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DOE OSTI2016

A Remark on Nonclassical Diffusion Equations with Memory

The nonclassical diffusion equation with hereditary memory u{sub t}−Δu{sub t}−Δu−∫{sub 0}{sup ∞}κ(s)Δu(t−s) ds+φ(u)=f on a 3D bounded domain is considered, for a very general class of memory kernels κ. Setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1997

Numerical Modeling of HgCdTe Solidification: Effects of Phase Diagram, Double-Diffusion Convection and Microgravity Level

Melt convection, along with species diffusion and segregation on the solidification interface are the primary factors responsible for species redistribution during HgCdTe crystal growth from the melt. As no direct information about convection velocity is available, numerical modeling is a logical approach to estimate convection. Furthermore influence of microgravity level, double-diffusion and material properties should be taken into account. In the present study, HgCdTe is considered as a binary alloy with melting temperature available from a phase diagram. The numerical model of convection and solidification of binary alloy is based on the general equations of heat and mass transfer in two-dimensional region. Mathematical modeling of binary alloy solidification is still a challenging numericial problem. A Rigorous mathematical approach to this problem is available only when convection is not considered at all. The proposed numerical model was developed using the finite element code FIDAP. In the present study, the numerical model is used to consider thermal, solutal convection and a double diffusion source of mass transport.

Bune, Andris V.↗

NASA NTRS1999

Experimental and Numerical Study of Ammonium Perchlorate Counterflow Diffusion Flames

Many solid rocket propellants are based on a composite mixture of ammonium perchlorate (AP) oxidizer and polymeric binder fuels. In these propellants, complex three-dimensional diffusion flame structures between the AP and binder decomposition products, dependent upon the length scales of the heterogeneous mixture, drive the combustion via heat transfer back to the surface. Changing the AP crystal size changes the burn rate of such propellants. Large AP crystals are governed by the cooler AP self-deflagration flame and burn slowly, while small AP crystals are governed more by the hot diffusion flame with the binder and burn faster. This allows control of composite propellant ballistic properties via particle size variation. Previous measurements on these diffusion flames in the planar two-dimensional sandwich configuration yielded insight into controlling flame structure, but there are several drawbacks that make comparison with modeling difficult. First, the flames are two-dimensional and this makes modeling much more complex computationally than with one-dimensional problems, such as RDX self- and laser-supported deflagration. In addition, little is known about the nature, concentration, and evolution rates of the gaseous chemical species produced by the various binders as they decompose. This makes comparison with models quite difficult. Alternatively, counterflow flames provide an excellent geometric configuration within which AP/binder diffusion flames can be studied both experimentally and computationally.

Smooke, M. D.↗

DOE OSTI2015

TiO 2 Microboxes with Controlled Internal Porosity for High‐Performance Lithium Storage

Abstract Titanium dioxide (TiO 2 ) is considered a promising anode material for high‐power lithium ion batteries (LIBs) because of its low cost, high thermal/chemical stability, and good safety performance without solid electrolyte interface formation. However, the poor electronic conductivity and low lithium ion diffusivity of TiO 2 result in poor cyclability and lithium ion depletion at high current rates, which hinder them from practical applications. Herein we demonstrate that hierarchically structured TiO 2 microboxes with controlled internal porosity can address the aforementioned problems for high‐power, long‐life LIB anodes. A self‐templating method for the synthesis of mesoporous microboxes was developed through Na 2 EDTA‐assisted ion exchange of CaTiO 3 microcubes. The resulting TiO 2 nanorods were organized into microboxes that resemble the microcube precursors. This nanostructured TiO 2 material has superior lithium storage properties with a capacity of 187 mAh g −1 after 300 cycles at 1 C and good rate capabilities up to 20 C.

