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DOE OSTI2022

Synchrotron X-ray based particle image velocimetry to measure multiphase streamflow and densitometry

Synchrotron-based radiation techniques are increasingly used in the field of flow condition measurements. As a complicated multiphase flow, cavitation has been investigated in a wide variety of industrial fields, to prevent damages on spillways at large dams or intensify the bacteria eradication rate during water treatments. To control the detrimental effects of cavitating flows, the dynamics of and mechanisms affecting cavitation development shall be visualized and identified in real-time. Due to the large reflection and scattering effects of multiphase flow, standard visible-light imaging technologies cannot penetrate vapor bubbles and provide extremely limited information. In this study, the synchrotron source emitted high-flux X-ray pulses are used to capture flow motion and visualize internal structures with sufficiently high spatial and temporal resolutions. The image processing procedures on raw x-ray images are developed using 2D Fourier transform (FT) and wavelet transform (WT) to implement the contrast enhancement and de-noising. Through the edge detection algorithm, tracking either seeded particles or phase interfaces inside the opaque multiphase flow can be achieved to perform the particle image velocimetry. Further, with decomposing particle-only and particle-eliminated subplots from the X-ray phase-contrast image, the time-resolved velocity and void fraction fields are obtained simultaneously, paving a way for further flow condition analysis such as densitometry.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Development of Modified Perturbation Solutions to the One-Phase Stefan Problems With a Convective Boundary

The classical Stefan problem is used to track the moving solid-liquid interface during the freezing process. Perturbation theory has often been applied to find an approximate analytical solution due to the nonlinearity of the moving interface. However, the Stefan number (i.e., the sensible over latent heat) must be small and usually less than 0.01 to assume the perturbation expansion, which in turn limits the thermal engineering applications. In this study, a modified perturbation solution is developed by adding a correction term after the leading-order solution to be valid for a much wider range of Stefan numbers (i.e., 0.01 less than or equal to Ste less than or equal to 1). Specifically, a one-phase Stefan problem is first formulated subjected to a convective boundary in the Cartesian, cylindrical, and spherical coordinate systems for inward solidification. The leading-order solution is calculated based on the regular perturbation theory, while the correction term is obtained using the Monte-Carlo method and a multi-variant regression. Results show that the correction term has a linear relationship with the Stefan number and is not significantly influenced by the Biot number. The proposed modified perturbation solution can accurately and rapidly predict the nonlinear moving interface motion for the freezing process.

DIRECT ENERGY CONVERSION,MATHEMATICS AND COMPUTING↗

DOE OSTI2018

A Numerical Study of Two-Phase Flow and Interfacial Mass Transfer in a Wetted Wall Column for Counter-Current Gas Absorption

We present a numerical study to investigate the effects of surface waves of falling films on the mass transfer from flowing gas phase into the liquid film. The model employs the volume of fluid method to explicitly track the gas-liquid phase interface and a one-fluid formulation to model the multiphase flow coupled with mass transfer between the two phases. The surface waves are generated by the time-varying injection rate of the falling film. The effects of surface wave frequencies and amplitudes on the overall mass transfer also are investigated for a counter-current gas-liquid flow of oxygen and water. Because of the solubility difference in the two phases, the model can capture the concentration discontinuity across the liquid-gas interface. Good agreement is obtained between the numerical results and experimental data reported in the literature. By systematically varying the frequencies and amplitudes of flow rate for the falling film, the surface waves generated on the film surface enhance mass transfer in general. Here, the model prediction shows a proportional increase in mass transfer for small frequencies of surface waves, but saturation at higher frequencies.

