resolution (i.e. LES filter width) normalized by flame thickness as that of an atmospheric flame. It is more likely that the spatial resolution remains constant and in this case, the modeling becomes much more challenging for higher pressures: the magnitudes of the sub-grid contributions increase for larger filter width to flame thickness ratio and thus the accuracy of sub-grid modeling is expected to play a more important role in determining the fidelity of the simulations. The flames considered in this work feature a relatively low ratio of turbulent velocity fluctuations to laminar flame speed, and under these conditions, positive values of the sub-grid curvature term and negative values of the strain rate term are observed toward the leading edge of the flame brush. This behavior cannot be captured by the well-established existing models for the sub-grid curvature term as these model expressions yield deterministically negative values. Similarly, existing models for the tangential strain term yield deterministically positive values. Detailed explanations have been provided for the observed behaviors of the unclosed terms of the FSD transport equations and their respective model predictions.

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

A-Priori Assessment of Interfacial Sub-grid Scale Closures in the Two-Phase Flow LES Context

Due to the continuous increase in available computing power, the Large Eddy Simulation (LES) of two-phase flows started to receive more attention in recent years. Well-established models from single-phase flows are often used to close the sub-grid scale convective momentum transport and recently some modifications have been suggested to account for the jump of density and viscosity at the interface of multi-phase flows. However, additional unclosed terms in multi-phase flows, which are absent in single-phase flows, often remain ignored. This paper focuses on the crucial gaps in literature, namely the modeling of volume fraction advection and surface tension effects on sub-grid level. An a-priori analysis has been conducted for this purpose, i.e. the Direct Numerical Simulation of an academic two-phase flow configuration (single wobbling bubble in a turbulent background flow) has been explicitly filtered (corresponding to implicit filtering in actual LES) for varying filter width and the corresponding sub-grid terms have been compared to potentially suitable model expressions. Besides other approaches, adequately formulated models based on the scale similarity principle emerged to be promising candidates for both sub-grid volume fraction advection as well as sub-grid surface tension effects. In this context, special attention has to be paid to the secondary filter. Owing to the nature of the quasi-singular surface tension term, surface-weighted filtering may be more appropriate and robust than standard volume filtering.

A priori tests of subgrid-scale models in an anisothermal turbulent channel flow at low mach number

The subgrid-scale modelling of a low Mach number strongly anisothermal turbulent flow is investigated using direct numerical simulations. The study is based on the filtering of the low Mach number equations, suited to low Mach number flows with highly variable fluid properties. The results are relevant to formulations of the filtered low Mach number equations established with the classical filter or the Favre filter. The two most significant subgrid terms of the filtered low Mach number equations are considered. They are associated with the momentum convection and the density-velocity correlation. We focus on eddy-viscosity and eddy-diffusivity models. Subgrid-scale models from the literature are analysed and two new models are proposed. The subgrid-scale models are compared to the exact subgrid term using the instantaneous flow field of the direct numerical simulation of a strongly anisothermal fully developed turbulent channel flow. There is no significant differences between the use of the classical and Favre filter regarding the performance of the models. We suggest that the models should take into account the asymptotic near-wall behaviour of the filter length. Eddy-viscosity and eddy-diffusivity models are able to represent the energetic contribution of the subgrid term but not its effect in the flow governing equations. The AMD and scalar AMD models are found to be in better agreement with the exact subgrid terms than the other investigated models in the a priori tests.

Stochastic flow approach to model the mean velocity profile of wall-bounded flows.

There is no satisfactory model to explain the mean velocity profile of the whole turbulent layer in canonical wall-bounded flows. In this paper, a mean velocity profile expression is proposed for wall-bounded turbulent flows based on a recently proposed stochastic representation of fluid flows dynamics. This original approach, called modeling under location uncertainty, introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone between the viscous sublayer and the turbulent sublayer. An expression of the small-scale velocity component is also provided in the viscous zone. Numerical assessments of the results are provided for turbulent boundary layer flows, pipe flows and channel flows at various Reynolds numbers.

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.

Estimation of physical parameters under location uncertainty using an ensemble2–expectation–maximization algorithm

Estimating the parameters of geophysical dynamic models is an important task in the data assimilation (DA) techniques used to forecast initialization and reanalysis. In the past, most parameter estimation strategies were derived by state augmentation, yielding algorithms that are easy to implement but may exhibit convergence difficulties. The expectation–maximization (EM) algorithm is considered advantageous because it employs two iterative steps to estimate the model state and the model parameter separately. In this work, we propose a novel ensemble formulation of the maximization step in EM that allows a direct optimal estimation of the physical parameters using iterative methods for linear systems. This departs from current EM formulations that are only capable of dealing with additive model error structures. This contribution shows how the EM technique can be used for dynamics identification problems with the model error parameterized as an arbitrary complex form. The proposed technique is used here for the identification of stochastic subgrid terms that account for processes unresolved by a geophysical fluid model. This method, together with the augmented state technique, is evaluated to estimate such subgrid terms through high‐resolution data. As compared to the augmented state technique, our method is shown to yield considerably more accurate parameters. In addition, in terms of prediction capacity, it leads to a smaller generalization error as a result of the overfitting of the trained model on the presented data and eventually better forecasts.

