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Physics-based numerical models often depend on several parameters to close. Some of them can be expressed using established theoretical or empirical closure formulations. However, some others aggregate complex physical processes and are hence left as tuneable parameters, and can only be calibrated by trial and error. Yet, calibration data are not always available to do so, which prevents these models from being applied to wide ranges of laboratory or river flows. We hence propose a machine learning-based methodology to close any group of unclosed and correlated parameters, applied here to a two-phase/two-layer (2P2L) morphodynamical model. The methodology combines a numerical experiment with a known theoretical solution and machine learning. It is applied to the considered model to close two friction parameters for which generalizable and vastly acknowledged closure formulations lack in the literature. The resulting hybrid model, combining the original 2P2L model and the closure models, is tested against two laboratory dam break test cases. Despite excessive smoothness and underestimation of the concentration in sediment, the hybrid model performed similarly to other models from the literature requiring trial and error calibration and showed high stability and accuracy regarding the estimation of the water-sediment mixture's inertia. HIGHLIGHTS A methodology to close tuneable parameters applicable to many morphodynamical models.; Methodology based on machine learning and self-generated data from a numerical experiment with theoretical solution.; Presentation of a two-phase/two-layer model for unsteady flows highly laden with sediment, used as a proof of concept of the methodology.; New model's performance is similar to evenly complex models requiring trial and error to calibrate tuneable parameters.;

Neural closure models have recently been proposed as a method for efficiently approximating small scales in multiscale systems with neural networks. The choice of loss function and associated training procedure has a large effect on the accuracy and stability of the resulting neural closure model. In this work, we systematically compare three distinct procedures: "derivative fitting", "trajectory fitting" with discretise-then-optimise, and "trajectory fitting" with optimise-then-discretise. Derivative fitting is conceptually the simplest and computationally the most efficient approach and is found to perform reasonably well on one of the test problems (Kuramoto-Sivashinsky) but poorly on the other (Burgers). Trajectory fitting is computationally more expensive but is more robust and is therefore the preferred approach. Of the two trajectory fitting procedures, the discretise-then-optimise approach produces more accurate models than the optimise-then-discretise approach. While the optimise-then-discretise approach can still produce accurate models, care must be taken in choosing the length of the trajectories used for training, in order to train the models on long-term behaviour while still producing reasonably accurate gradients during training. Two existing theorems are interpreted in a novel way that gives insight into the long-term accuracy of a neural closure model based on how accurate it is in the short term. • Analysis of neural closure models with "derivative fitting" and "trajectory fitting". • Trajectory fitting with discretise-then-optimise gives the best results. • Theoretical support is given by two theorems that bound the long-term error. • Two test problems are investigated: Burgers' and Kuramoto-Sivashinsky equations. [ABSTRACT FROM AUTHOR]

Nuclear thermal-hydraulic system analysis codes have been developed to simulate nuclear reactor systems, which solve simplified governing equations by replacing source terms with constitutive relations for simulating entire reactor systems with low computational resources. For half a century, many efforts have been made for wider versatility and higher accuracy of system codes, but various factors can affect the code analysis results, and it was difficult to isolate these factors and interpret them individually. In this study, two system codes, RELAP5 and TRACE, which have many users and are highly reliable, are selected to analyze only the effects of constitutive relations. The influence of constitutive relations is analyzed using in-house platforms that replicate constitute relations of RELAP5 and TRACE equally to exclude factors that may affect analysis results, such as governing equation solvers and user effects. Among the various constitutive relations, the analysis is performed on the wall variables expected to have the most influence on the analysis results. Part 1 paper presents the methodology and wall friction model comparison, while Part 2 paper shows wall heat transfer comparison of the two selected codes.

Nuclear thermal-hydraulic system analysis codes have been developed to comprehensively model nuclear reactor systems to evaluate the safety of a nuclear reactor system. For analyzing complex systems with finite computational resources, system codes usually solve simplified fluid equations for coarsely discretized control volumes with one-dimensional assumptions and replace source terms in the governing equations with constitutive relations. Wall boiling heat transfer models are regarded as essential models in nuclear safety evaluation among many constitutive relations. The wall boiling heat transfer models of two widely used nuclear system codes, RELAP5 and TRACE, are analyzed in this study. It is first described how wall heat transfer models are composed in the two codes. By utilizing the same method described in Part 1 paper, heat fluxes from the two codes are compared under the same thermal-hydraulic conditions. The significant factors for the differences are identified as well as at which conditions the non-negligible difference occurs. Steady-state simulations with both codes are also conducted to confirm how the difference in wall heat transfer models impacts the simulation results.

