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NASA NTRS2007

Determination of Black Hole Mass in Cyg X-1 by Scaling of Spectral Index-QPO Frequency Correlation

It is well established that timing and spectral properties of Galactic Black Hole (BH) X-ray binaries (XRB) are strongly correlated. In particular, it has been shown that low frequency Quasi-Periodic Oscillation (QPO) nu(sub low) - photon index GAMMA correlation curves have a specific pattern. In a number of the sources studied the shape of the index-low frequency QPO correlations are self-similar with a position offset in the nu(sub low) - GAMMA plane determined by a BH mass M(sub BH). Specifically, Titarchuk & Fiorito (2004) gave strong theoretical and observational arguments that the QPO frequency values in this nu(sub low) - GAMMA correlation should be inversely proportional to M(sub BH). A simple translation of the correlation for a given source along frequency axis leads to the observed correlation for another source. As a result of this translation one can obtain a scaling factor which is simply a BH mass ratio for these particular sources. This property of the correlations offers a fundamentally new method for BH mass determination in XRBs. Here we use the observed QPO-index correlations observed in three BH sources: GRO J1655-40, GRS 1915+105 and Cyg X-1. The BH mass of (6.3 plus or minus 0.5) solar mass in GRO J1655-40 is obtained using optical observations. RXTE observations during the recent 2005 outburst yielded sufficient data to establish the correlation pattern during both rise and decay of the event. We use GRO J1655-40 as a standard reference source to measure the BH mass in Cyg X-1. We also revisit the GRS 1915+105 data as a further test of our scaling method. We obtain the BH mass in Cyg X-1 in the range 7.6-9.9.

Shaposhnikov, Nickolai↗
NASA NTRS2007

Scaling Relations from Sunyaev-Zel'dovich Effect and Chandra X-ray Measurements of High-Redshift Galaxy Clusters

We present Sunyaev-Zel'dovich Effect (SZE) scaling relations for 38 massive galaxy clusters at redshifts 0.14 less than or equal to z less than or equal to 0.89, observed with both the Chandra X-ray Observatory and the centimeter-wave SZE imaging system at the BIMA and OVRO interferometric arrays. An isothermal ,Beta-model with central 100 kpc excluded from the X-ray data is used to model the intracluster medium and to measure global cluster properties. For each Cluster, we measure the X-ray spectroscopic temperature, SZE gas mass, total mass. and integrated Compton-gamma parameters within r(sub 2500). Our measurements are in agreement with the expectations based on a simple self-similar model of cluster formation and evolution. We compare the cluster properties derived from our SZE observations with and without Chandra spatial and spectral information and find them to be in good agreement: We compare our results with cosmological numerical simulations, and find that simulations that include radiative cooling, star formation and feedback match well both the slope and normalization of our SZE scaling relations.

Bonamente, Massimiliano↗
NASA NTRS2008

Scaling Relations from Sunyaev-Zel'dovich Effect and Chandra X-Ray Measurements of High-Redshift Galaxy Clusters

We present Sunyaev-Zel'dovich Effect (SZE) scaling relations for 38 massive galaxy clusters at redshifts 0.14 less than or equal to z less than or equal to 0.89, observed with both the Chandra X-ray Observatory and the centimeter-wave SZE imaging system at the BIMA and OVRO interferometric arrays. An isothermal Beta-model with the central 100 kpc excluded from the X-ray data is used to model the intracluster medium and to measure global cluster properties. For each cluster, we measure the X-ray spectroscopic temperature, SZE gas mass, total mass, and integrated Compton gamma-parameters within r(sub 2500). Our measurements are in agreement with the expectations based on a simple self-similar model of cluster formation and evolution. We compare the cluster properties derived from our SZE observations with and without Chandra spatial and spectral information and find them to be in good agreement. We compare our results with cosmological numerical simulations and find that simulations that include radiative cooling, star formation, and feedback match well both the slope and normalization of our SZE scaling relations.