LES of turbulent non-isothermal two-phase flows within a multifield approach

Safety issues in nuclear power plant involve complex turbulent bubbly flows. To predict the behavior of these flows, the two-fluid approach is often used. Nevertheless, this model has been developed for the simulation of small spherical bubbles, considered as a dispersed field. To deal with bubbles with a large range of sizes, a multifield approach based on this two-fluid model has been proposed. A special treatment, called the Large Bubble Model (LBMo), has been implemented and coupled to the dispersed model. However, only laminar and isothermal flows were considered in previous papers. Thus, here, Large Eddy Simulations (LES) are investigated to model turbulence effects. For this purpose, the two-fluid model equations are filtered to highlight the specific subgrid terms. Then, an a priori LES study using filtered Direct Numerical Simulation (DNS) results is detailed. This analysis allows classifying these terms according to their relative weight and then concentrating the modelling efforts on the predominant ones. Five different turbulence models are compared. These results are finally used to perform true LES on a turbulent two-phase flow. Moreover, in order to tackle non-isothermal flows occurring in industrial studies, a new heat transfer model is implemented and validated to deal with phase change at large interfaces using the Large Bubble Model.

Large eddy simulation of turbulent interfacial flows using Approximate Deconvolution Model

Large eddy simulation (LES) is currently widespread in turbulence modeling research. Although LES has reached a mature level in modelling single-phase turbulent flows even in industrial scales, the challenges with LES of turbulent interfacial flows are still remaining. The main issue arises in subgrid closure modelling where the small-scale physics of the flow, as well as the small-scale interfacial topological changes, must be accounted for in filtered governing equations. In this paper, the Approximate Deconvolution Model (ADM) (Stolz and Adams, 1999) is extended to the two-phase LES problems to model all the subgrid terms appearing in the filtered governing equations. Then, the ADM-VOF approach is developed comprehensively and implemented in the frame of C++ libraries of OpenFOAM. The ADM-VOF is then employed for an a posteriori LES on the phase inversion benchmark which represents a buoyancy-driven turbulent interfacial flow with several interfacial events such as coalescence and rupture. To investigate the performance of this approach, a series of quantitative comparisons with quasi-DNS data and conventional LES (i.e. with single-phase assumption) is conducted based on macroscopic characteristics of the flow including enstrophy and interfacial length. The results reveal that the structural ADM-VOF approach improves the prediction of macroscopic flow characteristics associated with unresolved contributions compared to the conventional LES. The dependency of ADM-VOF performance on the grid resolution is also investigated. It shows that ADM-VOF exhibits more potentials in predicting the interfacial scales on a finer grid. Additionally, our study unveils that the choice of functional methods for modeling relaxation term may not be extendable to the two-phase ADM.

Study of the large-eddy simulation subgrid terms of a low Mach number anisothermal channel flow

The subgrid terms of the low Mach number equations are investigated in a strongly anisothermal low Mach number flow. The filtered low Mach number equations are established in three formulations in order to compare the unweighted classical filter and the density-weighted Favre filter on the one hand, and the filtering of the momentum conservation equation and the velocity transport equation on the other hand. In the three formulations, we establish the filtered equations of mass conservation, momentum conservation, energy conservation, of the ideal gas law and of the resolved kinetic energy transport equation. The magnitude of all subgrid terms is assessed a priori in the three formulations using the results of direct numerical simulations of a strongly anisothermal fully developed turbulent channel flow. The classification of the subgrid terms gives the relevance of various effects of the temperature gradient.

A two-level finite element method with backtracking technique for the Navier-Stokes equations at high Reynolds numbers

A two-level finite element method with backtracking technique for the stationaryNavier-Stokes equations at high Reynolds numbers is considered. The basic idea of the method is to first solve a fully nonlinearNavier-Stokes equations with a subgrid stabilization term on a coarse grid, and then solve a subgrid stabilized linear fine grid problem based on one step of Newton iteration,and finally, solve a linear correction problem on the coarse mesh. The theoretical and numerical results show that, with suitable scalings of algorithmic parameters,the method can yield an optimal convergence rate. Numerical results are also given to verify theoretical predictions and demonstrate the effectiveness of the method.

