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DOE OSTI2022

The local wavenumber model for computation of turbulent mixing

We present an overview of the current status in the development of a two-point spectral closure model for turbulent flows, known as the local wavenumber (LWN) model. The model is envisioned as a practical option for applications requiring multi-physics simulations in which statistical hydrodynamics quantities such as Reynolds stresses, turbulent kinetic energy, and measures of mixing such as density-correlations and mix-width evolution, need to be captured with relatively high fidelity. In this review, we present the capabilities of the LWN model since it was first formulated in the early 1990s, for computations of increasing levels of complexity ranging from homogeneous isotropic turbulence, inhomogeneous and anisotropic single-fluid turbulence, to two-species mixing driven by buoyancy forces. The review concludes with a discussion of some of the more theoretical considerations that remain in the development of this model.

42 ENGINEERING↗

NASA NTRS1994

A stochastic precipitation disaggregation scheme for GCM applications

In the surface hydrologic parameterization of general circulation models (GCMs), it is commonly assumed that the precipitation processes are homogeneous over a GCM grid square and that the precipitation intensity is uniformly distributed. Based on evidence that the spatial distribution of precipitation within a GCM grid square is crucial for the land surface hydrology parameterization, a few researchers have explored the impacts of assuming that the precipitation is exponentially distributed. This paper explores the suitability of the aforementioned assumptions. First, a statistical analysis is conducted of historical precipitation data for three GCM grids in different regions of the United States. The analysis suggests that neither the uniform nor the exponential distribution assumption may be suitable at the GCM grid scale and, that instead, the spatial variability in precipitation is characterized by statistical patterns that are inhomogeneous. These patterns vary from grid to grid and are induced by the interaction between atmospheric conditions and various land surface characteristics, such as topographical features, surface properties, etc. Within the same grid square, however, the statistical patterns are generally constant from year to year. Based on this analysis, a computationally viable (i.e., usable with GCMs) stochastic precipitation disaggregation scheme that utilizes these stable statistical patterns is proposed. The method was used to generate spatially distributed hourly rainfall for a summer season in the southwestern region of the continental United States. Analysis of the results shows that the methodology preserves the seasonal characteristics of spatial variability in precipitation that is observed in the long-term historical data.

Gao, Xiaogang↗

DOE OSTI2021

Correlating Nanoscale Secondary Ion Mass Spectrometry and Atom Probe Tomography Analysis of Uranium Enrichment in Metallic Nuclear Fuel

Distribution and enrichment of 235U within nuclear fuels are essential to material performance and reactor safety. With the development of metallic U fuels, understanding 235U homogeneity has become increasingly important, however, it is difficult to map with high spatial, and mass resolution to analyze both matrix and secondary phase precipitates. Here, we analyze 235U enrichment in matrix and carbide phases in LEU-Mo via two chemical imaging modalities: nanoscale secondary ion mass spectrometry (nanoSIMS) and atom probe tomography (APT). NanoSIMS and APT provided consistent results, with no statistically significant difference between measured and nominal enrichment (19.75 % 235U).

U-10Mo, atom probe tomography, nanoscale secondary↗

DOE OSTI2024

Evaluating the Consistency of Cosmological Distances Using Supernova Siblings in the Near-infrared

Abstract The study of supernova (SN) siblings, supernovae with the same host galaxy, is an important avenue for understanding and measuring the properties of Type Ia SN Ia light curves (LCs). Thus far, sibling analyses have mainly focused on optical LC data. Considering that LCs in the near-infrared (NIR) are expected to be better standard candles than those in the optical, we carry out the first analysis compiling SN siblings with only NIR data. We perform an extensive literature search of all SN siblings and find six sets of siblings with published NIR photometry. We calibrate each set of siblings ensuring they are on homogeneous photometric systems, fit the LCs with the SALT3-NIR and SNooPy models, and find median absolute differences inμvalues between siblings of 0.248 and 0.186 mag, respectively. To evaluate the significance of these differences beyond measurement noise, we run simulations that mimic these LCs and provide an estimate for uncertainty on these median absolute differences of ∼0.052 mag, and we find that, statistically, our analysis rules out the nonexistence of intrinsic scatter in the NIR at the 99% level. When comparing the same sets of SN siblings, we observe a median absolute difference inμvalues between siblings of 0.177 mag when using optical data alone as compared to 0.186 mag when using NIR data alone. It is unclear if these results may be due to limited statistics or poor quality NIR data, all of which will be improved with the Nancy Grace Roman Space Telescope.

