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Results for “Separation of variables”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1977

Response of winter and spring wheat grain yields to meteorological variation

Mathematical models which quantify the relation of wheat yield to selected weather-related variables are presented. Other sources of variation (amount of applied nitrogen, improved varieties, cultural practices) have been incorporated in the models to explain yield variation both singly and in combination with weather-related variables. Separate models were developed for fall-planted (winter) and spring-planted (spring) wheats. Meteorological variation is observed, basically, by daily measurements of minimum and maximum temperatures, precipitation, and tabled values of solar radiation at the edge of the atmosphere and daylength. Two different soil moisture budgets are suggested to compute simulated values of evapotranspiration; one uses the above-mentioned inputs, the other uses the measured temperatures and precipitation but replaces the tabled values (solar radiation and daylength) by measured solar radiation and satellite-derived multispectral scanner data to estimate leaf area index. Weather-related variables are defined by phenological stages, rather than calendar periods, to make the models more universally applicable.

Feyerherm, A. M.↗
DOE OSTI2024

Perturbations of Gibbons-Maeda black holes in Einstein-Maxwell-dilaton theories

The study of perturbations around black hole backgrounds in general relativity and Einstein-Maxwell theory has a long history, going back to the work of Regge and Wheeler in the 1950s. As part of a broader investigation of perturbations around black holes in supergravity, we describe here our results for the perturbations around the Gibbons-Maeda static charged black holes in a class of Einstein-Maxwell-Dilaton theories. Our analysis follows the general strategy developed by Chandrasekhar and Xanthopoulos for the perturbations of the Reissner-Nordström black hole. Here, the analysis is considerably more involved, because of the presence of the dilaton field, which couples to the other polar modes. We nonetheless find that the problem is completely solvable, in the sense that one can separate variables and eventually describe all the perturbations in terms of diagonalised second-order radial equations. We are able to prove the mode stability of all the Gibbons-Maeda black hole solutions. Published by the American Physical Society 2024

Pope, C. N. (ORCID:0000000284012564)↗
NASA NTRS1972

On propagation of long waves in curved ducts.

A two-dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.

Rostafinski, W.↗
NASA NTRS1973

Computational methods for studying acoustic propagation in nonuniform waveguides.

A variational technique is used to model sound transmission through a nonuniform duct segment consisting of an axial variation in wall admittance or cross sectional area. The method involves the Ritz minimization of functionals which have the governing equations as stationary conditions. The variational method is verified by application to segments of variables-separable geometry for which eigenfunction expansion techniques offer an alternative solution procedure, and by comparison with the results of stepped duct approximations to the nonuniformity. Quantitative data are presented which indicate the boundary condition matching to be a suitable measure of the accuracy of the transmitted field.

Beckemeyer, R. J.↗
NASA NTRS1978

Solution of the subsynchronous whirl problem in the high-pressure hydrogen turbomachinery of the Space Shuttle Main Engine

Subsynchronous whirl of the high-pressure fuel turbopump limited operation of the Space Shuttle Main Engine for some months in early 1976. The means by which this problem was successfully attacked is of particular interest to the rotor-dynamics community, not only because this machinery was the highest power-to-weight ratio known (77,000 hp...760 pounds), but because of the multiple forcing functions involved and the means, both analytical and experimental, which were utilized in separating variables, pointing toward successful solutions, and evaluating results. The general means of identifying fundamental characteristics, analyzing data, and conducting computer investigations are delineated. The results of analysis and testing are discussed. Since whirl inception occurs at a shaft speed greater than twice the first system critical, this was increased by stiffening the shaft and bearing supports; the system damping and system stiffness was additionally increased by proper interstage seal design to the extent that the instability threshold is now beyond the operating range.

Ek, M. C.↗
NASA NTRS1982

ISEE work on collisionless shocks - CDAW-3, the meeting and the results July 23-26, 1979

At CDAW(Coordinated Data Analysis Workshop)-3, data from ISEE-1 and -2 were used, and 17 data sets were incorporated into a total of 44 megabytes of disk storage. The presence of ISEE-1 and -2 at variable separation distances has made it possible to settle the question of shock motion and to obtain shock thicknesses. A full range of wave instrumentation was available to study upstream waves and turbulence, and energetic particle investigations made it possible to obtain good observations of shock-induced particle acceleration. It is concluded that these improved observations have resulted in a revival of interest in the study of the collisionless shock, both as an experimental and theoretical research problem. It is expected that these investigations will lead to a better understanding of how to use the earth's bow shock to provide well-defined and theoretically useful examples for study.

