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Results for “DELTA FUNCTION”

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 19 records
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NASA NTRS1983

Delta function excitation of waves in the earth's ionosphere

Excitation of the earth's ionosphere by delta function current sheets is considered, and the temporal and spatial evolution of wave packets is analyzed for a two-component collisional F2 layer. Approximations of an inverse Fourier-Laplace transform via saddle point methods provide plots of typical wave packets. These illustrate cold plasma wave theory and may be used as a diagnostic tool since it is possible to relate specific features, e.g., the frequency of a modulation envelope, to plasma parameters such as the electron cyclotron frequency. It is also possible to deduce the propagation path length and orientation of a remote radio beacon.

Vidmar, R. J.↗
NASA NTRS1981

A new global operator for two-particle delta functions

A new type of global operator to be used in evaluating matrix elements of two-particle delta functions is introduced. It is based, like the Trivedi one-particle operator, on the Poisson equation and is easier to apply than the method of Hiller, Sucher and Feinberg. After a test in the helium isoelectronic sequence, the new method is applied successfully to the interesting problem of hyperfine structure in muonic helium.

Drachman, R. J.↗
NASA NTRS1979

Global operators for delta functions - Extension to scattering states and the inclusion of spin

The paper presents an extension of the identities of Hiller, Sucher, and Feinberg for the evaluation of bound-state matrix elements of delta function operators, to scattering states. Attention is given to the technical difficulties encountered in the application of the scattering-state identities and how they can be overcome, including infrared singularities which arise if Coulomb forces are present and the velocity of the incident particle is small. In addition, both the scattering and bound-state identities are generalized to include spin-dependent contact interactions and spin-dependent Hamiltonians. Finally, it is stressed that the identities can be used to improve the accuracy of calculations of interest for diverse physical phenomena, ranging from positron annihilation to hyperfine structure.

Sucher, J.↗
NASA NTRS2014

On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

Huynh, H. T.↗
NASA NTRS1986

Airfoil gust response and the sound produced by airifoil-vortex interaction

This paper contributes to the understanding of the noise generation process of an airfoil encountering an unsteady upwash. By using a fast Fourier transform together with accurate airfoil response functions, the lift-time waveform for an airfoil encountering a delta function gust (the indicial function) is calculated for a flat plate airfoil in a compressible flow. This shows the interesting property that the lift is constant until the generated acoustic wave reaches the trailing edge. Expressions are given for the magnitude of this constant and for the pressure distribution on the airfoil during this time interval. The case of an airfoil cutting through a line vortex is also analyzed. The pressure-time waveform in the far field is closely related to the left-time waveform for the above problem of an airfoil entering a delta function gust. The effects of varying the relevant parameters in the problem are studied, including the observed position, the core diameter of the vortex, the vortex orientation and the airfoil span. The far field sound varies significantly with observer position, illustrating the importance of non-compactness effects. Increasing the viscous core diameter tends to smooth the pressure-time waveform. For small viscous core radius and infinite span, changing the vortex orientation changes only the amplitude of the pressure-time waveform, and not the shape.

Amiet, R. K.↗
NASA NTRS1982

The delta-four-stream approximation for radiative flux transfer

An approximate technique for solving the radiative transfer problem for fluxes reflected, transmitted, and absorbed by a homogeneous scattering layer is presented. The technique is a straightforward delta-function modification of a solution to the transfer equation in closed form via the discrete-ordinates method with four streams. By the use of four streams, significant improvement in accuracy is obtained for anisotropic phase functions over that obtained by similarly employed two-stream approximations with delta-functions. However, the computational effort is increased only slightly. Equations are developed for the general case of a layer underlaid by a reflecting Lambert surface. Graphical comparisons are given of fractional error resulting from the use of this method with that resulting from the use of typical four-stream and delta-two-stream techniques.

Cuzzi, J. N.↗
NASA NTRS1974

Mirroring within the Fokker-Planck formulation of cosmic ray pitch angle scattering in homogeneous magnetic turbulence

The Fokker-Planck coefficient for pitch angle scattering, appropriate for cosmic rays in homogeneous, stationary, magnetic turbulence, is computed from first principles. No assumptions are made concerning any special statistical symmetries the random field may have. This result can be used to compute the parallel diffusion coefficient for high energy cosmic rays moving in strong turbulence, or low energy cosmic rays moving in weak turbulence. Becuase of the generality of the magnetic turbulence which is allowed in this calculation, special interplanetary magnetic field features such as discontinuities, or particular wave modes, can be included rigorously. The reduction of this results to previously available expressions for the pitch angle scattering coefficient in random field models with special symmetries is discussed. The general existance of a Dirac delta function in the pitch angle scattering coefficient is demonstrated. It is proved that this delta function is the Fokker-Planck prediction for pitch angle scattering due to mirroring in the magnetic field.

