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NASA NTRS1978

Third-order solution of an artificial-satellite theory

A third-order solution is developed for the motions of artificial satellites moving in the gravitational field of the earth, whose potential includes the second-, third-, and fourth-order zonal harmonics. Third-order periodic perturbations with fourth-order secular perturbations are derived by Hori's perturbations method. All quantities are expanded into power series of the eccentricity, but the solution is obtained so as to be closed with respect to the inclination. A comparison with the results of numerical integration of the equations of motion indicates that the solution can predict the position of a close-earth satellite with a small eccentricity with an accuracy of better than 1 cm over 1 month.

Kinoshita, H.↗

NASA NTRS1978

Time transformations and Cowell's method

The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variable s defined by an equation of the form dt = cr to the n-th power ds, where t represents time, r the radial distance from the center of attraction, c is a constant, and n is a parameter. This has been primarily motivated by the 'uniformizing' effects of such a transformation resulting in desirable 'analytic' stepsize control for elliptical orbits. This report discusses the 'proper' choice of the parameter n defining the independent variable s for various types of orbits and perturbation models, and develops a criterion for its selection.

Velez, C. E.↗

NASA NTRS1978

Odd nitrogen production by meteoroids

The process by which odd nitrogen species (atomic nitrogen and nitric oxide) are formed during atmospheric entry of meteoroids is analyzed theoretically. An ablating meteoroid is assumed to be a point source of mass with a continuum regime evolving in its wake. The amounts of odd nitrogen species, produced by high-temperature reactions of air in the continuum wake, are calculated by numerical integration of chemical rate equations. Flow properties are assumed to be uniform across the wake, and 29 reactions involving five neutral species and five singly ionized species are considered, as well as vibrational and electron temperature nonequilibrium phenomena. The results, when they are summed over the observed mass, velocity, and entry-angle distribution of meteoroids, provide odd-nitrogen-species annual global production rates as functions of altitude. The peak production of nitric oxide is found to occur at an altitude of about 85 km; atomic nitrogen production peaks at about 95 km. The total annual rate for nitric oxide is 40 million kg; for atomic nitrogen it is 170 million kg.

Park, C.↗

NASA NTRS1978

Development and evaluation of a hybrid averaged orbit generator

A rapid orbit generator based on a first-order application of the Generalized Method of Averaging has been developed for the Research and Development (R&D) version of the Goddard Trajectory Determination System (GTDS). The evaluation of the averaged equations of motion can use both numerically averaged and recursively evaluated, analytically averaged perturbation models. These equations are numerically integrated to obtain the secular and long-period motion. Factors affecting efficient orbit prediction are discussed and guidelines are presented for treatment of each major perturbation. Guidelines for obtaining initial mean elements compatible with the theory are presented. An overview of the orbit generator is presented and comparisons with high precision methods are given.

Mcclain, W. D.↗

NASA NTRS1978

Calculation of CO concentration for liquid fueled gas turbine combustor

The extensive computation time required for the numerical integration of the differential equations obtained in studies of the concentrations of pollutants emitted by gas turbine combustors, can be reduced significantly by assuming the heat releasing hydrocarbon reactions to be in local equilibrium. In determining the CO and NO concentrations produced in spray combustion, it is, therefore, tempting to use the same local equilibrium assumption in order to simplify the kinetic calculations. An investigation of the validity of the local equilibrium assumption is conducted by using a simple analytical model, and then by actually carrying out the kinetic and local equilibrium calculations for typical case.

Patil, P. B.↗

NASA NTRS1977

A real-time subsynoptic, meso, and microscale severe storm forecast system

A system of differential equations is integrated numerically in space and time over several different matrices in an effort to simulate the atmospheric wave structures which organize severe local storms. Preliminary results with case studies indicate that dynamical fields produced by the numerical simulations can be translated into very fine scale space and time zones where severe storm forecast indices can be developed. These fine scale indices are now available in real-time when run on Langley's STAR 100 computer system.

Kaplan, M. L.↗

NASA NTRS1979

Maintenance of quasi-stationary waves in a 2-level quasi-geostrophic spectral model with topography

The maintenance of the quasi-stationary waves obtained through numerically integrating a 2-level quasi-geostrophic spectral model on a beta-plane is investigated. An idealized topography which has only wavenumber n in the zonal direction and the first mode in the meridional direction is used to force the quasi-stationary waves. It is shown that the topographical forcing is not necessarily the mechanism for maintaining the quasi-stationary waves.

