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NASA NTRS1994

A Comparison of Metallographic Cooling Rate Methods Used in Meteorites

The primary objective of this study was to test the postulate that cooling rates acquired from metal grains in chondrites are consistent with those from iron meteorites. Both types of metal occur in some Group IAB meteorites, which are mixtures of massive metal with well-developed Widmanstatten structures and chondritic inclusions with dispersed metal grains. The grains have textures and compositions similar to chondritic metal, including negligible P. The meteorites studied show little or no sign of shock reheating and textural evidence indicates that silicates and metal were mixed before Widmanstatten patterns formed during cooling. Cooling rates were obtained by comparing measured to modeled taenite grain or lamellae dimensions and central Ni contents. Modeling entails solving diffusion equations using experimental diffusion coefficients, phase relations, and bulk or local Ni and P contents, taking into account geometry, undercooling, and impingement. There is one set of parameters for grains and another, quite different set for Widmanstatten lamellae, including a factor of 30 difference in diffusion coefficients. Yet cooling rates obtained from Widmanstatten structures and metal grains in chondritic inclusions of the same meteorite are consistent; uncertainties in the best data are +/- 10 K/Ma, equivalent to a factor of 1 +/- 0.25. This agreement implies that the data and models are correct or contain fortuitously offsetting errors, which is quite unlikely. Cooling rates range from 40 K/Ma to 70 K/Ma in IAB meteorites that contain both grains and Widmanstatten structures. Rates based on grains in Ni-poor and Ni-rich meteorites lacking Widmansatten patterns expand the range from 30 K/Ma to perhaps 200 K/Ma. Cooling rates correlate with Ni content; Ni-poor meteorites have slower rates than Ni-rich ones. Evidently, IAB meteorites were radially distributed over greater than 30km in a body with a radius less than 50km. A comparison of the available Ar ages with cooling times inferred from the cooling rates suggests that the parent body cooled more slowly after the metallographic cooling rates were established.

Herpfer, Marc A.↗

NASA NTRS2004

Numerical Simulation of Nanostructure Growth

Nanoscale structures, such as nanowires and carbon nanotubes (CNTs), are often grown in gaseous or plasma environments. Successful growth of these structures is defined by achieving a specified crystallinity or chirality, size or diameter, alignment, etc., which in turn depend on gas mixture ratios. pressure, flow rate, substrate temperature, and other operating conditions. To date, there has not been a rigorous growth model that addresses the specific concerns of crystalline nanowire growth, while demonstrating the correct trends of the processing conditions on growth rates. Most crystal growth models are based on the Burton, Cabrera, and Frank (BCF) method, where adatoms are incorporated into a growing crystal at surface steps or spirals. When the supersaturation of the vapor is high, islands nucleate to form steps, and these steps subsequently spread (grow). The overall bulk growth rate is determined by solving for the evolving motion of the steps. Our approach is to use a phase field model to simulate the growth of finite sized nanowire crystals, linking the free energy equation with the diffusion equation of the adatoms. The phase field method solves for an order parameter that defines the evolving steps in a concentration field. This eliminates the need for explicit front tracking/location, or complicated shadowing routines, both of which can be computationally expensive, particularly in higher dimensions. We will present results demonstrating the effect of process conditions, such as substrate temperature, vapor supersaturation, etc. on the evolving morphologies and overall growth rates of the nanostructures.

Hwang, Helen H.↗

NASA NTRSDate not supplied

Composite Bond Line Measurements Based on A Bayesian Analysis of Flash Thermography Data

For bonded composite materials, an accurate characterization of the adhesive bond line is needed to predict failure modes and fracture toughness. In this paper, bond line thickness was estimated from data obtained using through transmission flash thermography. The forward model that predicts back surface temperature is based on a three layer heat diffusion equation with varying diffusivity and flux boundary conditions. The corresponding inverse problem of estimating bond line thickness from measurement data was solved using a Bayesian approach that assumed Gaussian priors for the bond line thickness and thermal diffusivity of the adherends. Finally, the outputs of the thermography based method were compared to measurements that were collected using a micrometer and ultrasound testing.

