Moment Closure Model Research Articles

Effects of Reynolds Number on the Wake Characteristics of a Notchback Ahmed Body

Abstract This paper investigates the effects of Reynolds number on the wake characteristics of a notchback Ahmed body with effective backlight angle, βe=17.8 deg. The Reynolds number based on the body height was varied from 5×103 to 5×104. Prior to the Reynolds number investigation, a Reynolds-averaged Navier–Stokes (RANS) model assessment was performed using nine turbulence models consisting of one- and two-equation eddy-viscosity models and second moment closure models. The standard Spalart–Allmaras model was the only model that accurately predicted the asymmetric time-averaged wake topology, as reported in previous studies, for the βe=17.8 deg notchback Ahmed body at Re=5×104. The drag coefficient decreased with increasing Reynolds number, while the lift coefficient remained constant for Re≥1×104. The wake structure exhibited three regimes: symmetric (Re≤1×104), transitionally asymmetric (1×104<Re≤3.5×104), and fully asymmetric (Re>3.5×104) states. The wake asymmetry was attributed to an imbalance in entrainment from the sides and asymmetric separation from the roof and the C-pillars of the body. The tilting and stretching terms in the vorticity transport equation were used to provide insight into the source of asymmetry in the vorticity field around the body.

High-order moment closure models with random batch method for efficient computation of multiscale turbulent systems.

We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz '96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.

An Assessment of Second Moment Closure Modeling for Stratified Wakes Using Direct Numerical Simulations Ensembles

Abstract Buoyant wakes encountered in the ocean environment are characterized by high Reynolds (Re) and Froude (Fr) numbers, leading to significant space–time resolution requirements for turbulence resolving CFD models (i.e., direct numerical simulations (DNS), large eddy simulations (LES)). Therefore, Reynolds-averaged Navier–Stokes (RANS) based models are attractive for these configurations. The inherently complex dynamics of stratified systems render eddy-viscosity-based modeling inappropriate. RANS second-moment closure (SMC) based modeling is more suitable because it accounts for flow anisotropy by solving the transport equations of important second-moment terms. Accordingly, eleven transport equations are solved at the SMC level, and a range of submodels are implemented for diffusion, pressure strain and scrambling, and dissipation terms. This work studies nonstratified and stratified towed wakes using SMC and DNS. Submodels in the SMC are evaluated in terms of how well their exact Reynolds averaged form impacts the accuracy of the full RANS closure. An ensemble average of 40 and 80–100 DNS realizations are required and conducted for these temporally evolving nonstratified and stratified wakes, respectively, to obtain converged higher-order statistics. SMC over-predicts wake height by over a factor of 2, and under-predicts defect velocity, wake width, and turbulent kinetic and potential energies by factors ranging from 1.3 to 3.5. Also, SMC predicts a near isotropic decay of normal Reynolds stresses (a33→−0.25), in contrast to the anisotropic decay (a33→−0.64) returned by DNS. The DNS data also provide important insights related to the inaccuracy of the dissipation rate isotropy assumption and the non-negligible contribution of pressure diffusion terms. These results lead to several important recommendations for SMC modeling improvement.

Machine Learning Moment Closure Models for the Radiative Transfer Equation II: Enforcing Global Hyperbolicity in Gradient-Based Closures

This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.

Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds

This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation. In our previous work (Huang et al. in J Comput Phys 453:110941, 2022), we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional \(P_N\) closure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper (Huang et al., in: Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures, 2021. arXiv:2105.14410), we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. We design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.

