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NASA NTRS2014

CLARREO Approach for Reference Intercalibration of Reflected Solar Sensors: On-Orbit Data Matching and Sampling

The implementation of the Climate Absolute Radiance and Refractivity Observatory (CLARREO) mission was recommended by the National Research Council in 2007 to provide an on-orbit intercalibration standard with accuracy of 0.3% (k = 2) for relevant Earth observing sensors. The goal of reference intercalibration, as established in the Decadal Survey, is to enable rigorous high-accuracy observations of critical climate change parameters, including reflected broadband radiation [Clouds and Earth's Radiant Energy System (CERES)], cloud properties [Visible Infrared Imaging Radiometer Suite (VIIRS)], and changes in surface albedo, including snow and ice albedo feedback. In this paper, we describe the CLARREO approach for performing intercalibration on orbit in the reflected solar (RS) wavelength domain. It is based on providing highly accurate spectral reflectance and reflected radiance measurements from the CLARREO Reflected Solar Spectrometer (RSS) to establish an on-orbit reference for existing sensors, namely, CERES and VIIRS on Joint Polar Satellite System satellites, Advanced Very High Resolution Radiometer and follow-on imagers on MetOp, Landsat imagers, and imagers on geostationary platforms. One of two fundamental CLARREO mission goals is to provide sufficient sampling of high-accuracy observations that are matched in time, space, and viewing angles with measurements made by existing instruments, to a degree that overcomes the random error sources from imperfect data matching and instrument noise. The data matching is achieved through CLARREO RSS pointing operations on orbit that align its line of sight with the intercalibrated sensor. These operations must be planned in advance; therefore, intercalibration events must be predicted by orbital modeling. If two competing opportunities are identified, one target sensor must be given priority over the other. The intercalibration method is to monitor changes in targeted sensor response function parameters: effective offset, gain, nonlinearity, optics spectral response, and sensitivity to polarization. In this paper, we use existing satellite data and orbital simulationmethods to determinemission requirements for CLARREO, its instrument pointing ability, methodology, and needed intercalibration sampling and data matching for accurate intercalibration of RS radiation sensors on orbit.

radiometry↗
NASA NTRS1979

The simultaneous inference of stratospheric NO2-H2O and HNO3-CF2Cl2 using limb sounding radiometry

An inversion algorithm has been developed to simultaneously infer the concentrations of two gases with overlapping spectral signatures using limb emission measurements. The algorithm provides a solution in two or three iterations and has been tested in a simulation study for the inference of stratospheric NO2-H2O and HNO3-CF2Cl2. Existing satellite instrumentation was the basis for errors used in the calculations. These included noise, scale and bias errors, angular registration errors, spacecraft motion effects, and effects due to a finite instrument field of view. It is estimated that concentrations of all four gases can be measured globally from a satellite with errors of less than 20%.

Russell, J. M., III↗
NASA NTRS1982

IUE data reduction - The parameterization of the motion of the IUE reseau grids and spectral formats as a function of time and temperature

Improvements are made on IUE data reduction using corrected reseau positions and dispersion constants for temperature variations and secular effects. Models describing the motion of the high dispersion spectral format as a function of THDA alone reduce the scatter in the predicted position of a given wavelength, and scatter is further reduced when a linear dependence on time is also allowed. A correction technique is presented for early data which reduces errors in wavelength assignments from over 30 km/s to less than 3 km/s in high dispersion. The reseau temperature correction helps locate the minimum background signal, centered between the echelle orders. The residual uncertainty of 2 km/s for SWP and 2.7 km/s for LWR makes the wavelength accuracy of time and temperature corrected IUE spectra comparable to that of temperature corrected data from the Copernicus satellite.

Thompson, R. W.↗
NASA NTRS1983

Satellite remote sensing of meteorological parameters

Recent advances in remote atmospheric sensing are briefly reviewed, with particular attention given to vertical temperature and humidity profiles, cloud structure, and wind. Present capabilities and projections of future improvements in accuracy and resolution are given for the Microwave Sounding Unit, High Resolution Infrared Sounder, Defence Meteorological Satellite Project, and VISSR Atmospheric Sounder. It is noted that future sounding systems will require (1) high spectral resolution; (2) multispectral observations of the atmosphere and the surface in order to correct for most of the geophysical processes contaminating the outgoing radiance; and (3) a control algorithm capable of using information from multispectral channels to identify those parameters that have errors larger than a specified value.