Bonamente, Massimiliano↗
NASA NTRS2008

CFD Modeling of Superheated Fuel Sprays

An understanding of fuel atomization and vaporization behavior at superheat conditions is identified to be a topic of importance in the design of modern supersonic engines. As a part of the NASA aeronautics initiative, we have undertaken an assessment study to establish baseline accuracy of existing CFD models used in the evaluation of a ashing jet. In a first attempt towards attaining this goal, we have incorporated an existing superheat vaporization model into our spray solution procedure but made some improvements to combine the existing models valid at superheated conditions with the models valid at stable (non-superheat) evaporating conditions. Also, the paper reports some validation results based on the experimental data obtained from the literature for a superheated spray generated by the sudden release of pressurized R134A from a cylindrical nozzle. The predicted profiles for both gas and droplet velocities show a reasonable agreement with the measured data and exhibit a self-similar pattern similar to the correlation reported in the literature. Because of the uncertainty involved in the specification of the initial conditions, we have investigated the effect of initial droplet size distribution on the validation results. The predicted results were found to be sensitive to the initial conditions used for the droplet size specification. However, it was shown that decent droplet size comparisons could be achieved with properly selected initial conditions, For the case considered, it is reasonable to assume that the present vaporization models are capable of providing a reasonable qualitative description for the two-phase jet characteristics generated by a ashing jet. However, there remains some uncertainty with regard to the specification of certain initial spray conditions and there is a need for experimental data on separate gas and liquid temperatures in order to validate the vaporization models based on the Adachi correlation for a liquid involving R134A.

Raju, M. S.↗
NASA NTRS2011

A Model of Reduced Kinetics for Alkane Oxidation Using Constituents and Species for N-Heptane

The reduction of elementary or skeletal oxidation kinetics to a subgroup of tractable reactions for inclusion in turbulent combustion codes has been the subject of numerous studies. The skeletal mechanism is obtained from the elementary mechanism by removing from it reactions that are considered negligible for the intent of the specific study considered. As of now, there are many chemical reduction methodologies. A methodology for deriving a reduced kinetic mechanism for alkane oxidation is described and applied to n-heptane. The model is based on partitioning the species of the skeletal kinetic mechanism into lights, defined as those having a carbon number smaller than 3, and heavies, which are the complement of the species ensemble. For modeling purposes, the heavy species are mathematically decomposed into constituents, which are similar but not identical to groups in the group additivity theory. From analysis of the LLNL (Lawrence Livermore National Laboratory) skeletal mechanism in conjunction with CHEMKIN II, it is shown that a similarity variable can be formed such that the appropriately non-dimensionalized global constituent molar density exhibits a self-similar behavior over a very wide range of equivalence ratios, initial pressures and initial temperatures that is of interest for predicting n-heptane oxidation. Furthermore, the oxygen and water molar densities are shown to display a quasi-linear behavior with respect to the similarity variable. The light species ensemble is partitioned into quasi-steady and unsteady species. The reduced model is based on concepts consistent with those of Large Eddy Simulation (LES) in which functional forms are used to replace the small scales eliminated through filtering of the governing equations; in LES, these small scales are unimportant as far as the overwhelming part of dynamic energy is concerned. Here, the scales thought unimportant for recovering the thermodynamic energy are removed. The concept is tested by using tabular information from the LLNL skeletal mechanism in conjunction with CHEMKIN II utilized as surrogate ideal functions replacing the necessary functional forms. The test reveals that the similarity concept is indeed justified and that the combustion temperature is well predicted, but that the ignition time is over-predicted, a fact traced to neglecting a detailed description of the processes leading to the heavies chemical decomposition. To palliate this deficiency, functional modeling is incorporated into this conceptual reduction in addition to the modeling the evolution of the global constituent molar density, the enthalpy evolution of the heavies, the contribution to the reaction rate of the unsteady lights from other light species and from the heavies, the molar density evolution of oxygen and water, and the mole fractions of the quasisteady light species. The model is compact in that there are only nine species-related progress variables. Results are presented showing the performance of the model for predicting the temperature and species evolution. The model reproduces the ignition time over a wide range of equivalence ratios, initial pressure, and initial temperature.

