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At least 235 records · Page 13

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NASA NTRS1974

Evaluation of approximate relations for Delta /Q/ using a numerical solution of the Boltzmann equation

Data obtained from a numerical solution of the Boltzmann equation for shock-wave structure are used to test the accuracy of accepted approximate expressions for the two moments of the collision integral Delta (Q) for general intermolecular potentials in systems with a large translational nonequilibrium. The accuracy of the numerical scheme is established by comparison of the numerical results with exact expressions in the case of Maxwell molecules. They are then used in the case of hard-sphere molecules, which are the furthest-removed inverse power potential from the Maxwell molecule; and the accuracy of the approximate expressions in this domain is gauged. A number of approximate solutions are judged in this manner, and the general advantages of the numerical approach in itself are considered.

Nathenson, M.↗

NASA NTRS1975

Spline function approximation techniques for image geometric distortion representation

Least squares approximation techniques were developed for use in computer aided correction of spatial image distortions for registration of multitemporal remote sensor imagery. Polynomials were first used to define image distortion over the entire two dimensional image space. Spline functions were then investigated to determine if the combination of lower order polynomials could approximate a higher order distortion with less computational difficulty. Algorithms for generating approximating functions were developed and applied to the description of image distortion in aircraft multispectral scanner imagery. Other applications of the techniques were suggested for earth resources data processing areas other than geometric distortion representation.

Anuta, P. E.↗

NASA NTRS1974

Inelastic scattering in atom-diatomic molecule collisions. I - Rotational transitions in the sudden approximation

The impact parameter method and the sudden approximation are applied to determine the total probability of inelastic rotational transitions arising from a collision of an atom and a homonuclear diatomic molecule at large impact parameters. An analytical approximation to this probability is found for conditions where the electron exchange or overlap forces dominate the scattering. An approximate upper bound to the range of impact parameters for which rotational scattering can be important is determined. In addition, an estimate of the total inelastic cross section is found at conditions for which a statistical model describes the scattering well. The results of this analysis are applied to Ar-O2 collisions and may be readily applied to other combinations of atoms and molecules.

Stallcop, J. R.↗

NASA NTRS1975

Strong scintillations in astrophysics. I - The Markov approximation, its validity and application to angular broadening

The Markov approximation to the propagation of waves in an extended, irregular medium is discussed in an astrophysical context. A new derivation is presented which is simple and which shows that the assumption of Gaussian statistics used by previous authors is irrelevant. We discuss the relevance of the approximation and show that it may apply in many situations of interest, including interstellar scintillations of pulsar signals. The approximation does not require the assumption of weak scattering or Gaussian correlation functions. The Markov equation for the angular spectrum is particularly simple, and solutions are discussed for typical turbulence spectra. It is found that the equation for the angular spectrum is very nearly that used by previous authors, and the present discussion shows that these results are much more general than previously thought. A possible observational test for distinguishing between Gaussian and power-law interstellar density spectra is discussed.

Lee, L. C.↗

NASA NTRS1975

Approximations for the study of drift boundaries in the magnetosphere

An approximate relation between energy and equatorial radial distance for a particle of given equatorial pitch angle moving adiabatically in a dipole magnetic field is used to describe the Alfven layer for particles of arbitrary pitch angle. The convection electric field is assumed to be uniform in the equatorial plane of the magnetosphere, and magnetic field lines are taken to be equipotentials in the region traversed by the magnetospheric particles being studied. Approximate solutions in the high- and low-energy limits are given in a form directly applicable to the interpretation of measurements at fixed particle energy and pitch angle. These results are particularly useful for electrons, since the approximations provide lower bounds to the exact solutions. The results are used to interpret recent Explorer 45 electron measurements of Williams et al. (1974).

Kivelson, M. G.↗

NASA NTRS1975

Comparison of techniques for approximating ocean bottom topography in a wave-refraction computer model

A study of the effects of using different methods for approximating bottom topography in a wave-refraction computer model was conducted. Approximation techniques involving quadratic least squares, cubic least squares, and constrained bicubic polynomial interpolation were compared for computed wave patterns and parameters in the region of Saco Bay, Maine. Although substantial local differences can be attributed to use of the different approximation techniques, results indicated that overall computed wave patterns and parameter distributions were quite similar.