Flash thermography↗

NASA NTRSDate not supplied

Composite Bond Line Measurements Based on a Bayesian Analysis of Flash Thermography Data

Flash thermography↗

NASA NTRS1997

The Implicit and Explicit alpha-mu Schemes

Artificial numerical dissipation is an important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The method of space-time conservation element and solution element, currently under development, is a nontraditional numerical method for solving conservation laws. The method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. Explicit solvers for model and fluid dynamic conservation laws have previously been investigated. In this paper, we introduce a new concept in the design of implicit schemes, and use it to construct two highly accurate solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to purely inviscid (convective) equations. The stability and consistency of the schemes are analyzed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme. We also construct an explicit solver with the treatment of diffusion being based on that in the implicit solvers. The explicit solver has only a CFL stability limitation on the Courant number, yet it retains the second-order spatial accuracy of the implicit schemes.

Chang, Sin-Chung↗

NASA NTRS1999

Implicit Space-Time Conservation Element and Solution Element Schemes

Artificial numerical dissipation is in important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate implicit numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The new schemes presented are two highly accurate implicit solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to convection-dominated equations with very small viscosity. The stability and consistency of the schemes are analysed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme.

Chang, Sin-Chung↗

DOE OSTI2022

Mass Transport in Membrane Systems: Flow Regime Identification by Fourier Analysis

The numerical calculation of local mass distributions in membrane systems by computational fluid dynamics (CFD) offers indispensable benefits. However, the concept to calculate such distributions in response to separate variations of operation conditions (OCs) makes it difficult to address overall, flow-physics-related questions, which require the consideration of the collective interaction of OCs. It is shown that such understanding-related relationships can be obtained by the analytical solution of the advection–diffusion equation considered. A Fourier series model (FSM) is presented, which provides exact solutions of an advection–diffusion equation for a wide range of OCs. On this basis, a new zeroth-order model is developed, which is very simple and as accurate as the complete FSM for all conditions of practical relevance. Advection-dominated blocked and diffusion-dominated unblocked flow regimes are identified (depending on a Péclet number which compares the flow geometry with a length scale imposed by the flow), which implies relevant requirements for the use of lab results for pilot- and full-scale applications. Analyses reveal the equivalence of variations of OCs, which offers a variety of options to accomplish desired flow regime changes.

Heinz, Stefan (ORCID:0000000248712416)↗

DOE OSTI2019

Development of a Novel Accelerator for Neutron Transport Solution Using the Galerkin Spectral Element Methods (Final Report)

This report summarizes the results of a three-year research project sponsored by the U.S. Department of Energy (DOE) Nuclear Energy University Program (NEUP) to develop and implement advanced acceleration schemes for the DOE NEAMS neutronics code PROTEUSSN. The project team included the University of Massachusetts Lowell, The Ohio State University, University of Michigan, and Argonne National Laboratory. The PROTEUS code is a high-fidelity capable deterministic neutron transport code based on unstructured finite element meshes, which solves the steady-state and transient neutron transport problem using the 2nd-order discrete ordinate method (SN2ND), the method of characteristics (MOC), and the advanced nodal transport method (NODAL). The DSA scheme has been implemented in PROTEUS to speed up SN transport calculations. The existing DSA scheme employs a consistent finite element formulation on the same fine mesh structure as the SN solution While the use of consistent discretizations makes the DSA method effective, it does not necessarily make it efficient because the numerical solution of a discretized elliptic diffusion problem itself can be costly, particularly when the problem size becomes large. We developed a new discontinuous Galerkin (DG) discretization of the diffusion equation, called DG-DSA, which can effectively and efficiently accelerate the SN transport iterations. As compared with the previous work, the novelty of our method is that the diffusion equation is solved on a coarse-mesh grid using the DG methods, and the DG diffusion discretization incorporates local hp adaptation, i.e., local adaptation of mesh size and/or polynomial degree, based on local total cross section (or optical thickness). Therefore, the resulting number of degrees of freedom (DOF) of the DG discretization is much less than the conventional consistent DSA discretizations, and thus DG-DSA can achieve significant improvement in computational efficiency. We implemented this scheme in PROTEUS-SN. In addition, we developed a new stabilization scheme named linear prolongation CMFD (lpCMFD). A novel feature of this scheme is that the conventional flat flux ratio–based scaling approach is replaced with a linear interpolation of the scalar flux differences at the coarse-mesh cell edges between the neutron transport and CMFD calculations. Fourier convergence analysis and numerical results show that lpCMFD is unconditionally stable and more effective than other CMFD based on acceleration schemes such as pCMFD, odCMFD. We also developed a new nonlinear diffusion acceleration scheme for solving neutron transport equations. This scheme, called LR-NDA, employs a local refinement approach on the framework of CMFD by solving a local boundary value problem of the scalar flux on the coarse-mesh structure to replace the piecewise constant scalar flux obtained by CMFD. The refined flux is then used to update the scalar flux in the neutron transport source iteration. It has been demonstrated that that LR-NDA is much more effective and stable than CMFD for a wide range of optical thicknesses. LR-NDA is a local adaptive method, which means LR-NDA does not necessarily require local refinement for all the coarse-mesh cells on the problem domain, i.e., it can be used only for relatively optically thick regions where the standard CMFD scheme would encounter the convergence problem.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1976

