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Results for “high-order phase field models”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Here we discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use C 0 and C 1 virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

42 ENGINEERING↗
DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗
DOE OSTI2023

Large Eddy Simulation of Gasoline Sprays in a Lagrangian–Eulerian Framework Using the High-Order Spectral Element Method

Predicting the spray evolution using simulations requires accurate modeling of the turbulent gas-phase flow field. Here, in this study, the high-order spectral-element method (SEM), implemented in the code Nek5000, was used to provide highly resolved solutions to the turbulent flow field. Spray modeling capabilities were implemented into the Nek5000 code. The spray is modeled in a Lagrangian–Eulerian (LE) framework, where the liquid is represented by discrete parcels of droplets. The method for coupling liquid and gas in the context of SEM is described, which allows for very fine meshes to be used without affecting the stability of the solution. Large-eddy simulations (LES) of the eight-hole ECN Spray G gasoline injector were conducted. Numerical results are compared against experimental data for liquid penetration, droplet size and gas velocity. The morphology of the multiplume spray is compared against experimental data. The effect of different spray injection inputs is analyzed. It was found that using a plume direction of 33 deg and an injection cone angle of 30 deg produced the best results overall. This work shows the applicability of SEM for spray modeling applications, where use of a high-order flow solver can help us understand the multiplume spray aerodynamics and how it leads to plume collapse under certain conditions. Results also highlight the need for tuning spray input parameters in the LE framework, even when high-fidelity gas flow solutions are possible.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2021

Real-Time Lossless Compression for Ultra-High-Density Synchrophasor and Point on Wave Data

Modern advanced Phasor Measurement Units (PMUs) are developed with ultra-high reporting rates to meet the demand for monitoring the power systems dynamics in detail. Due to the large volume of data, the communication and storage systems are seriously challenged with the presence of Ultra-High-Density (UHD) synchrophasor and Point on Wave (POW) data. Therefore, it is an urgent task to compress the UHD data for more efficient communication and data storage. This paper proposes several methods to compress the synchrophasor and POW data in a lossless manner. First, an Improved-Time-Series-Special Compression (ITSSC) method is proposed to compress the UHD frequency data. Second, a Delta-difference Huffman method is combined with the TSSC algorithm to compress the UHD phase angle data. Finally, a cyclical high-order delta modulation method is proposed to compress the UHD POW data. The proposed models are extensively tested and compared with different existing lossless compression algorithms using the field-collected synchrophasor and POW data at different reporting rates. The results indicate that the proposed algorithms are efficient in performing lossless compression for the UHD synchrophasor and POW data in real time.

42 ENGINEERING↗
DOE OSTI2022

Large-Eddy Simulation of a wind turbine using a Filtered Actuator Line Model

When dealing with multirotor devices such as quadcopters or wind farms, the cost of blade-resolved large-eddy simulation (LES) becomes prohibitive. Combining LES with a family of lower-fidelity models, called actuator line models (ALMs), has grown in popularity in the past decade. ALM replaces full blade resolution with an array of actuator points or lines parameterized by aerodynamic lift/drag polar plots along the blades. Body forces computed based on these actuator points are then projected onto the LES flow mesh, mimicking the effect of rotating blades on the flow. However, the optimal projection radius and the associated LES grid size is often too restrictive for multirotor simulations. Recently, a new tip-correction-based filtered ALM (F-ALM) was proposed by Martinez-Tossas and Meneveau (2019), which allows coarser-than-optimal grids by avoiding the associated overprediction of thrust. In this work, F-ALM is implemented into a high-order, in-house LES code to simulate National Renewable Energy Laboratory Phase VI wind turbine. It is then followed by a comparison between the baseline ALM and the newly implemented F-ALM in terms of instantaneous and time-averaged flow fields and blade loads, revealing the advantages of F-ALM in preventing the overprediction of power on coarse grids. Finally, this encourages accurate and affordable simulations of multirotor devices in the future.

17 WIND ENERGY↗
DOE OSTI2022

High harmonic generation in mixed XUV and NIR fields at a free-electron laser

Abstract We present the results of an experiment investigating the generation of high-order harmonics by a femtosecond near-infrared (NIR) laser pulse in the presence of an extreme ultraviolet (XUV) field provided by a free-electron laser (FEL), a process referred to as XUV-assisted high-order harmonic generation (HHG). Our experimental findings show that the XUV field can lead to a small enhancement in the harmonic yield when the XUV and NIR pulses overlap in time, while a strong decrease of the HHG yield and a red shift of the HHG spectrum is observed when the XUV precedes the NIR pulse. The latter observations are in qualitative agreement with model calculations that consider the effect of a decreased number of neutral emitters but are at odds with the predicted effect of the correspondingly increased ionization fraction on the phase matching. Our study demonstrates the technical feasibility of XUV-assisted HHG experiments at FELs, which may provide new avenues to investigate correlation-driven electron dynamics as well as novel ways to study and control propagation effects and phase matching in HHG.