Yao, M. S.↗

NASA NTRS1978

Recent advances in analytical satellite theory

Recent work on analytical satellite perturbation theory has involved the completion of a revision to 4th order for zonal harmonics, the addition of a treatment for ocean tides, an extension of the treatment for the noninertial reference system, and the completion of a theory for direct solar-radiation pressure and earth-albedo pressure. Combined with a theory for tesseral-harmonics, lunisolar, and body-tide perturbations, these formulations provide a comprehensive orbit-computation program. Detailed comparisons with numerical integration and observations are presented to assess the accuracy of each theoretical development.

Gaposchkin, E. M.↗

NASA NTRS1978

The inverse problem: Ocean tides derived from earth tide observations

Indirect mapping ocean tides by means of land and island-based tidal gravity measurements is presented. The inverse scheme of linear programming is used for indirect mapping of ocean tides. Open ocean tides were measured by the numerical integration of Laplace's tidal equations.

Kuo, J. T.↗

NASA NTRS1979

The influence of sweep on the aerodynamic loading of an oscillating NACA 0012 airfoil. Volume 1: Technical report

Aerodynamic experiments were performed on an oscillating NACA 0012 airfoil utilizing a tunnel-spanning wing in both unswept and 30 degree swept configurations. The airfoil was tested in steady state and in oscillatory pitch about the quarter chord. The unsteady aerodynamic loading was measured using pressure transducers along the chord. Numerical integrations of the unsteady pressure transducer responses were used to compute the normal force, chord force, and moment components of the induced loading. The effects of sweep on the induced aerodynamic load response was examined. For the range of parameters tested, it was found that sweeping the airfoil tends to delay the onset of dynamic stall. Sweeping was also found to reduce the magnitude of the unsteady load variation about the mean response. It was determined that at mean incidence angles greater than 9 degrees, sweep tends to reduce the stability margin of the NACA 0012 airfoil; however, for all cases tested, the airfoil was found to be stable in pure pitch. Turbulent eddies were found to convect downstream above the upper surface and generate forward-moving acoustic waves at the trailing edge which move upstream along the lower surface.

St.hilaire, A. O.↗

NASA NTRS1979

Frost formation on an airfoil: A mathematical model 1

A computer model to predict the frost formation process on a flat plate was developed for application to most environmental conditions under which frost occurs. The model was analytically based on a generalized frost thermal conductivity expression, on frost density and thickness rate equations, and on modified heat and mass transfer coefficients designed to fit the available experimental data. The broad experimental ranges reflected by the extremes in ambient humidities, wall temperatures, and convective flow properties in the various publications which were examined served to severely test the flexibility of the model. An efficient numerical integration scheme was developed to solve for the frost surface temperature, density, and thickness under the changing environmental conditions. The comparison of results with experimental data was very encouraging.

Dietenberger, M.↗

NASA NTRS1979

Large eddy simulation of turbulent channel flow: ILLIAC 4 calculation

The three-dimensional time dependent equations of motion were numerically integrated for fully-developed turbulent channel flow. A large scale flow field was obtained directly from the solution of these equations, and small scale field motions were simulated through an eddy viscosity model. The calculations were carried out on the ILLIAC 4 computer. The computed flow patterns show that the wall layer consists of coherent structures of low speed and high speed streaks alternating in the spanwise direction. These structures were absent in the regions away from the wall. Hot spots, small localized regions of very large turbulent shear stress, were frequently observed. The profiles of the pressure velocity-gradient correlations show a significant transfer of energy from the normal to the spanwise component of turbulent kinetic energy in the immediate neighborhood of the wall ('the splatting effect').

