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NASA NTRS1991

Benchmark calculations of thermal reaction rates. I - Quantal scattering theory

The thermal rate coefficient for the prototype reaction H + H2 yields H2 + H with zero total angular momentum is calculated by summing, averaging, and numerically integrating state-to-state reaction probabilities calculated by time-independent quantum-mechanical scattering theory. The results are very carefully converged with respect to all numerical parameters in order to provide high-precision benchmark results for confirming the accuracy of new methods and testing their efficiency.

Chatfield, David C.↗

NASA NTRS1994

Hot-Gas-Slide and Coolant-Side Heat Transfer in Liquid Rocket Engine Combustors

The objectives of this article are to develop a multidisciplinary, computational methodology to predict the hot-gas-side and coolant-side heat transfer in film cooling assisted, regeneratively cooled liquid rocket engine combustors, and to use it in parametric studies to recommend optimized design of the coolant channels for a developmental combustor. An integrated numerical model which incorporates computational fluid dynamics (CFD) for the hot-gas thermal environment, and thermal analysis for the liner and coolant channels, was developed. This integrated CFD/thermal model was validated by comparing predicted heat fluxes with those of hot-firing test and industrial design methods for a 40-k calorimeter thrust chamber and the Space Shuttle Main Engine main combustion chamber. Parametric studies were performed for the advanced main combustion chamber to find a strategy for a proposed coolant channel design.

Ten-see Wang↗

NASA NTRS1985

Accumulation of the Terrestrial Planets

The validity of the two body approximation in calculations of planetary growth is examined. Use of this approximation is essential to practical 3 dimensional simulations of planetary growth. With regard to gravitational perturbations, the changes in semimajor axis, eccentricity, and inclination resulting from close planetesimal encounters (near 1 A.U.) out to 10 Tisserand sphere of influence radii were calculated by two and three dimensional numerical integration. These were compared with the results of treating the encounter as a two body problem, as is customary in Monte Carlo calculations of orbital evolution and in numerical and analytical studies of planetary accumulation. It is found that for values of (V/V sub e) approx. 0.35 (V = relative velocity, V = escape velocity of largest body), the two body approximation fails to describe the outcome of individual encounters. In this low velocity region, the two body gravitational focusing cross section is no longer valid.

Wetherill, G. W.↗

NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗

NASA NTRS1971

Comparison of various methods for mathematical analysis of the Foucault knife edge test pattern to determine optical imperfections

The linearized integral equation for the Foucault test of a solid mirror was solved by various methods: power series, Fourier series, collocation, iteration, and inversion integral. The case of the Cassegrain mirror was solved by a particular power series method, collocation, and inversion integral. The inversion integral method appears to be the best overall method for both the solid and Cassegrain mirrors. Certain particular types of power series and Fourier series are satisfactory for the Cassegrain mirror. Numerical integration of the nonlinear equation for selected surface imperfections showed that results start to deviate from those given by the linearized equation at a surface deviation of about 3 percent of the wavelength of light. Several possible procedures for calibrating and scaling the input data for the integral equation are described.

Gatewood, B. E.↗

NASA NTRS1996

Closed-Form Evaluation of Mutual Coupling in a Planar Array of Circular Apertures

The integral expression for the mutual admittance between circular apertures in a planar array is evaluated in closed form. Very good accuracy is realized when compared with values that were obtained by numerical integration. Utilization of this closed-form expression, for all element pairs that are separated by more than one element spacing, yields extremely accurate results and significantly reduces the computation time that is required to analyze the performance of a large electronically scanning antenna array.

Bailey, M. C.↗

NASA NTRS1992

Numerical analysis of the hot-gas-side and coolant-side heat transfer in liquid rocket engine combustors

The objective of this paper are to develop a multidisciplinary computational methodology to predict the hot-gas-side and coolant-side heat transfer and to use it in parametric studies to recommend optimized design of the coolant channels for a regeneratively cooled liquid rocket engine combustor. An integrated numerical model which incorporates CFD for the hot-gas thermal environment, and thermal analysis for the liner and coolant channels, was developed. This integrated CFD/thermal model was validated by comparing predicted heat fluxes with those of hot-firing test and industrial design methods for a 40 k calorimeter thrust chamber and the Space Shuttle Main Engine Main Combustion Chamber. Parametric studies were performed for the Advanced Main Combustion Chamber to find a strategy for a proposed combustion chamber coolant channel design.

