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A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples given.
A general technique for solving a system of rate equations describing the interaction of an electromagnetic field and a molecular system is presented. The method is used to obtain approximate time-dependent solutions for the upper-level populations of fluorescent dyes in the presence of a pump field.
Use of the stiffness matrix eigenmodes, instead of the vibration eigenmodes, as generalized coordinates is proposed for condensation of static load deflection equations in finite element stiffness method. The modes are selected by strain energy criteria and the resulting fast, approximate analysis technique is evaluated by applications to idealized built-up wings and a fuselage segment. The best results obtained are a two-order of magnitude reduction of the number of degrees of freedom in a high aspect ratio wing associated with less than one percent error in prediction of the largest displacement.
Previous observations of long-period Cepheids in the Crux-Centaurus-Circinus-Norma (CCCN) region (bounded by galactic longitudes of approximately 294 and 331 deg) are used to delineate spiral features that are consistent with available data for other spiral tracers, especially H I gas. The galactic distribution of the long-period Cepheids in the CCCN region is examined, and a possible interpretation of the spiral structure as viewed in the CCCN region is suggested. Specifically, it is concluded that: (1) the region that is often called the Centaurus link contains no Cepheids with periods longer than 11.25 days and may be part of a spiral feature characterized by older Population I material; (2) long-period Cepheids at large distances (4 kpc) are concentrated near 307 deg galactic longitude and are probably associated with the 'inner arm' identified by Kraft (1965); and (3) the velocities of the observed Cepheids differ significantly from the predictions of the Schmidt (1965) model.
A new diffusion model is developed for radiative transfer in particulate media. It includes the effects of higher Legendre moments while avoiding the mathematical complexities of solving multiple coupled differential moment equations and satisfying higher-order boundary conditions. The method accurately extrapolates the conventional Eddington approximation to problems involving large absorption. Although the simplifying assumptions limit the model to nonbeam incidence and spatial homogeneity of the scattering material, they are nonrestrictive with regard to the size and shape of the medium, the character of background reflections, and the type of phase function.
The study tests the accuracy of the Rayleigh-Gans-Debye (RGD) approximation against a rigorous scattering theory calculation for a simplified model of E. coli (about 1 micron in size) - a solid spheroid. A general procedure is formulated whereby the scattered field amplitude correlation function, for both polarized and depolarized contributions, can be computed for a collection of particles. An explicit formula is presented for the scattered intensity, both polarized and depolarized, for a collection of randomly diffusing or moving particles. Two specific cases for the intermediate scattering functions are considered: diffusing particles and freely moving particles with a Maxwellian speed distribution. The formalism is applied to microorganisms suspended in a liquid medium. Sensitivity studies revealed that for values of the relative index of refraction greater than 1.03, RGD could be in serious error in computing the intensity as well as correlation functions.
An approximate solution to the motion of the Orbiter during rollout is used to perform a parametric study of the rollout distance required by the Orbiter. The study considers the maximum expected dispersions in the landing speed and the touchdown point. These dispersions are assumed to be correlated so that a fast landing occurs before the nominal touchdown point. The maximum rollout distance is required by the maximum landing speed with a 10 knot tailwind and the center of mass at the forward limit of its longitudinal travel. The maximum weight that can be stopped within 15,000 feet on a hot day at Kennedy Space Center is 248,800 pounds. The energy absorbed by the brakes would exceed the limit for reuse of the brakes.
A highly accurate approximation and a rapidly convergent numerical procedure are developed for two dimensional steady supersonic flow over an airfoil. Examples are given for a symmetric airfoil over a range of Mach numbers. Several interesting features are found in the calculation of the tail shock and the flow behind the airfoil.
The paper addresses the problem of how to find the Greedy Triangulation (GT) efficiently in the average case. It is noted that the problem is open whether there exists an efficient approximation algorithm to the Optimum Triangulation. It is first shown how in the worst case, the GT may be obtained in time O(n to the 3) and space O(n). Attention is then given to how the algorithm may be slightly modified to produce a time O(n to the 2), space O(n) solution in the average case. Finally, it is mentioned that Gilbert has found a worst case solution using totally different techniques that require space O(n to the 2) and time O(n to the 2 log n).
