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DOE OSTI2022

FEDIS (The “Fresnel Equations” for Diffuse radiation on Inclined photovoltaic Surfaces) [SWR-22-64]

The well-known Fresnel equations solve for the reflection and transmission of light for precise incident angles. The transmission of diffuse radiation incident on a planar or domed surface is often needed for real-world applications. Due to the complexity of the Fresnel equations, the analytical solution of the integration has hitherto been unobtainable over the last centuries. Therefore, this problem was numerically solved by integrating the angular transmittances in space often leading to substantial computing burden and bias in the results. To efficiently estimate the solar energy resource for a glass-covered photovoltaic (PV) module, we derive an analytical solution of diffuse transmission based on the rigorous integration of an alternate form of the Fresnel equations. The approach leads to a simple yet accurate relative transmittance model that reconciles the solar energy sensed by pyranometers and PV panels. With limited and clearly stated approximations, the complex mathematical derivation resulted in an elegant solution.

Xie, Yu↗

NASA NTRS2004

Amplitude Scintillation due to Atmospheric Turbulence for the Deep Space Network Ka-Band Downlink

Fast amplitude variations due to atmospheric scintillation are the main concerns for the Deep Space Network (DSN) Ka-band downlink under clear weather conditions. A theoretical study of the amplitude scintillation variances for a finite aperture antenna is presented. Amplitude variances for weak scattering scenarios are examined using turbulence theory to describe atmospheric irregularities. We first apply the Kolmogorov turbulent spectrum to a point receiver for three different turbulent profile models, especially for an exponential model varying with altitude. These analytic solutions then are extended to a receiver with a finite aperture antenna for the three profile models. Smoothing effects of antenna aperture are expressed by gain factors. A group of scaling factor relations is derived to show the dependences of amplitude variances on signal wavelength, antenna size, and elevation angle. Finally, we use these analytic solutions to estimate the scintillation intensity for a DSN Goldstone 34-m receiving station. We find that the (rms) amplitude fluctuation is 0.13 dB at 20-deg elevation angle for an exponential model, while the fluctuation is 0.05 dB at 90 deg. These results will aid us in telecommunication system design and signal-fading prediction. They also provide a theoretical basis for further comparison with other measurements at Ka-band.

Ho, C.↗

DOE OSTI2020

Facile and quantitative estimation of strain in nanobubbles with arbitrary symmetry in 2D semiconductors verified using hyperspectral nano-optical imaging

When layers of van der Waals materials are deposited via exfoliation or viscoelastic stamping, nanobubbles are sometimes created from aggregated trapped fluids. Though they can be considered a nuisance, nanobubbles have attracted scientific interest in their own right owing to their ability to generate large in-plane strain gradients that lead to rich optoelectronic phenomena, especially in the semiconducting transition metal dichalcogenides. Determination of the strain within the nanobubbles, which is crucial to understanding these effects, can be approximated using elasticity theory. However, the Föppl–von Kármán equations that describe strain in a distorted thin plate are highly nonlinear and often necessitate assuming circular symmetry to achieve an analytical solution. Here, we present an easily implemented numerical method to solve for strain tensors of nanobubbles with arbitrary symmetry in 2D crystals. The method only requires topographic information from atomic force microscopy and the Poisson ratio of the 2D material. We verify that this method reproduces the strain for circularly symmetric nanobubbles that have known analytical solutions. Finally, we use the method to reproduce the Grüneisen parameter of the E' mode for 1L-WS2 nanobubbles on template-stripped Au by comparing the derived strain with measured Raman shifts from tip-enhanced Raman spectroscopy, demonstrating the utility of our method for estimating localized strain in 2D crystals.

2D materials↗

NASA NTRS2019

Application of System Identification to Parachute Modeling

A series of exact analytical solutions is developed for a capsule with two parachutes to verify CAPDYN and FAST results for terminal conditions. The case described herein is for the capsule and two parachutes acting as a fixed system in terminal descent. In this simplest case, the density is modeled as a constant defined by the initial release altitude. The analytical solution predicts the system terminal dynamic pressure, velocity, and altitude for comparison with CAPDYN and FAST simulation results.