In this paper, we consider modeling missing dynamics with a nonparametric non-Markovian model, constructed using the theory of kernel embedding of conditional distributions on appropriate reproducing kernel Hilbert spaces (RKHS), equipped with orthonormal basis functions. Depending on the choice of the basis functions, the resulting closure model from this nonparametric modeling formulation is in the form of parametric model. This suggests that the success of various parametric modeling approaches that were proposed in various domains of applications can be understood through the RKHS representations. When the missing dynamical terms evolve faster than the relevant observable of interest, the proposed approach is consistent with the effective dynamics derived from the classical averaging theory. In the linear Gaussian case without the time-scale gap, we will show that the proposed non-Markovian model with a very long memory yields an accurate estimation of the nontrivial autocovariance function for the relevant variable of the full dynamics. The supporting numerical results on instructive nonlinear dynamics show that the proposed approach is able to replicate high-dimensional missing dynamical terms on problems with and without the separation of temporal scales. [ABSTRACT FROM AUTHOR]

The Coupled-Eulerian-Lagrangian (CEL) method is an established numerical approach for the simulation of geotechnical boundary value problems involving soil–structure interaction and large deformations. A consequence of this approach is the existence of two overlapping numerical meshes, one Lagrangian and one Eulerian. The structure is usually modeled by the Lagrangian part while the soil is represented by the Eulerian part. The material within the Eulerian part of the mesh overlap may be considered a kind of 'virtual' or 'ghost' material, as it only exists due to the numerical approach and has no counterpart in reality. Up to now it has never been evaluated how much this ghost material influences the simulation results. In this study, two geotechnical applications of CEL are analyzed which contain ghost material in a zone of soil–structure-interaction. Each example implements several model variants by applying different material combinations and contact formulations. Additional simulations with a pure Arbitrary-Lagrangian-Eulerian (ALE) approach, in which no ghost material exists, are presented to give further insights. [ABSTRACT FROM AUTHOR]

Previous modeling studies have shown that observed micro-scale (mm) variability of nutrients and phytoplankton biomass can strongly impact the large-scale mean growth response of phytoplankton in ways that cannot be represented by typical models based on the mean field approximation. Also, models accounting for the flexible eco-physiology of phytoplankton predict quite different responses to changing environmental conditions compared to most current (inflexible) models. Combining these two ideas for nutrient-phytoplankton systems we have developed a new “Flexible NP closure model” to represent competition among the three typically observed phytoplankton size classes: pico-, nano-, and micro-phytoplankton. Both micro-scale variability and flexible eco-physiology are expected to impact the competition among these size classes. With this work we begin to address how both these factors determine the size structure and size diversity of phytoplankton in the ocean. Under eutrophic conditions, variability does not impact the modeled growth rate of any size class. On the other hand, under oligotrophic conditions, variability preferentially enhances the biomass of the largest typically observed micro-size class, and reduces the biomass of the smallest nano- and intermediate pico-size classes.

Carboniferous volcanic rocks are widely distributed in the Junggar Basin and may provide a unique insight into the tectonic history of this basin. In this paper, we focus on the tectonic evolution of the northern Junggar Basin (NJB) using geochronological and geochemical data to establish the diachronous closure model for the Keramaili oceanic basin. The Lower Carboniferous strata in the eastern part (well WC‐1) and the Upper Carboniferous strata in the western part (well Y‐2) of the NJB are composed of a mix of compositional and rock‐type forms. These volcanic rocks are mostly high‐K assemblages characterized by high K (1.73–6.48%), high alkali (Na2O + K2O = 5.04–7.45%), and low TiO2 (0.57–1.06%). They also exhibit various REE concentrations (68.44–168.19 ppm), an enrichment of LREE (LaN/YbN = 4.54–8.09), and small negative Eu anomalies (δEu = 0.74–0.90). The depletion of HFSE and relevant tectonic setting discrimination diagrams demonstrate that these two volcanic associations were formed in a subduction‐related environment. The data also indicate that the high‐K volcanic rocks are derived from a mix of melted subducted sediments and mantle metasomatized by silica‐rich fluids. Although these two sets of volcanic rocks formed in Early and Late Carboniferous, the similar petrographic and geochemical natures suggest similar magma source and tectonic setting. Along with different regional tectonic settings in the NJB during the Carboniferous, we propose an asynchronous subduction of Keramaili oceanic basin, resulting in the inconsistent temporal and spatial distribution of Upper Carboniferous volcanic rocks in the NJB. [ABSTRACT FROM AUTHOR]

