Power-law Extrapolation Research Articles

Spectral Fittings of Warm Coronal Radiation with High Seed Photon Temperature: Apparent Low-temperature and Flat Soft Excess in AGNs

A warm corona has been widely proposed to explain the soft excess (SE) in X-ray above the 2–10 keV power law extrapolation in Active Galactic Nuclei (AGNs). In actual spectral fittings, the warm coronal seed photon temperature (T s) is usually assumed to be far away from the soft X-ray, but kT s can reach close to 0.1 keV in the standard accretion disk model. In this study, we used Monte Carlo simulations to obtain radiation spectra from a slab-like warm corona and fitted the spectra using the spherical-geometry-based routine thcomp or a thermal component. Our findings reveal that high T s can influence the fitting results. A moderately high kT s (around 0.03 keV) can result in an apparent low-temperature and flat SE, while an extremely high kT s (around 0.07 keV) can even produce an unobserved blackbody-like SE. Our conclusions indicate that, for spectral fittings of the warm coronal radiation (SE in AGNs), kT s should be treated as a free parameter with an upper limit, and an accurate coronal geometry is necessary when kT s > 0.01 keV.

New-Generation Electron-Propagator Methods for Calculations of Electron Affinities and Ionization Energies: Tests on Organic Photovoltaic Molecules.

A new generation of ab initio electron-propagator self-energies recently superseded its antecedents' accuracy and computational efficiency in calculating vertical ionization energies (VIEs) of closed-shell molecules. (See J. Chem. Phys. 2021, 155, 204107, J. Chem. Theory Comput. 2022, 18, 4927, J. Chem. Phys. 2023, 159, 124109.) No adjustable parameters were introduced in the generation of reference orbitals or in the construction of self-energies. The same approach has been extended in this work to vertical electron affinities (VEAs). Calculations were performed on 24 conjugated, organic photovoltaic molecules with diverse functional groups. These molecules are considerably larger than those studied in previous tests on VIEs. Several new-generation self-energies produce mean absolute errors (MAEs) below 0.1 eV versus ΔCCSD(T) (i.e., total energy differences from the coupled-cluster singles, doubles, and perturbative triples method) VIEs and VEAs obtained with identical basis sets. A composite model employs cubically and quintically scaling algorithms and power-law basis-set extrapolations based on augmented double-triple or triple-quadruple ζ data. Its MAEs are near 0.05 eV versus benchmark values, with 0.03 eV error bars for the lowest VIE and the highest VEA of each molecule. A more efficient and equally accurate composite model for calculating VIEs avoids full transformations of electron repulsion integrals to the molecular orbital basis. High probability factors support the diagonal self-energy approximation, wherein Dyson orbitals are proportional to canonical, Hartree-Fock orbitals.

Inclination dependence of warm coronal flux in high-accretion-rate AGNs

ABSTRACT Most active galactic nucleus (AGN) spectra show a soft X-ray excess above the 2–10 keV power law extrapolation. A warm corona has been widely used to explain the excess, but its observed radiation flux in the actually physical environment has yet to be further studied. For the first time, we calculate the relativistic warm coronal flux under the finite disc-corona thickness in high-accretion-rate systems. The numerical results show that the warm coronal flux generally rises first and then drops with increasing inclination. The flux rise is more significant for a compact and low-temperature warm corona and can reach 1–2 dexes. Meanwhile, the flux drop is significant if and only if the warm corona is heavily obscured due to the finite thickness. Our model can successfully explain the soft excess variance and the X-ray weak fraction in a high-accretion-rate AGN sample. In conclusion, our study indicates that when fitting the soft X-ray spectra of AGNs, the relativistic inclination dependence of warm coronal flux is essential, especially for the high-accretion-rate systems with thick warm coronae.