47 OTHER INSTRUMENTATION↗
DOE OSTI2023

Quantum-Path-Resolved Attosecond High-Harmonic Spectroscopy

Strong-field ionization of molecules releases electrons which can be accelerated and driven back to recombine with their parent ion, emitting high-order harmonics. This ionization also initiates attosecond electronic and vibrational dynamics in the ion, evolving during the electron travel in the continuum. Revealing this subcycle dynamics from the emitted radiation usually requires advanced theoretical modeling. Here we show that this can be avoided by resolving the emission from two families of electronic quantum paths in the generation process. The corresponding electrons have the same kinetic energy, and thus the same structural sensitivity, but differ by the travel time between ionization and recombination-the pump-probe delay in this attosecond self-probing scheme. We measure the harmonic amplitude and phase in aligned CO 2 and N 2 molecules and observe a strong influence of laser-induced dynamics on two characteristic spectroscopic features: a shape resonance and multichannel interference. This quantum-path-resolved spectroscopy thus opens wide prospects for the investigation of ultrafast ionic dynamics, such as charge migration.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2020

Two-fluid and kinetic transport physics of Kelvin–Helmholtz instabilities in nonuniform low-beta plasmas

Hall-magnetohydrodynamic (Hall-MHD) theory, two-fluid simulations, and kinetic simulations are used in this work to investigate the cross-field transport properties of Kelvin–Helmholtz instabilities in nonuniform low-beta collisionless plasmas. Hall-MHD analysis shows how the linear properties of the instability are modified by density gradients and magnetization. High-order accurate two-fluid and kinetic simulations, with complete dynamics of finite-mass electrons and ions, are applied to a suite of parameter cases to systematically assess the effects of diamagnetic drift, magnetization, charge separation, and finite Larmor motion. Initialization of exact two-species kinetic equilibria facilitates the study of isolated physical effects and enables detailed cross-comparisons between two-fluid and kinetic simulations, including for cases where ion gyroradii are comparable to gradient scale lengths. For nonuniform plasmas with significant space charge, the results of two-fluid and kinetic simulations are found to disagree with Hall-MHD predictions. Kelvin–Helmholtz instability growth rates, per unit shear, are shown to be smaller when ion diamagnetic drift and $E \times B$ drift are parallel and larger when the two drifts are antiparallel. The effect is attributed to polarization drift in the shear layer, which leads to redistribution of charge, alters the electric field that drives plasma advection, and consequently modifies growth rates. Instability-induced mass transport for different parameters is characterized in terms of the flux across the shear layer and a simplified diffusion model. Distribution functions from kinetic simulations are shown to deviate substantially from Maxwellian reconstructions, indicating the importance of kinetic physics during the nonlinear phase of the instability.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Large-eddy simulation of non-vaporizing sprays using the spectral-element method

Predictive simulations of high-pressure sprays require accurate representation of the turbulent gaseous flow field generated by liquid jet. Typically, the accuracy that can be obtained with low-order numerical methods (e.g. finite volume, finite element) is limited by stability issues in fine grids and the order of convergence of the method. In this work, we resolve the turbulent flow field in an Eulerian manner using the high-order spectral element method, coupled with a Lagrangian parcels approach to model the atomizing liquid jet. Large eddy simulations of single-hole sprays under non-evaporative conditions were conducted and compared against experimental data from Margot et al. (2008) and Spray A data from the Engine Combustion Network. The sensitivity of liquid penetration and droplet sizes to different breakup model parameters was studied. The effect of different numerical parameters, such as polynomial order of the solution (grid resolution), on liquid penetration was also analyzed. The method achieved grid-independent results using p-refinement, achieving finer resolution (by a factor of x 1.7 - x 3.5) in the gas-phase solution than in state-of-the-art simulations using the finite-volume method. Here the results showed good agreement with experimental data, demonstrating the ability of the current method to accurately capture liquid penetration and the shape of the spray.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A duality-based coupling of Cosserat crystal plasticity and phase field theories for modeling grain refinement