Large eddy simulation of a turbulent supercritical hydrothermal flame with a novel direct moment closure model

A novel direct moment closure (DMC) model has been developed in the context of large eddy simulation (LES) and applied to simulate a supercritical flame. To evaluate the performance of the model, three-dimensional direct numerical simulation (DNS) of a supercritical hydrothermal non-premixed jet flame is conducted. The characteristics of the ignition and propagation of the supercritical hydrothermal flame are analyzed, and then the a-priori analysis of the DMC model is carried out based on the DNS model data. It is found that the prediction of reaction rate can be remarkably improved by the DMC model compared with the laminar chemistry closure (LCC) model. In the a-posteriori study, LES of the turbulent supercritical hydrothermal flame with the DMC and LCC models have been performed and compared with the DNS data respectively. It turns out that the LES-DMC model can successfully reproduce the complex behaviors of the turbulent supercritical hydrothermal flame, and the predicted mean and rms values of velocities, temperature, and species are more accurate than those of the LES-LCC model.

Prediction of turbulent flow characteristics and heat transfer in a dilute droplet-laden flow over a backward-facing step

We present numerical investigation of the influence of the thermophysical properties of droplets on turbulence modification, scattering of droplets, and heat transfer augmentation in a two-phase mist flow over a backward-facing step. Predictions are made for droplets of water, ethanol, acetone, and glycerol, with droplet diameters in the range d 1 =1–100 μm and mass fractions in the range ML 1=0.01–0.1 at the inlet. The RANS approach is used to simulate the gaseous phase, and the motion and heat transfer of the dispersed phase are computed using Eulerian model. Carrier phase turbulence is predicted using a second moment closure model. The effect of attenuation of the gas phase turbulent kinetic energy is shown to be minimal for acetone droplets (more than 7%), whereas the largest effects are seen for glycerol and water droplets (up to 15%). Heat transfer enhancement is shown to reach a maximal value with the use of ethanol (more than twice as high as for single-phase flow over the backward-facing step), while minimal intensification is obtained for the evaporation of acetone droplets (up to 25%).

Second Moment Closure Modeling and Direct Numerical Simulation of Stratified Shear Layers

AbstractBuoyant shear layers encountered in many engineering and environmental applications have been studied by researchers for decades. Often, these flows have high Reynolds and Richardson numbers, which leads to significant/intractable space–time resolution requirements for direct numerical simulation (DNS) or large eddy simulation (LES). On the other hand, many of the important physical mechanisms, such as stress anisotropy, wake stabilization, and regime transition, inherently render eddy viscosity-based Reynolds-averaged Navier–Stokes (RANS) modeling inappropriate. Accordingly, we pursue second-moment closure (SMC), i.e., full Reynolds stress/flux/variance modeling, for moderate Reynolds number nonstratified, and stratified shear layers for which DNS is possible. A range of submodel complexity is pursued for the diffusion of stresses, density fluxes and variance, pressure strain and scrambling, and dissipation. These submodels are evaluated in terms of how well they are represented by DNS in comparison to the exact Reynolds-averaged terms, and how well they impact the accuracy of full RANS closure. For the nonstratified case, SMC model predicts the shear layer growth rate and Reynolds shear stress profiles accurately. Stress anisotropy and budgets are captured only qualitatively. Comparing DNS of exact and modeled terms, inconsistencies in model performance and assumptions are observed, including inaccurate prediction of individual statistics, non-negligible pressure diffusion, and dissipation anisotropy. For the stratified case, shear layer and gradient Richardson number growth rates, and stress, flux and variance decay rates, are captured with less accuracy than corresponding flow parameters in the nonstratified case. These studies lead to several recommendations for model improvement.

Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. A variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.

Large Eddy Simulation of a Turbulent Supercritical Hydrothermal Flame with a Novel Direct Moment Closure Model

Large Eddy Simulation of a Turbulent Supercritical Hydrothermal Flame with a Novel Direct Moment Closure Model

Large eddy simulation and finite-volume conditional moment closure modelling of a turbulent lifted H2/N2 flame