It is shown that the inverse analytical solutions, provided separately by Fymat and Box-McKellar, for reconstructing particle size distributions from remote spectral transmission measurements under the anomalous diffraction approximation can be derived using a cosine and a sine transform, respectively. Sufficient conditions of validity of the two formulas are established. Their comparison shows that the former solution is preferable to the latter in that it requires less a priori information (knowledge of the particle number density is not needed) and has wider applicability. For gamma-type distributions, and either a real or a complex refractive index, explicit expressions are provided for retrieving the distribution parameters; such expressions are, interestingly, proportional to the geometric area of the polydispersion.
A method for constructing a direct functional relation between the computational and physical domains such that boundaries in the computational domain map into boundaries in the physical domain is presented. This method which is called the 'two-boundary technique' is based on generating functions that represent two disconnected boundaries in the physical domain and an independent function which connects one boundary with another to generate grids. The suggested connecting functions are simple linear and cubic parametric polynomials with parameters based on position and derivatives at the disconnected boundaries. The boundaries can be defined by analytical, approximate, and tension spline functions; the grid density in specified regions is controlled by embedding contraction functions in either the boundary representation or the connecting function.
A modification of an implicit approximate-factorization finite-difference algorithm applied to the two dimensional Euler and Navier-Stokes equations in general curvilinear coordinates is presented for supersonic free stream flow about and through inlets. The modification transforms the coupled system of equations into an uncoupled diagonal form which requires less computation work. For steady-state applications the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. Solutions are given for inviscid and laminar flow about a two dimensional wedge inlet configuration. Comparisons are made between computed results and exact theory.
A new method for calculating the coefficients of the numerator polynomial of the direct Routh approximation method (DRAM) using the least square error criterion is formulated. The necessary conditions have been obtained in terms of algebraic equations. The method is useful for low frequency as well as high frequency reduced-order models.
The 1(1)S yields 2(3)P electron-impact-excitation cross section for Li II is calculated in the distorted-wave approximation. Two forms of the distorted-wave method are examined; in the first form both the initial and final wave functions are distorted, while in the second form only the initial wave function is distorted. In both forms a partial-wave expansion of the scattered amplitude is made, and exchange is incorporated in a consistent manner. The effects of including more than one configuration in the target-state wave function are also examined. For incident energies greater than 90 eV, distorted-wave calculations agree moderately well with recent experimental and theoretical results.
The Charney problem for baroclinic instability involves the quasi-geostrophic instability of a zonal flow on a beta plane where the zonal flow is characterized by a constant vertical shear. The atmosphere is non-Boussinesq and continuous. The solution of this problem involves confluent hypergeometric functions, and the mathematical difficulty of the problem has precluded extracting simple results of generality. It is shown that there exists very simple, powerful approximate result for the growth rate of the most rapidly growing instability, viz., that this growth rate is linearly proportional to the surface meridional temperature gradient. The coefficient of proportionality is also easily determined. The result extends to substantially more general profiles than those in the Charney problem.
A two-stream accuracy study for internally (thermal) driven problems is presented by comparison with a recently developed 'exact' adding/doubling method. The resulting errors in external (or boundary) radiative intensity and flux are usually larger than those for the externally driven problems and vary substantially with the radiative parameters. Error predictions for a specific problem are difficult. An unexpected result is that the exact method is computationally as fast as the two-stream approximation for nonisothermal media.
An investigation was performed to determine the feasibility of using a two plate model to approximate the Tracking and Data Relay Satellite (TDRS) in orbit propagation, taking into account the effects of solar radiation pressure. The two plate model comprises one plate which always points to the Earth, and the other which is hinged to an axis normal to the orbital plane and is always rotated so that its normal makes a minimum angle with the direction of the sun. The results indicate that it is sufficient to take three parameters, the areas of the two plates and the reflectivity of the Earth pointing plate, to achieve an accuracy of one meter during a 24 hour orbit propagation.
An approximation to a classification algorithm incorporating spatial context information in a general, statistical manner is presented which is computationally less intensive. Classifications that are nearly as accurate are produced.