Gao, Xuehui↗

DOE OSTI2017

An Investigation into Solution Verification for CFD-DEM

This report presents the study of the convergence behavior of the computational fluid dynamicsdiscrete element method (CFD-DEM) method, specifically National Energy Technology Laboratory’s (NETL) open source MFiX code (MFiX-DEM) with a diffusion based particle-tocontinuum filtering scheme. In particular, this study focused on determining if the numerical method had a solution in the high-resolution limit where the grid size is smaller than the particle size. To address this uncertainty, fixed particle beds of two primary configurations were studied: i) fictitious beds where the particles are seeded with a random particle generator, and ii) instantaneous snapshots from a transient simulation of an experimentally relevant problem. Both problems considered a uniform inlet boundary and a pressure outflow. The CFD grid was refined from a few particle diameters down to 1/6 th of a particle diameter. The pressure drop between two vertical elevations, averaged across the bed cross-section was considered as the system response quantity of interest. A least-squares regression method was used to extrapolate the grid-dependent results to an approximate “grid-free” solution in the limit of infinite resolution. The results show that the diffusion based scheme does yield a converging solution. However, the convergence is more complicated than encountered in simpler, single-phase flow problems showing strong oscillations and, at times, oscillations superimposed on top of globally non-monotonic behavior. The challenging convergence behavior highlights the importance of using at least four grid resolutions in solution verification problems so that (over-determined) regression-based extrapolation methods may be applied to approximate the grid-free solution. The grid-free solution is very important in solution verification and VVUQ exercise in general as the difference between it and the reference solution largely determines the numerical uncertainty. By testing different randomized particle configurations of the same general problem (for the fictitious case) or different instances of freezing a transient simulation, the numerical uncertainties appeared to be on the same order of magnitude as ensemble or time averaging uncertainties. By testing different drag laws, almost all cases studied show that model form uncertainty in this one, very important closure relation was larger than the numerical uncertainty, at least with a reasonable CFD grid, roughly five particle diameters. In this study, the diffusion width (filtering length scale) was mostly set at a constant of six particle diameters. A few exploratory tests were performed to show that similar convergence behavior was observed for diffusion widths greater than approximately two particle diameters. However, this subject was not investigated in great detail because determining an appropriate filter size is really a validation question which must be determined by comparison to experimental or highly accurate numerical data. Future studies are being considered targeting solution verification of transient simulations as well as validation of the filter size with direct numerical simulation data.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Spatiotemporal Changes in Trace Metal Bioavailability in the Sediment Pore water of a Constructed Wetland Using Passive Pore water Samplers

Sediments in aquatic systems often act as a major sink for contaminants. Diffusive gradient in thin films (DGTs) and in situ equilibrium dialysis samplers (peepers) are two major in situ pore water sampling devices that overcome the problems associated with conventional pore water sampling methods. In the present study, DGTs and peepers were used to study the spatial and seasonal effects (cool months, October–February; warm months, May–September) on metal bioavailability in the H‐02 constructed wetland and the sink versus source role of the sediments by calculating the metal resupply capacity. Data showed similar seasonal trends in metal concentrations using passive samplers, peepers, and DGTs. Pooled Cu and Zn concentrations measured using DGTs were lower in warm months (1.67 ± 1.50 and 2.62 ± 0.68 μg L –1 , respectively, p < 0.001) versus in cool months (2.12 ± 0.65 and 5.58 ± 1.33 μg L –1 , respectively, p < 0.001; mean ± 95% confidence interval). Sulfate (SO 4 2− ) concentrations were significantly (p = 0.0139) lower in warm months (averaged at 0.22 ± 0.05 mg L –1 ) compared to in cool months (0.16 ± 0.05 mg L –1 ). The increase in SO 4 2− concentration is an indicator of the lower activity of sulfate‐reducing bacteria, which need SO 4 2− during anaerobic respiration, in which SO 4 2− is reduced to sulfide (S 2– ) that forms insoluble salts with Cu and Zn, which could partially explain the higher bioavailability of these metals in the cool season. Metal resupply capacity of the sediments was mostly <0.2 for Cu and Zn. Taken together, the H0‐2 wetland sediments mostly acted as a sink to both Cu and Zn over the course of the present study. Environ Toxicol Chem 2023;42:2726–2736. © 2023 The Authors. Environmental Toxicology and Chemistry published by Wiley Periodicals LLC on behalf of SETAC.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2015