Wind power in forested regions: Power law extrapolation vs. lidar observation

As the increasing hub height of modern wind turbine, wind shear model is generally adopted as a useful tool to extrapolate wind speed available to higher levels for wind power assessment. This paper conducts a data-driven study to examine the power law model in assessing the wind power in forested regions featured with seasonally-varied roughness. Specifically, wind speeds are observed using a laser lidar mounted in forested regions in the northeast of China, ranging from 80 m to 200 m with a constant interval of 20 m. Subsequently, comparisons are made between the wind profiles established based on the power law model and field observations. The exponents used include the empirical exponents and the fitted exponents, the former are derived from the standard codes, and the latter are fitted based on the seasonal and annual data collections. Results indicate that the power law model is unable to fully capture the vertical distribution of wind speed in forested regions, it is the exponent that plays a decisive role influencing the reliability representing the real wind profile. Compared with the empirical exponents, the seasonally-fitted exponents exhibits much better suitability for wind speed extrapolation in forested regions, as well as for the wind power assessment.

Regional variations in observed fracture abundance and associated predictions for smaller fractures on Europa with implications for regional plume origins

Large fractures on Europa's surface have been extensively studied for the insight they provide into the evolution of the satellite's ice shell. The presence of small fractures has not been characterized on a global or regional scale because of the resolution limits in the current image data. However, if smaller fractures are present they could significantly increase the overall fracture abundance and influence the mechanical properties of the ice shell. Regional variability in this abundance would also impact the spatial heterogeneity of ice shell characteristics. To investigate the possible contribution of smaller fractures to the influence of the fracture network on the behavior of the ice lithosphere, we measure the surface length of visible fractures in regions on Europa's surface and calculate their length distributions and fracture intensities. These outcomes are extrapolated, using best-fit functions, to consider the possible abundance and role of smaller fractures in these networks. The analysis of the visible fractures sampled different geologic terrains and geographic regions to consider the effect of terrain type and location. Fracture abundance in observable populations shows some variation with terrain type. However, observable abundance also changes with image resolution, indicating that current data limit the interpretation of visible fracture populations. Typical log-normal, exponential, and power-law fracture network distributions, based on analysis of terrestrial fracture networks, were considered as methods for extrapolating the observed fracture populations. Power-law was found to be the most appropriate extrapolation choice. It was also found to predict observed fracture abundance in one sample region with a smaller sub-region that had locally higher image resolution. Applying a power-law extrapolation for the fracture abundance as a function of length, we predict fracture abundance characteristics that are up to two orders of magnitude higher than abundance of observed fractures, and which vary by an order of magnitude between regions. Thus, the predicted fracture populations indicate that abundant small fractures would contribute significantly to the mechanical behavior of the ice lithosphere and create regional variations in fracture abundance. For example, where they are abundant, these smaller fractures could contribute to fracture networks providing localized migration pathways for plume activity on Europa. New image data from the Europa Clipper mission should enable fracture mapping at smaller scales to refine the roles of these networks and the smaller fractures on Europan lithospheric processes and rheology.

On the nature and propagation of errors in roughness parameters obtained from spectral analysis of atomic force microscopy topographic images

Scale-dependent surface roughness strongly affects critical surface properties of materials, including adhesion, wettability, and optical/thermal properties. As a particular example, tuning the ratio of the true to nominal area—a parameter that depends on the root mean square (RMS) local slope of the finest scales of topography—is an effective approach to tailor the wetting characteristics of solid surfaces. While power spectral density (PSD) analysis of atomic force microscopy (AFM) topographic images allows for directly assessing the scale-dependence of surface roughness, this approach to analyze AFM height maps requires power-law modeling and extrapolation of a PSD with inherently non-normal error distributions. Here, we use a Monte Carlo approach based on synthetic AFM images of known input power-law parameters to (1) evaluate the accuracy of fitting techniques based on the expected distribution of the PSD; (2) evaluate the error propagation from the standard errors of the fitted power-law parameters to the computed RMS slope and area ratio; and (3) evaluate the statistical power of various PSD regression techniques when differentiating surfaces. The results indicated that standard error for ordinary least squares on a log-log PSD (log OLS) underpredicts the observed variance by ∼50%. This underprediction can be eliminated by implementing a log-link gamma regression. Moreover, when propagating the standard error to derived parameters (e.g., the RMS slope), the propagated error is generally conservative relative to the observed variance and closely predicts the observed variance when extrapolating to the finest scale. This result demonstrates the possibility of accurately estimating roughness parameters that are critical for evaluating surface phenomena on the basis of fitting and extrapolating AFM data using self-affine models. Finally, our results provided evidence for the possibility of statistically differentiating surfaces with similar power-law parameters when using weighted gamma regression with a mean of 10 images, as opposed to unweighted log-OLS that requires as many as 10 000 images to differentiate images.