Kim, J.↗

NASA NTRS1979

System noise analysis of the dumbbell tethered satellite for gravity-gradient measurements

An analysis of the dumbbell gravity gradiometer concept for measuring short wavelength variations in the earth's gravity gradient is presented. Variations in the gradient are recorded by measuring tension variations in a vertically stabilized satellite consisting of heavy masses connected by a long wire or rod. Tension noise arises from the excitation of various mechanical oscillations of the system. The principal noise sources that were identified are fluctuations in atmospheric drag heating and drag force resulting from density variations and winds. Approximate analytical expressions are presented for the tension noise as a function of the system design parameters for various possible configurations. Computer simulations using numerical integration were performed to study the tension noise for several sample cases. Three designs consistent with Shuttle launch capabilities are discussed.

Colombo, G.↗

NASA NTRS1978

The motion of Jupiter XIII /Leda/, 1974-2000

A revised orbit and extended ephemeris are calculated for Leda, the thirteenth satellite of Jupiter, by means of numerical integration. Thirty-one observations between September 1974 and December 1977 satisfy the new orbit to 0.9 arcsec on the average.

Aksnes, K.↗

NASA NTRS1979

Shuttle entry trajectory reconstruction using inflight accelerometer and gyro measurements

An error analysis has been made of a Shuttle postflight entry trajectory reconstruction process to obtain trajectory state estimation errors and to assess the impact of these errors on Shuttle aerodynamic force coefficient extraction. In this analysis, the entry trajectory is assumed to be reconstructed via numerical integration of onboard accelerometer and gyro measurements and constrained to satisfy ground-based radio tracking. The trajectory state estimation errors are calculated using a Kalman-Schmidt sequential filter assuming various measurement error models and combinations of ground-based tracking. The resultant trajectory estimation errors are analyzed in a simplified perturbation process to establish the accuracy to which postflight aerodynamic force coefficients can be determined. Results are presented which show that the principal error sources affecting the trajectory reconstruction and thus the force coefficient extraction, assuming perfect atmospheric density knowledge, are the accelerometer and gyro resolution, acceleration-sensitive gyro drifts, and the alignment uncertainties associated with integration on the Shuttle.

Compton, H. R.↗

NASA NTRS1978

Viscous flow analysis in mixed flow rotors

A method for analyzing the viscous flow through turbomachine rotors is presented. The field analysis is based upon the solution of the full Navier-Stokes equations over the rotor blade-to-blade stream channels. An Alternating-Direction-Implicit method is employed to carry out the necessary numerical integration of the elliptic governing equations. The flow analysis may be applied to various types of turbomachine rotors. Preliminarily, only the case of laminar flows are considered in this paper. The flow characteristics within the rotors of a mixed flow turbine and a radial bladed compressor are investigated over a wide range of operating conditions. Excellent results are obtained when compared with existing experimental data. The method of this analysis is quite general and can deal with a wide range of applications. Possible modification of the present study to deal with turbulent flow cases are also identified.

Khalil, I.↗

NASA NTRS1979

Present obliquity oscillations of Mars - Fourth-order accuracy in orbital e and I

A long period analysis of solar system orbital evolution, correct to fourth order in orbital eccentricities and inclinations (Bretagnon, 1974), and an improved value of the planet's moment of inertia (Reasenberg, 1977) have been incorporated in a recalculation of the obliquity oscillations of Mars. A linearized solution predicts a maximum oscillation amplitude of 13.6 deg centered on a long-term average value of 24.4 deg. A numerical integration of the obliquity for the past 10,000,000 years is also presented. Epochs of minimal oscillation like the present occur at intervals of the order of 4 m.y.

Ward, W. R.↗

NASA NTRS1979

Some simple properties of stellar pulsation modes

Except for the lowest harmonics, small-amplitude stellar pulsation modes possess many simple properties whose evaluation does not require numerical integration of the fourth-order equations of motion. All antinodes tend to have the same total kinetic energy except for those lying near physical or geometric boundaries. However, when kinetic energy per unit volume is considered, order-of-magnitude enhancements are seen in antinodes lying near the center of the star, and factor-of-2 enhancements occur near the polar axis. The nodes are distributed very regularly along the radius. They follow an exponential law in g-regions, and their separation is proportional to the sound travel time in p-regions. A simple graphical procedure is described for surveying the oscillation frequencies of a new stellar model. A precise condition is derived giving the division of energy between radial and angular motion. Another condition gives the fractional contribution to the velocity field of its two sources, the divergence and the curl. Certain simplifying results of weak coupling among the linear modes are briefly described.

Wolff, C. L.↗