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At least 127 records · Page 7

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NASA NTRS1976

Conformal coordinates associated with uniformly accelerated motion

Specific problems in the theory of relativity are often simplified by an appropriate choice of the coordinate system. Restricted conformal coordinates provide an especially simple analysis of motion with uniform acceleration, known as hyperbolic motion. Conformal coordinates x', t' may be obtained from Cartesian coordinates x, t by the transformation x'+ct'=F(x+ct) and x'-ct'=G(x-ct), where c is the velocity of light. A variable motion of the x' system is determined by the choice of the functions F and G.

Jones, R. T.↗

NASA NTRS1976

On differential transformations between Cartesian and curvilinear (geodetic) coordinates

Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.

Soler, T.↗

NASA NTRS1975

Sensitivity analysis of short-arc station coordinate determinations from range data

The accurate determination of the geocentric coordinates of a tracking station is essential for most geodetic and geophysical satellite applications. Since most of these satellites are close to the earth, the geopotential model is a dominant source of error which significantly influences station coordinate determinations. Other sources, such as GM error and drag, also influence the accuracy of the station coordinate determination. One technique for reducing the effect of these errors is to use short-arcs consisting of a few passes of the satellite over the tracking station. This paper analyzes the sensitivity of short-arc station coordinate estimates to various errors in the physical model, to the number of observations, and to the station-satellite geometry using simulated as well as real data.

Schutz, B. E.↗

NASA NTRS1976

Protection coordination of the Kennedy Space Center electric distribution network

A computer technique is described for visualizing the coordination and protection of any existing system of devices and settings by plotting the tripping characteristics of the involved devices on a common basis. The program determines the optimum settings of a given set of protective devices and configuration in the sense of the best expected coordinated operation of these devices. Subroutines are given for simulating time versus current characteristics of the different relays, circuit breakers, and fuses in the system; coordination index computation; protection checks; plotting; and coordination optimation.

Source record↗

NASA NTRS1976

Sensitivity analysis of short-arc station coordinate determinations from range data

Schutz, B. E.↗

NASA NTRS1978

Forebody and afterbody solutions of the Navier-Stokes equations for supersonic flow over blunt bodies in a generalized orthogonal coordinate system

A coordinate transformation, which can approximate many different two-dimensional and axisymmetric body shapes with an analytic function, is used as a basis for solving the Navier-Stokes equations for the purpose of predicting 0 deg angle of attack supersonic flow fields. The transformation defines a curvilinear, orthogonal coordinate system in which coordinate lines are perpendicular to the body and the body is defined by one coordinate line. This system is mapped in to a rectangular computational domain in which the governing flow field equations are solved numerically. Advantages of this technique are that the specification of boundary conditions are simplified and, most importantly, the entire flow field can be obtained, including flow in the wake. Good agreement has been obtained with experimental data for pressure distributions, density distributions, and heat transfer over spheres and cylinders in supersonic flow. Approximations to the Viking aeroshell and to a candidate Jupiter probe are presented and flow fields over these shapes are calculated.

Gnoffo, P. A.↗

NASA NTRS1978

Determining crustal strain rates with a spaceborne geodynamics ranging system. 2: Station coordinate analysis

The use of a spaceborne geodynamics ranging system for determining crustal strain rates is analyzed. The use of site coordinates rather than intersite baseline distances for the strain rate determinations is emphasized. After discussing the analytical techniques which are to be employed, numerical results are presented which suggest that the use of site coordinates would result in a 20-70% improvement in the precision of the deduced values of straining. Precision of a few parts in 10 to the 9th power would be achievable with simple geometrics and a decade or two of measurements; precisions of a few parts in 10 to the 8th power would be achievable in a few years. A consideration of possible correlations among the derived target site coordinates leads to the conclusion that, with the proper choice of coordinate systems, the correlations can be made small and non-detrimental to the strain rate determinations.