Murri, Daniel G.↗

DOE OSTI2019

Correlation functions for a spin-(1/2) Ising-XYZ diamond chain: Further evidence for quasi-phases and pseudo-transitions

One-dimensional systems with short-range interactions cannot exhibit a long-range order at nonzero temperature. However, there are some particular one-dimensional models, such as the Ising–Heisenberg spin models with a variety of lattice geometries, which exhibit unexpected behavior similar to the discontinuous or continuous temperature-driven phase transition. Although these pseudo-transitions are not true temperature-driven transitions showing only abrupt changes or sharp peaks in thermodynamic quantities, they may be confused while interpreting experimental data. Here we consider the spin-(1/2) Ising-XYZ diamond chain in the regime when the model exhibits temperature-driven pseudo-transitions. We provide a detailed investigation of several correlation functions between distant spins that illustrates the properties of quasi-phases separated by pseudo-transitions. Inevitably, all correlation functions show the evidence of pseudo-transition, which are supported by the analytical solutions and, besides we provide a rigorous analytical investigation around the pseudo-critical temperature. It is worth to mention that the correlation functions between distant spins have an extremely large correlation length at pseudo-critical temperature.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1990

A three-dimensional approach for analysis of sidewall injector mixing and combustion

A three-dimensional (3-D) analytical approach to calculate the mixing and combustion efficiency along engine combustors that use sidewall fuel injectors is described. The analysis consists of three parts: (1) an empirical correlation for fuel penetration, (2) application of integral conservation laws following jet turning and pressure equilibration, and (3) an analytical solution for the simplified three-dimensional partial differential equations to describe the downstream mixing and combustion process. Numerical results are compared to some empirical and experimental work, and favorable agreements are shown. The importance of including anisotropic turbulence and three-dimensional effects are illustrated by comparison with data and other less comprehensive analyses. The analytical solution is suitable for rapid estimates of engine performance and parametric studies of combustor designs containing transverse or angled injector ports arranged in rows and/or columns.

Quan, Victor↗

DOE OSTI2018

The Homogeneous Cooling State as a Verification Test for Kinetic Theory-Based Continuum Models of Gas–Solid Flows

Granular and multiphase (gas–solids) kinetic theory-based models have emerged a leading modeling strategy for the simulation of particle flows. Similar to the Navier–Stokes equations of single-phase flow, although substantially more complex, kinetic theory-based continuum models are typically solved with computational fluid dynamic (CFD) codes. Under the assumptions of the so-called homogeneous cooling state (HCS), the governing equations simplify to an analytical solution describing the “cooling” of fluctuating particle velocity, or granular temperature. The HCS is used here to verify the implementation of a recent multiphase kinetic theory-based model in the open source mfix code. In this work, results from the partial verification test show that the available implicit (backward) Euler time integration scheme converges to the analytical solution with the expected first-order rate. A second-order accurate backward differentiation formula (BDF) is also implemented and observed to converge at a rate consistent with its formal accuracy.

Computational fluid dynamics↗

NASA NTRS1999

Study of Magnetic Damping Effect on Convection and Solidification Under G-Jitter Conditions

As shown by NASA resources dedicated to measuring residual gravity (SAMS and OARE systems), g-jitter is a critical issue affecting space experiments on solidification processing of materials. This study aims to provide, through extensive numerical simulations and ground based experiments, an assessment of the use of magnetic fields in combination with microgravity to reduce the g-jitter induced convective flows in space processing systems. We have so far completed asymptotic analyses based on the analytical solutions for g-jitter driven flow and magnetic field damping effects for a simple one-dimensional parallel plate configuration, and developed both 2-D and 3-D numerical models for g-jitter driven flows in simple solidification systems with and without presence of an applied magnetic field. Numerical models have been checked with the analytical solutions and have been applied to simulate the convective flows and mass transfer using both synthetic g-jitter functions and the g-jitter data taken from space flight. Some useful findings have been obtained from the analyses and the modeling results. Some key points may be summarized as follows: (1) the amplitude of the oscillating velocity decreases at a rate inversely proportional to the g-jitter frequency and with an increase in the applied magnetic field; (2) the induced flow approximately oscillates at the same frequency as the affecting g-jitter, but out of a phase angle; (3) the phase angle is a complicated function of geometry, applied magnetic field, temperature gradient and frequency; (4) g-jitter driven flows exhibit a complex fluid flow pattern evolving in time; (5) the damping effect is more effective for low frequency flows; and (6) the applied magnetic field helps to reduce the variation of solutal distribution along the solid-liquid interface. Work in progress includes numerical simulations and ground-based measurements. Both 2-D and 3-D numerical simulations are being continued to obtain further information on g-jitter driven flows and magnetic field effects. A physical model for ground-based measurements is completed and some measurements of the oscillating convection are being taken on the physical model. The comparison of the measurements with numerical simulations is in progress. Additional work planned in the project will also involve extending the 2-D numerical model to include the solidification phenomena with the presence of both g-jitter and magnetic fields.