Wang, Ten-See↗

NASA NTRS1993

Symmetric Galerkin boundary formulations employing curved elements

Accounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (RISP) numerical integration algorithm is shown to produce a Galerkin (i.e., double) integration strategy that is competitive with its counterpart (i.e., singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C 1 continuity, a fact that allows the employment of the popular continuous element technology. The behavior of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation-based BEA, for large-scale problems.

Kane, J. H.↗

NASA NTRS1976

Pines method for circular and elliptical orbits

The Pines method for trajectory integration is derived from the equations of two-body motion. The concept of the perturbation derivative is explained and a formal definition is presented. An algorithm for applying the Pines method to numerical integration of trajectories is included.

Semar, C.↗

NASA NTRS1980

A calculation procedure for viscous flow in turbomachines, volume 3

A method for analyzing the nonadiabatic viscous flow through turbomachine blade passages was developed. The field analysis is based upon the numerical integration of the full incompressible Navier-Stokes equations, together with the energy equation on the blade-to-blade surface. A FORTRAN IV computer program was written based on this method. The numerical code used to solve the governing equations employs a nonorthogonal boundary fitted coordinate system. The flow may be axial, radial or mixed and there may be a change in stream channel thickness in the through-flow direction. The inputs required for two FORTRAN IV programs are presented. The first program considers laminar flows and the second can handle turbulent flows. Numerical examples are included to illustrate the use of the program, and to show the results that are obtained.

Khalil, I.↗

NASA NTRS2021

Solving Two-Equation Turbulence Models With a Perspective on Solving Transport Equations

There are three principal objectives of this report. The first objective is to investigate frequently used two-equation models when solving the RANS equations. In this study, we consider the 2006Wilcox and the 2003 Menter Shear Stress Transport (SST) models. Also, a simple change of the Wilcox model is introduced to improve computational predictions for transonic flows. Computations for flows over two different airfoils are examined to compare these turbulence models. Effects on modeling the physics of each flow due to variations in the models, such as using either strainrate or vorticity in the turbulence production term, are considered and discussed. The second objective of this report is to not only explore but also evaluate the performance of the solution algorithm for the mean flow and the transport equations. The RANS and turbulence modeling equations aresolved in a weakly coupled manner with a diagonal implicit Runge-Kutta (DIRK) solution algorithm. Throughout this report, emphasis is given to reducing the residuals to machine zero in all flow calculations, so as to eliminate the error due to numerical integration of the discrete governing equations. The final objective is to provide a perspective on solving transport equations for turbulence modeling. Aspects of solving such stiff systems of equations, as well as various numerical difficulties and possible techniques to overcome them, are considered. Discussion is also provided concerning what is called ’numerical compatibility’, which is an essential requirement when designing a solution algorithm for solving transport equations.

turbulence modeling↗

NASA NTRS1993

The obliquity of Mars and 'climate friction'

A mathematical theory is presented which explains the increase of the mean obliquity of Mars over geologic time due to the so called 'climate friction' (i.e., the climatic changes associated with obliquity oscillations of Mars). The theory is compared with a 10 m.y. numerical integration of the equations performed for a hypothetically large amount of climate friction for two cases of the obliquity oscillations: (1) a single sinusoid and (2) a sum of three sinusoids. The theory and numerics agree for both cases within about 12 percent on the size of the secular increase in obliquity. One possible mechanism of climate friction investigated is 'postglacial rebound' on Mars. According to this theory, giant polar caps form when the obliquity is low, and slowly squeeze out an equatorial bulge. When the obliquity is high, the caps disappear, but the bulge takes some time to collapse, due to mantle viscosity, causing it to oscillate out of phase with the obliquity oscillations. This causes a secular increase in the average obliquity.