Constant-complexity stochastic simulation algorithm with optimal binning

At the molecular level, biochemical processes are governed by random interactions between reactant molecules, and the dynamics of such systems are inherently stochastic. When the copy numbers of reactants are large, a deterministic description is adequate, but when they are small, such systems are often modeled as continuous-time Markov jump processes that can be described by the chemical master equation. Gillespie’s Stochastic Simulation Algorithm (SSA) generates exact trajectories of these systems, but the amount of computational work required for each step of the original SSA is proportional to the number of reaction channels, leading to computational complexity that scales linearly with the problem size. The original SSA is therefore inefficient for large problems, which has prompted the development of several alternative formulations with improved scaling properties. We describe an exact SSA that uses a table data structure with event time binning to achieve constant computational complexity with respect to the number of reaction channels for weakly coupled reaction networks. We present a novel adaptive binning strategy and discuss optimal algorithm parameters. We compare the computational efficiency of the algorithm to existing methods and demonstrate excellent scaling for large problems. This method is well suited for generating exact trajectories of large weakly coupled models, including those that can be described by the reaction-diffusion master equation that arises from spatially discretized reaction-diffusion processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2017

Synergy of van der Waals and self-interaction corrections in transition metal monoxides

Density functional theory in principle predicts correct ground-state properties for all materials. However, the rocksalt structure of MnO has been obtained only by the high-level random phase approximation and diffusion Monte Carlo (DMC). In this work, we propose and test for MnO, FeO, CoO, and NiO that a semilocal density functional can solve this problem by properly including both self-interaction and van der Waals corrections. The importance of the latter was previously unanticipated. The MnO structural energy difference from SCAN+rVV10+U agrees well with that from DMC. Here SCAN is a recent semilocal exchange-correlation functional, rVV10 is the revised Vydrov-VanVoorhis long-range van der Waals correction, and the onsite Hubbard U is taken from linear response.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Control-Affine Schrödinger Bridge and Generalized Bohm Potential

From a stochastic control perspective, the Schrödinger bridge is a density-valued continuous curve parameterized by time that connects a given pair of initial and terminal probability densities via minimum effort controlled Brownian motion. The control-affine Schrödinger bridge extends this idea to a generic control-affine Itô diffusion, possibly with an additive state cost. Here, in this letter, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex potential. This complex-valued potential is a generalization of the real-valued Bohm potential in quantum mechanics. Our derived potential is akin to the optical potential in nuclear physics where the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function). The key takeaway is that the process noise that drives the evolution of probability densities induces an absorbing medium in the evolution of wave function. These results make new connections between control theory and non-equilibrium statistical mechanics through the lens of quantum mechanics.

Markov processes↗

DOE OSTI2022

A Statistical Interpolation Code for Ocean Analysis and Forecasting

Abstract We present a data assimilation package for use with ocean circulation models in analysis, forecasting, and system evaluation applications. The basic functionality of the package is centered on a multivariate linear statistical estimation for a given predicted/background ocean state, observations, and error statistics. Novel features of the package include support for multiple covariance models, and the solution of the least squares normal equations either using the covariance matrix or its inverse—the information matrix. The main focus of this paper, however, is on the solution of the analysis equations using the information matrix, which offers several advantages for solving large problems efficiently. Details of the parameterization of the inverse covariance using Markov random fields are provided and its relationship to finite-difference discretizations of diffusion equations are pointed out. The package can assimilate a variety of observation types from both remote sensing and in situ platforms. The performance of the data assimilation methodology implemented in the package is demonstrated with a yearlong global ocean hindcast with a 1/4° ocean model. The code is implemented in modern Fortran, supports distributed memory, shared memory, multicore architectures, and uses climate and forecasts compliant Network Common Data Form for input/output. The package is freely available with an open source license from www.tendral.com/tsis/ .