• A new explicit inter-layer mass exchange closure model is rigorously developed. • The new macroscopic model contains curvature, mass exchange and deformation terms • The new model can be used to understand unsaturated flow and transport mechanisms in thin porous media. • The model improves computational speeds allowing users to obtain faster and accurate simulations at lower cost. • The model can be used to solve interesting thin porous media-related problems of common interests. Unsaturated flow through thin layers of porous media are encountered in many industrial applications, including fuel cells and liquid-absorbing hygiene products such as wipes, paper towels, and diapers. Simplified and efficient closure models are very valuable for understanding fluid flow and transport mechanisms in thin porous media and for the development and design of these products. In this study, a new explicit inter-layer mass exchange closure model is rigorously developed and incorporated into the macroscopic mass balance model in order to obtain a complete set of equations which can be used to describe the absorbency processes in thin porous media and to predict the temporal and spatial behaviors of variables such as saturation, porosity, and layer thickness. The developed 2D averaged macroscopic model will enormously improve the computational speed, allowing one to develop a fast and reasonably accurate simulation of the unsaturated flow. Furthermore, the new closure model will account for the mass exchange between the interior adjacent layers and will include the liquid-absorption effect to better understand the transports mechanisms for characterizing absorbent materials and validating models. The developed closure model is validated for the water flooding problem arising in polymer electrolyte membrane fuel cells and can be adapted and used in the future to solve new interesting thin porous media-related problems of common interests. [ABSTRACT FROM AUTHOR]

We present the new discrete optimization-based interface-aware sub-scale dynamics (IA-SSD) closure model for multimaterial cells for Lagrangian cell-centered hydrodynamics. For the multimaterial cell, the kinematic and thermodynamic properties (e.g., velocity, density, pressure and internal energy) will typically vary between the materials. The discrete closure model is responsible for an accurate update of the thermodynamic states of the individual material components in the multimaterial cell, and for determining the nodal forces that move the vertices of the cell. The IA-SSD closure model consists of two stages — a bulk stage followed by a sub-scale stage. During the bulk stage, the total change in the volume of the cell, total force applied to the cell, and total work done on the cell are distributed between the materials to update their volume, velocity and total energy. This distribution is performed using volume fractions of the materials. During the second stage, sub-scale interactions of the materials inside the multimaterial cell are taken into account. At this stage, information about the topology of the materials inside the multimaterial cell is used, allowing the orientations of internal interfaces to be included in the model. Each material interacts in a pair-wise fashion with the materials with which it has a common boundary. The interactions are based on the solution of the acoustic Riemann problem between each pair of materials and are limited using physically justified constraints: positivity of volume, positivity of internal energy, and controlled rate of pressure relaxation. To determine the values of the limiter coefficients, a constrained-optimization framework is employed using a quadratic objective function with linear constraints. It is a first-of-its kind application of constrained optimization to develop discrete closure models in a more rigorous fashion. The pair-wise interaction between materials is essentially one dimensional in the direction that is normal to interface. For this reason, we demonstrate in this paper the performance of our new model on one dimensional numerical examples. [ABSTRACT FROM AUTHOR]

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.

Closure models started from Chou's work have been developed for more than 70 years, aiming at providing analytical tools to describe turbulent flows in the spectral space. In this study, a preliminary attempt is presented to introduce a closure model in the physical space, using the velocity structure functions as key parameters. The present closure model appears to qualitatively reproduce the asymptotic scaling behav-iors at small and large scales, despite some inappropriate behaviors such as oscillations. Therefore, further improvements of the present model are expected to provide appropriate descriptions of turbulent flows in the physical space. [ABSTRACT FROM AUTHOR]

In porous burners, the flame thickness can be smaller than the pore size, resulting in sharp and locally-anchored flame fronts. The presence of such steep gradients at the pore level is a major hurdle for the derivation of volume-averaged models, particularly for the highly non-linear reaction rates. With the intent to address the difficulties associated with volume-averaging for porous media combustion, this work makes use of 3D Pore-Level Direct Numerical Simulations including conjugate heat transfer and complex chemistry in burners of finite length. These detailed 3D simulations are compared to their 1D volume-averaged counterpart, with effective properties estimated directly on the computational domains and of identical thermo-chemical scheme. Discrepancies in terms of burning rate, profiles, and a priori analysis on the reaction rates are discussed. Various pore sizes and geometries are considered. At the pore level, it is shown that preheating, wrinkling and wall quenching are the three main factors driving the global burning rate. Importantly, hydrodynamic dispersion is shown to have an indirect role on combustion processes. From the observations of combustion at pore scale, a new closure for reaction rates based on a flamelet assumption is proposed. It accounts for flame wrinkling and eliminates the unwanted effect of hydrodynamic dispersion on burning rate. [ABSTRACT FROM AUTHOR]

In multiphase flows, the mathematical and physical nature of the phenomena leads to the postulation of closure models to describe the exchange of momentum across the interface as well as other properties such as heat and mass, and various models are available for this purpose . However, unlike dispersed flows, in the case of free surface flows there is a lack of correlations for the drag coefficient, which is an important issue in the management of this type of flow. In this paper, the interfacial density force was calculated dynamically without the use of any correlation through the modeling of the viscous tensor and the normal vector at the interface. The model was applied to test cases as vertical shear flow, dam break and Rayleigh–Taylor instability. The comparison of the results was satisfactory and good agreement with numerical and experimental data from the literature, as well as with the standard approach also employed was observed. These results indicate that the model is a good alternative for stratified flows, and it can also be used directly to calculate the drag coefficient when the standard formulation is preferred. [ABSTRACT FROM AUTHOR]

