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NASA NTRS2017

Assimilation of AERONET and MODIS AOT Observations Using Variational and Ensemble Data Assimilation Methods and Its Impact on Aerosol Forecasting Skill

Data assimilation of Aerosol Robotic Network (AERONET) and Moderate Resolution Imaging Spectroradiometer (MODIS) aerosol optical thickness (AOT) for aerosol forecasting was tested within the Navy Aerosol Analysis Prediction System (NAAPS) framework, using variational and ensemble data assimilation methods. Navy aerosol forecasting currently makes use of a deterministic NAAPS simulation coupled to Navy Variational Data Assimilation System for aerosol optical depth, a two-dimensional variational data assimilation system, for MODIS AOT assimilation. An ensemble version of NAAPS (ENAAPS) coupled to an ensemble adjustment Kalman filter (EAKF) from the Data Assimilation Research Testbed was recently developed, allowing for a range of data assimilation and forecasting experiments to be run with deterministic NAAPS and ENAAPS. The main findings are that the EAKF, with its flow-dependent error covariances, makes better use of sparse observations such as AERONET AOT. Assimilating individual AERONET observations in the two-dimensional variational system can increase the analysis errors when observations are located in high AOT gradient regions. By including AERONET with MODIS AOT assimilation, the magnitudes of peak aerosol events (AOT> 1) were better captured with improved temporal variability, especially in India and Asia where aerosol prediction is a challenge. Assimilating AERONET AOT with MODIS had little impact on the 24 h forecast skill compared to MODIS assimilation only, but differences were found downwind of AERONET sites. The 24 h forecast skill was approximately the same for forecasts initialized with analyses from AERONET AOT assimilation alone compared to MODIS assimilation, particularly in regions where the AERONET network is dense; including the United States and Europe, indicating that AERONET could serve as a backup observation network for over-land synoptic-scale aerosol events.

Rubin, Juli I.↗

NASA NTRS2018

Generation of Mars Helicopter Rotor Model for Comprehensive Analyses

The present research is aimed at providing a performance model for the Mars Helicopter (MH), to understand the complexity of the flow, and identify future regions of improvement. The low density of the Martian atmosphere and the relatively small MH rotor, result in very low chord-based Reynolds number flows. The low density and Reynolds numbers reduce the lifting force and lifting efficiency, respectively. The high drag coefficients in subcritical flow, especially for thicker sections, are attributed to laminar separation from the rear of the airfoil. In the absence of test data, efforts have been made to explore these effects using prior very low Reynolds number research efforts. The rotor chord-based Reynolds number range is observed to be subcritical, which makes boundary layer transition unlikely to occur. The state of the two-dimensional rotor boundary layer in hover is approximated by calculating the instability point, laminar separation point, and the transition location to provide understanding of the flow state in the high Mach-low Reynolds number regime. The results are used to investigate the need for turbulence modeling in Computational Fluid Dynamics (CFD) calculations afterwards. The goal is to generate a performance model for the MH rotor for a free wake analysis, because the computational budget for a complete Navier-Stokes solution for a rotating body-fitted rotor is substantial. In this study, a Reynolds-Averaged Navier-Stokes (RANS) based approach is used to generate the airfoil deck using C81Gen with stitched experimental data for very high angles of attack. A full Grid Resolution Study is performed and over 4,500 cases are completed to create the full airfoil deck. The laminar separation locations are predicted within the accuracy of the approximate method when compared with the CFD calculations. The model is presented through airfoil data tables (c81 files) that are used by comprehensive rotor analysis codes such as CAMRADII, or the mid-fidelity CFD solver RotCFD. Finally, the rotor performance is compared with experimental data from the 25ft Space Simulator at the NASA Jet Propulsion Laboratory (JPL) and shows good correlation for the rotor Figure of Merit over the available thrust range.