ANALYSIS OF THE TURBULENCE-RADIATION INTERACTION IN A METHANE-AIR DIFFUSION FLAME

The phenomenon of turbulence-radiation interaction (TRI) has been demonstrated experimentally, theoretically and numerically to be important in a great number of engineering applications. This paper presents a numerical study on the subject, focusing on a methane-air diffusion flame confined in a rectangular enclosure. An open source, Fortran-based code, Fire Dynamics Simulator, is used for the analysis. Large Eddy Simulation (LES) is adopted to model the turbulence, and to resolve the sub-grid scale terms the dynamic Smagorinsky model is employed. To solve the radiative heat transfer, the finite volume method is used alongside the Weighted-Sum-of-Gray-Gases model. The main objective of the present work is to assess the magnitude of TRI effects for the configuration proposed. For this purpose, the time-averaged wall heat fluxes and volumetric radiative heat source, calculated from the LES results, are compared with those same quantities obtained by independent simulations initialized using mean temperature and species concentration fields. TRI effects are found to be responsible for differences up to 30% between results considering and neglecting turbulent fluctuations. These differences are larger for the radiative heat source and for the radiative heat flux to the walls, smaller for the total heat flux, and almost negligible for the convective heat flux. The influence of the fuel stream Reynolds number on the TRI effects is also evaluated, and a slight decrease on the magnitude of TRI is observed with the increase of that parameter.

Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows

In this work, we study the performance of some local projection-based solvers in the Large Eddy Simulation (LES) of laminar and turbulent flows governed by the incompressible Navier–Stokes Equations (NSE). On one side, we focus on a high-order term-by-term stabilization Finite Element (FE) method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard Local Projection Stabilization (LPS) methods is replaced by an interpolation-stabilized structure. The interest of LPS methods is that they ensure a self-adapting high accuracy in laminar regions of turbulent flows, which turns to be of overall optimal high accuracy if the flow is fully laminar. On the other side, we propose a new Reduced Basis (RB) Variational Multi-Scale (VMS)-Smargorinsky turbulence model, based upon an empirical interpolation of the sub-grid eddy viscosity term. This method yields dramatical improvements of the computing time for benchmark flows. An overview about known results from the numerical analysis of the proposed methods is given, by highlighting the used mathematical tools. In the numerical study, we have considered two well known problems with applications in industry: the (3D) turbulent flow in a channel and the (2D/3D) recirculating flow in a lid-driven cavity.

Regularization method for large eddy simulations of shock-turbulence interactions

The rapid change in scales over a shock has the potential to introduce unique difficulties in Large Eddy Simulations (LES) of compressible shock-turbulence flows if the governing model does not sufficiently capture the spectral distribution of energy in the upstream turbulence. A method for the regularization of LES of shock-turbulence interactions is presented which is constructed to enforce that the energy content in the highest resolved wavenumbers decays as k−5/3, and is computed locally in physical-space at low computational cost. The application of the regularization to an existing subgrid scale model is shown to remove high wavenumber errors while maintaining agreement with Direct Numerical Simulations (DNS) of forced and decaying isotropic turbulence. Linear interaction analysis is implemented to model the interaction of a shock with isotropic turbulence from LES. Comparisons to analytical models suggest that the regularization significantly improves the ability of the LES to predict amplifications in subgrid terms over the modeled shockwave. LES and DNS of decaying, modeled post shock turbulence are also considered, and inclusion of the regularization in shock-turbulence LES is shown to improve agreement with lower Reynolds number DNS.

Approximate deconvolution model for the simulation of turbulent gas-solid flows: An a priori analysis

Highly resolved two-fluid model (TFM) simulations of gas-solid flows in vertical periodic channels have been performed to study closures for the filtered drag force and the Reynolds-stress-like contribution stemming from the convective terms. An approximate deconvolution model (ADM) for the large-eddy simulation of turbulent gas-solid suspensions is detailed and subsequently used to reconstruct those unresolved contributions in an a priori manner. With such an approach, an approximation of the unfiltered solution is obtained by repeated filtering allowing the determination of the unclosed terms of the filtered equations directly. A priori filtering shows that predictions of the ADM model yield fairly good agreement with the fine grid TFM simulations for various filter sizes and different particle sizes. In particular, strong positive correlation (ρ > 0.98) is observed at intermediate filter sizes for all sub-grid terms. Additionally, our study reveals that the ADM results moderately depend on the choice of the filters, such as box and Gaussian filter, as well as the deconvolution order. The a priori test finally reveals that ADM is superior compared to isotropic functional closures proposed recently [S. Schneiderbauer, “A spatially-averaged two-fluid model for dense large-scale gas-solid flows,” AIChE J. 63, 3544–3562 (2017)].

Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

We present here a new stochastic modelling approach in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unresolved small-scale velocity component. They bring an explicit subgrid term to the reduced system which enables us to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or non-stationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flow data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers $Re=100$ and $Re=3900$. The finite-dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects, as well as structured energy dissipation effects, take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.