COMAP Early Science. VI. A First Look at the COMAP Galactic Plane Survey

We present early results from the CO Mapping Array Project (COMAP) Galactic Plane Survey conducted between 2019 June and 2021 April, spanning 20° < ℓ < 40° in Galactic longitude and ∣b∣ < 1.°5 in Galactic latitude with an angular resolution of 4.′5. We present initial results from the first part of the survey, including the diffuse emission and spectral energy distributions of H ii regions and supernova remnants (SNRs). Using low- and high-frequency surveys to constrain free–free and thermal dust emission contributions, we find evidence of excess flux density at 30 GHz in six regions, which we interpret as anomalous microwave emission. Furthermore we model ultracompact H ii contributions using data from the 5 GHz CORNISH catalog and reject these as the cause of the 30 GHz excess. Six known SNRs are detected at 30 GHz, and we measure spectral indices consistent with the literature or show evidence of steepening. The flux density of the SNR W44 at 30 GHz is consistent with a power-law extrapolation from lower frequencies with no indication of spectral steepening in contrast with recent results from the Sardinia Radio Telescope. We also extract five hydrogen radio recombination lines (RRLs) to map the warm ionized gas, which can be used to estimate electron temperatures or to constrain continuum free–free emission. The full COMAP Galactic Plane Survey, to be released in 2023/2024, will span ℓ ∼ 20°–220° and will be the first large-scale radio continuum and RRL survey at 30 GHz with 4.′5 resolution.

Gravitational Wave Direct Detection does not Constrain the Tensor Spectral Index at CMB Scales

I discuss constraints on the power spectrum of primordial tensor perturbations from a combination of Cosmic Microwave Background (CMB) measurements and the gravitational wave direct detection experiments LIGO/Virgo and DECIGO. There are two main points: (1) Inflation predicts an approximately power-law form for the primordial tensor spectrum, but makes no prediction for its amplitude. Given that neither Planck nor LIGO/Virgo has actually detected primordial tensor modes, it is trivially true that no model-independent constraint on the slope of the tensor power spectrum is possible with current data. (2) CMB and LIGO/Virgo scales differ by more than 19 orders of magnitude, and 16 for DECIGO. I show that a power-law extrapolation from CMB to direct detection frequencies overestimates the amplitude of primordial tensor modes by as much as two orders of magnitude relative to an ensemble of realistic single-field inflation models. Moreover, the primordial tensor amplitude at direct detection scales is mostly uncorrelated with the tensor spectral index at CMB scales, and any constraint is strongly dependent on the specific form of the inflationary potential.