Astronomy & Astrophysics↗

NASA NTRS1985

Simulation and modeling of homogeneous, compressed turbulence

Low Reynolds number homogeneous turbulence undergoing low Mach number isotropic and one-dimensional compression was simulated by numerically solving the Navier-Stokes equations. The numerical simulations were performed on a CYBER 205 computer using a 64 x 64 x 64 mesh. A spectral method was used for spatial differencing and the second-order Runge-Kutta method for time advancement. A variety of statistical information was extracted from the computed flow fields. These include three-dimensional energy and dissipation spectra, two-point velocity correlations, one-dimensional energy spectra, turbulent kinetic energy and its dissipation rate, integral length scales, Taylor microscales, and Kolmogorov length scale. Results from the simulated flow fields were used to test one-point closure, two-equation models. A new one-point-closure, three-equation turbulence model which accounts for the effect of compression is proposed. The new model accurately calculates four types of flows (isotropic decay, isotropic compression, one-dimensional compression, and axisymmetric expansion flows) for a wide range of strain rates.

Wu, C. T.↗

NASA NTRS1989

High-resolution microwave spectra of solar bursts

A phenomenological and statistical study of flares observed in total power with the frequency-agile interferometer at the Owens Valley Radio Observatory during several months of high solar activity in 1981 is reported. Roughly 80 percent of the events have a complex spectrum consisting of more than one spectral component, implying that the microwave radiation of a burst usually does not come from a single homogeneous source. The presence of more than one component can lead to significant errors when data with low spectral resolution are used to determine the low-side spectral index. The low-frequency slope of a single spectra component is often steeper than expected, and the peak frequency stays nearly constant throughout a microwave event.

Stahli, M.↗

DOE OSTI2020

Effects of Atwood and Reynolds numbers on the evolution of buoyancy-driven homogeneous variable-density turbulence

The evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at Atwood numbers up to 0.75 and large Reynolds numbers is studied by using high-resolution direct numerical simulations. To help understand the highly non-equilibrium nature of buoyancy-driven HVDT, the flow evolution is divided into four different regimes based on the behaviour of turbulent kinetic energy derivatives. The results show that each regime has a unique type of dependence on both Atwood and Reynolds numbers. It is found that the local statistics of the flow based on the flow composition are more sensitive to Atwood and Reynolds numbers compared to those based on the entire flow. It is also observed that, at higher Atwood numbers, different flow features reach their asymptotic Reynolds-number behaviour at different times. The energy spectrum defined based on the Favre fluctuations momentum has less large-scale contamination from viscous effects for variable-density flows with constant properties, compared to other forms used previously. The evolution of the energy spectrum highlights distinct dynamical features of the four flow regimes. Thus, the slope of the energy spectrum at intermediate to large scales evolves from -7/3 to -1 , as a function of the production-to-dissipation ratio. The classical Kolmogorov spectrum emerges at intermediate to high scales at the highest Reynolds numbers examined, after the turbulence starts to decay. Finally, the similarities and differences between buoyancy-driven HVDT and the more conventional stationary turbulence are discussed and new strategies and tools for analysis are proposed.

