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NASA NTRS1991

A simplified treatment of SiB's land surface albedo parameterization

The earlier presented surface albedo parameterization is simplified by assuming that the reflectance of direct solar radiation is a simple function of solar zenith angle. The function chosen contains three parameters that vary with vegetation type, greenness, and leaf area index. Tables of parameter values are presented. Using these tables, SiB's (Simple Biosphere model) absorbances of direct solar radiation can be reproduced with an average relative error of less than 0.5 percent. Finally, the direct reflectance function is integrated over zenith angle to produce an equation for the surface reflectance of diffuse radiation.

Koster, Randal D.↗

NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗

NASA NTRS1991

Convective instability in superposed fluid and anisotropic porous layers

The onset of thermal convection due to heating from below in a system consisting of a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity is studied. The linear perturbation equations are solved numerically. It is found that the effects of anisotropy on the onset of thermal convection are most profound for small values of the depth ratio xi (ratio of fluid layer thickness to porous layer thickness), since in that case, the onset of convection corresponds to significant motion in both layers. For fixed vlues of the vertical permeability in the porous medium, decreasing the value of zeta (ratio of horizontal to vertical permeability) leads to stabilization of the superposed layer configuration because of increased resistance to motion in the porous medium. For larger values of xi, the onset of motion is increasingly confined to the fluid layer, with the transport of heat through the porous layer occurring primarily by conduction. Accordingly, the influence of zeta on the stability characteristics for larger xi is less significant than the effects of an anisotropic thermal conductivity.

Chen, Falin↗

NASA NTRS1992

Magnetic field decay in isolated neutron stars

Three mechanisms that promote the loss of magnetic flux from an isolated neutron star - Ohmic decay, ambipolar diffusion, and Hall drift - are investigated. Equations of motions are solved for charged particles in the presence of a magnetic field and a fixed background of neutrons, while allowing for the creation and destruction of particles by weak interactions. Although these equations apply to normal neutrons and protons, the present interpretations of their solutions are extended to cover cases of neutron superfluidity and proton superconductivity. The equations are manipulated to prove that, in the presence of a magnetic force, the charged particles cannot be simultaneously in magnetostatic equilibrium and chemical equilibrium with the neutrons. The application of the results to real neutron stars is discussed.

Goldreich, Peter↗

NASA NTRS1995

Solar structure inversion with LOWL data

Inversion results for the radial hydrostatic structure of the Sun, using six months of oscillation data obtained with the LOWL instrument, are presented. Both low and intermediate degree modes are used, thus avoiding the systematic errors that might have occurred in previous inversions by merging more than one data set. Using modes of between 0 deg and 90 deg and frequencies of between 1.5 mHz and 3.5 mHz, the variations with depth of the speed of sound, the density and the pressure were inferred for radii of between 0.05 and 0.85 stellar radius. It was found that in this region, the sound speed was within 0.15% of that of a model constructed using an equation of state that incorporated helium diffusion. The density difference between the Sun and the model was less than 0.8%. Given the small error bars on the inversion results, these differences are considered as being significant.

Basu, Sarbani↗

NASA NTRS1985

Diffusion and solubility of oxygen in silver

The diffusion and solubility of oxygen in Ag in the temperature range between 412 and 862 C was determined. The following interpolation formula was found for the solubility: L = 8.19.1/100.exp(-11 860/RT)Mol O2/g.At.Ag.at 1/.5. The process obeys the Sieverts square root law within the limits of error. The dissolution of oxygen in Ag may be accompanied by the dissociation of the oxygen molecules into atoms. The tests on Ag-foils reveal that below a temperature of about 500 C a higher solubility is simulated by the adsorption of oxygen. The diffusion coefficient of oxygen in silver obeys the following equation: D = 2.72.1/100.exp(-11 000/RT)sq cm/s. The relatively low activation energy of 11 kcal/g.At suggests that the diffusion of oxygen takes places over interstitial sites.