Large eddy simulations with three-dimensional finite-volume Conditional Moment Closure (CMC) model are performed for a hydrogen/nitrogen lifted flame with detailed chemical mechanism. The emphasis is laid on the influences of mesh resolution and convection scheme of finite-volume CMC equations on predictions of reactive scalars and unsteady flame dynamics. The results show that the lift-off height is underestimated and the reactive scalars are over-predicted with coarser CMC mesh. It is also found that further refinement of the CMC mesh would not considerably improve the results. The time sequences of the most reactive and stoichiometric hydroxyl radical mass fractions indicate that finer CMC mesh can capture more unsteady details than the coarser CMC mesh. Moreover, the coarse CMC mesh has lower conditional scalar dissipation rate, which would promote the earlier auto-ignition of the flame base. Besides, the effects of the convection scheme for the CMC equations (i.e., upwind, central differencing and their blends) on the lifted flame characteristics are also investigated. It is shown that different convection schemes lead to limited differences on the time-averaged temperature, mixture fraction and species mass fractions. Moreover, the root-mean square values of hydrogen and hydroxyl mass fractions show larger deviation from the measurements with hybrid upwind and central differencing scheme, especially around the flame base. Furthermore, the distributions of the numerical fluxes on the CMC faces also show obvious distinctions between the upwind and blending schemes. The budget analysis of the individual CMC terms shows that a sequence of CMC faces has comparable contributions with upwind scheme. However, with the hybrid schemes, the instantaneous flux is dominantly from limited CMC faces. The reactivity of a CMC cell is more easily to be affected by its neighbors when the upwind scheme is used.

Localized conditional source-term estimation model for turbulent combustion

A novel functional form for approximating the conditional scalars in turbulent reacting flows is introduced based on the Bernstein polynomial. Multi-scalar measurement data of turbulent premixed and non-premixed flames are used to demonstrate that the new functional form provides an excellent reduced-order model for the conditional scalars. This model order reduction technique can be used to improve the accuracy, reduce the computational cost and enhance the spatial localization of the Conditional Source-term Estimation (CSE) model. CSE is a turbulence-chemistry interaction model similar to the Conditional Moment Closure (CMC) model, except that the conditional scalars are estimated from the filtered field in an ensemble of LES cells using an integral equation. An a priori analysis using the DNS data of a series of statistically planar turbulent premixed flames shows that using Bernstein polynomials as the presumed functional form for the conditional scalars provides better regularization than the conventional CSE approach. Furthermore, the ensemble size – that was previously kept on the order of thousands of LES cells in the singly-conditioned CSE – can be reduced to as low as 16 LES cells. This enhanced localization approach reduces the modelling error and the computational cost compared to the conventional CSE approach for tabulated and reduced chemistry models.

Numerical investigation of the autoignition of underexpanded methane jets

This work constitutes a numerical investigation of the autoignition of underexpanded methane jets at high-pressure conditions. Injection pressures from 125 to 500bar, back pressures from 40 to 125bar, and pressure ratios between 2.5 and 10 have been targeted. The aim is to identify the effect of the main control variables of gas injection on the autoignition delay and location. To this end, Reynolds-Averaged Navier-Stokes simulations with the k-ωSST model have been carried out for five broad parametric variations. The computational domain represents a Constant Volume Cell with a prototype gas injector. A two-stage workflow enables proper thermodynamic treatment of the conservation equations with real-gas modeling, sufficient resolution for shock structures in the near nozzle area, detailed kinetics described by the San Diego mechanism and treatment of turbulence-chemistry interaction with elliptic Conditional Moment Closure model. The results are interpreted conceptually as an interplay of jet reactivity, effectively described by an exponential dependence on ambient temperature and a power law dependence on ambient pressure, and of jet aerodynamics, empirically described by a quadratic dependence on pressure ratio. Injection temperature is introduced by defining an appropriate characteristic system temperature and a correlation is constructed, whose predictions are juxtaposed against a modified Arrhenius model and measurements from independent experimental studies in literature. The effect of injection variations on ignition location in physical space is also examined.