Chahine, M. T.↗
NASA NTRS1985

Solar absolute spectral irradiance 118-300 nm - July 25, 1983

The full-disk solar absolute spectral irradiance in the spectral range 118-300 nm was obtained from a rocket observation above White Sands Missile Range, NM, on July 25, 1983, halfway in time between solar maximum and solar minimum. Comparison with measurements made in May 1982 shows no changes within the absolute errors of the experiment. Comparison with measurements made during solar maximum in 1979 and 1980 indicates a large decrease in the absolute solar irradiance at wavelengths below 190 nm. The absolute calibration of the instruments for this flight was accomplished at the National Bureau of Standards Synchrotron Radiation Facility.

Mount, G. H.↗
NASA NTRS1993

Growth of zinc selenide single crystals by physical vapor transport in microgravity

The goals of this research were the optimization of growth parameters for large (20 mm diameter and length) zinc selenide single crystals with low structural defect density, and the development of a 3-D numerical model for the transport rates to be expected in physical vapor transport under a given set of thermal and geometrical boundary conditions, in order to provide guidance for an advantageous conduct of the growth experiments. In the crystal growth studies, it was decided to exclusively apply the Effusive Ampoule PVT technique (EAPVT) to the growth of ZnSe. In this technique, the accumulation of transport-limiting gaseous components at the growing crystal is suppressed by continuous effusion to vacuum of part of the vapor contents. This is achieved through calibrated leaks in one of the ground joints of the ampoule. Regarding the PVT transport rates, a 3-D spectral code was modified. After introduction of the proper boundary conditions and subroutines for the composition-dependent transport properties, the code reproduced the experimentally determined transport rates for the two cases with strongest convective flux contributions to within the experimental and numerical error.

Rosenberger, Franz↗
NASA NTRS2006

Simulating Visible/Infrared Imager Radiometer Suite Normalized Difference Vegetation Index Data Using Hyperion and MODIS

The success of MODIS (the Moderate Resolution Imaging Spectrometer) in creating unprecedented, timely, high-quality data for vegetation and other studies has created great anticipation for data from VIIRS (the Visible/Infrared Imager Radiometer Suite). VIIRS will be carried onboard the joint NASA/Department of Defense/National Oceanic and Atmospheric Administration NPP (NPOESS (National Polar-orbiting Operational Environmental Satellite System) Preparatory Project). Because the VIIRS instruments will have lower spatial resolution than the current MODIS instruments 400 m versus 250 m at nadir for the channels used to generate Normalized Difference Vegetation Index data, scientists need the answer to this question: how will the change in resolution affect vegetation studies? By using simulated VIIRS measurements, this question may be answered before the VIIRS instruments are deployed in space. Using simulated VIIRS products, the U.S. Department of Agriculture and other operational agencies can then modify their decision support systems appropriately in preparation for receipt of actual VIIRS data. VIIRS simulations and validations will be based on the ART (Application Research Toolbox), an integrated set of algorithms and models developed in MATLAB(Registerd TradeMark) that enables users to perform a suite of simulations and statistical trade studies on remote sensing systems. Specifically, the ART provides the capability to generate simulated multispectral image products, at various scales, from high spatial hyperspectral and/or multispectral image products. The ART uses acquired ( real ) or synthetic datasets, along with sensor specifications, to create simulated datasets. For existing multispectral sensor systems, the simulated data products are used for comparison, verification, and validation of the simulated system s actual products. VIIRS simulations will be performed using Hyperion and MODIS datasets. The hyperspectral and hyperspatial properties of Hyperion data will be used to produce simulated MODIS and VIIRS products. Hyperion-derived MODIS data will be compared with near-coincident MODIS collects to validate both spectral and spatial synthesis, which will ascertain the accuracy of converting from MODIS to VIIRS. MODIS-derived VIIRS data is needed for global coverage and for the generation of time series for regional and global investigations. These types of simulations will have errors associated with aliasing for some scene types. This study will help quantify these errors and will identify cases where high-quality, MODIS-derived VIIRS data will be available.