Li, Ben Q.↗

DOE OSTI2020

Improvements to the New CTH Code Verification & Validation Test Suite (FY2020)

The CTH multiphysics hydrocode, which is used for a wide range of important calculations, has undertaken in recent years to overhaul its software quality and testing processes. A key part of this effort entailed building a new, robust V&V test suite made up of traditional hydrocode verification problems, such as those listed in the ASC Tri-Lab Test Suite and the Enhanced Tri-Lab Test Suite, as well as validation problems for some of CTHs most frequently used equations of state, materials models, and other key capabilities. Substantial progress towards this goal was made in FY19. In FY20, this test suite has been expanded to include verification and validation tests of the Sesame and JWL equation of state models as well as the Mader verification problem from the Tri-Lab Test Suite and the Blake verification problem - a linear elastic analog to the Hunter problem from the Enhanced Tri-Lab Test Suite. This report documents CTH performance on the new test suite problems. Verification test results are compared to analytic solutions and, for most tests, convergence results are presented. Validation test results are compared to experimental data and mesh refinement studies are included. CTH performs well overall on the new test problems. Convergence rates for the Blake and Mader problems are comparable to those for similar ASC codes. The JWL and Sesame verification tests show good agreement with analytic solutions. Likewise, CTH simulation results show good agreement with experimental validation data for the Sesame and JWL equations of state for the materials tested. Future V&V work will focus on adding tests for other key capabilities like fracture and high explosive models.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

The Thermalization Verification Problem for EMPIRE-PIC and EMPIRE-Fluid

We report on the verification of elastic collisions in EMPIRE-PIC and EMPIRE-Fluid in support of the ATDM L2 V&V Milestone. The thermalization verification problem and the theory behind it is presented along with an analytic solution for the temperature of each species over time. The problem is run with both codes under multiple parameter regimes. The temperature over time is compared between the two codes and the theoretical results. A preliminary convergence analysis is performed on the results from EMPIRE-PIC and EMPIRE-Fluid showing the rate at which the codes converge to the analytic solution in time (EMPIRE-Fluid) and particles (EMPIRE-PIC).

42 ENGINEERING↗

NASA NTRS2010

Propellant Sloshing Parameter Extraction from CFD Analysis

Propellant slosh is a potential source of disturbance critical to the stability of space vehicle. The sloshing dynamics is typically represented by a mechanical model of spring mass damper. This mechanical model is then included in the equation of motion of the entire vehicle for Guidance, Navigation and Control analysis. The typical parameters required by the mechanical model include natural frequency of the sloshing, sloshing mass, sloshing mass center coordinates, and critical damping coefficient. During the 1960 s US space program, these parameters were either computed from analytical solution for simple geometry or by experimental testing for the sub-scaled configurations. The purpose of this work is to demonstrate the soundness of a CFD approach in modeling the detailed fluid dynamics of tank sloshing and the excellent accuracy in extracting mechanical properties for different tank configurations and at different fill levels. The validation studies included straight cylinder against analytical solution, and sub-scaled Centaur LOX and LH2 tanks with and without baffles against experimental results. This effort shows that CFD technology can provide accurate mechanical parameters for any tank configuration, and is especially valuable to the future design of propellant tanks, as there is no previous experimental data available for the same size and configuration.

Yang, H. Q.↗

DOE OSTI2021

Topological or rotational non-Abelian gauge fields from Einstein-Skyrme holography

We report analytically known states at non-zero temperature which may serve as a powerful tool to reveal common topological and thermodynamic properties of systems ranging from the QCD phase diagram to topological phase transitions in condensed matter materials. In the holographically dual gravity theory, these are analytic solutions to a five-dimensional non-linear-sigma (Skyrme) model dynamically coupled to Einstein gravity. This theory is shown to be holographically dual to \( \mathcal{N} \) = 4 Super-Yang-Mills theory coupled to an SU(2)-current. All solutions are fully backreacted asymptotically Anti-de Sitter (AdS) black branes or holes. One family of global AdS black hole solutions contains non-Abelian gauge field configurations with positive integer Chern numbers and finite energy density. Larger Chern numbers increase the Hawking-Page transition temperature. In the holographically dual field theory this indicates a significant effect on the deconfinement phase transition. Black holes with one Hawking temperature can have distinct Chern numbers, potentially enabling topological transitions. A second family of analytic solutions, rotating black branes, is also provided. These rotating solutions induce states with propagating charge density waves in the dual field theory. We compute the Hawking temperature, entropy density, angular velocity and free energy for these black holes/branes. These correspond to thermodynamic data in the dual field theory. For these states the energy-momentum tensor, (non-)conserved current, and topological charge are interpreted.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis

Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a “minimax” problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Finally, several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Electron Boltzmann equation in nonthermal plasmas

Numerical and analytical solutions of the electron Boltzmann equation for a two-temperature steady-state He plasma are examined in a broad range of conditions, i.e., atom temperature ranging from 5000 K to 20,000 K; electron temperature ranging from 10,000 K to 20,000 K; and atom density ranging from 10 to the 10th to 10 to the 18th per cu cm. The WKB analytical solution is shown to be satisfactory in most situations. Attention is also given to the deviation of the electron distribution from Maxwellian, and to the possibility of raising the tail of the distribution.