Influencing factors on creep displacement assessment of bonded fasteners in concrete

The long-term sustained load behaviour of bonded fasteners in concrete is tested and assessed in accordance with European Assessment Documents (EADs) 330087 and 330499. In these EADs, the measured displacements during sustained loading are projected to a 50-year working life using a power-law regression and extrapolation combining an initial displacement δ0 and a time-dependent creep term a∙tb. The projection of sustained load displacement data is sensitive to methods for determining and employing the initial displacement, data sampling choices for regression, and the length of testing. To illustrate these sensitivities, six sustained load tests with an extended test duration of 300 days and high displacement measurement frequency were conducted and numerically investigated. The aim of this paper is to address and verify the influence of various parameters on the displacement projection. In particular, the determination of the initial displacement, sampling window, and their proper use in the application of the Findley power-law method for creep displacement extrapolations are evaluated. To that end, different scenarios are assessed varying the time step between the displacement measurements, the size of the time window of measured displacements used for the regression analysis and use of the initial displacement in regression. A final check on the appropriateness and conservatism of the use of a power-law regression is proposed. Based on this analysis, proposals for the assessment of long-term displacement of bonded fasteners in concrete under sustained load are made.

Slope movement classification and new insights into failure prediction based on landslide deformation evolution

This study aims to investigate the features of landslide evolution from the perspectives of statistics and nonlinear systems. First, to understand the complex processes of landslide evolution intuitively, a new landslide classification based on deformation patterns is presented. Then, frequency-size statistics of velocity events are performed in five historical landslides, including four failure cases and one non-failure case. The results show that velocity events follow power-law distributions within a certain size range. The power-law behaviors are usually associated with a ubiquitous phenomenon of self-organized criticality (SOC). A statistical test indicates that some outliers with extreme sizes significantly exceed power-law extrapolation. One possible explanation is that they are “dragon-kings” (DKs), possessing different formation mechanisms from their peers in power laws. Herein, the role of SOC and DKs in landslide evolution is analyzed in detail. The two theories provide new ways to understand the underlying mechanisms of landslide evolution. Meanwhile, power-law distributions and DKs provide new insights to predict landslide deformation.

A comparison of quasar emission reconstruction techniques for z ≥ 5.0 Lyman α and Lyman β transmission

ABSTRACT Reconstruction techniques for intrinsic quasar continua are crucial for the precision study of Lyman α (Ly α) and Lyman β (Ly β) transmission at z &amp;gt; 5.5, where the λ &amp;lt; 1215 Å emission of quasars is nearly completely absorbed. While the number and quality of spectroscopic observations have become theoretically sufficient to quantify Ly α transmission at 5.0 &amp;lt; z &amp;lt; 6.0 to better than $1{{\ \rm per\ cent}}$, the biases and uncertainties arising from predicting the unabsorbed continuum are not known to the same level. In this paper, we systematically evaluate eight reconstruction techniques on a unified testing sample of 2.7 &amp;lt; z &amp;lt; 3.5 quasars drawn from the Extended Baryon Oscillation Spectroscopic Survey. The methods include power-law extrapolation, stacking of neighbours, and six variants of principal component analysis (PCA) using direct projection, fitting of components, or neural networks to perform weight mapping. We find that power-law reconstructions and the PCA with fewest components and smallest training sample display the largest biases in the Ly α forest (${-}9.58{{\ \rm per\ cent}}/{+}8.22{{\ \rm per\ cent}}$, respectively). Power-law extrapolations have larger scatters than previously assumed of ${+}13.1{{\ \rm per\ cent}}/{-}13.2{{\ \rm per\ cent}}$ over Ly α and ${+}19.9{{\ \rm per\ cent}}/{-}20.1{{\ \rm per\ cent}}$ over Ly β. We present two new PCAs that achieve the best current accuracies of $9{{\ \rm per\ cent}}$ for Ly α and $17{{\ \rm per\ cent}}$ for Ly β. We apply the eight techniques after accounting for wavelength-dependent biases and scatter to a sample of 19 quasars at z &amp;gt; 5.7 with IR X-Shooter spectroscopy, obtaining well-characterized measurements for the mean flux transmission at 4.7 &amp;lt; z &amp;lt; 6.3. Our results demonstrate the importance of testing and, when relevant, training, continuum reconstruction techniques in a systematic way.