High-rate deformation processes of metals entail intense grain refinement and special attention needs to be paid to capture the evolution of microstructure. In this article, a new formulation for coupling Cosserat crystal plasticity and phase field is developed. A common approach is to penalize kinematic incompatibility between lattice orientation and displacement-based elastic rotation. However, this can lead to significant solution sensitivity to the penalty parameter, resulting in low accuracy and convergence rates. To address these issues, a duality-based formulation is developed which directly imposes the rotational kinematic compatibility. A weak inf-sup-based skew-symmetric stress projection is introduced to suppress instabilities present in the dual formulation. An additional least squares stabilization is introduced to suppress the spurious lattice rotation with a suitable parameter range derived analytically and validated numerically. The required high-order continuity is attained by the reproducing kernel approximation. It is observed that equal order displacement-rotation-phase field approximations are stable, which allows efficient employment of the same set of shape functions for all independent variables. The proposed formulation is shown to yield superior accuracy and convergence with marginal parameter sensitivity compared to the penalty-based approach and successfully captures the dominant rotational recrystallization mechanism including block dislocation structures and grain boundary migration.

42 ENGINEERING↗
DOE OSTI2022

High-order ( N = 4–6) multiphoton absorption and mid-infrared Kerr nonlinearity in GaP, ZnSe, GaSe, and ZGP crystals

We report a study of high-order multiphoton absorption, nonlinear refraction, and their anisotropy in four notable mid-infrared χ (2) crystals: GaP, ZnSe, GaSe, and ZGP using femtosecond pulses at 2.35 μm with peak intensity in excess of 200 GW/cm 2 . We found that the nonlinear absorption obeys a perturbation model with multiphoton absorption (MPA) orders from N = 4–6, in agreement with the bandgaps of the crystals. A study of the role of free carrier absorption, performed by changing the pulse duration from 30 to 70 fs while maintaining a constant peak intensity, showed that, at our intensity levels, free carriers generated during the MPA process absorb much more strongly than would be expected from their known linear absorption cross section. Possible mechanisms for this anomalous behavior are high-field effects, such as intravalley scattering in the conduction band and absorption to higher lying bands. Nonlinear refractive indices were measured using (i) closed aperture Z-scan and (ii) spectral broadening due to self-phase modulation, both methods agreeing with each other.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Higher symmetry breaking and nonreciprocity in a driven-dissipative Dicke model

Higher symmetries in interacting many-body systems often give rise to new phases and unexpected dynamical behavior. Here, we theoretically investigate a variant of the Dicke model with higher-order discrete symmetry, resulting from complex-valued coupling coefficients between quantum emitters and a bosonic mode. We propose a driven-dissipative realization of this model focusing on optomechanical response of a driven atom tweezer array comprised of 𝑛 subensembles and placed within an optical cavity, with the phase of the driving field advancing stepwise between subensembles. Examining stationary points and their dynamical stability, we identify a phase diagram for 𝑛≥3 with three distinctive features: a ℤ 𝑛 (ℤ 2⁢𝑛 ) symmetry-breaking superradiant phase for even (odd) 𝑛, a normal unbroken-symmetry phase that is dynamically unstable due to nonreciprocal forces between emitters, and a first-order phase transition separating these phases. This 𝑛-phase Dicke model may be equivalently realized in a variety of optomechanical or optomagnonic settings, where it can serve as a test bed for studying high-order symmetry breaking and nonreciprocal interactions in open systems.

Non-reciprocal propagation↗
DOE OSTI2025

High harmonic generation in Haldane model quantum dots

We study theoretically the nonlinear electron dynamics of Haldane model quantum dots placed in the field of an ultrashort optical pulse. One of the tuning parameters of the Haldane model is the phase accumulated by an electron during its transfer between the next nearest neighbor sites. We study how this parameter affect the nonlinear electron dynamics and the generation of high optical harmonics. With increasing the phase from its zero value to 90°, the low-energy electron states in the conduction and valence bands become more localized near the edges of the quantum dot resulting in suppression of the band gap, enhancement of the dipole inter-band coupling, and strong suppression of the average low-energy electron density of the states. As a result, the nonlinear response of the electron system of a Haldane model quantum dot is the strongest at intermediate values of the phase, ≈ 30− 40°. At these values of the phase, the electron dynamics is irreversible and a few first high-order harmonics have the largest intensities. Here, when the phase approaches 90° value the generation of high harmonics is strongly suppressed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Two-color-driven enhanced high-order harmonic generation in solids