Presumed Joint-PDF Modelling for Turbulent Stratified Flames

Turbulence–chemistry interaction models such as the conditional moment closure and various flamelet models require a presumed Probability Density Function (PDF) model of the conditioning variables. In turbulent stratified flames, a joint-PDF model for a reaction progress variable and the mixture fraction is required. The joint-PDF is often modelled by two beta functions using the first and second moments of the two conditioning variables. In this work, the performance of a joint-PDF model based on the flamelet PDF approach is compared to the double beta-PDF model. The conditional PDF of the reaction progress variable conditioned on mixture fraction is modelled with the flamelet PDF. Unlike the beta PDF, the flamelet functional form for the conditional PDF varies with the local mixture fraction. The two PDF models are coupled with a two-dimensional premixed flamelet-generated manifold chemistry model. Two stratified flames of the Cambridge–Sandia bluff-body burner are simulated using a Reynolds-Averaged Navier–Stokes (RANS) approach. The results indicate that the flamelet PDF model has a superior, albeit marginal, predictions of the temperature, major and minor species mass fractions.

A-priori and a-posteriori studies of a direct moment closure approach for turbulent combustion using DNS data of a premixed flame

The direct moment closure (DMC) approach for modeling the filtered chemical source terms in large eddy simulation of turbulent combustion is comprehensively assessed by using direct numerical simulation data of a turbulent premixed flame with a skeletal chemistry. In this approach, closure is achieved by linearizing the chemical source terms using the Taylor series expansions. Different from traditional method, the exponential term is treated as a single variable in the expansion, and the filtered chemical reaction rate can be readily modeled with the first-order and second-order moments, plus higher-order correlations. The a-priori investigations demonstrate that the DMC model is able to reasonably represent the chemical reaction rates, and the second-order correlation of reactive scalars can be well modeled with the algebraic gradient-type model. Compared to the first-order moment closure, the second-order one makes a significant improvement on the predictions. These improvements are also observed in the a-posteriori study in which the premixed flame can be better resolved by the second-order moment closure model. Consequently, the second-order moment closure model provides an alternative for turbulent combustion modeling and is recommended to be used as an improvement to the laminar chemistry closure model. The sensitivity of the DMC approach to the grid size needs further investigation.

Large eddy simulation of turbulent partially premixed flames with inhomogeneous inlets using the dynamic second-order moment closure model

The Sydney piloted partially premixed flames with both homogeneous and inhomogeneous fuel/air inlets are studied using a dynamic second-order moment closure model in the framework of large eddy simulation with a four-step global reaction mechanism. The filtered reaction rates are directly closed using the second-order correlation moments of reactive scalars. Good agreements between LES and experimental data are obtained for time-averaged radial profiles, scatter plots and conditional profiles for mixture fraction, temperature and major species, demonstrating the capability of the model to reproduce the partially premixed flames. Further studies of the combustion characteristics of the two flames suggest that the coincident occurrence of high dissipation and reaction can weaken the flame stability for the homogeneous case, while the separation in the inhomogeneous case can help to enhance the stability. In the inhomogeneous case, the premixed flame is dominant upstream with a proportion of 70% to heat release and then decreases to about 40% downstream. The inhomogeneous conditions can establish a near-stoichiometric reactant belt with lower dissipation near the nozzle exit, which connects the hot pilot and central cold fuel/air mixture. Through this belt, the hot pilot gas can ignite the near-stoichiometric reactant in premixed mode by releasing significant heat and the flame can propagate upstream to form a stable flame.