Suitability of power-law extrapolation for wind speed estimation on a tropical island

In this study, the ability of the power-law wind model to reproduce a tropical vertical wind profile is investigated. Data from a one-year observation on an offshore island of Singapore, obtained using a lidar, are used to show that the power-law model profiles are unable to capture the pertinent features of the observed wind profiles. Two alternative profile models, one based on the free convection limit scaling and the other based on the arctangent function, are introduced and tested. The latter model, although lacking physical basis, is found to be superior in predicting the measured profiles. A systematic study of the seasonal and diurnal variations of the model parameters is presented and some practical implications of the results are also discussed.

Decreasing wind speed extrapolation error via domain-specific feature extraction and selection

Abstract. Model uncertainty is a significant challenge in the wind energy industry and can lead to mischaracterization of millions of dollars' worth of wind resources. Machine learning methods, notably deep artificial neural networks (ANNs), are capable of modeling turbulent and chaotic systems and offer a promising tool to produce high-accuracy wind speed forecasts and extrapolations. This paper uses data collected by profiling Doppler lidars over three field campaigns to investigate the efficacy of using ANNs for wind speed vertical extrapolation in a variety of terrains, and it quantifies the role of domain knowledge in ANN extrapolation accuracy. A series of 11 meteorological parameters (features) are used as ANN inputs, and the resulting output accuracy is compared with that of both standard log-law and power-law extrapolations. It is found that extracted nondimensional inputs, namely turbulence intensity, current wind speed, and previous wind speed, are the features that most reliably improve the ANN's accuracy, providing up to a 65 % and 52 % increase in extrapolation accuracy over log-law and power-law predictions, respectively. The volume of input data is also deemed important for achieving robust results. One test case is analyzed in depth using dimensional and nondimensional features, showing that the feature nondimensionalization drastically improves network accuracy and robustness for sparsely sampled atmospheric cases.

Radiation spectra of warm and optically thick coronae in AGNs

A soft X-ray excess above the 2–10 keV power-law extrapolation is generally observed in the X-ray spectra of active galactic nuclei. The origin of this excess is still not well understood. Presently there are two competitive models: blurred ionized reflection and warm Comptonization. In the case of warm Comptonization, observations suggest a corona temperature in the range 0.1–2 keV and a corona optical depth of about 10–20. Moreover, radiative constraints from spectral fits with Comptonization models suggest that most of the accretion power should be released in the warm corona and the disk below is basically non-dissipative, radiating only the reprocessed emission from the corona. However, the true radiative properties of such a warm and optically thick plasma are not well known. For instance, the importance of the Comptonization process, the potential presence of strong absorption and/or emission features, and the spectral shape of the output spectrum have been studied only very recently. Here, we present simulations of warm and optically thick coronae using the TITAN radiative transfer code coupled with the NOAR Monte-Carlo code, the latter fully accounting for Compton scattering of continuum and lines. Illumination from above by hard X-ray emission and from below by an optically thick accretion disk are taken into account, as well as (uniform) internal heating. Our simulations show that for a large part of the parameter space, the warm corona with sufficient internal mechanical heating is dominated by Compton cooling and neither strong absorption nor emission lines are present in the outgoing spectra. In a smaller part of the parameter space, the calculated emission agrees with the spectral shape of the observed soft X-ray excess. Remarkably, this also corresponds to the conditions of radiative equilibrium of an extended warm corona covering a non-dissipative accretion disk almost entirely. These results confirm that warm Comptonization is a valuable model that can explain the origin of the soft X-ray excess.

Tantalum strength at extreme strain rates from impact-driven Richtmyer-Meshkov instabilities.