42 ENGINEERING↗

NASA NTRS2001

Design and Calibration of Autonomous Coherent Doppler Lidar for Space Missions

Developed a new algorithm for the simulation of three dimensional homogeneous turbulent velocity fields. For typical atmospheric conditions it is impossible to produce a simulated velocity field that simultaneously satisfy a given spatial correlation and the corresponding spatial spectrum because of spectral aliasing. The new algorithms produce a turbulent velocity field which has accurate spatial correlations which is required for performance predictions from space-based systems. Developed a new algorithm for extracting the spatial statistics of the atmospheric velocity field using coherent Doppler lidar. The performance of the algorithm was compared with past methods and the new algorithm produces useful results for space-based data, which was not possible before. Developed new methods for verification of the errors in ground-based and space-based Doppler lidar wind measurements. These new methods do not require independent in situ data. This is an important issue for the verification of space-based Doppler lidar measurements of the global wind field. The performance of the new algorithm was compared with past results for both space-based and ground-based operation. The new algorithm has the best performance and is the only algorithm that performed satisfactory for spacebased operation. The performance of coherent Doppler lidar for a space missions with various scanning geometries was determined using computer simulation which contained the effects of random instrumental velocity errors, wind shear, wind variability along the range-gate and from shot-to-shot, and random variations in atmospheric aerosol backscatter over the measurement volume. The bias in the velocity estimates was small and the accuracy in the is typically less than 0.5 m/s for high signal conditions. For a large number of shot per velocity estimate, the threshold signal level for acceptable estimates is proportional to the number of shots to the minus one half power. This agrees with previous results determined for ground-based measurements. The use of multi-element optical detectors for autonomous operation of coherent Doppler lidar was shown to be a very promising technique. Optimal detector geometries were determined by computer simulation of performance: for ground-based testing with a fixed calibration target and for space-based operation using the random surface returns. The effects of refractive turbulence on ground-based calibration of coherent Doppler lidar was determined by computer simulations and compared with theoretical predictions. New techniques were required to correctly predict performance for the focused beam geometry commonly used for verification of space-based operation. An improved velocity estimator was evaluated for space-based applications were signal shot measurements are used to produce vector wind measurements. This permits more accurate measurements when the signal level is not known a priori or not available from multiple shot measurements. The average Doppler lidar signal spectrum including the effects of velocity turbulence was derived and calculated. This permits new estimation algorithms for turbulence based on spectral estimates. In situ atmospheric measurements were conducted and analyzed using an instrumented kite-platform. This work helps provide the required in situ data for verification of Doppler lidar velocity statistics.

Frehlich, Rod G.↗

NASA NTRS2016

On the Susceptibility of Cold Tropical Cirrus to Ice Nuclei Abundance

Numerical simulations of cirrus formation in the tropical tropopause layer (TTL) are used to evaluate the impact heterogeneous ice nuclei (IN) abundance on cold cloud microphysical properties. The model includes homogeneous and heterogeneous ice nucleation, deposition growthsublimation, and sedimentation. Reanalysis temperature and wind fields are used to force the simulations, with addition of high-frequency wave temperature variability based on in situ measurements. The model results are constrained by recent in situ observations of TTL cirrus and relative humidity, as well as satellite measurements of cirrus bulk properties and occurrence frequencies. Temperature variability driven by high-frequency waves has a dominant influence on TTL cirrus microphysical properties and occurrence frequencies, and inclusion of these waves is required to produce agreement between the simulated and observed cloud properties as well as the abundance of TTL cirrus. With a composition-independent supersaturation threshold for homogeneous freezing of aqueous aerosols, the model produces excessive ice concentrations compared with in situ observations. Inclusion of relatively numerous heterogeneous nuclei ( 100L) in the simulations improves the agreement with observed ice concentrations. However, when IN contribute significantly to TTL cirrus ice nucleation, the occurrence frequency of large supersaturations with respect to ice is less than indicated by in situ measurements. The simulated TTL cirrus extinction statistics agree with observations (within uncertainties) from both in situ and remote-sensing measurements. We find that the sensitivity of TTL cirrus extinction and ice water content statistics to heterogeneous ice nuclei abundance is relatively weak. Likewise, the simulated occurrence frequencies of TTL cirrus are insensitive to ice nuclei abundance, both in terms of cloud frequency height distribution and regional distribution throughout the tropics.

occurences↗

DOE OSTI2025

Spectral distortions to momentum and scalar exchanges by non-turbulent motion and patchy landscape variability

