Machine Learning Closure Modeling for Reduced-Order Models of Dynamical Systems.
Abstract not provided.
Parish, Eric Joshua↗
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Learning closure models for Turbulence: Calibration of BHR using Machine Learning techniques
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Djibrilla Boureima, Ismael↗
Discrete models of fluids: spatial averaging, closure and model reduction
We consider semidiscrete ODE models of single-phase fluids and two-fluid mixtures. In the presence of multiple fine-scale heterogeneities, the size of these ODE systems can be very large. Spatial averaging is then a useful tool for reducing computational complexity of the problem. The averages satisfy exact balance equations of mass, momentum, and energy. These equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying ODEs. To produce continuum equations that can be simulated without resolving microscale dynamics, we recently proposed a closure method based on the use of regularized deconvolution. Here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods. The structure of the fine-scale particle solvers is reminiscent of molecular dynamics. For this reason we use nonlinear averaging introduced for atomistic systems by Noll, Hardy, and Murdoch-Bedeaux. We also consider a simpler linear averaging originally developed in large eddy simulation of turbulence. We present several simple but representative examples of spatially averaged ODEs, where the closure error can be analyzed. Based on this analysis we suggest a general strategy for reducing the relative error of approximate closure. For problems with periodic highly oscillatory material parameters we propose a spectral boosting technique that augments the standard deconvolution and helps to correctly account for dispersion effects. We also conduct several numerical experiments, one of which is a complete mesoscale simulation of a stratified two-fluid flow in a channel. In this simulation, the operation count per coarse time step scales sublinearly with the number of particles.
Multiscale modeling, discrete models, fluid flow↗
CFD-DEM and PR-DNS studies of low-temperature densely packed beds
Over the past few decades, granular media is gaining attention as a viable option for heat transfer fluids (HTFs). Several research efforts are studying the use of particle-based heat transfer fluids in a wide variety of applications. With this motivation, the current work focusses on analyzing the different heat transfer mechanisms in low-temperature mono-sized densely packed granular media. To study the heat transfer behavior of granular media at different scales, the current work employs a two-way coupled computational strategy. The motion of particles is solved using the Discrete Element Method (DEM) and the interstitial air is solved using a Finite-Volume (CFD) approach. The Open-Source library CFDEM Coupling® is used in the current study to join the Finite Volume PISO solver of OpenFOAM® and the DEM solver of LIGGGHTS®. Typically, particle-particle contact conduction and particle-air convection are the most popular closure models. But recent research identified a different heat transfer phenomenon in packed beds that cannot be identified by conduction or convection models. While closure models were developed to implement this on a CFD-DEM framework, they did not capture the effect of intra-particulate thermal gradients on this phenomenon. Hence the current work also employs Particle-Resolved Direct Numerical Simulations (PR-DNS) to gain valuable insights allowing for the modification of existing models. A new closure model is then proposed here and is implemented in the CFD-DEM framework. This model provides key insights into the different heat transfer mechanism of packed beds.
42 ENGINEERING↗
Predictive RANS simulations via Bayesian Model-Scenario Averaging
The turbulence closure model is the dominant source of error in most Reynolds-Averaged Navier–Stokes simulations, yet no reliable estimators for this error component currently exist. Here we develop a stochastic, a posteriori error estimate, calibrated to specific classes of flow. It is based on variability in model closure coefficients across multiple flow scenarios, for multiple closure models. The variability is estimated using Bayesian calibration against experimental data for each scenario, and Bayesian Model-Scenario Averaging (BMSA) is used to collate the resulting posteriors, to obtain a stochastic estimate of a Quantity of Interest (QoI) in an unmeasured (prediction) scenario. The scenario probabilities in BMSA are chosen using a sensor which automatically weights those scenarios in the calibration set which are similar to the prediction scenario. The methodology is applied to the class of turbulent boundary-layers subject to various pressure gradients. For all considered prediction scenarios the standard-deviation of the stochastic estimate is consistent with the measurement ground truth. Furthermore, the mean of the estimate is more consistently accurate than the individual model predictions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
The effect of capturing the correct turbulence dissipation rate in BHR
In this manuscript, we discuss the shortcoming of a quasi-equilibrium assumption made in the BHR closure model. Turbulence closure models generally assume fully developed turbulence, which is not applicable to 1) non-equilibrium turbulence (e.g. change in mean pressure gradient) or 2) laminar-turbulence transition flows. Based on DNS data, we show that the current BHR dissipation equation [modeled based on the fully developed turbulence phenomenology] does not capture important features of nonequilibrium flows. To demonstrate our thesis, we use the BHR equations to predict a non-equilibrium flow both with the BHR dissipation and the dissipation from DNS. We find that the prediction can be substantially improved, both qualitatively and quantitatively, with the correct dissipation rate. We conclude that a new set of nonequilibrium phenomenological assumptions must be used to develop a new model equation for the dissipation to accurately predict the turbulence time scale used by other models.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Simulation of Boiling Two-Phase Flow in a Helical Coil Steam Generator Using the Spectral Element Code Nek-2P