Harstad, Kenneth G.↗
NASA NTRS2011

First Results from ARTEMIS, a New Two-Spacecraft Lunar Mission: Counter-Streaming Plasma Populations in the Lunar Wake

We present observations from the first passage through the lunar plasma wake by one of two spacecraft comprising ARTEMIS (Acceleration, Reconnection, Turbulence, and Electrodynamics of the Moon's Interaction with the Sun), a new lunar mission that re-tasks two of five probes from the THEMIS magnetospheric mission. On Feb 13, 2010, ARTEMIS probe P1 passed through the wake at 3.5 lunar radii downstream from the Moon, in a region between those explored by Wind and the Lunar Prospector, Kaguya, Chandrayaan, and Chang'E missions. ARTEMIS observed interpenetrating proton, alpha particle, and electron populations refilling the wake along magnetic field lines from both flanks. The characteristics of these distributions match expectations from self-similar models of plasma expansion into vacuum, with an asymmetric character likely driven by a combination of a tilted interplanetary magnetic field and an anisotropic incident solar wind electron population. On this flyby, ARTEMIS provided unprecedented measurements of the interpenetrating beams of both electrons and ions naturally produced by the filtration and acceleration effects of electric fields set up during the refilling process. ARTEMIS also measured electrostatic oscillations closely correlated with counter-streaming electron beams in the wake, as previously hypothesized but never before directly measured. These observations demonstrate the capability of the comprehensively instrumented ARTEMIS spacecraft and the potential for new lunar science from this unique two spacecraft constellation.

Halekas, J. S.↗
NASA NTRS2011

The Atacama Cosmology Telescope: Sunyaev-Zel'dovich-Selected Galaxy Clusters AT 148 GHz in the 2008 Survey

We report on 23 clusters detected blindly as Sunyaev-Zel'dovich (SZ) decrements in a 148 GHz, 455 deg (exp 2) map of the southern sky made with data from the Atacama Cosmology Telescope 2008 observing season. All SZ detections announced in this work have confirmed optical counterparts. Ten of the clusters are new discoveries. One newly discovered cluster, ACT-CL 10102-4915, with a redshift of 0.75 (photometric), has an SZ decrement comparable to the most massive systems at lower redshifts. Simulations of the cluster recovery method reproduce the sample purity measured by optical follow-up. In particular, for clusters detected with a signal-to-noise ratio greater than six, simulations are consistent with optical follow-up that demonstrated this subsample is 100% pure, The simulations further imply that the total sample is 80% complete for clusters with mass in excess of 6 x 10(exp 14) solar masses referenced to the cluster volume characterized by 500 times the critical density. The Compton gamma-X-ray luminosity mass comparison for the 11 best-detected clusters visually agrees with both self-similar and non-adiabatic, simulation-derived scaling laws,

Marriage, Tobias A.↗
NASA NTRS2007

On the Propagation and Interaction of Spherical Blast Waves

The characteristics and the scaling laws of isolated spherical blast waves have been briefly reviewed. Both self-similar solutions and numerical solutions of isolated blast waves are discussed. Blast profiles in the near-field (strong shock region) and the far-field (weak shock region) are examined. Particular attention is directed at the blast overpressure and shock propagating speed. Consideration is also given to the interaction of spherical blast waves. Test data for the propagation and interaction of spherical blast waves emanating from explosives placed in the vicinity of a solid propellant stack are presented. These data are discussed with regard to the scaling laws concerning the decay of blast overpressure.

Kandula, Max↗
NASA NTRS2013

Modeling of Turbulent Free Shear Flows

The modeling of turbulent free shear flows is crucial to the simulation of many aerospace applications, yet often receives less attention than the modeling of wall boundary layers. Thus, while turbulence model development in general has proceeded very slowly in the past twenty years, progress for free shear flows has been even more so. This paper highlights some of the fundamental issues in modeling free shear flows for propulsion applications, presents a review of past modeling efforts, and identifies areas where further research is needed. Among the topics discussed are differences between planar and axisymmetric flows, development versus self-similar regions, the effect of compressibility and the evolution of compressibility corrections, the effect of temperature on jets, and the significance of turbulent Prandtl and Schmidt numbers for reacting shear flows. Large eddy simulation greatly reduces the amount of empiricism in the physical modeling, but is sensitive to a number of numerical issues. This paper includes an overview of the importance of numerical scheme, mesh resolution, boundary treatment, sub-grid modeling, and filtering in conducting a successful simulation.