42 ENGINEERING↗

DOE OSTI2021

Tribocatalytically-activated formation of protective friction and wear reducing carbon coatings from alkane environment

Minimizing the wear of the surfaces exposed to mechanical shear stresses is a critical challenge for maximizing the lifespan of rotary mechanical parts. In this study, we have discovered the anti-wear capability of a series of metal nitride-copper nanocomposite coatings tested in a liquid hydrocarbon environment. The results indicate substantial reduction of the wear in comparison to the uncoated steel substrate. Analysis of the wear tracks indicates the formation of carbon-based protective films directly at the sliding interface during the tribological tests. Raman spectroscopy mapping of the wear track suggests the amorphous carbon (a-C) nature of the formed tribofilm. Further analysis of the tribocatalytic activity of the best coating candidate, MoN-Cu, as a function of load (0.25–1 N) and temperature (25 °C and 50 °C) was performed in three alkane solutions, decane, dodecane, and hexadecane. Results indicated that elevated temperature and high contact pressure lead to different tribological characteristics of the coating tested in different environments. The elemental energy dispersive x-ray spectroscopy analysis and Raman analysis revealed formation of the amorphous carbon film that facilitates easy shearing at the contact interface thus enabling more stable friction behavior and lower wear of the tribocatalytic coating. These findings provide new insights into the tribocatalysis mechanism that enables the formation of zero-wear coatings.

36 MATERIALS SCIENCE↗

DOE OSTI2017

Dynamics and unsteady morphologies at ice interfaces driven by D 2 O–H 2 O exchange

The growth dynamics of D 2 O ice in liquid H 2 O in a microfluidic device were investigated between the melting points of D 2 O ice (3.8 °C) and H 2 O ice (0 °C). As the temperature was decreased at rates between 0.002 °C/s and 0.1 °C/s, the ice front advanced but retreated immediately upon cessation of cooling, regardless of the temperature. This is a consequence of the competition between diffusion of H 2 O into the D 2 O ice, which favors melting of the interface, and the driving force for growth supplied by cooling. Raman microscopy tracked H/D exchange across the solid H 2 O–solid D 2 O interface, with diffusion coefficients consistent with transport of intact H 2 O molecules at the D 2 O ice interface. At fixed temperatures below 3 °C, the D 2 O ice front melted continuously, but at temperatures near 0 °C a scalloped interface morphology appeared with convex and concave sections that cycled between growth and retreat. Finally, this behavior, not observed for D 2 O ice in contact with D 2 O liquid or H 2 O ice in contact with H 2 O liquid, reflects a complex set of cooperative phenomena, including H/D exchange across the solid–liquid interface, latent heat exchange, local thermal gradients, and the Gibbs–Thomson effect on the melting points of the convex and concave features.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Chapter 18: Learning and Tracking Ad Hoc Fiducial Markers in Spatial Augmented Reality

We describe a spatial augmented reality system with a tangible user interface used to control computer simulations of complex systems. In spatial augmented reality, the user’s physical space is augmented with projected imagery, blending real objects with projected information, and a tangible user interface enables users to manipulate physical objects as controllers for interactive visualizations. Our system learns ad hoc objects in the user’s environment as fiducial markers (i.e., objects that are visually recognized and tracked). When combined with simulation and visualization tools, these interfaces allow the user to control simulations or ensembles of simulations via physical objects using apt metaphors. While other research has leveraged the use of depth cameras, our system enables the use of standard cameras in readily available smartphones and webcams and has an implementation that runs completely in JavaScript in the web browser. We discuss the prerequisite object-recognition requirements for such tangible user interfaces and describe computer-vision and machine-learning algorithms meeting those requirements. We conclude by presenting example applications, which are also available online.

energy distribution↗

DOE OSTI2025

Ultrafast low-temperature metal–insulator interface phonon dynamics and heat transport in a Pt/Gd 3 Fe 5 O 12 heterostructure