We theoretically investigate the emission of high-harmonic (HH) radiation in model crystals by bichromatic few-cycle driving pulses that are composed as the phase-coherent superposition of a mid-infrared fundamental pulse and its second harmonic. Adjusting the model-crystal parameters to reproduce the lowest band gap of MgO, we examine the extent to which distinct domains of the HH spectrum can be controlled and enhanced by tuning the temporal profile of the bichromatic driving laser electric field. We change the driving-pulse shape by varying its fundamental-versus-second-harmonic pulse amplitude ratio and delay, while keeping the energy of the driving laser pulse fixed. For suitable amplitude ratios and delays, we find an up to fivefold enhancement of the spectral HH yield and significant shifts of the HH cutoff frequency.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

Phase transitions of the ferroelectric (ND 4 ) 2 FeCl 5 ·D 2 O under magnetic field

Due to the strong coupling between magnetism and ferroelectricity, (ND 4 ) 2 FeCl 5 ·D 2 O exhibits several intriguing magnetic and electric phases. In this Letter, we include higher-order onsite anisotropic spin interactions to explain the ferroelectric phase transitions of (ND 4 ) 2 FeCl 5 ·D 2 O in a magnetic field and to produce the large weights of high-order harmonic components in the cycloid structure that are observed from neutron diffraction experiments. Moreover, we predict a new ferroelectric phase sandwiched between the FE II and FE III phases in a magnetic field. Furthermore, by emphasizing the importance of the higher-order spin anisotropic interactions, our work provides a framework to understand multiferroic materials with rich phase diagrams.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

MEUMAPPS (C++ Version)

Many materials, metal alloys in particular, have features on the on micrometer or nanometer scale that have a large impact on the properties of the material. These features are known as the microstructure of the material. Understanding why and how the microstructure forms in a material is of fundamental scientific interest as well as of significant technological interest. The capability to predict microstructure evolution in a material allows the intentional design of microstructures and hence the intentional design of material properties. The phase-field method is one of the leading methods for predicting microstructure evolution. One of the most significant problems for phase-field models is their computational expense. Even limited phase-field simulations can easily require thousands of CPU core-hours to complete, which significantly limits their use. This code provides both a general framework for creating scalable, GPU-accelerated phase-field model applications as well as several applications themselves. The code is capable of using hundreds of GPUs efficiently, which greatly reduces the time required to perform simulations. The code is written with an emphasis on performance portability, that is the ability for the code to run efficiently on a number of different computing architectures without modification of the source code. The performance portability of this code is primarily enabled through the use of two libraries, Kokkos (performance portable data structures and execution patterns) and heFFTe (performance portable distributed 3D fast Fourier transforms). The code consists of a core library, applications, and tests. The core library includes shared functionality between applications. This includes interfaces with fast Fourier transform (FFT) libraries such as heFFTe, data structures based on Kokkos, file input and output capabilities, and a solver for infinitesimal strain mechanical equilibrium problems. Five applications are included in the code. The flagship application is the MEUMAPPS-SS application, which implements the Kim-Kim-Suzuki phase-field model for precipitation for an arbitrary number of phases and components in a metal alloy. Five simpler applications are also included that solve the Eshelby inclusion problem, Allen-Cahn equation, the coupled Allen-Cahn and diffusion equations, and the Cahn-Hilliard equation. The code includes two applications to solve the Cahn-Hilliard equation, one with constant-step-size first-order time integration and the second with adaptive high-order time integration.

DeWitt, Stephen [Oak Ridge National Lab. (ORNL), O↗
DOE OSTI2020

Hydrodynamic instabilities and heat transfer characteristics in the duct flow of a fluid in the supercritical thermodynamic regime