To enable the design of a light water small modular reactor, the boiling flow inside a helical coil steam generator has been simulated with the two-fluid model in Nek-2P. Nek-2P is the multiphase branch of the spectral element code Nek5000. Details of the implementation of the two-fluid model and the included closure models are discussed. The presented closure models include interactions for momentum, heat, and mass transfer between phases. Models for the drag, lift, and turbulent dispersion forces are included. The complete model is fully consistent in the limits of both phasic volume fractions approaching zero and is able to simulate flows of dispersed vapor, continuous liquid, dispersed liquid, continuous vapor, or any combination thereof. The closure models and their implementation in Nek-2P have been validated by comparing to experimental data for a boiling flow, demonstrating excellent agreement. Results from the simulation of the helical coil indicate strong phasic separation driven by the effects of buoyancy and inertia. Finally, significant differences were observed in the results compared to simulations performed using Star-CCM+, although these differences were somewhat expected.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Modeling the Interactions Between Multiple Crack Closure Mechanisms at Threshold
A fatigue crack closure model is developed that includes interactions between the three closure mechanisms most likely to occur at threshold; plasticity, roughness, and oxide. This model, herein referred to as the CROP model (for Closure, Roughness, Oxide, and Plasticity), also includes the effects of out-of plane cracking and multi-axial loading. These features make the CROP closure model uniquely suited for, but not limited to, threshold applications. Rough cracks are idealized here as two-dimensional sawtooths, whose geometry induces mixed-mode crack- tip stresses. Continuum mechanics and crack-tip dislocation concepts are combined to relate crack face displacements to crack-tip loads. Geometric criteria are used to determine closure loads from crack-face displacements. Finite element results, used to verify model predictions, provide critical information about the locations where crack closure occurs.
Newman, John A.↗
Equipping Neural Network Surrogates with Uncertainty for Propagation in Physical Systems
Coarse-grained or filtered models typically rely on closure models to account for unresolved scales. For instance, large eddy simulation for modeling turbulent fluid flows explicitly resolves the largest scales, but requires modeling closure terms to account for the sub-filter scales. With the vast amount of data available from high-fidelity simulations, there are unique opportunities to leverage data-driven modeling techniques to formulate expressive and flexible closure models. Despite their flexibility, data-driven models struggle in domain shift settings, i.e. when deployed in configurations not captured in the training dataset. In particular, the efficacy of neural network surrogates is difficult to assess a priori due to the deterministic, point-estimate nature of predictions. In high-consequence applications, such models require reliable uncertainty estimates in the data-informed and out-of-distribution regimes. To quantify uncertainties in both regimes, we employ Bayesian neural networks which are able to capture both epistemic and aleatoric uncertainties. We will discuss challenges associated with the training and evaluation of these networks. Furthermore, we will discuss uncertainty embedding strategies to enable efficient sampling and propagation of uncertainty through high-fidelity simulations.
Bayesian neural networks↗
Modeling of near-wall turbulence
An improved k-epsilon model and a second order closure model is presented for low Reynolds number turbulence near a wall. For the k-epsilon model, a modified form of the eddy viscosity having correct asymptotic near wall behavior is suggested, and a model for the pressure diffusion term in the turbulent kinetic energy equation is proposed. For the second order closure model, the existing models are modified for the Reynolds stress equations to have proper near wall behavior. A dissipation rate equation for the turbulent kinetic energy is also reformulated. The proposed models satisfy realizability and will not produce unphysical behavior. Fully developed channel flows are used for model testing. The calculations are compared with direct numerical simulations. It is shown that the present models, both the k-epsilon model and the second order closure model, perform well in predicting the behavior of the near wall turbulence. Significant improvements over previous models are obtained.
Shih, T. H.↗
A Second-Order Closure Turbulence Model: New Heat Flux Equations and No Critical Richardson Number
We formulate a new second-order closure turbulence model by employing a recent closure for the pressure-temperature correlation at the equation level. As a result, we obtain new heat flux equations that avoid the long-standing issue of a finite critical Richardson number. The new, structurally simpler model improves on the Mellor-Yamada 1982 and Galperin et al. 1988 models; key feature includes enhanced mixing under stable conditions facilitating agreement with observational, experimental and high-resolution numerical data sets. The model predicts a planetary boundary layer height deeper than predicted by models with low critical Richardson numbers, as demonstrated in single column model runs of the GISS ModelE general circulation model.
second-order closure turbulence model↗
Advancements in engineering turbulence modeling
Some new developments in two-equation models and second order closure models are presented. Two-equation models (k-epsilon models) have been widely used in computational fluid dynamics (CFD) for engineering problems. Most of low-Reynolds number two-equation models contain some wall-distance damping functions to account for the effect of wall on turbulence. However, this often causes the confusion and difficulties in computing flows with complex geometry and also needs an ad hoc treatment near the separation and reattachment points. A set of modified two-equation models is proposed to remove the aforementioned shortcomings. The calculations using various two-equation models are compared with direct numerical simulations of channel flow and flat boundary layers. Development of a second order closure model is also discussed with emphasis on the modeling of pressure related correlation terms and dissipation rates in the second moment equations. All the existing models poorly predict the normal stresses near the wall and fail to predict the 3-D effect of mean flow on the turbulence (e.g. decrease in the shear stress caused by the cross flow in the boundary layer). The newly developed second order near-wall turbulence model is described and is capable of capturing the near-wall behavior of turbulence as well as the effect of 3-D mean flow on the turbulence.