Interfacial thermal and acoustic phenomena have an important role in quantum science and technology, including in spintronic and spincaloritronic materials and devices. Simultaneous measurements of the low-temperature thermal and acoustic properties of a metal/insulator heterostructure reveal distinct dynamics in the characteristic phonon frequency ranges of acoustic and thermal transport. The measurements probed a heterostructure consisting of a thin film of Pt on the ferrimagnetic insulator gadolinium iron garnet (Gd 3 Fe 5 O 12 , GdIG) grown epitaxially on a gadolinium gallium garnet substrate. Ultrafast structural dynamics within the Pt layer were tracked using time-resolved ultrafast x-ray diffraction and analyzed to probe interfacial acoustic and thermal properties. The rapid heating of the Pt layer by a 400 nm wavelength femtosecond-duration optical pulse produced transient structural changes that provided the stimulus for these measurements. Rapid heating produced a broadband acoustic pulse that was partially reflected by the Pt/GdIG interface. Temporal frequencies up to 740 GHz, corresponding to angular frequencies of several THz, were detected in a wavelet analysis of the acoustic oscillations of the strain in the Pt layer. The structural results were analyzed to determine (i) the acoustic damping coefficient and phonon mean free path in Pt at frequencies of hundreds of GHz and (ii) the Grüneisen anharmonicity parameter. The thermal conductance of the Pt/GdIG interface was tracked using the slower, tens-of-picosecond-scale, dynamics of the initial cooling of the heated Pt layer. Analysis using a model based on the Boltzmann transport equation shows that the phonon transmission is lower at the phonon frequencies relevant to thermal transport than for subterahertz regime acoustics.

Acoustic phenomena↗

DOE OSTI2019

Monovalent and Divalent Cations at the a-Al2O3(0001)/Water Interface: How Cation Identity Affects Interfacial Ordering and Vibrational Dynamics

Mineral oxide-water interfaces are important for a wide range of industrial, geochemical, and biological processes. The reactivity of these interfaces is strongly impacted by the presence of ions. Thus, it is critical to understand how ions alter the interfacial environment. This can be achieved by measuring the changes in the structure and vibrational dynamics of interfacial water induced by the presence of ions in close vicinity to the mineral surface. The a-Al2O3(0001) surface represents a flexible platform to study the effect of ions on interfacial aqueous environments at positive, neutral and negative surface charge. Using vibrational sum frequency generation (vSFG) in the frequency and time domain, we investigate how monovalent and divalent cations affect the hydrogen bonding environment of the first few layers of interfacial water next to a-Al2O3(0001). Our results indicate that monovalent cations, such as Li+, Na+, K+, and Cs+, appear to have lower binding affinities for the interface compared to Ca2+, Sr2+, and Ba2+. This leads to an interfacial region that is structured in a cation valence dependent manner. Addition of divalent cations at the negatively charged interface (pH 10) increases the spectral intensity in the 3400 cm-1 region compared to neat pH 10 H2O, in contrast to monovalent cations that only attenuate the vSFG signal. Time resolved vSFG measurements reveal that the O-H vibrational lifetime (T1) of interfacial species at pH 10 in the presence of NaCl and BaCl2 remains similar, but restructuring of the surface seen in steady state vSFG is manifested in the degree to which strongly hydrogen bonded species recover to their original populations post excitation. By tracking the accumulation of ions at the interface via the vSFG response, we can characterize the unique surface arrangements of interfacial water molecules induced by a range of monovalent and divalent cations at the a-Al2O3(0001)/water interface.

Piontek, Stefan↗

DOE OSTI2019

Production Operations Management System

The Production Operations Management System (POMS) is a set of software tools which allows production teams and analysis groups across multiple Fermilab experiments to launch, modify and monitor large scale campaigns of related Monte Carlo or data processing jobs. POMS provides a web service interface that enables automated jobs submission on distributed resources according to customers’ requests and subsequent monitoring and recovery of failed submissions, debugging and record keeping. POMS interfaces with existing HEP data access, processing, movement and monitoring tools at Fermilab, including Jobsub, dCache, SAM, and FIFEmon, and in combination with them handles the creation and tracking of unique filenames for multistage campaigns. A flexible, interactive GUI interface exists as part of the system to visualize campaign stages, parallelization tracks and data flow. An important feature of POMS is a one-to-one connection between the GUI campaign visualizer and a text based representation of the complete campaign configuration. An extensible library of template campaign configurations is available. The templates are user modifiable and map cleanly to and from the GUI description of the campaign. Future releases of POMS will interface with Rucio for file movement.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

FY25 Theory and Simulation Performance Target: Development of an integrated modeling framework for fusion reactor design and assessment (Final Report)