Goldstein, M. L.↗
NASA NTRS1960

The Interactions Between Nitrogen and Oxygen Molecules

Lippincott's delta-function model for atomic interactions is analyzed, both physically and mathematically, and extended, by differentiation between K- and L-shell electrons and the introduction of a variable parameter in the expression for the delta-function strength, to cover homonuclear molecules more complex than hydrogen. In addition, modifications are made which allow treatments of diatomic, heteronuclear molecules. This theory, in conjunction with a reasonably extensive study of resonance, dispersion, and configuration interaction phenomena, as well as the use of simple quantum mechanical arguments, is then applied to the N2-N2, N2-O2, and O2-O2 interactions.

Meador, Willard E., Jr.↗
NASA NTRS1994

New description of charged particle propagation in random magnetic fields

When charged particles spiral along a large constant magnetic field, their trajectories are scattered by random components that are superposed on the guiding field. In the simplest analysis of this situation, scattering causes the particles to diffuse parallel to the guiding field. At the next level of approximation, moving pulses that correspond to a coherent mode of propagation are present, but they are represented by delta-functions whose infinitely narrow width makes no sense physically and is inconsistent with the finite duration of coherent pulses observed in solar energetic particle events. To derive a more realistic description, the transport problem is formulated in terms of 4 x 4 matrices, which derive from a representation of the particle distribution function in terms of eigenfunctions of the scattering operator, and which lead to useful approximations that give explicit predictions of the detailed evolution not only of the coherent pulses, but also of the diffusive wake. More specifically, the new description embodies a simple convolution of a narrow Gaussian with the solutions above that involve delta-functions, but with a slightly reduced coherent velocity. The validity of these approximations, which can easily be calculated on a desktop computer, has been exhaustively confirmed by comparison with results of Monte Carlo simulations which kept track of 50 million particles and which were carried out on the Maspar computer at Goddard Space Flight Center.

Earl, James A.↗
NASA NTRS1995

Thermophotovoltaic Generators Using Selective Metallic Emitters

In the literature to date on thermophotovoltaic (TPV) generators, two types of infrared emitter's have been emphasized : gray body emitters and rare earth oxide selective emitters. The gray body emitter is defined as an emitter with a spectral emissivity independent of wavelength whereas the rare earth oxide selective emitter is idealized as a delta function emitter with a high emissivity at a select wavelength and a near zero emissivity at all other wavelengths. Silicon carbide is an example of a gray body emitter and ER-YAG is an example of a selective emitter. The Welsbach mantle in a common lantern is another example of an oxide selective emitter. Herein, we describe an alternative type of selective emitter, a selective metallic emitter. These metallic emitters are characterized by a spectral emissivity curve wherein the emissivity monotonically increases with shorter infrared wavelengths as is shown. The metal of curve "A", tungsten, typifies this class of selective metallic emitter's. In a thermophotovoltaic generator, a photovoltaic cell typically converts infrared radiation to electricity out to some cut-off wavelength. For example, Gallium Antimonide (GaSb) TPV cells respond out to 1.7 microns. The problem with gray body emitters is that they emit at all wavelengths. Therefore, a large fraction of the energy emitted will be outside of the response band of the TPV cell. The argument for the selective emitter is that, ideally, all the emitted energy can be in the cells response band. Unfortunately, rare earth oxide emitters are not ideal. In order to suppress the emissivity toward zero away from the select wavelength, the use of thin fiber's is necessary. This leads to a fragile emitter typical of a lantern mantle. Even given a thin ER-YAG emitter, the measured emissivity at the select wavelength of 1.5 microns has been reported to be 0.6 while the off wavelength background emissivity falls to only 0.2 at 5 microns. This gives a selectivity ratio of only 3. Another problem with a delta function selective emitter is its low power density at practical temperatures because of its narrow emission bandwidth. The concept of selectivity can be generalized by noting that we simply wish to maximize the ratio of in-cell-band power to out-of-cell-band power. Using this generalized selectivity concept and assuming a GaSb cell covered by a simple dielectric filter, we note that the emissivity selectivity ratio for tungsten is 0.3 (at 1.5 microns) / 0.07 (at 5 microns) = 4.3. In the folloy4ng sections, we note that the selective metallic emitters can be valuable in both radioisotope TPV generators in space and in hydrocarbon fired TPV generators here on earth.

Fraas, Lewis M.↗
NASA NTRS1980

Orthogonality of spherical harmonic coefficients

Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.

Mcleod, M. G.↗
NASA NTRS1985

Molecular processes in a high temperature shock layer

The development of techniques for the calculation of electron capture widths, electronic wave functions, cross sections and rates needed for the description of the dissociative recombination (DR) of molecular ions with electrons were described. The cross sections and rates were calculated by using harmonic oscillator wave functions for the ion and a delta function approximation for the continuum vibrational wave function in the repulsive dissociative channel. In order to obtain DR cross sections of quantitative accuracy, a computer program which solves the one dimensional nuclear motion wave equation was revised to calculate the cross sections and rates. The program and the new results are described. Included is a discussion of large windows found in the dissociative recombination cross sections from excited ion vibrational levels. These windows have not been previously reported in the literature. The magnitude of the DR cross sections for several dissociative routes are sensitive to the location of the crossing of the neutral and ion potential curves. Studies of the effects of basis set and CI wave function size on vertical excitation energies are described. Preliminary studies on N2 and O2 using large scale wave functions are also reported.

Guberman, S. L.↗