Srinivasan, Ashwanth↗

DOE OSTI2017

Evolution of Binary Supermassive Black Holes in Rotating Nuclei

The interaction of a binary supermassive black hole with stars in a galactic nucleus can result in changes to all the elements of the binary’s orbit, including the angles that define its orientation. If the nucleus is rotating, the orientation changes can be large, causing large changes in the binary’s orbital eccentricity as well. We present a general treatment of this problem based on the Fokker–Planck equation for f , defined as the probability distribution for the binary’s orbital elements. First- and second-order diffusion coefficients are derived for the orbital elements of the binary using numerical scattering experiments, and analytic approximations are presented for some of these coefficients. Solutions of the Fokker–Planck equation are then derived under various assumptions about the initial rotational state of the nucleus and the binary hardening rate. We find that the evolution of the orbital elements can become qualitatively different when we introduce nuclear rotation: (1) the orientation of the binary’s orbit evolves toward alignment with the plane of rotation of the nucleus and (2) binary orbital eccentricity decreases for aligned binaries and increases for counteraligned ones. We find that the diffusive (random-walk) component of a binary’s evolution is small in nuclei with non-negligible rotation, and we derive the time-evolution equations for the semimajor axis, eccentricity, and inclination in that approximation. The aforementioned effects could influence gravitational wave production as well as the relative orientation of host galaxies and radio jets.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1990

Diffusive shock acceleration of energetic electrons and the method of generations

The diffusive shock acceleration of energetic electrons at a plane shock is studied by means of the method of generations developed by Petukhov et al. (1986), where the particles are binned according to the number of particle acceleration cycles. The method emphasizes the cyclic character of the shock acceleration process, and takes into account the particle energy increment per cycle, the energy loss per cycle, and the escape probability of particles downstream of the shock. In the limit of a small fractional momentum gain per cycle, the method reproduces the known steady-state semianalytic solution for constant diffusion coefficients on both sides of the shock. It is demonstrated that the method may also be applied to time-dependent shock-acceleration problems.

Webb, G. M.↗

NASA NTRS1992

Impeller tandem blade study with grid embedding for local grid refinement

Flow non-uniformity at the discharge of high power density impellers can result in significant unsteady interactions between impeller blades and downstream diffuser vanes. These interactions result in degradation of both performance and pump reliability. The MSFC Pump Technology Team has recognized the importance of resolving this problem and has thus initiated the development and testing of a high head coefficient impeller. One of the primary goals of this program is to improve impeller performance and discharge flow uniformity. The objective of the present work is complimentary. Flow uniformity and performance gains were sought through the application of a tandem blade arrangement. The approach adopted was to numerically establish flow characteristics at the impeller discharge for the baseline MSFC impeller and then parametrically evaluate tandem blade configurations. A tandem design was sought that improves both impeller performance and discharge uniformity. The Navier-Stokes solver AEROVISC was used to conduct the study. Grid embedding is used to resolve local gradients while attempting to minimize model size. Initial results indicate that significant gains in flow uniformity can be achieved through the tandem blade concept and that blade clocking rather than slot location is the primary driver for flow uniformity.

Bache, George↗

NASA NTRS1992

Steady induction effects in geomagnetism. Part 1A: Steady motional induction of geomagnetic chaos

Geomagnetic effects of magnetic induction by hypothetically steady fluid motion and steady magnetic flux diffusion near the top of Earth's core are investigated using electromagnetic theory, simple magnetic earth models, and numerical experiments with geomagnetic field models. The problem of estimating a steady fluid velocity field near the top of Earth's core which induces the secular variation indicated by broad-scale models of the observed geomagnetic field is examined and solved. In Part 1, the steady surficial core flow estimation problem is solved in the context of the source-free mantle/frozen-flux core model. In the first paper (IA), the theory underlying such estimates is reviewed and some consequences of various kinematic and dynamic flow hypotheses are derived. For a frozen-flux core, fluid downwelling is required to change the mean square normal magnetic flux density averaged over the core-mantle boundary. For surficially geostrophic flow, downwelling implies poleward flow. The solution of the forward steady motional induction problem at the surface of a frozen-flux core is derived and found to be a fine, easily visualized example of deterministic chaos. Geomagnetic effects of statistically steady core surface flow may well dominate secular variation over several decades. Indeed, effects of persistent, if not steady, surficially geostrophic core flow are described which may help explain certain features of the present broad-scale geomagnetic field and perhaps paleomagnetic secular variation.