Ogilvie, K. W.↗
NASA NTRS1983

Theory of hydromagnetic turbulence

The present state of MHD turbulence theory as a possible solar wind research tool is surveyed. The theory is statistical, and does not make statements about individual events. The ensembles considered typically have individual realizations which differ qualitatively, unlike equilibrium statistical mechanics. Most of the theory deals with highly symmetric situations; most of these symmetries have yet to be tested in the solar wind. The applicability of MHD itself to solar wind parameters is highly questionable; yet it has no competitors, as a potentially comprehensive dynamical description. The purpose of solar wind research require sharper articulation. If they are to understand radial turbulent plasma flows from spheres, laboratory experiments and numerical solution of equations of motion may be cheap alternative to spacecraft. If "real life" information is demanded, multiple spacecraft with variable separation may be necessary to go further. The principal emphasis in the theory so far has been on spectral behavior for spatial covariances in wave number space. There is no respectable theory of these for highly anisotropic situations. A rather slow development of theory acts as a brake on justifiable measurement, at this point.

Montgomery, D.↗
NASA NTRS1983

On the interpretation of kernels - Computer simulation of responses to impulse pairs

A method is presented for the use of a unit impulse response and responses to impulse pairs of variable separation in the calculation of the second-degree kernels of a quadratic system. A quadratic system may be built from simple linear terms of known dynamics and a multiplier. Computer simulation results on quadratic systems with building elements of various time constants indicate reasonably that the larger time constant term before multiplication dominates in the envelope of the off-diagonal kernel curves as these move perpendicular to and away from the main diagonal. The smaller time constant term before multiplication combines with the effect of the time constant after multiplication to dominate in the kernel curves in the direction of the second-degree impulse response, i.e., parallel to the main diagonal. Such types of insight may be helpful in recognizing essential aspects of (second-degree) kernels; they may be used in simplifying the model structure and, perhaps, add to the physical/physiological understanding of the underlying processes.

Hung, G.↗
NASA NTRS1983

Difference equation technique for determining the evolution of radially symmetric ultrasonic fields

A simplified difference equation technique has been developed to calculate the three-dimensional evolution of radially symmetric, complex ultrasonic fields. The parabolic approximation to the Helmholtz equation is obtained by assuming that localized changes in the magnitude of the field with respect to distance along the axis of propagation are small compared to those with respect to radial displacement. The Helmholtz equation in cylindrical coordinates may then be approximated, after separating variables, using truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane using corresponding adjacent locations in the preceding plane. Stability criteria which determine acceptable values for various parameters are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two-dimensional focusing and non-focusing, uniform and Gaussian profile, piezoelectric transducers in both profiled and non-profiled media.

Dockery, G. D.↗
NASA NTRS1995

Magnetohydrodynamic turbulence in the solar wind

The fluctuations in magnetic field and plasma velocity in solar wind, which possess many features of fully developed magnetohydrodynamic (MHD) turbulence, are discussed. Direct spacecraft observations from 0.3 to over 20 AU, remote sensing radio scintillation observations, numerical simulations, and various models provide complementary methods that show that the fluctuations in the wind parameters undergo significant dynamical evolution independent of whatever turbulence might exist in the solar photosphere and corona. The Cluster mission, with high time resolution particle and field measurements and its variable separation strategies, should be able to provide data for answering many questions on MHD turbulence.

Goldstein, Melvyn L.↗
NASA NTRS2004

Cloud Coverage and Height Distribution from the GLAS Polar Orbiting Lidar: Comparison to Passive Cloud Retrievals

The Geoscience Laser Altimeter System (GLAS) began full on orbit operations in September 2003. A main application of the two-wavelength GLAS lidar is highly accurate detection and profiling of global cloud cover. Initial analysis indicates that cloud and aerosol layers are consistently detected on a global basis to cross-sections down to 10(exp -6) per meter. Images of the lidar data dramatically and accurately show the vertical structure of cloud and aerosol to the limit of signal attenuation. The GLAS lidar has made the most accurate measurement of global cloud coverage and height to date. In addition to the calibrated lidar signal, GLAS data products include multi level boundaries and optical depth of all transmissive layers. Processing includes a multi-variable separation of cloud and aerosol layers. An initial application of the data results is to compare monthly cloud means from several months of GLAS observations in 2003 to existing cloud climatologies from other satellite measurement. In some cases direct comparison to passive cloud retrievals is possible. A limitation of the lidar measurements is nadir only sampling. However monthly means exhibit reasonably good global statistics and coverage results, at other than polar regions, compare well with other measurements but show significant differences in height distribution. For polar regions where passive cloud retrievals are problematic and where orbit track density is greatest, the GLAS results are particularly an advance in cloud cover information. Direct comparison to MODIS retrievals show a better than 90% agreement in cloud detection for daytime, but less than 60% at night. Height retrievals are in much less agreement. GLAS is a part of the NASA EOS project and data products are thus openly available to the science community (see http://glo.gsfc.nasa.gov).