Cohen, S. C.↗

NASA NTRS1978

The influence of earth tides on earth's coordinates

The importance of the Earth's tides on Earth coordinates were examined for the following reasons: (1) the precision for obtaining the Earth's coordinates shows that the effects of Earth tides appear on the values obtained for the coordinates; (2) the possibility of determining the values of the Earth tides; and (3) the consideration of theoretical models that can compute the values of Earth tides. The astronomical and geodetic coordinates of a point at the Earth's surface are described.

Vincente, R. O.↗

NASA NTRS1979

Useful coordinate transformations for antenna applications

General coordinate transformations which are commonly encountered in many antenna applications are presented. Neither the feed coordinates nor the far-field pattern coordinates in general coincide with the antenna coordinates. Transformations discussed allow one to relate the spherical and cartesian components of one system to the spherical and cartesian components of the other system. In particular, attempts are made to use unified notations to assist in a straightforward application of the transformations.

Rahmat-Samii, Y.↗

NASA NTRS1979

A simultaneous coordinate relaxation algorithm for large, sparse matrix eigenvalue problems

An algorithm is proposed for a scheme of simultaneous coordinate relaxation. A variant of root-shifting coordinate relaxation, this procedure consists of iterating several vectors at the same time, instead of one at a time. Results of application of the algorithm to test matrices are discussed. For many matrix eigenvalue problems for which coordinate relaxation is a viable approach, the present algorithm is more effective than previous implementations of coordinate relaxation. Total central processor operations should be decreased due to significantly improved convergence.

Raffenetti, R. C.↗

NASA NTRS1980

A three-dimensional turbulent compressible subsonic duct flow analysis for use with constructed coordinate systems

An approximate analysis, applicable to nonorthogonal coordinate systems having a curved centerline and planar transverse coordinate surfaces normal to the centerline, is presented for computation of three-dimensional subsonic flow in straight and curved diffusers. The formulation is intended to facilitate the use of constructed coordinates in circumstances where it is difficult to maintain smooth behavior in higher derivatives; the use of local Cartesian variables and fluxes leads to governing equations which require only first derivatives of the coordinate transformation. The analysis is applied to a particular family of duct and diffuser geometries having curved centerlines and superelliptic cross sections. Qualitative agreement with experimental measurements is observed with regard to streamwise vortices and distortion of the primary flow.

Levy, R.↗

NASA NTRS1980

Generations of orthogonal surface coordinates

Two generation methods were developed for three dimensional flows where the computational domain normal to the surface is small. With this restriction the coordinate system requires orthogonality only at the body surface. The first method uses the orthogonal condition in finite-difference form to determine the surface coordinates with the metric coefficients and curvature of the coordinate lines calculated numerically. The second method obtains analytical expressions for the metric coefficients and for the curvature of the coordinate lines.

Blottner, F. G.↗

NASA NTRS1980

A three-dimensional body-fitted coordinate system for flow field calculations on asymmetric nosetips

A three dimensional body-fitted coordinate system developed for use in the calculation of inviscid flows over ablated, asymmetric reentry vehicle nosetips is described. Because of the potential geometric asymmetries, no standard coordinate system (e.g., spherical, axisymmetric reference surface-normal) is capable of being closely aligned with the nosetip surface. To generate a 3-D, body-fitted coordinate system an analytic mapping procedure is applied that is conformal within each meridional plane of the nosetip; these transformations are then coupled circumferentially to yield a three dimensional coordinate system. The mappings used are defined in terms of hinge points, which are points selected to approximate the body contours in each meridional plane. The selection of appropriate hinge points was automated to facilitate the use of the resulting nosetip flow field code.