Kunc, J. A.↗

DOE OSTI2014

Uncertainty quantification for the impact of injection rate fluctuation on the geomechanical response of geological carbon sequestration

We present an analysis of the geomechanical effects of injection rate fluctuations for geological sequestration of carbon dioxide (CO2). Initially, we present analytical solutions for the effects of injection rate fluctuations on CO2 fluid pressure spatial distribution and temporal evolution for a typical injection scenario. Numerical calculations are performed using a finite element method to investigate the effects of injection rate fluctuations on geomechanical deformation, stresses, and potential failure of the aquifer and caprock layers. The numerical method was first validated by the fluid pressure distribution’s good agreement with the analytical solution. It was shown that for any Gaussian fluctuations of injection rate Q with given mean Q and variance ε_Q, the coefficients of variance for fluid pressure (ϵ_p=ε_p ), deformation (ϵ_u=ε_u ), and stresses (ϵ_σ=ε_σ ) increase linearly with the coefficient of variance for injection rate (ϵ_Q=ε_Q ). The proportional constants are identified, and the fluctuations have the most pronounced effect on the geomechanical stresses, and, therefore, on the potential failure of the aquifer and caprock layers. Instead of expensive computational simulation, this study provides an efficient tool to estimate the geomechanical response variance to injection rate fluctuation. A failure analysis was presented based on the numerical results, where probability of failure was estimated for fluctuating injection rates with different mean and variance during the entire injection period. It was found that with increasing injection rate fluctuation, the failure probability increases significantly. Therefore, the risk associated with injection rate fluctuations should be carefully evaluated.

58 GEOSCIENCES↗

DOE OSTI2015

A boundary condition to the Khokhlov-Zabolotskaya equation for modeling strongly focused nonlinear ultrasound fields

An equivalent source model was proposed as a boundary condition to the nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation to simulate high intensity focused ultrasound (HIFU) fields generated by medical ultrasound transducers with the shape of a spherical shell. The boundary condition was set in the initial plane; the aperture, the focal distance, and the initial pressure of the source were chosen based on the best match of the axial pressure amplitude and phase distributions in the Rayleigh integral analytic solution for a spherical transducer and the linear parabolic approximation solution for the equivalent source. Analytic expressions for the equivalent source parameters were derived. It was shown that the proposed approach allowed us to transfer the boundary condition from the spherical surface to the plane and to achieve a very good match between the linear field solutions of the parabolic and full diffraction models even for highly focused sources with F-number less than unity. The proposed method can be further used to expand the capabilities of the KZ nonlinear parabolic equation for efficient modeling of HIFU fields generated by strongly focused sources.

42 ENGINEERING↗

DOE OSTI2021

Rocking response of liquid-filled cylindrical tanks

The hydrodynamic response of liquid-filled storage tanks subjected to translational earthquake shaking has been the focus of studies for more than 75 years. In service, tanks experience six components of seismic input, with rocking due to support flexibility and/or rotational components of ground motion. Published theory for predicting hydrodynamic responses in rigid cylindrical tanks due to rotational seismic input is examined. Analytical solutions for pressure in the fluid and on the tank wall, and base moment are modified to account for the effects of angular displacements at the base. A finite element model of a water-filled cylindrical tank using the Arbitrary Lagrangian and Eulerian (ALE) solver in LS-DYNA is used to demonstrate the efficacy of the proposed expressions for three base rocking motions. Results from the modified analytical solutions and the finite element model are in excellent agreement.

58 GEOSCIENCES↗

NASA NTRS2007

Analytical Approach Validation for the Spin-Stabilized Satellite Attitude

An analytical approach for spin-stabilized spacecraft attitude prediction is presented for the influence of the residual magnetic torques and the satellite in an elliptical orbit. Assuming a quadripole model for the Earth s magnetic field, an analytical averaging method is applied to obtain the mean residual torque in every orbital period. The orbit mean anomaly is used to compute the average components of residual torque in the spacecraft body frame reference system. The theory is developed for time variations in the orbital elements, giving rise to many curvature integrals. It is observed that the residual magnetic torque does not have component along the spin axis. The inclusion of this torque on the rotational motion differential equations of a spin stabilized spacecraft yields conditions to derive an analytical solution. The solution shows that the residual torque does not affect the spin velocity magnitude, contributing only for the precession and the drift of the spin axis of the spacecraft. The theory developed has been applied to the Brazilian s spin stabilized satellites, which are quite appropriated for verification and comparison of the theory with the data generated and processed by the Satellite Control Center of Brazil National Research Institute. The results show the period that the analytical solution can be used to the attitude propagation, within the dispersion range of the attitude determination system performance of Satellite Control Center of Brazil National Research Institute.

Zanardi, Maria Cecilia F. P. S.↗