A closure model for flow-induced fiber orientation is proposed that is based on a reconstruction of the orientation distribution function (ODF). An a priori analysis by using Monte-Carlo simulation was performed. The main results are as follows. For strong rotary Brownian motion (low Péclet numbers), the model performs well even by using only the second moment to reconstruct the ODF. By considering the fourth moment in the reconstruction procedure, the accuracy of the model is improved and the model is applicable to higher Péclet numbers. At very high Péclet numbers and for non-Brownian fibers, the solution predicted by the proposed model deviates from the reference solution. [ABSTRACT FROM AUTHOR]

We investigate the scaling behaviour of a turbulent kinetic energy (TKE) closure model for stably stratified conditions. The mixing length scale for stable stratification is proportional to the ratio of the square root of the TKE and the local Brunt–Väisälä frequency, which is a commonly applied formulation. We analyze the scaling behaviour of our model in terms of traditional Monin–Obukov Similarity Theory and local scaling. From the model equations, we derive expressions for the stable limit behaviour of the flux–gradient relations and other scaling quantities. It turns out that the scaling behaviour depends on only a few model parameters and that the results obey local scaling theory. The analytical findings are illustrated with model simulations for the second GABLS intercomparison study. We also investigate solutions for the case in which an empirical correction function is used to express the eddy diffusivity for momentum as a function of the Richardson number (i.e. an increasing turbulent Prandtl number with stability). In this case, it seems that for certain parameter combinations the model cannot generate a steady-state solution. At the same time, its scaling behaviour becomes unrealistic. This shows that the inclusion of empirical correction functions may have large and undesired consequences for the model behaviour. [ABSTRACT FROM AUTHOR]

Abstract: The two-stream model (TSM; Chatfield and Brost, J. Geophys. Res. 92 (1987) 13263–13276) was developed to account for advection-like vertical mixing features of a passive tracer in the convective boundary layer (CBL). In the TSM, the tracer is advected by two-streams (that is, mean updraft and downdraft), of which the vertical profiles are prescribed as functions of the CBL similarity scales, that is, the CBL height and velocity scales. Compared to the results from large-eddy simulation (LES) and laboratory model, the TSM shows similar plume propagation behavior, but the plume propagation speed appears to be too slow. This drawback of the TSM is found to be mainly due to too weak updraft and downdraft. To improve the TSM, therefore, we modify the prescribed vertical profiles of the mean updraft and downdraft using the LES data. Compared to the original up and down draft velocity profiles, the modified profiles have significantly larger magnitudes in the lower CBL while having similar magnitudes in the upper CBL. The realism of the modified TSM simulation of vertical mixing in the CBL is tested for near surface and elevated tracer sources against the LES results of which validity has been well verified with laboratory experiments and observation. For comparison purpose, we introduce other non-local closure models such as the Blackadar model, the asymmetric convective model, and the transilient turbulence parameterization as well as a local K-theory-based scheme. The modified TSM not only predicts much more improved tracer propagation than the original TSM but also yields the most superior results among the models introduced in this study, showing a remarkably realistic description of the temporal behavior for the concentration distributions and the ground and source level concentrations. [Copyright &y& Elsevier]

Three different kinds of closure model of fiber orientation tensors were applied to simulate numerically the hydrodynamic stability of fiber suspensions in a channel flow. The effects of closure models and three-dimensional ( 3-D) orientation distribution of fibers on the results of stability analysis were examined. It is found that the relationship of the behavior in hydrodynamic stability and the parameter of the fiber given by all the three models are the same. However, the attenuation of flow instability is most distinct using 3-D hybrid model because the orientation of the fiber departures from the flow direction, and least apparent using its 2-D counterpart for the the fibers show a tendency towards alignment with the flow direction in this case. [ABSTRACT FROM AUTHOR]

Infiltration of foreign materials into a fatigue crack has previously been shown to be able to retard the crack and extend fatigue life. Most of the related studies were empirical and phenomenological in nature. To aid engineering decisions, it would be advantageous if the possible outcome of a repair can be evaluated beforehand. To this end, a crack closure model taking into account the additional closure effect of the infiltrant has been developed and verified against experimental results. With this model, the sensitivity of the repair effect to various parameters such as mechanical properties of the infiltrant, depth of penetration and infiltration load level can be assessed. [ABSTRACT FROM AUTHOR]

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable's trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model's stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system's mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model. [ABSTRACT FROM AUTHOR]