Eichenauer, W.↗

NASA NTRS1987

Downstream boundary effects on the frequency of self-excited oscillations in transonic diffuser flows

An investigation of downstream boundary effects on the frequency of self-excited oscillations in two-dimensional, separated transonic diffuser flows has been conducted numerically by solving the compressible, Reynolds-averaged, thin-layer Navier-Stokes equation with a two-equation turbulence model. It was found that the unsteady diffuser flowfields are very sensitive to the location of the downstream boundary. Extension of the diffuser downstream boundary significantly reduces the frequency and amplitude of oscillations for pressure, velocity and shock. Computational results suggest that the mechanism causing the self-excited oscillation changes from viscous convective wave dominated oscillations to inviscid acoustic wave dominated oscillations when the location of downstream boundary varies from 8.66 to 134.7 throat height. The existence of a suction slot in the experimental setup obscures the physical downstream boundary and, therefore, presents a difficulty for quantitative comparisons between computation and experiment.

Hsieh, T.↗

DOE OSTI2015

Solutal separation in a binary nanofluid due to thermodiffusion

Transport phenomena in porous media have received considerable attention due to an increasing interest in geothermal processes, chemical catalytic reactors, waste storage (especially geological or ocean storage of carbon dioxide), etc. Among others, oil industry has shown an increasing interest in studying diffusion phenomenon. Nanofluid is a term used to describe the suspension of low concentration of metallic and non-metallic nanoparticles in a base fluid. The size of a nanoparticle ranges from 10 to 100nm, and the conventional fluids used are water, ethylene glycol (C{sub 2}H{sub 6}O{sub 2}) or engine oil. Various studies have proven that nanoparticles improve the heat transfer of a base fluid. However, using various nanofluids it has been shown that the results could vary depending on different initial concentrations. The main objective of this paper is to study the diffusion and the thermodiffusion effect in a nanofluid for different fluid/porous media configurations. In this configuration, a liquid layer surrounds a porous layer. The full Brinkman equation coupled with the heat and mass transfer equations have been solved numerically for the porous layer using the finite element technique. The full Navier stokes equation coupled with heat and mass transfer equations have been solved for the liquid layer using the finite element method. A constraint between the liquid and porous layer has been applied to ensure heat flow and mass transfer continuity is maintained. A square cavity filled with hydrocarbon nanofluid of a mixture of fullerene-toluene with varying concentration of fullerene has been subject to different heating conditions. The entire cavity has been considered to be fully wetted with nanofluid. Results have confirmed that in the presence of a nanofluid a heat transfer enhancement is present up to certain initial concentration of the fullerene. The heat convection coefficient has been found to be 16% higher when a nanofluid is used as the working fluid.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

We present that second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage. Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal. In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. Finallyl, we show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Thermodynamic calculations of oxygen self-diffusion in mixed-oxide nuclear fuels

Mixed-oxide fuels containing uranium with thorium and/or plutonium may play an important part in future nuclear fuel cycles. There are, however, significantly less data available for these materials than conventional uranium dioxide fuel. In the present study, we employ molecular dynamics calculations to simulate the elastic properties and thermal expansivity of a range of mixed oxide compositions. These are then used to support equations of state and oxygen self-diffusion models to provide a self-consistent prediction of the behaviour of these mixed oxide fuels at arbitrary compositions.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1984

Investigation of Aeronomic Processes on the Basis of Ionospheric Sporadic E

A method by means of which the characteristics of atmospheric turbulence can be determined is discussed. Turbulent diffusivity deduced from ionospheric sporadic E in different altitudes above the ionospheric stations at Bekescsaba and Juliusrah are given. Seasonal variations in different altitudes above Bekescsaba are shown. An attempt was made to determine the variation of the turbulent diffusivity during and after geomagnetic disturbances by computing the horizontal wind velocity from the thermal wind equation. It is argued that the change of the turbulent diffusivity below the turbopause is opposed to that above the turbopause.

Bencze, P.↗

DOE OSTI2013

A NUMERICAL STUDY OF DIFFUSIVE COSMIC-RAY TRANSPORT WITH ADIABATIC FOCUSING

Focused particle transport in a nonuniform large-scale magnetic field is investigated numerically in the case of isotropic pitch-angle scattering. Evolving particle density profiles and distribution moments are computed from solutions of a system of stochastic differential equations, equivalent to the original Fokker-Planck equation for the particle distribution. Conflicting analytical predictions for the transport coefficients in the diffusion limit, independently calculated by Beeck and Wibberenz and Shalchi, are compared with the numerical results. The reasons for the discrepancies among the analytical and numerical treatments, as well as the general limitations of the diffusion model, are discussed. The telegraph equation, derived in a higher-order expansion of the particle distribution function, is shown to describe the particle transport much more accurately than the diffusion model, especially ahead of a moving density pulse.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1974