Recently, Richtmyer-Meshkov instability (RMI) experiments driven by high explosives and fielded with perturbations on a free surface have been used to study strength at extreme strain rates and near zero pressure. The RMI experiments reported here used impact loading, which is experimentally simpler, more accurate to analyze, and which also allows the exploration of a wider range of conditions. Three experiments were performed on tantalum at shock stresses from 20 to 34 GPa, with six different perturbation sizes at each shock level, making this the most comprehensive set of strength-focused RMI experiments reported to date on any material. The resulting estimated average strengths of 1200-1400 MPa at strain rates of 10^{7}/s exceeded, by 40% or more, a common power law extrapolation from data at strain rates below 10^{4}/s. Taken together with other data in the literature that show much higher strength at simultaneous high rates and high pressure, these RMI data isolated effects and indicated that, in the range of conditions examined, the pressure effects are more significant than rate effects.

A High-Resolution Airborne Color-Infrared Camera Water Mask for the NASA ABoVE Campaign

The airborne AirSWOT instrument suite, consisting of an interferometric Ka-band synthetic aperture radar and color-infrared (CIR) camera, was deployed to northern North America in July and August 2017 as part of the NASA Arctic-Boreal Vulnerability Experiment (ABoVE). We present validated, open (i.e., vegetation-free) surface water masks produced from high-resolution (1 m), co-registered AirSWOT CIR imagery using a semi-automated, object-based water classification. The imagery and resulting high-resolution water masks are available as open-access datasets and support interpretation of AirSWOT radar and other coincident ABoVE image products, including LVIS, UAVSAR, AIRMOSS, AVIRIS-NG, and CFIS. These synergies offer promising potential for multi-sensor analysis of Arctic-Boreal surface water bodies. In total, 3167 km2 of open surface water were mapped from 23,380 km2 of flight lines spanning 23 degrees of latitude and broad environmental gradients. Detected water body sizes range from 0.00004 km2 (40 m2) to 15 km2. Power-law extrapolations are commonly used to estimate the abundance of small lakes from coarser resolution imagery, and our mapped water bodies followed power-law distributions, but only for water bodies greater than 0.34 (±0.13) km2 in area. For water bodies exceeding this size threshold, the coefficients of power-law fits vary for different Arctic-Boreal physiographic terrains (wetland, prairie pothole, lowland river valley, thermokarst, and Canadian Shield). Thus, direct mapping using high-resolution imagery remains the most accurate way to estimate the abundance of small surface water bodies. We conclude that empirical scaling relationships, useful for estimating total trace gas exchange and aquatic habitats on Arctic-Boreal landscapes, are uniquely enabled by high-resolution AirSWOT-like mappings and automated detection methods such as those developed here.

Proactive Contamination Control for Sub 10nm Particle in Advanced Semiconductor Manufacturing

Advanced Semiconductor Manufacturing has reached the limits of metrology measuring contamination and defects on both the wafers and in the high purity materials (i.e., killer size particles in liquid chemicals and ultrapure water). Furthermore, ability of the most advanced filters to control particles in sub-10nm range is also limited. High purity materials used in liquid delivery systems have never been qualified for sub-10nm size particles. Finally, particle shedding of sub-10nm particles from high-purity components located post-filter cannot be directly measured and only be estimated using power-law extrapolation. Yield Enhancement IFT (international focus team) of IRDS (International Roadmap for Devices and Systems) is working together with the Liquid Chemical Committee of SEMI Standards to establish the methodology to enable proactive technology management in the space of contamination control. This presentation will provide the most recent experimental data generated by UPW IRDS team and SEMI Standards Liquid Chemical task forces, demonstrating the issues related to the contamination occurrence. The presentation will also include information about the SEMI standards and corresponding systematic approach that supports supply chain effort for developing and commercialization of new solutions to proactively address the contamination issues.