Laws of effluent dispersion in the steady-state atmospheric surface layer in stable and unstable conditions

The two-dimensional diffusion equation has been solved by an integral method to obtain the distribution of ground-level concentration of an inert effluent emitted from a semi-infinite area source in a steady-state and horizontally homogeneous atmospheric surface layer. Mean wind velocity and eddy diffusivity profiles derived from empirically determined flux-profile relations of Businger et al. (1971) for stable and unstable surface layers were used. It is found that concentration as a function of downwind distance can be described by a simple formula over distances of practical interest in surface layer dispersion. Corresponding results for a cross-wind infinite line source are obtained by simple differentiation. The concentration distribution is completely determined by the friction velocity, the Monin-Obukhov length, the roughness length, and the effluent source strength. The generalization of the integral method needed to obtain accurate solutions of the diffusion equation with the given wind velocity and diffusivity profiles is discussed in an appendix.

Lebedeff, S. A.↗

DOE OSTI2022

Divergence of Velocity Fields in Electrochemical Systems

The passage of current through electrochemical systems results in the development of concentration gradients in the electrolytic phase that can be modeled using concentrated solution theory. Application of this theory requires knowledge of three concentration-dependent transport coefficients, which are often taken to be conductivity, diffusion coefficient, and the cation transference number with respect to the solvent velocity. The governing diffusion equation for molar concentration contains two additional terms - the thermodynamic factor which is related to activity of the electrolytic species and the solvent velocity. The main advance in this paper is the derivation of an expression for the divergence of the solvent velocity. Solving this equation requires knowledge of the partial molar volume of the electrolyte. Analogous expressions are derived for the mass average and molar average velocities. These velocities occur naturally in the diffusion equation if concentration is expressed as weight fraction and mole fraction of the electrolytic species, respectively.

diffusion↗

NASA NTRS1995

The second-moment climatology of the GATE rain rate data

The first part of this paper presents the description of the GARP (Global Atmospheric Research Program) Atlantic Tropical Experiment (GATE) 1 rain-rate data and its two-dimensional spectral and correlation characteristics, which has made it possible to accomplish the following: to show the concentration of a significant power along the frequency axis in the spatiotemporal spectra; to detect a diurnal cycle (which has a range of variation of about 3.4-5.4 mm/n) as one of the sources of bias in the rain statistics of satellite data; to study the distinction between the north-south and east-west transport of spatial rain-rate field and character of its anisotropy; to evaluate the scales of the distinction between second-moment estimates associated with ground and satellite samples; and to determine the appropriate spatial and temporal scales of simple linear stochastic models fitted to averaged rain-rate fields. The second part of this paper is devoted to an analysis of the diffusion of the rain rate by establishing a relationship between the parameters of the multivariate autoregressive model and the coefficients of a diffusion equation. This analysis led to the use of rain data to estimate the rain advection velocity as well as other coefficients of the diffusion equation of the corresponding field. The results obtained can be used for comparison with corresponding estimates of other sources of data (satellite, Tropical Oceans Global Atmosphere Coupled Ocean - Atmosphere Response Experiment (TOGA, COARE) or simulated by physical models), for generating multiple samples of any size, for solving the inverse problems of some of the hydrodynamic equations, and in some other areas of rain data analysis and modeling.