Model↗

NASA NTRS1993

Grand challenge scientific questions in coupled modeling

Most convective field experiments in the past (e.g., SESAME, CCOPE, CINDE) have attempted to resolve only the immediate scales of moist convection using network arrays that spanned two or three atmospheric scales at most. Furthermore, these scales have been defined more on practical considerations (cost, manpower, etc.) than on a clear understanding of their theoretical significance. Unfortunately, this has precluded a description of the entire life cycle of MCS's and their interaction with larger scale systems, the land surface, and trace species. Fortunately, the following factors now make it possible to attempt to simulate scale contraction processes from the synoptic scale down to the cloud scale, as well as interactions between complex meteorological, land surface, precipitation, chemical, and hydrologic processes with coupled, multiscale models: the availability of new technology to sample meteorological fields at high temporal and spatial resolution over a broad region made possible by the weather observing modernization program; increased computer power and improved numerical approaches to run limited area models with nonhydrostatic precipitation physics so as to explicitly resolve MCS (Mesoscale Convective System) processes; and four dimensional assimilation of non-conventional data to provide dynamically consistent datasets for diagnostic analysis of nonlinear scale-interactive dynamics. Several examples of scale-interactive processes which present grand challenges for coupled, multiscale modeling were presented.

Koch, Steven↗

NASA NTRS2003

Gene Expression Dynamics Inspector (GEDI): for integrative analysis of expression profiles

Genome-wide expression profiles contain global patterns that evade visual detection in current gene clustering analysis. Here, a Gene Expression Dynamics Inspector (GEDI) is described that uses self-organizing maps to translate high-dimensional expression profiles of time courses or sample classes into animated, coherent and robust mosaics images. GEDI facilitates identification of interesting patterns of molecular activity simultaneously across gene, time and sample space without prior assumption of any structure in the data, and then permits the user to retrieve genes of interest. Important changes in genome-wide activities may be quickly identified based on 'Gestalt' recognition and hence, GEDI may be especially useful for non-specialist end users, such as physicians. AVAILABILITY: GEDI v1.0 is written in Matlab, and binary Matlab.dll files which require Matlab to run can be downloaded for free by academic institutions at http://www.chip.org/~ge/gedihome.html Supplementary information: http://www.chip.org/~ge/gedihome.html.

Database Management Systems↗

NASA NTRS2002

Effect of Crystal Orientation on Fatigue Failure of Single Crystal Nickel Base Turbine Blade Superalloys

High cycle fatigue (HCF) induced failures in aircraft gas turbine and rocket engine turbopump blades is a pervasive problem. Single crystal nickel turbine blades are being utilized in rocket engine turbopumps and jet engines throughout industry because of their superior creep, stress rupture, melt resistance, and thermomechanical fatigue capabilities over polycrystalline alloys. Currently the most widely used single crystal turbine blade superalloys are PWA 1480/1493, PWA 1484, RENE' N-5 and CMSX-4. These alloys play an important role in commercial, military and space propulsion systems. Single crystal materials have highly orthotropic properties making the position of the crystal lattice relative to the part geometry a significant factor in the overall analysis. The failure modes of single crystal turbine blades are complicated to predict due to the material orthotropy and variations in crystal orientations. Fatigue life estimation of single crystal turbine blades represents an important aspect of durability assessment. It is therefore of practical interest to develop effective fatigue failure criteria for single crystal nickel alloys and to investigate the effects of variation of primary and secondary crystal orientation on fatigue life. A fatigue failure criterion based on the maximum shear stress amplitude /Delta(sub tau)(sub max))] on the 24 octahedral and 6 cube slip systems, is presented for single crystal nickel superalloys (FCC crystal). This criterion reduces the scatter in uniaxial LCF test data considerably for PWA 1493 at 1200 F in air. Additionally, single crystal turbine blades used in the alternate advanced high-pressure fuel turbopump (AHPFTP/AT) are modeled using a large-scale three-dimensional finite element model. This finite element model is capable of accounting for material orthotrophy and variation in primary and secondary crystal orientation. Effects of variation in crystal orientation on blade stress response are studied based on 297 finite element model runs. Fatigue lives at critical points in the blade are computed using finite element stress results and the failure criterion developed. Stress analysis results in the blade attachment region are also presented. Results presented demonstrates that control of secondary and primary crystallographic orientation has the potential to significantly increase a component S resistance to fatigue crack growth with- out adding additional weight or cost. [DOI: 10.1115/1.1413767]