Turbulent diffusion with memories and intrinsic shear

The first part of the present theory is devoted to the derivation of a Fokker-Planck equation. The eddies smaller than the hydrodynamic scale of the diffusion cloud form a diffusivity, while the inhomogeneous, bigger eddies give rise to a nonuniform migratory drift. This introduces an eddy-induced shear which reflects on the large-scale diffusion. The eddy-induced shear does not require the presence of a permanent wind shear and is intrinsic to the diffusion. Secondly, a transport theory of diffusivity is developed by the method of repeated-cascade and is based upon a relaxation of a chain of memories with decreasing information. The full range of diffusion consists of inertia, composite, and shear subranges, for which variance and eddy diffusivities are predicted. The coefficients are evaluated. Comparison with experiments in the upper atmosphere and oceans is made.

Tchen, C. M.↗

NASA NTRS1978

An integral solution to a nonlinear diffusion problem

The generalized Galerkin method (or the method of integral relations) is applied to the type of problem described by quasilinear parabolic equations. As an example the problem of nonlinear transient slab diffusion with a general reservoir boundary condition is worked out. The integral relations are given for an arbitrary number of strips, and solutions using up to seven strips have been obtained in order to investigate the convergence of the method.

Hussaini, M. Y.↗

DOE OSTI2013

Second-order discretization in space and time for radiation hydrodynamics

We present a method for solving the equations of radiation hydrodynamics that is second-order accurate in space and time. This method combines the MUSCL-Hancock method for solving the Euler equations with the TR/BDF2 scheme in time for solving the equations of radiative transfer. We use an LDFEM to discretize the radiative transfer equations in space, which, though uncommon for radiation diffusion calculations, is a standard for radiation transport applications. We address the challenges inherent to using different spatial discretizations for the hydrodynamics and radiation and demonstrate how these may be overcome. We define our method for a 1-D model of compressible fluid dynamics coupled with grey radiation diffusion. Using the method of manufactured solutions, we show that the method is second-order accurate in space and time for both the equilibrium diffusion and streaming limit. (authors)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s{sup 2} times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. This includes variable coefficient parabolic equations that might give rise to skew symmetric terms. The RKC1 and RKC2 schemes do not share this convex monotonicity preserving property. One-dimensional and two-dimensional von Neumann stability analyses of RKC1, RKC2, RKL1 and RKL2 are also presented, showing that the latter two have some advantages. The paper includes several details to facilitate implementation. A detailed accuracy analysis is presented to show that the methods reach their design accuracies. A stringent set of test problems is also presented. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Development of a Physical-Topological Model for the Response of a High-Power Vertical DMOS Transistor to the Effect of Pulsed Gamma-Radiation

A method for developing models of semiconductor devices with two-dimensional nonuniform concentration profiles of donors and acceptors in working regions of a semiconductor device structure is for the first time proposed based on a complex of physical-topological modeling of charge carrier transport and process simulation of the forming processes of the device structure. The application of process simulation is due to the need to correctly define the parameters of the semiconductor device structure, which are used as initial data to calculate the electron transport according to the physical-topological model. The parameters of production processes of ion implantation, diffusion, and lithography, which were refined during process simulation, are determined for a high-power metal—oxide—semiconductor (MOS) transistor forming by the double diffusion method in accordance with known electrical characteristics and measured geometrical sizes of the structure. This results in two-dimensional distribution profiles of donors and acceptors in p–n junctions necessary to calculate the transistor breakdown under the effect of pulsed γ radiation. Breakdown processes are modeled with the help of the physical-topological model based on the Poisson and continuity equations as well as expressions for the diffusion and drift current densities in the transistor. Accounting for carriers formed at the instant of γ irradiation is implemented by the introduction of the dependence of the generation coefficient of electron–hole pairs on the radiation-dose power. The results of calculations correlate well with the experimental data, which makes it possible to give a conclusion regarding the adequacy of the proposed complex model.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. Here, we obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

97 MATHEMATICS AND COMPUTING↗