Modifications to the spectra of turbulent velocity and scalars and co-spectra of vertical fluxes of momentum and scalars due to patchy landscape heterogeneity and non-stationarity are explored for a Mediterranean ecosystem. About 9 months of high frequency measurements of the three velocity components, water vapor concentration, carbon dioxide concentration, and air temperature were analyzed for different seasons (spring/summer) and prevalent wind directions (southeast/northwest). The two wind directions sampled a contrast of clumped and patchy landscape comprised of olive trees (southeast) and wall bounded flow disturbed by the presence of few upwind trees (northwest). The measured spectra and co-spectra were also compared to theoretical scaling forms from stationary, planar homogeneous flow, in the absence of subsidence as derived from the Kansas experiment. To assess the role of low frequency non-turbulent motion on the spectral and co-spectral content, a 5-min Fourier cutoff was introduced and the analysis was limited to near-neutral conditions where the boundary layer depth is shallow compared to its unstable counterpart. It was shown that the velocity statistics were not appreciably impacted by the low-frequency motion causing non-stationarity. Moreover, the turbulent scalar fluxes were also shown not to be significantly impacted by such low frequency motion. The scalar variances were impacted, especially the water vapor variance and its concomitant spectral shape. When the non-turbulent motion was filtered, the scalar spectra at low wavenumbers followed expectations from the so-called attached eddy hypothesis (i.e. exhibited a $k^{-1}_x$ scaling with $k_x$ defining the longitudinal wavenumber) applicable for near-neutral conditions. For momentum co-spectra, the canonical shapes from the Kansas experiment appear to describe well the measurements here and in both dominant directions and seasons with some adjustment to the integral time scales based on wind direction. For the scalar co-spectra, deviations from the Kansas experiment were prevalent. The most noticeable and surprising deviations were their slow decay with increased sampling frequency at inertial subrange scales. This slow decay was shown not to contribute appreciably to the overall scalar fluxes. At those fine scales, predictions from local isotropy were expected to hold. The scalar co-spectral deviations from local isotropy were then discussed using a simplified co-spectral budget model where scalar–scalar co-spectra naturally emerged and the interplay between landscape heterogeneity and a scale-dependent pressure-scalar de-correlation time was postulated. It is also envisaged that the findings here offer a preliminary template for analyzing eddy-covariance data in situations that deviate from ideal conditions, especially regarding low-frequency modulations of scalar spectra and vertical scalar flux co-spectra.

Canopy turbulence↗

NASA NTRS1995

North-south asymmetry of the interplanetary magnetic helicity

Previous analyses of the north-south asymmetry of the interplanetary magnetic helicity have used the omnitape dataset and have shown that there exists a persistent and statistically significant asymmetry in the handedness of the magnetic fluctuations at 1 AU. This asymmetry is concentrated in fluctuations with spacecraft frame frequencies less than 10-5 Hz (periods greater than 30 hours) at 1 AU. Attempts to extend these analyses to include data collected in the outer heliosphere require that we consider spacecraft frame periods many times greater than 30 hours. This raises interesting questions regarding homogeneity and stationarity of the datasets at this scale and brings into question the possible breakdown of the computed correlation functions due to the sector structure of the solar wind. The likely geometry of magnetic fluctuations in the outer heliosphere provides yet another complication in the analysis. These issues will be discussed in detail and the latest results from our studies of the Voyager 1 & 2 and Pioneer 10 & 11 datasets will be presented. The analysis of Pioneer-Venus Orbiter observations will be shown as well. The potential asymmetry between the magnetic helicity of the two hemispheres has significant and measurable implications for cosmic ray propagation in the heliosphere and these implications will be reviewed in light of the new results.

Smith, Charles W.↗

DOE OSTI2021

Stakeholder Support for Wildlife Conservation Funding Policies

State wildlife management agencies in the United States have depended on a “user-pay” funding model for conservation efforts that relies on revenue from hunting license sales and a federal excise tax on firearms, ammunition, and archery equipment. Declines in hunting participation, however, jeopardize sustainability of the current funding model. Ensuring support among stakeholders for wildlife management and conservation may require expanding sources of funding and incorporating the perspectives and values of a diversifying constituency into decision making processes. We used a web-based survey of wildlife-associated recreationists in Michigan, USA to evaluate support for a range of conservation funding policies. Respondents self-identified primarily as hunters (n = 2,558) or wildlife watchers (n = 942). We used binary logistic regression to evaluate support for four conservation funding policy options: state sales tax, lottery proceeds, extractive industry revenue, and a user-based tax on outdoor gear (i.e., “backpack tax”). Determinants of support varied by type of policy and stakeholder characteristics. We found no statistically significant differences between hunters and wildlife watchers in their support for conservation funding policies when accounting for other variables such as wildlife value orientations, engagement in stewardship behaviors, age, and gender. The industry-based policy achieved the greatest level of approval, while the backpack tax had the lowest. Respondents were mixed in their support of the sales tax and lottery proceeds options. Cluster analysis revealed three homogenous groups related to conservation funding policies: “strong support,” “mixed/opposed,” and “anti-backpack tax.” Clusters differed in their support for conservation funding policies and on psychological and demographic variables. The “strong support” and “anti-backpack tax” groups differed in their levels of stewardship engagement, knowledge of conservation funding mechanisms, and support for the backpack tax option. The “mixed-opposed” group tended to be older, less educated, and less likely to be a member of a conservation organization. Results suggest support for conservation funding differs by policy type and social and psychological characteristics of stakeholders. Based on differences in policy support revealed in this study, we suggest a multi-tiered approach to funding conservation and building on support among wildlife stakeholders to mitigate the looming funding crisis for state wildlife agencies.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