This report documents the FY25 Theory and Simulation Performance Target (TSPT) of developing an integrated modeling framework for fusion reactor design and assessment (FREDA). Over Q1-Q4, new capabilities were developed across both plasma and engineering domains and demonstrated on an example representation of a Compact Advanced Tokamak with a Dual Cooled Lead Lithium blanket. This represents a first-of-a-kind demonstration of coupled core-to-wall-to-engineering for a reactor. Self-consistent CESOL workflows were applied to provide core, pedestal, and SOL prediction; new modules were developed for energetic particle stability (FAR3D) and transport (TGLF-EP) analysis; and boundary plasma modeling (SOLPS-ITER, BOUT++/Hermes-3) was expanded to evaluate wall and divertor heat fluxes and interface with engineering thermal analysis. A parameterized CAD tool, TRACER, was expanded to generate medium-fidelity divertor, blanket, and coil geometries; OpenFOAM and Diablo workflows were applied for first-wall and divertor thermal analyses with helium cooling; and reduced-order models were created for high-mass-flux divertor cooling. Magnet multiphysics capabilities were verified between Elmer, Diablo, and a new MFEM-based solver, and workflows enable stress, thermal, and neutron-fluence analysis of TF coils with neutronics-driven heating. Nuclear and blanket analysis workflows were demonstrated, including tritium breeding, transport, and CFD-informed thermo-mechanical assessment. Preliminary multi-fidelity uncertainty quantification workflows were applied to boundary modeling codes and shown to achieve variance reductions with fewer high-fidelity boundary simulations. Key findings highlight the challenges of resolving the ITEP gap to find suitable balance between wall and divertor loads, neutron heating, and practical limits of PFC cooling. Next step priorities are to develop automated workflows to check boundary code convergence and detachment, implement tighter physics-engineering CAD provenance tracking, and inclusion of plasma-material interface models for SLAG and tungsten cracking behavior. Collectively, these developments establish sophisticated capabilities for predictive, multi-fidelity, whole-device modeling that integrates plasma physics, materials, magnets, and nuclear engineering to guide pathways to viable Fusion Pilot Plant design points.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2017

Three-Dimensional Tracking of Interfacial Hopping Diffusion

Theoretical predictions have suggested that molecular motion at interfaces—which influences processes including heterogeneous catalysis, (bio)chemical sensing, lubrication and adhesion, and nanomaterial self-assembly—may be dominated by hypothetical “hops” through the adjacent liquid phase, where a diffusing molecule readsorbs after a given hop according to a probabilistic “sticking coefficient.” Here, we use three-dimensional (3D) single-molecule tracking to explicitly visualize this process for human serum albumin at solid-liquid interfaces that exert varying electrostatic interactions on the biomacromolecule. Following desorption from the interface, a molecule experiences multiple unproductive surface encounters before readsorption. An average of approximately seven surface collisions is required for the repulsive surfaces, decreasing to approximately two and a half for surfaces that are more attractive. The hops themselves are also influenced by long-range interactions, with increased electrostatic repulsion causing hops of longer duration and distance. These findings explicitly demonstrate that interfacial diffusion is dominated by biased 3D Brownian motion involving bulk-surface coupling and that it can be controlled by influencing short- and long-range adsorbate-surface interactions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

TOPAS-nBio: An Extension to the TOPAS Simulation Toolkit for Cellular and Sub-cellular Radiobiology