Ross, Kenton W.↗
NASA NTRS2009

Recovering Aerodynamic Side Loads on Rocket Nozzles using Quasi-Static Strain-Gage Measurements

During over-expanded operation of rocket nozzles, which is defined to be when the exit pressure is greater than internal pressure over some part of the nozzle, the nozzle will experience a transverse forcing function due to the pressure differential across the nozzle wall. Over-expansion occurs during the nozzle start-up and shutdown transient, even in high-altitude engines, because most test facilities cannot completely reproduce the near-vacuum pressures at those altitudes. During this transient, the pressure differential moves axially down the nozzle as it becomes pressurized, but this differential is never perfectly symmetric circumferentially. The character of the forcing function is highly complex and defined by a series of restricted and free shock separations. The subject of this paper is the determination of the magnitude of this loading during sub-scale testing via measurement of the structural dynamic response of the nozzle and its support structure. An initial attempt at back-calculating this load using the inverse of the transfer function was performed, but this attempt was shown to be highly susceptible to numerical error. The final method chosen was to use statically calibrated strain data and to filter out the system fundamental frequency such that the measured response yields close to the correct dynamic loading function. This method was shown to capture 93% of the pressure spectral energy using controlled load shaker testing. This method is one of the only practical ways for the inverse determination of the forcing function for non-stationary excitations, and, to the authors' knowledge, has not been described in the literature to date.

Brown, Andrew↗
NASA NTRS2007

Advanced Systems for Monitoring Underwater Sounds

The term "Passive Acoustic Monitoring System" (PAMS) describes a developmental sensing-and-data-acquisition system for recording underwater sounds. The sounds (more precisely, digitized and preprocessed versions from acoustic transducers) are subsequently analyzed by a combination of data processing and interpretation to identify and/or, in some cases, to locate the sources of those sounds. PAMS was originally designed to locate the sources such as fish of species that one knows or seeks to identify. The PAMS unit could also be used to locate other sources, for example, marine life, human divers, and/or vessels. The underlying principles of passive acoustic sensing and analyzing acoustic-signal data in conjunction with temperature and salinity data are not new and not unique to PAMS. Part of the uniqueness of the PAMS design is that it is the first deep-sea instrumentation design to provide a capability for studying soniferous marine animals (especially fish) over the wide depth range described below. The uniqueness of PAMS also lies partly in a synergistic combination of advanced sensing, packaging, and data-processing design features with features adapted from proven marine instrumentation systems. This combination affords a versatility that enables adaptation to a variety of undersea missions using a variety of sensors. The interpretation of acoustic data can include visual inspection of power-spectrum plots for identification of spectral signatures of known biological species or artificial sources. Alternatively or in addition, data analysis could include determination of relative times of arrival of signals at different acoustic sensors arrayed at known locations. From these times of arrival, locations of acoustic sources (and errors in those locations) can be estimated. Estimates of relative locations of sources and sensors can be refined through analysis of the attenuation of sound in the intervening water in combination with water-temperature and salinity data acquired by instrumentation systems other than PAMS. A PAMS is packaged as a battery-powered unit, mated with external sensors, that can operate in the ocean at any depth from 2 m to 1 km. A PAMS includes a pressure housing, a deep-sea battery, a hydrophone (which is one of the mating external sensors), and an external monitor and keyboard box. In addition to acoustic transducers, external sensors can include temperature probes and, potentially, underwater cameras. The pressure housing contains a computer that includes a hard drive, DC-to- DC power converters, a post-amplifier board, a sound card, and a universal serial bus (USB) 4-port hub.