Yoder, Dennis A.↗
NASA NTRS2012

Probing Pre-Galactic Metal Enrichment with High-Redshift Gamma-Ray Bursts

We explore high-redshift gamma-ray bursts (GRBs) as promising tools to probe pre-galactic metal enrichment. We utilize the bright afterglow of a Population III (Pop III) GRB exploding in a primordial dwarf galaxy as a luminous background source, and calculate the strength of metal absorption lines that are imprinted by the first heavy elements in the intergalactic medium (IGM). To derive the GRB absorption line diagnostics, we use an existing highly resolved simulation of the formation of a first galaxy which is characterized by the onset of atomic hydrogen cooling in a halo with virial temperature approximately greater than10(exp 4) K.We explore the unusual circumburst environment inside the systems that hosted Pop III stars, modeling the density evolution with the self-similar solution for a champagne flow. For minihalos close to the cooling threshold, the circumburst density is roughly proportional to (1 + z) with values of about a few cm(exp −3). In more massive halos, corresponding to the first galaxies, the density may be larger, n approximately greater than100 cm(exp −3). The resulting afterglow fluxes are weakly dependent on redshift at a fixed observed time, and may be detectable with the James Webb Space Telescope and Very Large Array in the near-IR and radio wavebands, respectively, out to redshift z approximately greater than 20. We predict that the maximum of the afterglow emission shifts from near-IR to millimeter bands with peak fluxes from mJy to Jy at different observed times. The metal absorption line signature is expected to be detectable in the near future. GRBs are ideal tools for probing the metal enrichment in the early IGM, due to their high luminosities and featureless power-law spectra. The metals in the first galaxies produced by the first supernova (SN) explosions are likely to reside in low-ionization stages (C II, O I, Si II and Fe II). We show that, if the afterglow can be observed sufficiently early, analysis of the metal lines may distinguish whether the first heavy elements were produced in a pair-instability supernova or a core-collapse (Type II) SN, thus constraining the initial mass function of the first stars.

Metal↗
NASA NTRS2013

The 0.5-2.22 micrometer Scattered Light Spectrum of the Disk around TW Hya: Detection of a Partially Filled Disk Gap at 80 AU*

We present a 0.5-2.2 micrometer scattered light spectrum of the circumstellar disk around TW Hya from a combination of spatially resolved Hubble Space Telescope STIS spectroscopy and NICMOS coronagraphic images of the disk. We investigate the morphology of the disk at distances greater than 40 AU over this wide range of wavelengths, and identify the presence of a depression in surface brightness at approximately 80 AU that could be caused by a gap in the disk. Additionally, we quantify the surface brightness, azimuthal symmetry, and spectral character of the disk as a function of radius. Our analysis shows that the scattering efficiency of the dust is largely neutral to blue over the observed wavelengths. We model the disk as a steady α-disk with an ad hoc gap structure. The thermal properties of the disk are selfconsistently calculated using a three-dimensional radiative transfer code that uses ray tracing to model the heating of the disk interior and scattered light images. We find a good fit to the data over a wide range of distances from the star if we use a model disk with a partially filled gap of 30% depth at 80 AU and with a self-similar truncation knee at 100 AU. The origin of the gap is unclear, but it could arise from a transition in the nature of the disk's dust composition or the presence of a planetary companion. Based on scalings to previous hydrodynamic simulations of gap-opening criteria for embedded proto-planets, we estimate that a planetary companion forming the gap could have a mass between 6 and 28 solar mass.

radiative transfer↗
NASA NTRS2014

First Results from ARTEMIS, A New Two-Spacecraft Lunar Mission: Counter-Streaming Plasma Populations in the Lunar Wake

We present observations from the first passage through the lunar plasma wake by one of two spacecraft comprising ARTEMIS (Acceleration, Reconnection, Turbulence, and Electrodynamics of the Moon's Interaction with the Sun), a new lunar mission that re-tasks two of five probes from the THEMIS magnetospheric mission. On Feb 13, 2010, ARTEMIS probe P1 passed through the wake at approximately 3.5 lunar radii downstream from the Moon, in a region between those explored by Wind and the Lunar Prospector, Kaguya, Chandrayaan, and Chang'E missions. ARTEMIS observed interpenetrating proton, alpha particle, and electron populations refilling the wake along magnetic field lines from both flanks. The characteristics of these distributions match expectations from self-similar models of plasma expansion into vacuum, with an asymmetric character likely driven by a combination of a tilted interplanetary magnetic field and an anisotropic incident solar wind electron population. On this flyby, ARTEMIS provided unprecedented measurements of the interpenetrating beams of both electrons and ions naturally produced by the filtration and acceleration effects of electric fields set up during the refilling process. ARTEMIS also measured electrostatic oscillations closely correlated with counter-streaming electron beams in the wake, as previously hypothesized but never before directly measured. These observations demonstrate the capability of the comprehensively instrumented ARTEMIS spacecraft and the potential for new lunar science from this unique two spacecraft constellation.