Poole, L. R.↗

NASA NTRS1975

Attenuation of sound in ducts with acoustic treatment: A generalized approximate equation

A generalized approximate equation for duct lining sound attenuation is presented. The specification of two parameters, the maximum possible attenuation and the optimum wall acoustic impedance is shown to completely determine the sound attenuation for any acoustic mode at any selected wall impedance. The equation is based on the nearly circular shape of the constant attenuation contours in the wall acoustic impedance plane. For impedances far from the optimum, the equation reduces to Morse's approximate expression. The equation can be used for initial acoustic liner design. Not least important is the illustrative nature of the solutions which provide an understanding of the duct propagation problem usually obscured in the exact calculations. Sample calculations using the approximate attenuation equation show that the peak and the bandwidth of the sound attenuation spectrum can be represented by quite simple functions of the ratio of actual wall acoustic resistance to optimum resistance.

Rice, E. J.↗

NASA NTRS1976

The Investigation of Optimal Discrete Approximations for Real Time Flight Simulations

The results are presented of an investigation of discrete approximations for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of approximation of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade approximation to the solution of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.

Parrish, E. A.↗

NASA NTRS1975

Validity of approximate methods in molecular scattering - Thermal HCl-He collisions

Accurate close coupling scattering calculations are presented for thermal energy HCl-He collisions. The interaction potential is obtained from the Gordon-Kim electron gas model, adjusted to have the correct long-range multipole form. A variety of phenomenological cross sections are computed from the close coupling S matrix, and these are compared with results from several commonly employed approximate methods. In particular, it is found that the total integral, total differential, and gas kinetic cross sections are accurately predicted by the central field approximation which retains just the spherical average of the interaction. Integral inelastic cross sections are represented quite accurately by the coupled states approximation of McGuire and Kouri, but only qualitatively by the effective potential method of Rabitz.

Green, S.↗

NASA NTRS1976

Single-electron Born approximations for charge transfer from multielectron atoms to protons

The Born approximation, including the internuclear interaction, is used to compute cross sections for the transfer of one electron from a multielectron atom to an incident proton. When the full internuclear interaction is included, the results lie far above high-energy experimental K-shell data for protons and argon. However, when only enough internuclear interaction is included so that the total projectile-target interaction goes to zero asymptotically in accordance with plane-wave functions actually used, fair agreement at high energies is obtained. The latter form of the Born approximation is compared with data on K-shell capture from helium as well as argon and with the simpler approximation of Oppenheimer and of Brinkman and Kramers (OBK), where no internuclear interaction is included. The OBK results typically lie a factor of 2 to 8 above the data, while the present Born results are within a factor of 2 of most of the existing high-velocity K-shell data.

Omidvar, K.↗

NASA NTRS1976

On the use of Pade approximants to represent unsteady aerodynamic loads for arbitrarily small motions of wings

The general behavior of unsteady airloads in the frequency domain is explained. Based on this, a systematic procedure is described whereby the airloads, produced by completely arbitrary, small, time-dependent motions of a thin lifting surface in an airstream, can be predicted. This scheme employs as raw materials any of the unsteady linearized theories that have been mechanized for simple harmonic oscillations. Each desired aerodynamic transfer function is approximated by means of an appropriate Pade approximant, that is, a rational function of finite degree polynomials in the Laplace transform variable. Although these approximations have many uses, they are proving especially valuable in the design of automatic control systems intended to modify aeroelastic behavior.

Vepa, R.↗

NASA NTRS1975

Attenuation of sound in ducts with acoustic treatment - A generalized approximate equation

Rice, E. J.↗

NASA NTRS1977

An approximate method for design and analysis of an ALOHA system

An approximate method for the design and performance prediction of a multiaccess communication system which employs the ALOHA packet-switching technique is developed, based on the use of a diffusion process approximation of an ALOHA-like system (with or without time-slotting). A simple closed-form solution for the variable Q(t), a variant of the number of backlog messages at time t, is given in terms of a few system and user parameters. Final results are expressed in terms of ordinary performance measures such as throughput and average delay. Several numerical examples are given to demonstrate the usefulness of the approximation technique developed.