Arakere, N. K.↗

NASA NTRS1989

Space Shuttle Main Engine structural analysis and data reduction/evaluation. Volume 2: High pressure oxidizer turbo-pump turbine end bearing analysis

The high-pressure oxidizer turbopump (HPOTP) consists of two centrifugal pumps, on a common shaft, that are directly driven by a hot-gas turbine. Pump shaft axial thrust is balanced in that the double-entry main inducer/impeller is inherently balanced and the thrusts of the preburner pump and turbine are nearly equal but opposite. Residual shaft thrust is controlled by a self-compensating, non-rubbing, balance piston. Shaft hang-up must be avoided if the balance piston is to perform properly. One potential cause of shaft hang-up is contact between the Phase 2 bearing support and axial spring cartridge of the HPOTP main pump housing. The status of the bearing support/axial spring cartridge interface is investigated under current loading conditions. An ANSYS version 4.3, three-dimensional, finite element model was generated on Lockheed's VAX 11/785 computer. A nonlinear thermal analysis was then executed on the Marshall Space Flight Center Engineering Analysis Data System (EADS). These thermal results were then applied along with the interference fit and bolt preloads to the model as load conditions for a static analysis to determine the gap status of the bearing support/axial spring cartridge interface. For possible further analysis of the local regions of HPOTP main pump housing assembly, detailed ANSYS submodels were generated using I-DEAS Geomod and Supertab (Appendix A).

Sisk, Gregory A.↗

NASA NTRSDate not supplied

Simulation and Modeling of Hypersonic Turbulent Boundary Layers Subject to Adverse Pressure Gradients due to Concave Streamline Curvature

Direct numerical simulations (DNS) of adverse-pressure-gradient turbulent boundary layers over a planar concave wall are presented for a nominal freestream Mach number of 5, with the objective of assessing the limitations of the currently available Reynolds-averaged Navier-Stokes (RANS) models. The wall geometry and flow conditions of the DNS are representative of the experimental data for a Mach 4.9 turbulent boundary layer that was tested on a two-dimensional planar concave wall model in the high-speed blow-down wind tunnel located at the National Aerothermochemistry Laboratory at Texas A&M University (TAMU). The DNS was validated against the experimental results of TAMU for the same flow conditions and wall geometry. An analysis of the DNS datasets was also conducted to provide an assessment of the validity of Morkovin’s hypothesis and the strong Reynolds analog for turbulence subject to mechanical nonequilibrium. In addition to the DNS results, RANS predictions are obtained by using the Baldwin-Lomax (BL), Spalart-Allmaras (SA), and the k - w SST turbulence models. The comparisons between RANS and DNS showed little impact of an adverse pressure gradient on the accuracy of these models, at least up to an incompressible Clauser pressure gradient parameter of beta(sub inc) 1.22. While the Boussinesq assumption provided reasonable predictions for the Reynolds shear stress, it failed to adequately predict the normal components of the Reynolds stress.

turbulent boundary layers↗

NASA NTRS2024

Lunar Terrain Mapping Using 3D Software and Modeling Techniques for the Glenn Research Center Communication Analysis Suite

As NASA prepares to return humans to the Moon as part of upcoming Artemis missions, engineers must examine the surface features and safety of lunar terrain at various locations. The Glenn Research Center Communication Analysis Suite (GCAS) includes visualization tools developed to aid scientists in this quest to comprehend lunar terrain data, as well as spatial communication information. Visual information and understanding will be highly useful to engineers as they use the GCAS to compare the topography of regions of interest for the Artemis III and future space travel missions. Not only can three-dimensional (3D) visualization display the intricacies of the lunar terrain, but it can also demonstrate relationships such as the presence or absence of communication links between Earth ground stations, lunar crew members, and rovers, and the presence of shadows on the lunar surface. In the future, the GCAS has the potential to support the planning of other space exploration missions as well. The emerging possibility of incorporating more planetary bodies becomes especially important as NASA plans not only to return to the Moon but to press onward to Mars.

Visualization↗

NASA NTRS1993

Can cluster environment modify the dynamical evolution of spiral galaxies?