Voorhies, Coerte V.↗

NASA NTRS1993

A numerical analysis of high-temperature heat pipe startup from the frozen state

Continuum and rarefied vapor flows co-exist along the heat pipe length for most of the startup period. A two-region model is proposed in which the vapor flow in the continuum region is modeled by the compressible Navier-Stokes equations, and the vapor flow in the rarefied region is simulated by a self-diffusion model. The two vapor regions are linked with appropriate boundary conditions, and heat pipe wail, wick, and vapor flow are solved as a conjugate problem. The numerical solutions for the entire heat pipe startup process from the frozen state are compared with the corresponding experimental data with good agreement.

Cao, Y.↗

NASA NTRS1994

Evolution of protoplanetary disks with dynamo magnetic fields

The notion that planetary systems are formed within dusty disks is certainly not a new one; the modern planet formation paradigm is based on suggestions made by Laplace more than 200 years ago. More recently, the foundations of accretion disk theory where initially developed with this problem in mind, and in the last decade astronomical observations have indicated that many young stars have disks around them. Such observations support the generally accepted model of a viscous Keplerian accretion disk for the early stages of planetary system formation. However, one of the major uncertainties remaining in understanding the dynamical evolution of protoplanetary disks is the mechanism responsible for the transport of angular momentum and subsequent mass accretion through the disk. This is a fundamental piece of the planetary system genesis problem since such mechanisms will determine the environment in which planets are formed. Among the mechanisms suggested for this effect is the Maxwell stress associated with a magnetic field treading the disk. Due to the low internal temperatures through most of the disk, even the question of the existence of a magnetic field must be seriously studied before including magnetic effects in the disk dynamics. On the other hand, from meteoritic evidence it is believed that magnetic fields of significant magnitude existed in the earliest, PP-disk-like, stage of our own solar system's evolution. Hence, the hypothesis that PP disks are magnetized is not made solely on the basis of theory. Previous studies have addressed the problem of the existence of a magnetic field in a steady-state disk and have found that the low conductivity results in a fast diffusion of the magnetic field on timescales much shorter than the evolutionary timescale. Hence the only way for a magnetic field to exist in PP disks for a considerable portion of their lifetimes is for it to be continuously regenerated. In the present work, we present results on the self-consistent evolution of a turbulent PP disk including the effects of a dynamo-generated magnetic field.

Reyes-Ruiz, M.↗

NASA NTRS1995

Galactic cosmic-ray mediation of a spherical solar wind flow. 1: The steady state cold gas hydrodynamical approximation

Realistic models of the outer heliosphere should consider that the interstellar cosmic-ray pressure becomes comparable to pressures in the solar wind at distances more than 100 AU from the Sun. The cosmic-ray pressure dynamically affects solar wind flow through deceleration. This effect, which occurs over a scale length of the order of the effective diffusion length at large radial distances, has important implications for cosmic-ray modulation and acceleration. As a first step toward solution of this nonlinear problem, a steady state numerical model was developed for a relatively cold spherical solar wind flow which encounters the confining isotropic pressure of the surrounding Galactic medium. This pressure is assumed to be dominated by energetic particles (Galactic cosmic rays). The system of equations, which are solved self-consistently, includes the relevant hydrodynamical equations for the solar wind flow and the spherical cosmic-ray transport equation. To avoid the closure parameter problem of the two-fluid model, the latter equation is solved for the energy-dependent cosmic-ray distribution function.

Le Roux, J. A.↗

DOE OSTI2014

Multilevel Monte Carlo simulation of Coulomb collisions

We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau–Fokker–Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε{sup −2}) or O(ε{sup −2}(lnε){sup 2}), depending on the underlying discretization, Milstein or Euler–Maruyama respectively. This is to be contrasted with a cost of O(ε{sup −3}) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε=10{sup −5}. We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