Lane, Michael↗
NASA NTRS2014

Estimation of Crop Gross Primary Production (GPP): I. Impact of MODIS Observation Footprint and Impact of Vegetation BRDF Characteristics

Accurate estimation of gross primary production (GPP) is essential for carbon cycle and climate change studies. Three AmeriFlux crop sites of maize and soybean were selected for this study. Two of the sites were irrigated and the other one was rainfed. The normalized difference vegetation index (NDVI), the enhanced vegetation index (EVI), the green band chlorophyll index (CIgreen), and the green band wide dynamic range vegetation index (WDRVIgreen) were computed from the moderate resolution imaging spectroradiometer (MODIS) surface reflectance data. We examined the impacts of the MODIS observation footprint and the vegetation bidirectional reflectance distribution function (BRDF) on crop daily GPP estimation with the four spectral vegetation indices (VIs - NDVI, EVI, WDRVIgreen and CIgreen) where GPP was predicted with two linear models, with and without offset: GPP = a × VI × PAR and GPP = a × VI × PAR + b. Model performance was evaluated with coefficient of determination (R2), root mean square error (RMSE), and coefficient of variation (CV). The MODIS data were filtered into four categories and four experiments were conducted to assess the impacts. The first experiment included all observations. The second experiment only included observations with view zenith angle (VZA) ≤ 35◦ to constrain growth of the footprint size,which achieved a better grid cell match with the agricultural fields. The third experiment included only forward scatter observations with VZA ≤ 35◦. The fourth experiment included only backscatter observations with VZA ≤ 35◦. Overall, the EVI yielded the most consistently strong relationships to daily GPP under all examined conditions. The model GPP = a × VI × PAR + b had better performance than the model GPP = a × VI × PAR, and the offset was significant for most cases. Better performance was obtained for the irrigated field than its counterpart rainfed field. Comparison of experiment 2 vs. experiment 1 was used to examine the observation footprint impact whereas comparison of experiment 4 vs. experiment 3 was used to examine the BRDF impact. Changes in R2, RMSE,CV and changes in model coefficients "a" and "b" (experiment 2 vs. experiment 1; and experiment 4 vs. experiment 3) were indicators of the impacts. The second experiment produced better performance than the first experiment, increasing R2 (↑0.13) and reducing RMSE (↓0.68 g C m−2 d−1) and CV (↓9%). For each VI, the slope of GPP = a × VI × PAR in the second experiment for each crop type changed little while the slope and intercept of GPP = a × VI × PAR + b varied field by field. The CIgreen was least affected by the MODIS observation footprint in estimating crop daily GPP (R2, ↑0.08; RMSE, ↓0.42 g C m−2 d−1; and CV, ↓7%). Footprint most affected the NDVI (R2, ↑0.15; CV, ↓10%) and the EVI (RMSE, ↓0.84 g C m−2 d−1). The vegetation BRDF impact also caused variation of model performance and change of model coefficients. Significantly different slopes were obtained for forward vs. backscatter observations, especially for the CIgreen and the NDVI. Both the footprint impact and the BRDF impact varied with crop types, irrigation options, model options and VI options.

Chlorophyll↗
NASA NTRS1978

26.3 MHz radio source survey. III - Correlation with extragalactic X-ray sources

The correlation between extragalactic X-ray sources and radio sources in a 26.3-MHz catalog is studied. A list of reliably identified sources is developed by examining X-ray, optical, and radio data for those candidate objects that are in or near the X-ray error boxes. The source 3UR 0432+05 is identified with 3C 120, 3UR 1555+27 (identified with A2142) is shown to be a steep-spectrum radio source, and it is found that 86% of the high-galactic-latitude sources can be reliably identified when the X-ray source error areas are no more than 1 sq deg. The results also indicate that: (1) X-ray sources identified with clusters of galaxies as a whole, with individual galaxies in clusters, and with separate isolated galaxies have similar decametric properties in that their spectral indices and radio luminosities fall in the same range; (2) there is a Bautz-Morgan dependence of both the X-ray and the decametric luminosities of clusters and of individual objects in clusters; and (3) the X-ray luminosity of all except compact sources appears to be approximately proportional to the decametric luminosity.

Erickson, W. C.↗
NASA NTRS1978

Annual Conference on Nuclear and Space Radiation Effects, 15th, University of New Mexico, Albuquerque, N. Mex., July 18-21, 1978, Proceedings

Radiation effects in MOS devices and circuits are considered along with radiation effects in materials, space radiation effects and spacecraft charging, SGEMP, IEMP, EMP, fabrication of radiation-hardened devices, radiation effects in bipolar devices and circuits, simulation, energy deposition, and dosimetry. Attention is given to the rapid anneal of radiation-induced silicon-sapphire interface charge trapping, cosmic ray induced errors in MOS memory cells, a simple model for predicting radiation effects in MOS devices, the response of MNOS capacitors to ionizing radiation at 80 K, trapping effects in irradiated and avalanche-injected MOS capacitors, inelastic interactions of electrons with polystyrene, the photoelectron spectral yields generated by monochromatic soft X radiation, and electron transport in reactor materials.