Shih, T.-H.↗
Advances in engineering turbulence modeling
Some new developments in two equation models and second order closure models are presented. In this paper, modified two equation models are proposed to remove shortcomings such as computing flows over complex geometries and the ad hoc treatment near the separation and reattachment points. The calculations using various two equation models are compared with direct numerical solutions of channel flows and flat plate boundary layers. Development of second order closure models will also be discussed with emphasis on the modeling of pressure related correlation terms and dissipation rates in the second moment equations. All existing models poorly predict the normal stresses near the wall and fail to predict the three dimensional effect of mean flow on the turbulence. The newly developed second order near-wall turbulence model to be described in this paper is capable of capturing the near-wall behavior of turbulence as well as the effect of three dimension mean flow on the turbulence.
Shih, T.-H.↗
Scaling laws for homogeneous turbulent shear flows in a rotating frame
The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri=-2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.
Speziale, Charles G.↗
Scaling laws for homogeneous turbulent shear flows in a rotating frame
The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri = -2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.
Speziale, Charles G.↗
Three-dimensional structures and turbulence closure of the wake developing in a wall shear layer
The turbulent wake interacting with the rotating wall shear layer is investigated analytically and numerically. The turbulent wakes of the rotating blades in a compressor which are interacting with the rotating hub-wall boundary layer are analyzed. A modified version of the closure model of the pressure-strain correlation term in the Reynolds stress transport equation is developed to predict the effect of rotation, which is appreciable for the present flow because the thick hub-wall boundary layer is interacting with the rotor wake. It is noted that the Poisson type equation for the pressure-strain correlation has an extra rotation term when the entire flow field is rotating. This extra rotation term is modeled to accommodate the effect of rotation. In addition, the standard correction for the wall effect is incorporated for the utilized Reynolds stress closure model. The rotation-modified Reynolds stress closure model is used to predict the present flow, and the predictions are compared with the experimental data. The experimental data reveal that the characteristics of the three-dimensional turbulent wake interacting with the wall shear layer are considerably altered by the effects of the wall and the rotation. These features are predicted with good accuracy by the turbulence closure model developed.
Hah, C.↗
Development of Numerical Extended Hydrodynamics for Transition-Regime Non-Equilibrium Flows Encountered in Semiconductor Manufacturing Processes
Six months of funding was received for the proposed three year research program (funding for the period from March 1, 1997 to August 31, 1997). Although the official starting date for the project was March 1, 1997, no funding for the project was received until July 1997. In the funded research period, considerable progress was made on Phase I of the proposed research program. The initial research efforts concentrated on applying the 10-, 20-, and 35-moment Gaussian-based closures to a series of standard two-dimensional non-reacting single species test flow problems, such as the flat plate, couette, channel, and rearward facing step flows, and to some other two-dimensional flows having geometries similar to those encountered in chemical-vapor deposition (CVD) reactors. Eigensystem analyses for these systems for the case of two spatial dimensions was carried out and efficient formulations of approximate Riemann solvers have been formulated using these eigenstructures. Formulations to include rotational non-equilibrium effects into the moment closure models for the treatment of polyatomic gases were explored, as the original formulations of the closure models were developed strictly for gases composed of monatomic molecules. The development of a software library and computer code for solving relaxing hyperbolic systems in two spatial dimensions of the type arising from the closure models was also initiated. The software makes use of high-resolution upwind finite-volumes schemes, multi-stage point implicit time stepping, and automatic adaptive mesh refinement (AMR) to solve the governing conservation equations for the moment closures. The initial phase of the code development was completed and a numerical investigation of the solutions of the 10-moment closure model for the simple two-dimensional test cases mentioned above was initiated. Predictions of the 10-moment model were compared to available theoretical solutions and the results of direct-simulation Monte Carlo (DSMC) calculations. The first results of this study were presented at a meeting last year.
Groth, Clinton P. T.↗
Progress Report on SAM Reduced-Order Model Development for Thermal Stratification and Mixing during Reactor Transients
This report documents the initial progress on the reduced-order flow model developments in SAM for thermal stratification and mixing modeling. Two different modeling approaches are pursued. The first one is based on one-dimensional fluid equations with additional terms accounting for the thermal mixing from both flow circulations and turbulent mixing. The second approach is based on three-dimensional coarse-grid CFD approach, in which the full three-dimensional fluid conservation equations are modeled with closure models to account for the effects of turbulence.
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