Assessment of Tabulated Premixed Conditional Moment Closure Model Using Direct Numerical Simulation

ABSTRACT The Conditional Moment Closure (CMC) is a proven methodology to model non-premixed flames, but an emerging technique for analyzing turbulent premixed flames. However, solving a full set of CMC equations is a time-consuming process. The Tabulated Premixed CMC (TPCMC) model is a modification of the premixed CMC model that seeks to reduce the time consumed in solving equations by making the premixed CMC model amenable to the tabulation of chemistry. TPCMC model considers the effect of small scale mixing of reactants and products using the scalar dissipation rate and effect of large scale turbulence on the flame using the beta probability density function. The validation of TPCMC model using Direct Numerical Simulation (DNS) data has been attempted in this paper. DNS of growth of an ignition kernel in a turbulent medium was performed for this purpose. Two sets of simulations, namely 3D DNS with three-step reduced chemistry and 2D DNS with multi-step chemistry, have been carried out in this study using PENCIL code. The Favre averaging of this DNS data was performed on the surfaces of constant radius to assess the accuracy of TPCMC model. The variations in turbulence intensity and integral length scale were considered to generate DNS database. TPCMC model accurately predicted the variation of fuel mass fraction and temperature for all the cases considered in this study and captured the qualitative trends in the CO mass fraction where a negligible deviation from the actual values was observed. The influence of initial parameters related to turbulence considered in this study was negligible on the accuracy of predictions obtained from the model.

A note on modeling mixing in the upper layers of the Bay of Bengal: Importance of water type, water column structure and precipitation

Turbulent mixing in the upper layers of the northern Bay of Bengal is affected by a shallow layer overlying the saline waters of the Bay, which results from the huge influx of freshwater from major rivers draining the Indian subcontinent and from rainfall over the Bay during the summer monsoon. The resulting halocline inhibits wind-driven mixing in the upper layers. The brackish layer also alters the optical properties of the water column. Air-sea interaction in the Bay is expected to play a significant role in the intraseasonal variability of summer monsoons over the Indian subcontinent, and as such the sea surface temperature (SST) changes during the summer monsoon are of considerable scientific and societal importance. In this study, data from the heavily instrumented Woods Hole Oceanographic Institution (WHOI) mooring, deployed at 18oN, 89.5oE in the northern Bay from December 2014 to January 2016, are used to drive a one-dimensional mixing model, based on second moment closure model of turbulence, to explore the intra-annual variability in the upper layers. The model results highlight the importance of the optical properties of the upper layers (and hence the penetration of solar insolation in the water column), as well as the temperature and salinity in the upper layers prescribed at the start of the model simulation, in determining the SST in the Bay during the summer monsoon. The heavy rainfall during the summer monsoon also plays an important role. The interseasonal and intraseasonal variability in the upper layers of the Bay are contrasted with those in the Arabian Sea, by the use of the same model but driven by data from an earlier deployment of a WHOI mooring in the Arabian Sea at 15.5 oN, 61.5 oE from December 1994 to December 1995.

OpenFOAM based conditional moment closure (CMC) model for solving non-premixed turbulent combustion: Integration and validation

This article presents a direct integration of conditional moment closure (CMC), an advanced combustion model with a computational fluid dynamics (CFD) package, OpenFOAM (Open-source Field Operation And Manipulation) to solve non-premixed turbulent combustion. Earlier our group (Roy and Sreedhara, 2016) had coupled CMC code with FLUENT to model non-premixed turbulent combustion, but it was through manual two-way coupling method where CFD and CMC codes were sequentially solved one after the other until the steady state is reached. The extensible and flexible framework of OpenFOAM encouraged us to build a CFD-CMC solver called cmcFoam. This fully-coupled solver solves CMC equations internally in OpenFOAM along with flow-field equations. Major advantages of this solver are (1) it is built in the open-source framework, (2) it can predict the transient behavior of flames, and (3) one can benefit from all features of OpenFOAM. The validity of the proposed solver has been thoroughly demonstrated by accurately predicting experimental data published for the Sandia syngas (CO/H2/N2) flame. Two-dimensional axisymmetric simulations were performed with a modified k-ε turbulence model and using a chemical kinetic scheme having 21 species and 84 chemical reactions (DRM 19). Results predicted by the solver showed an excellent agreement with the published data, both in mixture fraction and physical space. This benchmarked coupled code may be useful to analyze fairly complex industrial problems.