Simons, M.↗
NASA NTRS2000

Optical Performance Modeling of FUSE Telescope Mirror

We describe the Metrology Data Processor (METDAT), the Optical Surface Analysis Code (OSAC), and their application to the image evaluation of the Far Ultraviolet Spectroscopic Explorer (FUSE) mirrors. The FUSE instrument - designed and developed by the Johns Hopkins University and launched in June 1999 is an astrophysics satellite which provides high resolution spectra (lambda/Delta(lambda) = 20,000 - 25,000) in the wavelength region from 90.5 to 118.7 nm The FUSE instrument is comprised of four co-aligned, normal incidence, off-axis parabolic mirrors, four Rowland circle spectrograph channels with holographic gratings, and delay line microchannel plate detectors. The OSAC code provides a comprehensive analysis of optical system performance, including the effects of optical surface misalignments, low spatial frequency deformations described by discrete polynomial terms, mid- and high-spatial frequency deformations (surface roughness), and diffraction due to the finite size of the aperture. Both normal incidence (traditionally infrared, visible, and near ultraviolet mirror systems) and grazing incidence (x-ray mirror systems) systems can be analyzed. The code also properly accounts for reflectance losses on the mirror surfaces. Low frequency surface errors are described in OSAC by using Zernike polynomials for normal incidence mirrors and Legendre-Fourier polynomials for grazing incidence mirrors. The scatter analysis of the mirror is based on scalar scatter theory. The program accepts simple autocovariance (ACV) function models or power spectral density (PSD) models derived from mirror surface metrology data as input to the scatter calculation. The end product of the program is a user-defined pixel array containing the system Point Spread Function (PSF). The METDAT routine is used in conjunction with the OSAC program. This code reads in laboratory metrology data in a normalized format. The code then fits the data using Zernike polynomials for normal incidence systems or Legendre-Fourier polynomials for grazing incidence systems. It removes low order terms from the metrology data, calculates statistical ACV or PSD functions, and fits these data to OSAC models for the scatter analysis. In this paper we briefly describe the laboratory image testing of FUSE spare mirror performed in the near and vacuum ultraviolet at John Hopkins University and OSAC modeling of the test setup performed at NASA/GSFC. The test setup is a double-pass configuration consisting of a Hg discharge source, the FUSE off-axis parabolic mirror under test, an autocollimating flat mirror, and a tomographic imaging detector. Two additional, small fold flats are used in the optical train to accommodate the light source and the detector. The modeling is based on Zernike fitting and PSD analysis of surface metrology data measured by both the mirror vendor (Tinsley) and JHU. The results of our models agree well with the laboratory imaging data, thus validating our theoretical model. Finally, we predict the imaging performance of FUSE mirrors in their flight configuration at far-ultraviolet wavelengths.

Saha, Timo T.↗
NASA NTRS2002

Retrieval of Atmospheric and Surface Parameters from AIRS/AMSU/HSB Data Under Cloudy Conditions

New state of the art methodology is described to analyze AIRS/AMSU/HSB data in the presence of multiple cloud formations. The methodology forms the basis for the AIRS Science Team algorithm which will be used to analyze AIRS/AMSU/HSB data on EOS Aqua. The cloud clearing methodology requires no knowledge of the spectral properties of the clouds. The basic retrieval methodology is general and extracts the maximum information from the radiances, consistent with the channel noise covariance matrix. The retrieval methodology minimizes the dependence of the solution on the first guess field and the first guess error characteristics. Results are shown for AIRS Science Team simulation studies with multiple cloud formations. These simulation studies imply that clear column radiances can be reconstructed under partial cloud cover with an accuracy comparable to single spot channel noise in the temperature and water vapor sounding regions, temperature soundings can be produced under partial cloud cover with RMS errors on the order of, or better than, 1deg K in 1 km thick layers from the surface to 700 mb, 1 km layers from 700 mb to 300 mb, 3 km layers from 300 mb to 30 mb, and 5 km layers from 30 mb to 1 mb, and moisture profiles can be obtained with an accuracy better than 20% absolute errors in 1 km layers from the surface to nearly 200 mb.