Hall, D. W.↗

NASA NTRS1981

Transformation from proper time on earth to coordinate time in solar system barycentric space-time frame of reference. I. II

It is noted that in order to obtain accurate computed values of earth-based range and Doppler observables of a deep space probe, an expression is required for the difference between coordinate time (t) in the solar system barycentric space-time frame of reference and proper time (tau) recorded on a fixed atomic clock on earth. This is part 1 of a two-part article which obtains an expression for the coordinate time/proper time difference that can be used in deriving computed values of observations of a spacecraft or celestial body located anywhere in the solar system. The expression can also be used in computing Very Long Baseline Interferometry data types. In part 1, expression for the coordinate time/proper time difference is obtained that is a function of position and velocity vectors of the major celestial bodies of the solar system and the atomic clock on earth which reads proper time. This expression is transformed in part 2 to a function of time and the earth-fixed coordinates of the atomic clock.

Moyer, T. D.↗

NASA NTRS1983

A comparison of methods for computing the sigma-coordinate pressure gradient force for flow over sloped terrain in a hybrid theta-sigma model

In connection with the employment of the sigma coordinates introduced by Phillips (1957), problems can arise regarding an accurate finite-difference computation of the pressure gradient force. Over steeply sloped terrain, the calculation of the sigma-coordinate pressure gradient force involves computing the difference between two large terms of opposite sign which results in large truncation error. To reduce the truncation error, several finite-difference methods have been designed and implemented. The present investigation has the objective to provide another method of computing the sigma-coordinate pressure gradient force. Phillips' method is applied for the elimination of a hydrostatic component to a flux formulation. The new technique is compared with four other methods for computing the pressure gradient force. The work is motivated by the desire to use an isentropic and sigma-coordinate hybrid model for experiments designed to study flow near mountainous terrain.

Johnson, D. R.↗

NASA NTRS1985

Users manual for coordinate generation code CRDSRA

Generation of a viable coordinate system represents an important component of an isolated airfoil Navier-Stokes calculation. The manual describes a computer code for generation of such a coordinate system. The coordinate system is a general nonorthogonal one in which high resolution normal to the airfoil is obtained in the vicinity of the airfoil surface, and high resolution along the airfoil surface is obtained in the vicinity of the airfoil leading edge. The method of generation is a constructive technique which leads to a C type coordinate grid. The method of construction as well as input and output definitions are contained herein. The computer code itself as well as a sample output is being submitted to COSMIC.

Shamroth, S. J.↗

NASA NTRS1985

Generation of Surface Coordinates by Elliptic Partial Differential Equations

The problem of generating spatial coordinates by numerical methods through carefully selected mathematical models is of current interest both in mechanics and physics. The problem of generation of a desired system of coordinates in a given surface was considered, which essentially is an effort directed to the problem of grid generation in a two-dimensional non-Euclidean space. The mathematical model selected for this purpose is based on the formulae of Gauss for a surface. The proposed equations can be used to generate a new coordinate system from the data of an already given coordinate system in a surface. If the coefficients of the first and second fundamental forms have been given, then the proposed equations can be used to generate a surface satisfying the given data (surface fitting). The proposed equations can also be used to generate surfaces in the space between two arbitrary given surfaces, thus providing 3D grids in an Euclidean space.

Warsi, Z. U. A.↗

NASA NTRS1983

Computer programs for smoothing and scaling airfoil coordinates

Detailed descriptions are given of the theoretical methods and associated computer codes of a program to smooth and a program to scale arbitrary airfoil coordinates. The smoothing program utilizes both least-squares polynomial and least-squares cubic spline techniques to smooth interatively the second derivatives of the y-axis airfoil coordinates with respect to a transformed x-axis system which unwraps the airfoil and stretches the nose and trailing-edge regions. The corresponding smooth airfoil coordinates are then determined by solving a tridiagonal matrix of simultaneous cubic-spline equations relating the y-axis coordinates and their corresponding second derivatives. A technique for computing the camber and thickness distribution of the smoothed airfoil is also discussed. The scaling program can then be used to scale the thickness distribution generated by the smoothing program to a specific maximum thickness which is then combined with the camber distribution to obtain the final scaled airfoil contour. Computer listings of the smoothing and scaling programs are included.

Morgan, H. L., Jr.↗