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Results for “spherical coordinates”

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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Analytical expression of a finite, long, conical canted-cosine-theta coil for particle collider interaction regions

Magnets in the accelerator interaction region (IR) present significant challenges because of high field requirements and limited available space. Conical-shaped magnets offer advantages in these environments by allowing closer placement to the interaction point while maintaining clearance from synchrotron radiation. Interestingly, numerical studies have shown that conical canted-cosine-theta (CCT) designs produce a constant field distribution along the axial direction in the IR quadrupoles for the Electron-Ion Collider (EIC) at Brookhaven National Laboratory. However, the field harmonics generated by conical CCT windings are not yet fully understood. This paper presents an analytical approach to describe the magnetic field produced by a conical surface current and proposes a method for designing conical CCT magnets for accelerator applications. First, we begin with a surface current sheet having a general cosine-theta distribution in spherical coordinates and solve the vector potential using the Green’s function. The magnetic fields generated by the conical current sheet are expressed using associated Legendre polynomials. These results are then related to circular field harmonics and integral field harmonics for designing a coil that produces a pure multipole field. Next, a single layer of the conical CCT winding path is produced based on the cosine-theta current distribution. Finally, the magnetic field quality of dipole and quadrupole conical CCT coils with multiple layers is verified using the Biot-Savart law.

Yang, Ye↗
DOE OSTI2025

Linear Velocity Problems Part 2: Thermodynamic Considerations

A compressible polytropic gas, in one dimensional planar, cylindrical, or spherical coordinate systems, is examined through four thermodynamic properties: mass density, pressure, specific internal energy (SIE), and entropy, under the assumptions that the fluid velocity is linearly proportional to the radial coordinate and that the Euler gas dynamics equations assert a self similar solution class. The behavior of the thermodynamics is almost entirely determined by an arbitrary function that results from the derivation of the density distribution. Through physical and other arguments, the arbitrary function can be plotted spatially with time profiles, providing a deeper understanding of the system and how the arbitrary function affects the behavior of the compressible gas.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1969

LM lookangle program

Program computes the spacecraft look angles and the slant range, which define a spherical coordinate system located in the spacecraft body. The program is designed to reduce data from the Lunar Module Missions and to output desired information.

Agee, W. E.↗
NASA NTRS1973

Acoustic wave propagation in an axisymmetric swirling jet.

An analysis has been developed to study the acoustic wave propagation in an axisymmetric swirling subsonic jet flow. The governing convected wave equation derived in the spherical coordinates includes mean shears, shear gradients and pressure gradients. The directivity patterns for various spinning and non-spinning modes due to the influence of the mean jet swirl were obtained by numerically integrating the governing wave equation. The mean flow field used in the computation was that obtained semiempirically for subsonic swirling turbulent jet and is completely specified once the degree of swirl is known. The dependence of sound directivity on jet Mach number, swirl ratio and frequency are discussed.

Yu, J. C.↗
NASA NTRS1974

A finite element solution algorithm for the Navier-Stokes equations

A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame.

Baker, A. J.↗
NASA NTRS1976

The theory of the gravitational potential applied to orbit prediction

A complete derivation of the geopotential function and its gradient is presented. Also included is the transformation of Laplace's equation from Cartesian to spherical coordinates. The analytic solution to Laplace's equation is obtained from the transformed version, in the classical manner of separating the variables. A cursory introduction to the method devised by Pines, using direction cosines to express the orientation of a point in space, is presented together with sample computer program listings for computing the geopotential function and the components of its gradient. The use of the geopotential function is illustrated.

Kirkpatrick, J. C.↗
NASA NTRS1976

Study of a fail-safe abort system for an actively cooled hypersonic aircraft: Computer program documentation

The Fail-Safe Abort System TEMPerature Analysis Program, (FASTEMP), user's manual is presented. This program was used to analyze fail-safe abort systems for an actively cooled hypersonic aircraft. FASTEMP analyzes the steady state or transient temperature response of a thermal model defined in rectangular, cylindrical, conical and/or spherical coordinate system. FASTEMP provides the user with a large selection of subroutines for heat transfer calculations. The various modes of heat transfer available from these subroutines are: heat storage, conduction, radiation, heat addition or generation, convection, and fluid flow.

Haas, L. A., Sr.↗
NASA NTRS1974

Automation of computer-aided astrometric reduction of photographs of satellites

Programs supporting the calculation of topocentric spherical coordinates of satellites from satellite images on negatives are discussed. Particular attention is given to a program for automatic identification of negatives obtained by the AFU-75 camera, and to the minimization of the amount of input information.

Iurova, L. A.↗
NASA NTRS1976

Heat transfer evaluation in a plasma core reactor

Numerical evaluations of heat transfer in a fissioning uranium plasma core reactor cavity, operating with seeded hydrogen propellant, was performed. A two-dimensional analysis is based on an assumed flow pattern and cavity wall heat exchange rate. Various iterative schemes were required by the nature of the radiative field and by the solid seed vaporization. Approximate formulations of the radiative heat flux are generally used, due to the complexity of the solution of a rigorously formulated problem. The present work analyzes the sensitivity of the results with respect to approximations of the radiative field, geometry, seed vaporization coefficients and flow pattern. The results present temperature, heat flux, density and optical depth distributions in the reactor cavity, acceptable simplifying assumptions, and iterative schemes. The present calculations, performed in cartesian and spherical coordinates, are applicable to any most general heat transfer problem.