Susskind, Joel↗
NASA NTRS2015

Time Accurate Unsteady Pressure Loads Simulated for the Space Launch System at a Wind Tunnel Condition

Using the Fully Unstructured Three-Dimensional (FUN3D) computational fluid dynamics code, an unsteady, time-accurate flow field about a Space Launch System configuration was simulated at a transonic wind tunnel condition (Mach = 0.9). Delayed detached eddy simulation combined with Reynolds Averaged Naiver-Stokes and a Spallart-Almaras turbulence model were employed for the simulation. Second order accurate time evolution scheme was used to simulate the flow field, with a minimum of 0.2 seconds of simulated time to as much as 1.4 seconds. Data was collected at 480 pressure taps at locations, 139 of which matched a 3% wind tunnel model, tested in the Transonic Dynamic Tunnel (TDT) facility at NASA Langley Research Center. Comparisons between computation and experiment showed agreement within 5% in terms of location for peak RMS levels, and 20% for frequency and magnitude of power spectral densities. Grid resolution and time step sensitivity studies were performed to identify methods for improved accuracy comparisons to wind tunnel data. With limited computational resources, accurate trends for reduced vibratory loads on the vehicle were observed. Exploratory methods such as determining minimized computed errors based on CFL number and sub-iterations, as well as evaluating frequency content of the unsteady pressures and evaluation of oscillatory shock structures were used in this study to enhance computational efficiency and solution accuracy. These techniques enabled development of a set of best practices, for the evaluation of future flight vehicle designs in terms of vibratory loads.

Alter, Stephen J.↗
NASA NTRS2018

The Challenges of Measuring Methane from Space with a Lidar

The global and regional quantification of methane fluxes and identification of its sources and sinks has been highlighted as one of the goals of the NASA 2017 Earth Science Decadal Survey. Detecting methane from space and airborne platforms with an active (laser) remote sensing instrument presents several unique technology and measurement challenges. The instrument must have a single frequency, narrow-linewidth light source, and photon-sensitive detector at the right spectral region to make continuous measurements from orbit, day and night, all seasons and at all latitudes. It must have a high signal to noise ratio and must be relatively immune to biases from aerosol/cloud scattering, spectroscopic and meteorological data uncertainties, and instrument systematic errors. At Goddard Space Flight Center (GSFC), in collaboration with industry, we have developed an airborne instrument to measure methane. Our instrument is a nadir-viewing lidar that uses Integrated Path Differential Absorption (IPDA), to measure methane near 1.65 µm. We sample the absorption line using multiple wavelengths from a narrow linewidth laser source and a sensitive photodetector. This measurement approach provides maximum information content about the CH4 column, and minimizes biases in the XCH4 retrieval. In this paper, we will review our progress to date and discuss the technology challenges, options and tradeoffs to measure methane from space and airborne platforms.

airborne instruments↗
NASA NTRS1975

A comparison of Skylab S-193 and aircraft views of surface roughness and a look toward SEASAT

An extensive aircraft underflight program was conducted along the Skylab groundpath for the purpose of documenting wind, wave, and atmospheric conditions affecting the amplitude of the active and passive microwave signatures. The S-193 microwave system senses a roughness parameter at the ocean surface that is proportional to the surface windspeed. The aircraft program is described, and some comparisons of satellite and aircraft results are presented. The principal result of the comparison of active radar is that direct inferences of the surface windspeed are possible, but subject to considerable scatter, and that this scatter appears to be due to interaction between long gravity and short Bragg waves and backscatter due to rain as well as errors in correcting for azimuth dependence. An unforeseen opportunity to observe a Pacific hurricane by both Skylab and NOAA aircraft has contributed to the development of a simplified wave forecasting scheme applicable to hurricanes, and more general conditions, which combines the better qualities of both spectral and height/period forecasting techniques. Horizontal polarization data obtained by the aircraft in Hurricane Ava, and in other experiments, which led to this development are presented.

Ross, D.↗
NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