Over the past decade many effects of the cluster environment on member galaxies have been established. These effects are manifest in the amount and distribution of gas in cluster spirals, the luminosity and light distributions within galaxies, and the segregation of morphological types. All these effects could indicate a specific dynamical evolution for galaxies in clusters. Nevertheless, a more direct evidence, such as a different mass distribution for spiral galaxies in clusters and in the field, is not yet clearly established. Indeed, Rubin, Whitmore, and Ford (1988) and Whitmore, Forbes, and Rubin (1988) (referred to as RWF) presented evidence that inner cluster spirals have falling rotation curves, unlike those of outer cluster spirals or the great majority of field spirals. If falling rotation curves exist in centers of clusters, as argued by RWF, it would suggest that dark matter halos were absent from cluster spirals, either because the halos had become stripped by interactions with other galaxies or with an intracluster medium, or because the halos had never formed in the first place. Even if they didn't disagree with RWF, other researchers pointed out that the behaviour of the slope of the rotation curves of spiral galaxies (in Virgo) is not so clear. Amram, using a different sample of spiral galaxies in clusters, found only 10% of declining rotation curves (2 declining vs 17 flat or rising) in opposition to RWF who find about 40% of declining rotation curves in their sample (6 declining vs 10 flat or rising), we will hereafter briefly discuss the Amram data paper and compare it to the results of RWF. We have measured the rotation curves for a sample of 21 spiral galaxies in 5 nearby clusters. These rotation curves have been constructed from detailed two-dimensional maps of each galaxy's velocity field as traced by emission from the Ha line. This complete mapping, combined with the sensitivity of our CFHT 3.60 m. + Perot-Fabry + CCD observations, allows the construction of high-quality rotation curves. Details concerning the acquisition and reduction procedures of the data are given in Amram. We present and discuss our preliminary analysis and compare them with RWF's results.

Amram, P.↗

NASA NTRS1980

Numerical simulation of supersonic inlets using a three-dimensional viscous flow analysis

A three dimensional fully viscous computer analysis was evaluated to determine its usefulness in the design of supersonic inlets. This procedure takes advantage of physical approximations to limit the high computer time and storage associated with complete Navier-Stokes solutions. Computed results are presented for a Mach 3.0 supersonic inlet with bleed and a Mach 7.4 hypersonic inlet. Good agreement was obtained between theory and data for both inlets. Results of a mesh sensitivity study are also shown.

Anderson, B. H.↗

NASA NTRS2000

Mapping Environmental Contaminants at Ray Mine, AZ

Airborne Visible and InfraRed Imaging Spectrometer (AVIRIS) data was collected over Ray Mine as part of a demonstration project for the Environmental Protection Agency (EPA) through the Advanced Measurement Initiative (AMI). The overall goal of AMI is to accelerate adoption and application of advanced measurement technologies for cost effective environmental monitoring. The site was selected to demonstrate the benefit to EPA in using advanced remote sensing technologies for the detection of environmental contaminants due to the mineral extraction industry. The role of the Jet Propulsion Laboratory in this pilot study is to provide data as well as performing calibration, data analysis, and validation of the AVIRIS results. EPA is also interested in developing protocols that use commercial software to perform such work on other high priority EPA sites. Reflectance retrieval was performed using outputs generated by the MODTRAN radiative transfer model and field spectra collected for the purpose of calibration. We are presenting advanced applications of the ENVI software package using n-Dimensional Partial Unmixing to identify image-derived endmembers that best match target materials reference spectra from multiple spectral libraries. Upon identification of the image endmembers the Mixture Tuned Match Filter algorithm was applied to map the endmembers within each scene. Using this technique it was possible to map four different mineral classes that are associated with mine generated acid waste.