Polyak, Ilya↗

DOE OSTI2018

A quasilinear operator retaining magnetic drift effects in tokamak geometry

The interaction of radio frequency waves with charged particles in a magnetized plasma is usually described by the quasilinear operator that was originally formulated by Kennel and Engelmann (Phys. Fluids vol. 9, 1966, pp. 2377-2388). In their formulation the plasma is assumed to be homogenous and embedded in a uniform magnetic field. In tokamak plasmas the Kennel-Engelmann operator does not capture the magnetic drifts of the particles that are inherent to the non-uniform magnetic field. To overcome this deficiency a combined drift and gyrokinetic derivation is employed to derive the quasilinear operator for radio frequency heating and current drive in a tokamak with magnetic drifts retained. The derivation requires retaining the magnetic moment to higher order in both the unperturbed and perturbed kinetic equations. The formal prescription for determining the perturbed distribution function then follows a novel procedure in which two non-resonant terms must be evaluated explicitly. The systematic analysis leads to a diffusion equation that is compact and completely expressed in terms of the drift kinetic variables. The equation is not transit averaged, and satisfies the entropy principle, while retaining the full poloidal angle variation without resorting to Fourier decomposition. As the diffusion equation is in physical variables, it can be implemented in any computational code. In the Kennel-Engelmann formalism, the wave-particle resonant delta function is either for the Landau resonance or the Doppler shifted cyclotron resonance. In the combined gyro and drift kinetic approach, a term related to the magnetic drift modifies the resonance condition.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Physics-Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems

In this paper, we present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.

42 ENGINEERING↗

DOE OSTI2021

The scaling of kinetic and transport behaviors in the solution-phase chemistry of a plasma–liquid interface

The reactions at a plasma–liquid interface often involve species such as the solvated electron or the hydroxyl radical, which initiate the reduction or oxidation of solution-phase reactants (so-called scavengers) or are consumed by their own second-order recombination. Here, the mathematical scaling of the reaction–diffusion equations at the interface is used to obtain a characteristic time that can be used to determine the transition from highly efficient scavenger reduction or oxidation to lower efficiencies due to transport limitations. The characteristic time (tc) is validated using numerical solutions of the reaction–diffusion equations. When the scavenger kinetics are faster than second-order recombination, this characteristic transition time scales proportionally with the scavenger diffusivity (Ds) and the square of the scavenger bulk concentration (SB) and inversely proportional to the electron flux (J) squared; that is, t c = D s S B 2 F 2 /J 2 , where F is Faraday’s constant. However, when the scavenger kinetics are comparable or slower than second-order recombination, this scaling does not hold. Extending this analysis to three-dimensional systems shows that the profile of the electron flux on the surface affects the spatial location where reactions are most effective. Finally, the assessment of the implications of these behaviors for the reactor design highlights how effectively controlling the electron flux and solution transport may be necessary to improve the efficiency of scavenger reactions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1994

Multivariate space - time analysis of PRE-STORM precipitation

This paper presents the methodologies and results of the multivariate modeling and two-dimensional spectral and correlation analysis of PRE-STORM rainfall gauge data. Estimated parameters of the models for the specific spatial averages clearly indicate the eastward and southeastward wave propagation of rainfall fluctuations. A relationship between the coefficients of the diffusion equation and the parameters of the stochastic model of rainfall fluctuations is derived that leads directly to the exclusive use of rainfall data to estimate advection speed (about 12 m/s) as well as other coefficients of the diffusion equation of the corresponding fields. The statistical methodology developed here can be used for confirmation of physical models by comparison of the corresponding second-moment statistics of the observed and simulated data, for generating multiple samples of any size, for solving the inverse problem of the hydrodynamic equations, and for application in some other areas of meteorological and climatological data analysis and modeling.