Spinhime, J. D.↗
NASA NTRS2006

On Space-Time Inversion Invariance and its Relation to Non-Dissipatedness of a CESE Core Scheme

The core motivating ideas of the space-time CESE method are clearly presented and critically analyzed. It is explained why these ideas result in all the simplifying and enabling features of the CESE method. A thorough discussion of the a scheme, a two-level non-dissipative CESE solver of a simple advection equation with two independent mesh variables and two equations per mesh point is also presented. It is shown that the scheme possesses some rather intriguing properties such as: (i) its two independent mesh variables separately satisfy two decoupled three-level leapfrog schemes and (ii) it shares with the leapfrog scheme the same amplification factors, even though the a scheme and the leapfrog scheme have completely different origins and structures. It is also explained why the leapfrog scheme is not as robust as the a scheme. The amplification factors/matrices of several non-dissipative schemes are carefully studied and the key properties that contribute to their non-dissipatedness are clearly spelled out. Finally we define and establish space-time inversion (STI) invariance for several non-dissipative schemes and show that their non-dissipatedness is a result of their STI invariance.

Chang, Sin-Chung↗
NASA NTRS2011

Properties of a Variable-Delay Polarization Modulator

We investigate the polarization modulation properties of a variable-delay polarization modulator (VPM). The VPM modulates polarization via a variable separation between a polarizing grid and a parallel mirror. We find that in the limit where the wavelength is much larger than the diameter of the metal wires that comprise the grid, the phase delay derived from the geometric separation between the mirror and the grid is sufficient to characterize the device. However, outside of this range, additional parameters describing the polarizing grid geometry must be included to fully characterize the modulator response. In this paper, we report test results of a VPM at wavelengths of 350 micron and 3 mm. Electromagnetic simulations of wire grid polarizers were performed and are summarized using a simple circuit model that incorporates the loss and polarization properties of the device.

Chuss, David T.↗
NASA NTRS2014

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

acceleration of particles↗
NASA NTRS2018

The Advantages of Synergy-Quantitative Earth Science Data Visualization and Analysis with Giovanni, Panoply, and Excel

The NASA Giovanni data analysis system provides a multitude of basic analysis capabilities for numerous Earth science data products which are available in the NASA Goddard Earth Sciences Data and Information Services Center (GES DISC) archive, as well as for additional selected data products provided by other NASA Distributed Active Archive Center (DAAC) archives. In Giovanni, users can easily generate time-averaged data maps, area-averaged time-series, Latitude-Time and Longitude-Time Hovmoeller diagrams, correlation maps, accumulation maps, and map animations (22 analysis options are available in total). While ASCII text output is available for time-series plots, it is not included as an option for data maps. In order to provide a quantitative, easy-to-use numerical output in ASCII text form, the NetCDF file output from a Giovanni visualization is downloaded and then opened with the free NASA visualization software package Panoply. Panoply provides the capability of translating the Giovanni file into comma-separated-variable (CSV) output. Panoply also provides additional visualization options, including the facile calculation of difference maps and quasi-anomaly maps using Giovanni output files. The CSV files from Panoply can then be imported into an Excel spreadsheet, where an Excel macro converts the CSV files. The output consists of latitude-longitude-data value triads in text form for maps, and either longitude-time-data value or latitude-time-data value triads in text form for Hovmoeller diagrams. This presentation will explicate the basic procedure for the conversion, and then provide several examples where the procedure is applied to Giovanni output from different analysis options.

Earth science↗
NASA NTRS1972

Diffraction of a plane pulse by a three-dimensional corner.

The conical solutions for the incidence of a plane pulse on a three-dimensional corner are presented. The corner is represented by a trihedron with one edge perpendicular to the other two. Both the boundary condition of the first kind and that of the second kind are considered. Outside the characteristic sphere of the vertex of the corner, the solution is represented by the well-known conical solutions in two variables. Inside the characteristic sphere, the problem involves three conical variables. By the separation of variables, the problem is reduced to that of an eigenvalue problem with an irregular boundary which is in turn reduced to a system of homogeneous algebraic equations. The eigenvalues are then determined numerically. By the superposition of the conical solutions for plane pulses, the solution for the incidence of a plane wave is obtained. Numerical examples simulating the incidence of a sonic boom on the corner of a structure are presented.

Ting, L.↗