Plasma↗
NASA NTRS2015

Periodic Density Structures and the Origin of the Slow Solar Wind

The source of the slow solar wind has challenged scientists for years. Periodic density structures (PDSs), observed regularly in the solar wind at 1 AU (Astronomical Unit), can be used to address this challenge. These structures have length scales of hundreds to several thousands of megameters and frequencies of tens to hundreds of minutes. Two lines of evidence indicate that PDSs are formed in the solar corona as part of the slow solar wind release and/or acceleration processes. The first is corresponding changes in compositional data in situ, and the second is PDSs observed in the inner Heliospheric Imaging data on board the Solar Terrestrial Relations Observatory (STEREO)/Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) suite. The periodic nature of these density structures is both a useful identifier as well as an important physical constraint on their origin. In this paper, we present the results of tracking periodic structures identified in the inner Heliospheric Imager in SECCHI back in time through the corresponding outer coronagraph (COR2) images. We demonstrate that the PDSs are formed around or below 2.5 solar radii-the inner edge of the COR2 field of view. We compute the occurrence rates of PDSs in 10 days of COR2 images both as a function of their periodicity and location in the solar corona, and we find that this set of PDSs occurs preferentially with a periodicity of approximately 90 minutes and occurs near streamers. Lastly, we show that their acceleration and expansion through COR2 is self-similar, thus their frequency is constant at distances beyond 2.5 solar radii.

sun: corona↗
NASA NTRS2016

Identifying and Analyzing Uncertainty Structures in the TRMM Microwave Imager Precipitation Product over Tropical Ocean Basins

Despite continuous improvements in microwave sensors and retrieval algorithms, our understanding of precipitation uncertainty is quite limited, due primarily to inconsistent findings in studies that compare satellite estimates to in situ observations over different parts of the world. This study seeks to characterize the temporal and spatial properties of uncertainty in the Tropical Rainfall Measuring Mission Microwave Imager surface rainfall product over tropical ocean basins. Two uncertainty analysis frameworks are introduced to qualitatively evaluate the properties of uncertainty under a hierarchy of spatiotemporal data resolutions. The first framework (i.e. 'climate method') demonstrates that, apart from random errors and regionally dependent biases, a large component of the overall precipitation uncertainty is manifested in cyclical patterns that are closely related to large-scale atmospheric modes of variability. By estimating the magnitudes of major uncertainty sources independently, the climate method is able to explain 45-88% of the monthly uncertainty variability. The percentage is largely resolution dependent (with the lowest percentage explained associated with a 1 deg x 1 deg spatial/1 month temporal resolution, and highest associated with a 3 deg x 3 deg spatial/3 month temporal resolution). The second framework (i.e. 'weather method') explains regional mean precipitation uncertainty as a summation of uncertainties associated with individual precipitation systems. By further assuming that self-similar recurring precipitation systems yield qualitatively comparable precipitation uncertainties, the weather method can consistently resolve about 50 % of the daily uncertainty variability, with only limited dependence on the regions of interest.

Climate↗
NASA NTRS2016

Evolution of Intrinsic Scatter in the SFR-Stellar Mass Correlation at 0.5 less than z Less Than 3

We present estimates of intrinsic scatter in the star formation rate (SFR)--stellar mass (M*) correlation in the redshift range 0.5 less than z less than 3.0 and in the mass range 10(exp 7) less than M* less than 10(exp 11) solar mass. We utilize photometry in the Hubble Ultradeep Field (HUDF12) and Ultraviolet Ultra Deep Field (UVUDF) campaigns and CANDELS/GOODS-S and estimate SFR, M* from broadband spectral energy distributions and the best-available redshifts. The maximum depth of the UDF photometry (F160W 29.9 AB, 5 sigma depth) probes the SFR--M* correlation down to M* approximately 10(exp 7) solar mass, a factor of 10-100 x lower in M* than previous studies, and comparable to dwarf galaxies in the local universe. We find the slope of the SFR-M* relationship to be near unity at all redshifts and the normalization to decrease with cosmic time. We find a moderate increase in intrinsic scatter with cosmic time from 0.2 to 0.4 dex across the epoch of peak cosmic star formation. None of our redshift bins show a statistically significant increase in intrinsic scatter approximately 100 Myr. Our results are consistent with a picture of gradual and self-similar assembly of galaxies across more than three orders of magnitude in stellar mass from as low as 10(exp 7) solar mass.