The TOPAS Monte Carlo (MC) system is used in radiation therapy and medical imaging research, having played a significant role in making Monte Carlo simulations widely available for proton therapy related research. While TOPAS provides detailed simulations of patient scale properties, the fundamental unit of the biological response to radiation is a cell. Thus, our goal was to develop TOPAS-nBio, an extension of TOPAS dedicated to advance understanding of radiobiological effects at the (sub-)cellular, (i.e., the cellular and sub-cellular) scale. TOPAS-nBio was designed as a set of open source classes that extends TOPAS to model radiobiological experiments. TOPAS-nBio is based on and extends Geant4-DNA, which extends the Geant4 toolkit, the basis of TOPAS, to include very low-energy interactions of particles down to vibrational energies, explicitly simulates every particle interaction (i.e., without using condensed histories) and propagates radiolysis products. To further facilitate the use of TOPAS-nBio, a graphical user interface was developed. TOPAS-nBio offers full track-structure Monte Carlo simulations, integration of chemical reactions within the first millisecond, an extensive catalogue of specialized cell geometries as well as sub-cellular structures such as DNA and mitochondria, and interfaces to mechanistic models of DNA repair kinetics. We compared TOPAS-nBio simulations to measured and published data of energy deposition patterns and chemical reaction rates (G values). Our simulations agreed well within the experimental uncertainties. Additionally, we expanded the chemical reactions and species provided in Geant4-DNA and developed a new method based on independent reaction times (IRT), including a total of 72 reactions classified into 6 types between neutral and charged species. Chemical stage simulations using IRT were a factor of 145 faster than with step-by-step tracking. Finally, we applied the geometric/chemical modeling to obtain initial yields of double-strand breaks (DSBs) in DNA fibers for proton irradiations of 3 and 50 MeV and compared the effect of including chemical reactions on the number and complexity of DSB induction. Over half of the DSBs were found to include chemical reactions with approximately 5% of DSBs caused only by chemical reactions. In conclusion, the TOPAS-nBio extension to the TOPAS MC application offers access to accurate and detailed multiscale simulations, from a macroscopic description of the radiation field to microscopic description of biological outcome for selected cells. TOPAS-nBio offers detailed physics and chemistry simulations of radiobiological experiments on cells simulating the initially induced damage and links to models of DNA repair kinetics.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2017

Effect of Slotted Anode on Gas Bubble Behaviors in Aluminum Reduction Cell

In the aluminum reduction cells, gas bubbles are generated at the bottom of the anode which eventually reduces the effective current contact area and the system efficiency. To encourage the removal of gas bubbles, slotted anode has been proposed and increasingly adopted by some industrial aluminum reduction cells. Nonetheless, the exact gas bubble removal mechanisms are yet to be fully understood. A three-dimensional (3D) transient, multiphase flow mathematical model coupled with magnetohydrodynamics has been developed to investigate the effect of slotted anode on the gas bubble movement. The Eulerian volume of fluid approach is applied to track the electrolyte (bath)–molten aluminum (metal) interface. Meanwhile, the Lagrangian discrete particle model is employed to handle the dynamics of gas bubbles with considerations of the buoyancy force, drag force, virtual mass force, and pressure gradient force. The gas bubble coalescence process is also taken into account based on the O’Rourke’s algorithm. The two-way coupling between discrete bubbles and fluids is achieved by the inter-phase momentum exchange. Numerical predictions are validated against the anode current variation in an industrial test. Comparing the results using slotted anode with the traditional one, the time-averaged gas bubble removal rate increases from 36 to 63 pct; confirming that the slotted anode provides more escaping ways and shortens the trajectories for gas bubbles. Furthermore, the slotted anode also reduces gas bubble’s residence time and the probability of coalescence. Moreover, the bubble layer thickness in aluminum cell with slotted anode is reduced about 3.5 mm (17.4 pct), so the resistance can be cut down for the sake of energy saving and the metal surface fluctuation amplitude is significantly reduced for the stable operation due to the slighter perturbation with smaller bubbles.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Structure-guided utilization of lignocellulose for catalysis, energy, and biomaterials

As a complex composite of cellulose, hemicellulose, and lignin, plant lignocellulose has long served as a major resource for biomass conversion, materials engineering, and bio-based product development. High-resolution structural insights enabled by solid-state nuclear magnetic resonance (ssNMR) now allow the mapping of polymer interfaces, identification of functional group accessibility, and tracking of molecular organization during processing, all of which are critical factors for optimizing catalytic strategies. These insights could drive transformative progress in lignocellulose-based applications, including selective depolymerization, improved pretreatment design, and efficient upcycling of lignin into resins, plastics, and biomedical materials. In industry-relevant contexts, such as biofuel generation and renewable material manufacturing, understanding the hydration dynamics, cross-linking patterns, and structural heterogeneity is also essential. The ability to visualize these features in native biomass presents a unique opportunity to develop new strategies for sustainability and performance. As the structural toolbox continues to expand, it is becoming a central enabler for innovations in renewable energy, green chemistry, and advanced bioproducts.