Central exclusive production of J/\u03c8 and \u03c8(2S) mesons in pp collisions at sqrt{s}=13 TeV

Measurements are reported of the central exclusive production of J/ψ and ψ(2S) mesons in pp collisions at a centre-of-mass energy of 13 TeV. Backgrounds are significantly reduced compared to previous measurements made at lower energies through the use of new forward shower counters. The products of the cross-sections and the branching fractions for the decays to dimuons, where both muons are within the pseudorapidity range 2.0 < η < 4.5, are measured to beσJ/ψ→μ+μ−=435±18±11±17pbσψ2S→μ+μ−=11.1±1.1±0.3±0.4pb.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{array}{c}\\hfill {\\sigma}_{J/\\psi \\to {\\mu}^{+}{\\mu}^{-}} = 435 \\pm 18 \\pm 11 \\pm 17\\ \\mathrm{p}\\mathrm{b}\\hfill \\\\ {}\\hfill {\\sigma}_{\\psi (2S)\\to {\\mu}^{+}{\\mu}^{-}} = 11.1 \\pm 1.1 \\pm 0.3 \\pm 0.4\\ \\mathrm{p}\\mathrm{b}.\\hfill \\end{array} $$\\end{document}The first uncertainties are statistical, the second are systematic, and the third are due to the luminosity determination. The cross-sections are also measured differentially for meson rapidities between 2.0 and 4.5. Good agreement is observed with theoretical predictions. Photoproduction cross-sections are derived and compared to previous experiments, and a deviation from a pure power-law extrapolation of lower energy data is observed.

Evaluation of the Power-Law Wind-Speed Extrapolation Method with Atmospheric Stability Classification Methods for Flows over Different Terrain Types

The atmospheric stability and ground topography play an important role in shaping wind-speed profiles. However, the commonly used power-law wind-speed extrapolation method is usually applied, ignoring atmospheric stability effects. In the present work, a new power-law wind-speed extrapolation method based on atmospheric stability classification is proposed and evaluated for flows over different types of terrain. The method uses the wind shear exponent estimated in different stability conditions rather than its average value in all stability conditions. Four stability classification methods, namely the Richardson Gradient (RG) method, the Wind Direction Standard Deviation (WDSD) method, the Wind Speed Ratio (WSR) method and the Monin–Obukhov (MO) method are applied in the wind speed extrapolation method for three different types of terrain. Tapplicability is analyzed by comparing the errors between the measured data and the calculated results at the hub height. It is indicated that the WSR classification method is effective for all the terrains investigated while the WDSD method is more suitable in plain areas. Moreover, the RG and MO methods perform better in complex terrains than the other methods, if two-level temperature data are available.

A Contribution for the Assessment of Discretization Error Estimators Based on Grid Refinement Studies

This paper presents grid refinement studies for statistically steady, two-dimensional (2D) flows of an incompressible fluid: a flat plate at Reynolds numbers equal to 107, 108, and 109 and the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg with Re = 6 × 106. Results are based on the numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations supplemented by one of three eddy-viscosity turbulence models of choice: the one-equation model of Spalart and Allmaras and the two-equation models k – ω SST and k−kL. Grid refinement studies are performed in sets of geometrically similar structured grids, permitting an unambiguous definition of the typical cell size, using double precision and an iterative convergence criterion that guarantees a numerical error dominated by the discretization error. For each case, different grid sets with the same number of cells but different near-wall spacings are used to generate a data set that allows more than one estimation of the numerical uncertainty for similar grid densities. The selected flow quantities include functional (integral), surface, and local flow quantities, namely, drag/resistance and lift coefficients; skin friction and pressure coefficients at the wall; and mean velocity components and eddy viscosity at specified locations in the boundary-layer region. An extra set of grids significantly more refined than those proposed for the estimation of the numerical uncertainty is generated for each test case. Using power law extrapolations, these extra solutions are used to obtain an approximation of the exact solution that allows the assessment of the performance of the numerical uncertainty estimations performed for the basis data set. However, it must be stated that with grids up to 2.5 (plate) and 8.46 (airfoil) million cells in two dimensions, the asymptotic range is not attained for many of the selected flow quantities. All this data is available online to the community.