McCubbin, Ian↗

NASA NTRSDate not supplied

M2-SCREAM: A Stratospheric Composition Reanalysis of Aura MLS

Stratospheric composition and transport patterns are changing because of increasing greenhouse gas concentrations and declining levels of ozone depleting substances. Superimposed on forced trends are: dynamical variability at a range of time scales and unforeseen perturbations such as those resulting from injections of volcanic material and smoke from PyroCb events. Chemical reanalyses can be powerful tools for studying the composition of the stratosphere and disentangling forced responses from internal variability. This presentation introduces a new chemical reanalysis of stratospheric constituents developed and produced at NASA’s Global Modeling and Assimilation Office (GMAO) using the recently developed GMAO Constituent Data Assimilation System. Called the MERRA-2 Stratospheric Composition Reanalysis with Aura MLS (M2-SCREAM), this reanalysis consists of assimilated global three-dimensional fields of stratospheric ozone, water vapor, hydrogen chloride (HCl), nitric acid (HNO3) and nitrous oxide (N2O) mixing ratios and covers the period since the beginning of MLS observations in September 2004 to the present. M2-SCREAM assimilates version 4.2 MLS profiles of the five constituents alongside total ozone column from the Ozone Monitoring Instrument. The dynamics and tropospheric water vapor are constrained by assimilated meteorological fields from MERRA-2. The assimilated constituent fields and meteorology are provided to users at a 50-km horizontal resolution and a three-hourly frequency. Monthly analysis uncertainties and flags are also provided to the users. M2-SCREAM is an accurate and dynamically consistent high-resolution data record of the five constituents, all of which are of primary importance to stratospheric chemistry and transport studies. We will present a description of M2-SCREAM and selected results of a process-based evaluation of this product using independent data. As an illustration, we will discuss the perturbation to stratospheric composition from the eruption of Hunga Tonga–Hunga Ha'apai in January 2022 as seen in the reanalysis. We will also propose potential scientific applications of M2-SCREAM and outline plans for an upcoming comprehensive composition reanalysis that is being developed at NASA GMAO.

MERRA-2↗

NASA NTRS1990

An analysis of the vapor flow and the heat conduction through the liquid-wick and pipe wall in a heat pipe with single or multiple heat sources

A numerical analysis is presented for the overall performance of heat pipes with single or multiple heat sources. The analysis includes the heat conduction in the wall and liquid-wick regions as well as the compressibility effect of the vapor inside the heat pipe. The two-dimensional elliptic governing equations in conjunction with the thermodynamic equilibrium relation and appropriate boundary conditions are solved numerically. The solutions are in agreement with existing experimental data for the vapor and wall temperatures at both low and high operating temperatures.

Chen, Ming-Ming↗

NASA NTRS1997

Distortion Representation of Forecast Errors for Model Skill Assessment and Objective Analysis

We proposed a novel characterization of errors for numerical weather predictions. In its simplest form we decompose the error into a part attributable to phase errors and a remainder. The phase error is represented in the same fashion as a velocity field and is required to vary slowly and smoothly with position. A general distortion representation allows for the displacement and amplification or bias correction of forecast anomalies. Characterizing and decomposing forecast error in this way has two important applications, which we term the assessment application and the objective analysis application. For the assessment application, our approach results in new objective measures of forecast skill which are more in line with subjective measures of forecast skill and which are useful in validating models and diagnosing their shortcomings. With regard to the objective analysis application, meteorological analysis schemes balance forecast error and observational error to obtain an optimal analysis. Presently, representations of the error covariance matrix used to measure the forecast error are severely limited. For the objective analysis application our approach will improve analyses by providing a more realistic measure of the forecast error. We expect, a priori, that our approach should greatly improve the utility of remotely sensed data which have relatively high horizontal resolution, but which are indirectly related to the conventional atmospheric variables. In this project, we are initially focusing on the assessment application, restricted to a realistic but univariate 2-dimensional situation. Specifically, we study the forecast errors of the sea level pressure (SLP) and 500 hPa geopotential height fields for forecasts of the short and medium range. Since the forecasts are generated by the GEOS (Goddard Earth Observing System) data assimilation system with and without ERS 1 scatterometer data, these preliminary studies serve several purposes. They (1) provide a testbed for the use of the distortion representation of forecast errors, (2) act as one means of validating the GEOS data assimilation system and (3) help to describe the impact of the ERS 1 scatterometer data.