Smith, D. E.↗
NASA NTRS1979

Non-force-free solar magnetic fields in magnetohydrostatic equilibrium

The objective of the paper is to examine a class of non-force-free fields analytically. Specifically, magnetic fields in magnetohydrostatic equilibrium with a plasma in a gravitational field are treated in an approximation of two independent variables but three vector components. Spherical coordinates are emphasized, although the formal results for cylindrical and Cartesian coordinates are presented in the appendices. Formal solutions for the magnetic-field components are obtained in terms of the plasma variables, and field line equations are derived. A final equation governing the plasma variables is then obtained. Procedures are developed for analyzing this equation and obtaining sets of self-consistent particular solutions to the governing equations; a number of such sets of solutions are presented. As an example, one solution set is examined, illustrating the application of the results to the analysis of solar observational data.

Comfort, R. H.↗
NASA NTRS1979

Explicit large time-step schemes for the shallow water equations

Modifications to explicit finite difference schemes for solving the shallow water equations for meteorological applications by increasing the time step for the fast gravity waves are analyzed. Terms associated with the gravity waves in the shallow water equations are treated on a coarser grid than those associated with the slow Rossby waves, which contain much more of the available energy and must be treated with higher accuracy, enabling a several-fold increase in time step without degrading the accuracy of the solution. The method is presented in Cartesian and spherical coordinates for a rotating earth, using generalized leapfrog, frozen coefficient, and Fourier filtering finite difference schemes. Computational results verify the numerical stability of the approach.

Turkel, E.↗
NASA NTRS1980

Protostellar formation in rotating interstellar clouds. I - Numerical methods and tests

Attention is given to numerical methods and tests of a series of gravitational hydrodynamics computer codes constructed in order to numerically follow the dynamic collapse of spherically symmetric (1-D), axisymmetric (2-D), and non-axisymmetric (3-D) isothermal interstellar clouds. A spherical harmonic expansion is used to solve the Poisson equation for the gravitational potential. It is shown that the use of explicit donor-cell hydrodynamics on a moving spherical coordinate grid ensures mass and momentum conservation and allows the grid to follow the collapse of the fluid. Finally, the performance of the codes is examined.

Boss, A. P.↗
NASA NTRS1980

Analytical satellite theory in extended phase space

It is noted that a satellite theory, based on extended phase space and on the true anomaly, was introduced by Scheifele (1970). In the present paper a simple canonical transformation is shown that makes the transition from the classical Delaunay elements to the Scheifele variables. It is stressed that neither spherical coordinates nor Hamilton-Jacobi theory is used. Finally, attention is given to the meaning of the new variables, especially the use of the true anomaly as one of the variables.

Bond, V.↗
NASA NTRS1982

The electromagnetic force field, fluid flow field and temperature profiles in levitated metal droplets

A mathematical representation was developed for the electromagnetic force field, the flow field, the temperature field (and for transport controlled kinetics), in a levitation melted metal droplet. The technique of mutual inductances was employed for the calculation of the electromagnetic force field, while the turbulent Navier - Stokes equations and the turbulent convective transport equations were used to represent the fluid flow field, the temperature field and the concentration field. The governing differential equations, written in spherical coordinates, were solved numerically. The computed results were in good agreement with measurements, regarding the lifting force, and the average temperature of the specimen and carburization rates, which were transport controlled.

El-Kaddah, N.↗
NASA NTRS1982

Numerical model of a two-dimensional, non-plane transient magnetohydrodynamic flow

The equations describing two-dimensional three-component magnetohydrodynamic (MHD) transient flows are formulated for a system of spherical coordinates. With the numerical code based on Implicit Continuous Fluid Eulerian (ICE) scheme, MHD flows resulting from a sudden energy release in a stratified medium are examined. Because of the inclusion of out-of-plane components of velocity and magnetic fields, MHD transverse waves are observed in addition to fast, slow and entropy waves. Numerical results for compressible MHD shocks are found in satisfactory agreement with the theoretical predictions.

Han, S. M.↗
NASA NTRS1983

Numerical simulation of the leading-edge separation vortex for a wing and strake-wing configuration

In the present investigation, the 'thin layer' Navier-Stokes equations are used to compute the flow about a delta wing and a strake-delta wing configuration. Both configurations possess blunt noses and rounded leading edges. The computational grid about these configurations is generated using a newly developed three-dimensional grid-generation code which solves a set of Poisson equations with spherical coordinate variables using an alternating direction implicit (ADI) scheme. Computational results are obtained for an isolated wing with 60 deg sweep and a strake-wing configuration with 80-60 deg sweep. Computations for angles of attack in the range from 6 to 30 deg are presented. Attention is given to the effect of the angle of attack, the Mach number, and the Reynolds number. Theoretical results are compared with experimental data.

Fujii, K.↗