NASA NTRS1990

The influence of Lewis number and nonhomogeneous mixture on premixed turbulent flame structure

The structure of a premixed flame front in two-dimensional turbulence is investigated using full numerical simulation including heat release, variable fluid properties, and one-step Arrhenius chemistry. Non-unity Lewis number (Le) effects are described by comparing the local instantaneous turbulent flame structure to the steady one dimensional laminar flame structure for the same thermochemical parameters. Flame surface area, mean reactant consumption rate per unit area of flame, turbulent flame speed, and statistical descriptions of the flame geometry are also reported. Principal findings are that the local flame structure correlates strongly with the local flame curvature, while global properties depend both on strain and on curvature. Preliminary results for cases with nonhomogeneous reactant mixture strength are reported. Here, the emphasis is on the ability of a propagating flame to recover after encountering a fuel-lean pocket in an otherwise homogeneous mixture. Both one dimensional and two dimensional cases are described.

Haworth, D. C.↗

NASA NTRS1994

Homogeneous turbulence subjected to mean flow with elliptic streamlines

Direct numerical simulations are performed for homogeneous turbulence with a mean flow having elliptic streamlines. This flow combines the effects of rotation and strain on the turbulence. Qualitative comparisons are made with linear theory for cases with high Rossby number. The nonlinear transfer process is monitored using a generalized skewness. In general, rotation turns off the nonlinear cascade; however, for moderate ellipticities and rotation rates the nonlinear cascade is turned off and then reestablished. Turbulence statistics of interest in turbulence modeling are calculated, including full Reynolds stress budgets.

Blaisdell, G. A.↗

DOE OSTI2026

Tuning the Mechanical Properties of Crosslinked Copolymers via Sequence and Solvent‐Selective Swelling for Vat Photopolymerization

Block copolymers (BCPs) offer distinct advantages for vat photopolymerization by enabling mechanically programmable network structures through microphase-separated morphologies that can be kinetically trapped during curing, yielding properties unattainable in homogeneous resins. However, the respective roles of repeat-unit sequence and solvent environment, together with their interplay in directing network formation and mechanical performance, remain unclear. Here, we synthesize a series of CO 2 -based polycarbonate copolymers comprising a crosslinkable glassy poly(vinyl cyclohexene carbonate) (PVCHC, A block) and a non-crosslinkable soft poly(propylene carbonate) (PPC, B block). The polymer sequence is systematically varied (ABA, BAB, and statistical), and solvent choice controls block-selective swelling to jointly control gelation behavior, microphase morphology, and mechanical response through changes in the accessibility and local environment of photocrosslinkable vinyl groups during network formation, as revealed by photorheology and small angle x-ray scattering. By tuning polymer sequence and curing solvent, we transform nominally identical formulations from brittle to highly ductile materials, achieving a three-orders-of-magnitude range in toughness (0.003 to 9.1 MJ m −3 ). These results establish clear structure–processing–property relationships and identify polymer sequence and selective solvation as powerful strategies for programming both printability and performance of block copolymer resins for additive manufacturing.