Hoffman, Ross N.↗

NASA NTRS2006

UWB Tracking System Design with TDOA Algorithm

This presentation discusses an ultra-wideband (UWB) tracking system design effort using a tracking algorithm TDOA (Time Difference of Arrival). UWB technology is exploited to implement the tracking system due to its properties, such as high data rate, fine time resolution, and low power spectral density. A system design using commercially available UWB products is proposed. A two-stage weighted least square method is chosen to solve the TDOA non-linear equations. Matlab simulations in both two-dimensional space and three-dimensional space show that the tracking algorithm can achieve fine tracking resolution with low noise TDOA data. The error analysis reveals various ways to improve the tracking resolution. Lab experiments demonstrate the UWBTDOA tracking capability with fine resolution. This research effort is motivated by a prototype development project Mini-AERCam (Autonomous Extra-vehicular Robotic Camera), a free-flying video camera system under development at NASA Johnson Space Center for aid in surveillance around the International Space Station (ISS).

Ni, Jianjun↗

NASA NTRSDate not supplied

Analysis of Pyrolysis Products from Ablative TPS

NASA’s state-of-the-art ablative materials are composed of a three-dimensional network of carbon fibers impregnated with polymers that dissipate thermal energy through pyrolysis. A fundamental understanding of the decomposition mechanisms and pyrolysis product distributions of various classes of polymers is instrumental in the design of new ablative materials. Furthermore, innovative experiments are essential to the continuous modernization of material response models by providing high-fidelity data. Thus, an apparatus has been designed to measure pyrolysis products from polymers and composite materials by implementing in-situ mass spectrometric techniques. Initial results from experiments performed on siloxane resins and a common phenolic resin will be discussed. Both classes of polymers exhibit heating-rate-dependent decomposition mechanisms. At the onset of heating, phenolic polymers decompose through competitive reactions to form gaseous products and a carbonaceous char. Gas phase products of phenolic resins are typically composed of molecular hydrogen, water, and aromatic hydrocarbons. Pyrolysis products from siloxane polymers include molecular hydrogen, small molecules, and cyclic oligomers from the polymer.

Pyrolysis↗

NASA NTRSDate not supplied

Utilizing 3D-DIC on Mars 2020 Rover Wheel Assembly: Test-Analysis Correlation

Following the successful implementation of full-field photogrammetry, more specifically three-dimensional Digital Image Correlation (3D-DIC), on the Mars 2020 Heat Shield Structural Failure Review assessment, 3D-DIC was selected as one of the primary measurement techniques for the Mars 2020 Rover wheel assembly qualification test at the NASA Jet Propulsion Laboratory (JPL). To validate the Rover wheel landing loads simulations, it was extremely important to have high confidence in the wheel models. Due to the large deformations and strains that the wheel would be subject to during landing, traditional instrumentation such as linear variable displacement transducers (LVDTs), electrical-resistance strain gages and string potentiometers, would not be sufficient on their own to provide all the necessary validation data. Therefore, the NASA Engineering and Safety Center (NESC) provided the 3D-DIC expertise and support to measure the high deformation and strain in the wheel flexures and qualify the overall structural response of the Mars 2020 rover wheel assembly. There were two key objectives for the photogrammetry technique: (1) monitor the wheel response in real-time, guarding against anomalous behavior and failure, and (2) provide test data for test-analysis correlation to validate and/or improve the high-fidelity computational model. The contents of this paper will focus on the challenges of applying 3D-DIC to the Mars 2020 Rover wheel assembly and how these challenges were overcome. Examples of test-analysis correlation during the stiffness characterization and structural qualification will be presented and discussed in detail. Experimental results were compared with the analysis and showed excellent agreement between the predicted behavior and helped validate the high-fidelity models.

Mars 2020 Rover Wheel↗

NASA NTRS1994

LSENS, a general chemical kinetics and sensitivity analysis code for homogeneous gas-phase reactions. 2: Code description and usage

LSENS, the Lewis General Chemical Kinetics Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part 2 of a series of three reference publications that describe LSENS, provide a detailed guide to its usage, and present many example problems. Part 2 describes the code, how to modify it, and its usage, including preparation of the problem data file required to execute LSENS. Code usage is illustrated by several example problems, which further explain preparation of the problem data file and show how to obtain desired accuracy in the computed results. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions. Part 1 (NASA RP-1328) derives the governing equations describes the numerical solution procedures for the types of problems that can be solved by lSENS. Part 3 (NASA RP-1330) explains the kinetics and kinetics-plus-sensitivity-analysis problems supplied with LSENS and presents sample results.

Radhakrishnan, Krishnan↗