Kurczynski, Peter↗
NASA NTRS2017

Mixture-Fraction Measurements with Femtosecond-Laser Electronic-Excitation Tagging

Tracer-free mixture-fraction measurements were demonstrated in a jet using femtosecond-laser electronic-excitation tagging. Measurements were conducted across a turbulent jet at several downstream locations both in a pure-nitrogen jet exiting into an air-nitrogen mixture and in a jet containing an air-nitrogen mixture exiting into pure nitrogen. The signal was calibrated with known concentrations of oxygen in nitrogen. The spatial resolution of the measurement was approx.180 microns. The measurement uncertainty ranged from 5% to 15%, depending on the mixture fraction and location within the beam, under constant temperature and pressure conditions. The measurements agree with a mixture fraction of unity within the potential core of the jet and transition to the self-similar region.

Halls, Benjamin R.↗
NASA NTRS2017

Wavenumber-frequency Spectra of Pressure Fluctuations on a Generic Space Vehicle Measured via Unsteady Pressure-Sensitive Paint

Time histories of pressure fluctuations on a generic, hammerhead space vehicle model were measured using unsteady Pressure-Sensitive Paint (uPSP). The test was conducted in the 11-foot transonic wind tunnel of NASA Ames Research Center over a Mach number range of 0.6 M 1.2, and angles of attack of -4 4. The model was coated with a porous binder and PtTFPP-based porous polymer paint. An elaborate system of four high-speed cameras, and forty LED lamps was used for image acquisition. Various steps for image registration, reduction of shot noise, photogrammetry procedure to map images from the four cameras on a grid for the model, and finally a calibration procedure to convert the measured fluctuations in light intensity to fluctuating pressure, are discussed in the paper. The calibration process using a set of unsteady pressure sensors mounted on the model, was found to overcome some of the inherent problems of the fast response paint, such as rapid photo-degradation, non-linearity in pressure response, and significant temperature sensitivity. Comparison of spectra of pressure fluctuations between UPSP and pressure sensors demonstrated the ability of the paint to faithfully follow fluctuations up to 10 kHz, the maximum attempted. It was also found that the camera bit-depth and the illumination level limited the lowest measurable levels of pressure fluctuations to around 140dB. The large data set exposed various critical transonic flow physics not seen before, such as a coupling of the shock motion on the Payload Fairing (PF) with the separated flow region on the upper stage of the launch vehicle, and upstream convection of pressure fluctuation on PF at certain Mach numbers. The data also confirmed the expectation of a general lowering of the coefficient of pressure fluctuation with Mach number. The availability of the data set on a dense, regularly-spaced, surface grid allowed for the calculation of wavenumber-frequency (k-) spectra via straightforward applications of Fourier transform. The k- spectra were compared for the separated flow regions on the Second Stage, and the shock-boundary layer interactions on PF. The former showed self-similarity with Mach number while the latter was distinctly different, and confirmed the upstream propagation of pressure fluctuations. The k- spectra were dominated by the convected fluctuations; the acoustic domain was not discernable. These data, valuable for the vibro-acoustics analysis of aerospace vehicles, are believed to be the first obtained for the transonic flight regime, and pave the path for application on production models of aerospace vehicles.

buffet↗
NASA NTRS2019

A New Recycling Method to Generate Turbulent Inflow Profiles

The accuracy of the scale-resolving simulations for practical geometries strongly depends on the inflow boundary conditions. Imposing experimentally observed turbulent inflow profiles for the numerical simulations is a major challenge. Existing methods available in the literature assume self-similar behavior, which is not true for most of the experiments. In the present work, we formulate the turbulent inflow profile generation technique as an optimization problem. An adjoint technique is exploited to evaluate the sensitivities of multiple input parameters for the present problem. The present formulation is then tested to generate a laminar boundary layer profile, turbulent boundary layer profile, and turbulent jet profile.

Garai, Anirban↗