additive manufacturing↗

DOE OSTI2025

Molecular fluctuations inhibit intermittency in compressible turbulence

In the standard picture of fully developed turbulence, highly intermittent hydrodynamic fields are nonlinearly coupled across scales, where local energy cascades from large scales into dissipative vortices and large density gradients. Microscopically, however, constituent fluid molecules are in constant thermal (Brownian) motion, but the role of molecular fluctuations in large-scale turbulence is largely unknown, and with rare exceptions, it has historically been considered irrelevant at scales larger than the molecular mean free path. Recent theoretical and computational investigations have shown that molecular fluctuations can impact energy cascade at Kolmogorov length scales. Here, we show that molecular fluctuations not only modify energy spectrum at wavelengths larger than the Kolmogorov length in compressible turbulence, but also significantly inhibit spatio-temporal intermittency across the entire dissipation range. Using large-scale direct numerical simulations of computational fluctuating hydrodynamics, we demonstrate that the extreme intermittency characteristic of turbulence models is replaced by nearly Gaussian statistics in the dissipation range. These results demonstrate that the compressible Navier–Stokes equations should be augmented with molecular fluctuations to accurately predict turbulence statistics across the dissipation range. Our findings have significant consequences for turbulence modelling in applications such as astrophysics, reactive flows and hypersonic aerodynamics, where dissipation-range turbulence is approximated by closure models.

compressible turbulence↗

NASA NTRS1984

Analysis of normalized radar cross section (sigma-O) signature of Amazon rain forest using SEASAT scatterometer data

The normalized radar cross section (NRCS) signature of the Amazon rain forest was SEASAT scatterometer data. Statistics of the measured (NRCS) values were determined from multiple orbit passes for three local time periods. Plots of mean normalized radar cross section, dB against incidence angle as a function of beam and polarization show that less than 0.3 dB relative bias exists between all beams over a range of incidence angle from 30 deg to 53 deg. The backscattered measurements analyzed show the Amazon rain forest to be relatively homogeneous, azimuthally isotropic and insensitive to polarization. The return from the rain forest target appears relatively consistent and stable, except for the small diurnal variation (0.75 dB) that occurs at sunrise. Because of the relative stability of the rain forest target and the scatterometer instrument, the response of versus incidence angle was able to detect errors in the estimated yaw altitude angle. Also, small instrument gain biases in some of the processing channels were detected. This led to the development of an improved NRCS algorithm, which uses a more accurate method for estimating the system noise power.

Bracalente, E. M.↗

DOE OSTI2021

Unraveling the implications of finite specimen size on the interpretation of dynamic experiments for polycrystalline aluminum through direct numerical simulations

Normal and Pressure-shear plate impact (NPI and PSPI) experiments are popular experimental techniques for studying the mean-field macroscopic behavior of polycrystalline metals under high-rate dynamic loading. However, since both configurations rely upon geometry for subjecting the specimen to high strain rates, these experiments often involve a limited specimen size. Moreover, because of the inherent heterogeneities present within polycrystalline metals, it is difficult to ascertain if the size of the specimen and/or regions where measurements are made are sufficiently large for making representative inferences about the mean-field macroscopic properties from single-point velocity measurements. In the present study, we quantify the expected measurement variability on observable point measurements in NPI and PSPI experiments by carrying out direct numerical simulations (DNS) of statistically representative polycrystalline microstructures subjected to dynamic compression and compression-shear loading. In particular, we consider the role of specific material heterogeneities (e.g. the grain-to-grain difference in size, crystallographic orientation) on dispersion in the normal and transverse particle velocity records and on local fluctuations in key state variables (e.g. velocity, accumulated plastic strain) by incorporating these effects directly into a representative synthetic microstructure geometry and crystalline description of pure polycrystalline aluminum. The form of the present study is a large parametric investigation, consisting of ten ensembles of one hundred simulations. Each of the thousand simulations reflects a randomly realized synthetic microstructure in one of five cases of decreasing average grain size for the two loading configurations. Our analysis of the DNS results demonstrates that for both of these experimental configurations, the grain size directly correlates with the coefficient of variation (CV) in simulated point measurements, showing a convergent decrease in CV to zero (i.e. particle velocity record approaches the mean-field value) with decreasing grain size. Remarkably, the magnitude of variations in the particle velocity record is shown to be largest where the deviatoric stresses are most significant. In the case of NPI, this occurs at the elastic and plastic wavefront, whereas, in the case of PSPI, the magnitude of fluctuations are approximately constant throughout the experimental window time. The reasoning for the scatter in particle velocity due to the heterogeneous microstructure is demonstrated to be dependent on the mechanisms for accommodating deformation and on the interaction of reflection waves generated at sites of heterogeneities occurring at the scale of grains. Lastly, we develop a power-law description for the magnitude of scattering versus characteristic length, which provides a statistical framework for assessing the required number of grains per characteristic specimen dimension for minimizing scatter within these two experimental configurations (NPI, PSPI).

36 MATERIALS SCIENCE↗