Kobayashi, H.↗

NASA NTRS1977

Total Born-approximation cross sections for single-electron loss by atoms and ions colliding with atoms

The first Born approximation (FBA) is applied to the calculation of single-electron-loss cross sections for various ions and atoms containing from one to seven electrons. Screened hydrogenic wave functions are used for the states of the electron ejected from the projectile, and Hartree-Fock elastic and incoherent scattering factors are used to describe the target. The effect of the target atom on the scaling of projectile ionization cross sections with respect to the projectile nuclear charge is explored in the case of hydrogenlike ions. Also examined is the scaling of the cross section with respect to the target nuclear charge for electron loss by Fe(25+) in collision with neutral atoms ranging from H to Fe. These results are compared with those of the binary-encounter approximation (BEA) and with the FBA for the case of ionization by completely stripped target ions. Electron-loss cross sections are also calculated for the ions O(i+) (i = 3-7) and N(i+) (i = 0-6) in collision with He targets in the energy range of approximately 0.1 to 100 MeV/nucleon. These results are found to be in excellent agreement with the available data near the peak of the ionization cross section.

Rule, D. W.↗

NASA NTRS1977

Multimodal far-field acoustic radiation pattern - An approximate equation

The far-field sound radiation theory for a circular duct was studied for both single mode and multimodal inputs. The investigation was intended to develop a method to determine the acoustic power produced by turbofans as a function of mode cut-off ratio. This information is essential for the design of acoustic suppressors in engine ducts. With reasonable simplifying assumptions the single mode radiation pattern was shown to be reducible to a function of mode cut-off ratio only (modal indices removed). With modal cut-off ratio as the dominant variable, multimodal radiation patterns can be reduced to a simple explicit expression. This approximate expression provides excellent agreement with an exact calculation of the sound radiation pattern using equal acoustic power per mode. Radiation patterns for cases other than equal modal power are presented using the approximate radiation equation. An approximate expression for the duct termination losses as a function of cut-off ratio is also included.

Rice, E. J.↗

NASA NTRS1977

The parabolic approximation for sound propagation in a stratified moving medium

Propagation of sound in a stratified moving medium is discussed through an extension of the parabolic approximation to the acoustic equations of motion for short wavelengths. The parabolic approximation is related to the theory of geometric acoustics, and it is shown that it yields an improvement in accuracy over geometric theory. Also, the approximation corrects cumulative failures of geometric theory which occur when sound propagates many wavelengths from its source. The theory is illustrated by application to simple examples of quasi-plane wave propagation.

Myers, M. K.↗

NASA NTRS1978

Approximate dynamic model of a turbojet engine

An approximate dynamic nonlinear model of a turbojet engine is elaborated on as a tool in studying the aircraft control loop, with the turbojet engine treated as an actuating component. Approximate relationships linking the basic engine parameters and shaft speed are derived to simplify the problem, and to aid in constructing an approximate nonlinear dynamic model of turbojet engine performance useful for predicting aircraft motion.

Artemov, O. A.↗

NASA NTRS1978

Derivation and evaluation of an approximate analysis for three-dimensional viscous subsonic flow with large secondary velocities

An approximate analysis is presented for calculating three-dimensional, low Mach number, laminar viscous flows in curved passages with large secondary flows and corner boundary layers. The analysis is based on the decomposition of the overall velocity field into inviscid and viscous components with the overall velocity being determined from superposition. An incompressible vorticity transport equation is used to estimate inviscid secondary flow velocities to be used as corrections to the potential flow velocity field. A parabolized streamwise momentum equation coupled to an adiabatic energy equation and global continuity equation is used to obtain an approximate viscous correction to the pressure and longitudinal velocity fields. A collateral flow assumption is invoked to estimate the viscous correction to the transverse velocity fields. The approximate analysis is solved numerically using an implicit ADI solution for the viscous pressure and velocity fields. An iterative ADI procedure is used to solve for the inviscid secondary vorticity and velocity fields. This method was applied to computing the flow within a turbine vane passage with inlet flow conditions of M = 0.1 and M = 0.25, Re = 1000 and adiabatic walls, and for a constant radius curved rectangular duct with R/D = 12 and 14 and with inlet flow conditions of M = 0.1, Re = 1000, and adiabatic walls.

Anderson, O. L.↗