Polyak, Ilya↗

DOE OSTI2022

Finite flame thickness effects on Kolmogorov-Petrovsky-Piskunov turbulent burning velocities

KPP (Kolmogorov-Petrovsky-Piskunov) solutions of the reaction-diffusion equation have application in various physical phenomena occurring in biology, ecology, and reacting flows. In particular, these solutions are commonly used in turbulent combustion to scale turbulent burning velocities. Subject to certain conditions on reaction rate profile through the flame brush and turbulent diffusivity, this theory relates the turbulent burning velocity to the derivative of the reaction rate ($\tilde{ω}$) at the leading edge of the flame brush ($d\tilde{ω}$/$d\tilde{c}$ | $\tilde{c} =$0 ). Such waves are often referred to as “pulled fronts.” However, turbulent flames never actually satisfy the KPP conditions for a pulled front, as the turbulent flame brush, parametrized here by the thickness δ t , consists of an ensemble of laminar flamelets of thickness δ, where ε=δ/δ t $\ll$ 1 is very small, but nonzero, and $d\tilde{ω}$ /$d\tilde{c}$ tends to zero at the brush leading edge for high activation energy, combustion-type kinetics. Here, this paper analyzes these effects on KPP wave solutions, parametrized by ε=δ/δ t and Zeldovich number Ze focusing on whether turbulent flames retain their pulled front character and what the correction to the KPP wave speed is. Variational solutions of the reaction-diffusion equation show that the solution can be expanded in powers of 1/|ln⁡ε|. Both numerical and asymptotic results are presented, showing that the wave still exhibits pulled front solutions but with significant corrections to the KPP result. The leading order correction is of the form |ln⁡ε| –2 and independent of Ze. Higher order corrections are function of both ε and Ze. However, the dominant factor influencing the wave speed correction is due to the finite ε, with Ze exhibiting a weaker effect.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2008

Numerical Determination of Critical Conditions for Thermal Ignition

The determination of ignition or thermal explosion in an oxidizing porous body of material, as described by a dimensionless reaction-diffusion equation of the form .tu = .2u + .e-1/u over the bounded region O, is critically reexamined from a modern perspective using numerical methodologies. First, the classic stationary model is revisited to establish the proper reference frame for the steady-state solution space, and it is demonstrated how the resulting nonlinear two-point boundary value problem can be reexpressed as an initial value problem for a system of first-order differential equations, which may be readily solved using standard algorithms. Then, the numerical procedure is implemented and thoroughly validated against previous computational results based on sophisticated path-following techniques. Next, the transient nonstationary model is attacked, and the full nonlinear form of the reaction-diffusion equation, including a generalized convective boundary condition, is discretized and expressed as a system of linear algebraic equations. The numerical methodology is implemented as a computer algorithm, and validation computations are carried out as a prelude to a broad-ranging evaluation of the assembly problem and identification of the watershed critical initial temperature conditions for thermal ignition. This numerical methodology is then used as the basis for studying the relationship between the shape of the critical initial temperature distribution and the corresponding spatial moments of its energy content integral and an attempt to forge a fundamental conjecture governing this relation. Finally, the effects of dynamic boundary conditions on the classic storage problem are investigated and the groundwork is laid for the development of an approximate solution methodology based on adaptation of the standard stationary model.

Luo, W.↗

DOE OSTI2024

An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model

A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density $f$ = $f$ ($x, v, t$) converges to an isotropic function $M (v)$$ρ$$(x, t)$, called the drift-diffusion limit, where $M$ is a Maxwellian and the physical density $ρ$ satisfies a second-order parabolic PDE known as the drift-diffusion equation. Numerical approximations that mirror this property are said to be asymptotic preserving. In this paper we build a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in $ε$, where 1/$ε$ is the scale of the collision frequency, and asymptotic preserving. Here in particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in $ε$ to an accurate $h$-approximation of the drift-diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to $ε$ and the spacial resolution are also included.

97 MATHEMATICS AND COMPUTING↗