The behavior of fluids at supercritical thermodynamic conditions is inherently complex due to large variations in thermodynamic and transport properties. Recent numerical and experimental investigations illustrate ongoing interest for these fluids, especially supercritical CO 2 and supercritical water, for a variety of applications. For example, supercritical water reactors (SCWR) operate in this extreme condition of high-pressure and temperature, resulting in highly dynamic flow fields and unexpected heat transfer regimes. The potential heat transfer benefits in this regime are directly associated with the extreme variations in thermodynamic and transport properties, which occur at, and above, the critical point. This work characterizes the hydrodynamic instabilities that arise for fluids at supercritical thermodynamic conditions when buoyancy forces are significant. Two specific configurations are considered, a natural convection cavity flow, and a mixed convection, heated, horizontal channel flow. Natural convection flow in a cavity is a classical configuration with expected behavior below the critical point. This configuration aids in characterizing the effect of the variable properties in the supercritical thermodynamic regime. Further, limited studies in the existing literature have been conducted for low-Reynolds and intermediate-Rayleigh numbers, mixed-convection channel flows for supercritical water, which is the focus of the channel flow configuration. To investigate the thermally driven hydrodynamic instabilities in this regime, a high-order fully-implicit numerical method is used. Such strong variations in thermophysical properties (in particular, density) are difficult to simulate and an altogether compressible framework is needed. Therefore, the compressible Navier-Stokes equations are solved without any additional assumptions. The fully implicit, high-order in space and time, reconstructed discontinuous Galerkin method as implemented within the multi-physics code called ALE3D (Arbitrary Lagrangian and Eulerian in 2D and 3D), developed at Lawrence Livermore National Laboratory (LLNL), is used. This fully implicit, L-stable method accurately captures the compressible nature of the ow in the limit of very low Mach number. It has been widely accepted that above the critical point, only one phase is observed. However, recent research has indicated the existence of the distinct gas-like and liquid-like regions separated by the Widom line, the locus of the maxima of the specific heat. Along the Widom line, density decreases 6-fold, viscosity drops by a factor of 2, while specific heat spikes by an order of magnitude. These variations, specifically in density and viscosity, produce a thick pseudo-interface and flow dynamics behavior akin to film boiling. A pseudo-film at the heated wall of the cavity and the horizontal channel is observed where buoyancy forces induce mixing through the specific configurations. Further the local Rayleigh and Richardson numbers provide maps of the flow field and the buoyancy forces driving the microscopic mixing. In the first chapter, I describe a background of supercritical fluid and the various applications. The second chapter focuses on the mathematical model and numerical method used for simulations, where a description of the equation of state for supercritical water is described. The third chapter focuses on the natural convection cavity with a heated bottom wall. In this cavity a gas-like and a liquid-like flow within the supercritical thermodynamic regime are observed. The fourth chapter focuses on a forced convection, horizontal channel, distinguishing between the gas-like, liquid-like, and mixed flow regimes. Mixed convection flow, with the addition of gravitational forces in the horizontal channel show the influence of variable properties on the hydrodynamic development, heat transfer, and rising instabilities. The last chapter of this research focuses on characterizing the unstable hydrodynamics through time-averaging processes and analysis of the movement of energy through the developing plumes.

42 ENGINEERING↗
DOE OSTI2022

High Fidelity CFD Simulations Supporting the KP-FHR

Kairos Power, LLC, is developing its version of the Fluoride-cooled High-temperature Reactor, the KP-FHR. The design uses a pebble bed core with fluoride salt as a coolant. The pebbles used in the KP-FHR have a diameter of 4 cm, with a shell fuel region where TRISO particles are embedded. A Pebble bed core design is adopted by several Gen IV reactors, They boast many benefits, such as fuel integrity, highly efficient heat transfer, and passive safety. However, it is challenging to accurately predict temperature and flow inside a pebble bed. Traditional approaches use the porous media model, which regards the pebble bed as a continuous medium, but with different temperature fields representing different levels, such as the fluid temperature, pebble surface temperature, and pebble center temperature. Empirical heat transfer correlations are adopted to calculate the heat transfer coefficient between different phases. However, empirical correlations are usually validated with experimental data, which usually lacks detail inside the pebble bed. The available experimental data is also generally at a high Reynolds number, which falls outside of the conditions of KP-FHR. Explicit computational fluid dynamics (CFD) simulations of randomly packed pebble beds have only become feasible recently. This is thanks to the rapid development of computational power and scalable algorithms. In this work, we used the Spectral Element Method (SEM) CFD code NekRS to simulate the randomly packed pebble bed in a cylindrical container. NekRS, which is the GPU variant of Nek5000, but refactored to utilize the computational power of GPUs using the OCCA library to run on hybrid architecture high performance computing systems. It was initially developed with the libParamunal library, but truncated and tuned for large-scale turbulence simulation. As a result, the SEM reaches higher precision with the same degrees of freedom by using a high-order Lagrange polynomial basis distributed on Gauss-Lobatto-Legendre quadrature inside each element, compared to lower-order methods, such the Finite Volume Method and Finite Element Method. The report is divided into five parts. We start with a general discussion of the pebble bed reactor, along with a specific investigation into the KP-FHR. The second part presents the numerical methodology. In the third part, we study a modular pebble bed with 1741 pebbles in a container of 7 pebble-diameter radius. Beyond LES simulations done by NekRS, we also leveraged the thermal radiation model in OpenFOAM to study heat transfer under no-forced-flow scenarios. Then, in the fourth part we simulated a pebble bed similar to the size of the Hermes Test Reactor. The total number of pebbles is in these simulations is 34,374. The container radius is 14 pebble-diameters. Finally, the report concludes in part five, with a discussion of future work.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