bioproduct↗

DOE OSTI2019

Quantifying Wavelength-Dependent Plasmonic Hot Carrier Energy Distributions at Metal/Semiconductor Interfaces

Hot carriers generated from the nonradiative decay of localized surface plasmons are capable of driving charge-transfer reactions at the surfaces of metal nanostructures. Photocatalytic devices utilizing plasmonic hot carriers are often based on metal nanoparticle/semiconductor heterostructures owing to their efficient electron–hole separation ability. The rapid thermalization of hot carriers generated at the metal nanoparticles yields a distribution of carrier energies that determines the capability of the photocatalytic device to drive redox reactions. Here in this paper, we quantify the thermalized hot carrier energy distribution generated at Au/TiO 2 nanostructures using wavelength-dependent scanning electrochemical microscopy and a series of molecular probes with different redox potentials. We determine the quantum efficiencies and oxidizing power of the hot carriers from wavelength-dependent reaction rates and photocurrent across the metal/semiconductor interface. The wavelength-dependent reaction efficiency tracks the surface plasmon resonance spectrum of the Au nanoparticles, showing that the reaction is facilitated by plasmon excitation, while the responses from molecules with different redox potentials shed light on the energy distribution of the hot holes generated at metal nanoparticle/semiconductor heterostructures. The results provide important insight into the energies of the plasmon-generated hot carriers and quantum efficiencies of plasmonic photocatalytic devices.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

JGI Plant Gene Atlas: an updateable transcriptome resource to improve functional gene descriptions across the plant kingdom

Abstract Gene functional descriptions offer a crucial line of evidence for candidate genes underlying trait variation. Conversely, plant responses to environmental cues represent important resources to decipher gene function and subsequently provide molecular targets for plant improvement through gene editing. However, biological roles of large proportions of genes across the plant phylogeny are poorly annotated. Here we describe the Joint Genome Institute (JGI) Plant Gene Atlas, an updateable data resource consisting of transcript abundance assays spanning 18 diverse species. To integrate across these diverse genotypes, we analyzed expression profiles, built gene clusters that exhibited tissue/condition specific expression, and tested for transcriptional response to environmental queues. We discovered extensive phylogenetically constrained and condition-specific expression profiles for genes without any previously documented functional annotation. Such conserved expression patterns and tightly co-expressed gene clusters let us assign expression derived additional biological information to 64 495 genes with otherwise unknown functions. The ever-expanding Gene Atlas resource is available at JGI Plant Gene Atlas (https://plantgeneatlas.jgi.doe.gov) and Phytozome (https://phytozome.jgi.doe.gov/), providing bulk access to data and user-specified queries of gene sets. Combined, these web interfaces let users access differentially expressed genes, track orthologs across the Gene Atlas plants, graphically represent co-expressed genes, and visualize gene ontology and pathway enrichments.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Ephemeral antibubbles: Spatiotemporal evolution from direct numerical simulations

Antibubbles, which consist of a shell of a low-density fluid inside a high-density fluid, have several promising applications. We show, via extensive direct numerical simulations (DNSs), in both two and three dimensions, that the spatiotemporal evolution of antibubbles can be described naturally by the coupled Cahn-Hilliard-Navier-Stokes (CHNS) equations for a binary fluid. Our DNSs capture elegantly the gravity-induced thinning and breakup of an antibubble via the time evolution of the Cahn-Hilliard scalar-order-parameter field $\phi$, which varies continuously across interfaces, so we do not have to enforce complicated boundary conditions at the moving antibubble interfaces. To ensure that our results are robust, we supplement our CHNS simulations with sharp-interface volume-of-fluid DNSs. We track the thickness of the antibubble and calculate the dependence of the lifetime of an antibubble on several parameters; we show that our DNS results agree with various experimental results; in particular, the velocity with which the arms of the antibubble retract after breakup scales as σ 1/2 , where σ is the surface tension.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Truchas-Flow

The truchas flow algorithm is designed for simulating the behavior of incompressible (but not necessarily constant density) fluids on unstructured meshes while allowing for solid and void phases to serve as moving boundaries. The interface between fluids, solids and void is tracked via a geometric volume of fluid approach. This allows for a single fluid formulation of the governing equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