Rubincam, David P.↗

NASA NTRS1979

A semi-analytic theory for the motion of a close-earth artificial satellite with drag

A semi-analytic method is used to estimate the decay history/lifetime and to generate orbital ephemeris for close-earth satellites perturbed by the atmospheric drag and earth oblateness due to the spherical harmonics J2, J3, and J4. The theory maintains efficiency through the application of the theory of a method of averaging and employs sufficient numerical emphasis to include a rather sophisticated atmospheric density model. The averaged drag effects with respect to mean anomaly are evaluated by a Gauss-Legendre quadrature while the averaged variational equations of motion are integrated numerically with automatic step size and error control.

Liu, J. J. F.↗

NASA NTRS1985

Advanced three-dimensional dynamic analysis by boundary element methods

Advanced formulations of boundary element method for periodic, transient transform domain and transient time domain solution of three-dimensional solids have been implemented using a family of isoparametric boundary elements. The necessary numerical integration techniques as well as the various solution algorithms are described. The developed analysis has been incorporated in a fully general purpose computer program BEST3D which can handle up to 10 subregions. A number of numerical examples are presented to demonstrate the accuracy of the dynamic analyses.

Banerjee, P. K.↗

NASA NTRS1980

Computer prediction of large reflector antenna radiation properties

A FORTRAN program for calculating reflector antenna radiation patterns was rewritten and extended to include reflectors composed of a number of panels. These individual panels must be analytic surfaces. The theoretical foundation for the program is as follows: Geometrical optics techniques are used to trace rays from a feed antenna to the reflector surface and back to a mathematical plane just in front of the reflector. The resulting tangential electric field distribution, assumed to be the only source of forward radiation, is integrated numerically to calculate the radiation pattern for a desired set of angles. When the reflector is composed of more than one panel, each panel is treated as a separated antenna, the ray-tracing procedure and integration being repeated for each panel. The results of the individual aperture plane integrations are stored and summed to yield the relative electric field strength over the angles of interest. An example and several test cases are included to demonstrate the use of the program and verify the new method of computation.

Botula, A.↗

NASA NTRS2003

Updated Integrated Mission Program

Integrated Mission Program (IMP) is a computer program for simulating spacecraft missions around the Earth, Moon, Mars, and/or other large bodies. IMP solves the differential equations of motion by use of a Runge-Kutta numerical-integration algorithm. Users control missions through selection from a large menu of events and maneuvers. Mission profiles, time lines, propellant requirements, feasibility analyses, and perturbation analyses can be computed quickly and accurately. A prior version of IMP, written in FORTRAN 77, was reported in Program Simulates Spacecraft Missions (MFS-28606), NASA Tech Briefs, Vol. 17, No. 4 (April 1993), page 60. The present version, written in double-precision Lahey FORTRAN 90, incorporates a number of improvements over the prior version. Some of the improvements modernize the code to take advantage of today's greater central-processing-unit speeds. Other improvements render the code more modular; provide additional input, output, and debugging capabilities; and add to the variety of maneuvers, events, and means of propulsion that can be simulated. The IMP user manuals (of which there are now ten, each addressing a different aspect of the code and its use) have been updated accordingly.

Dauro, Vincent A., Sr.↗

NASA NTRS1991

Surface albedo from bidirectional reflectance

The validity of integrating over discrete wavelength bands is examined to estimate total shortwave bidirectional reflectance of vegetated and bare soil surfaces. Methods for estimating albedo from multiple angle, discrete wavelength band radiometer measurements are studied. These methods include a numerical integration technique and the integration of an empirically derived equation for bidirectional reflectance. It is concluded that shortwave albedos estimated through both techniques agree favorably with the independent pyranometer measurements. Absolute rms errors are found to be 0.5 percent or less for both grass sod and bare soil surfaces.

Ranson, K. J.↗

NASA NTRS1972

A study of nozzle and ejector flow problems by the method of integral relations

The application of the method of integral relations to nozzle and ejector flow problems was examined. For nozzle flow problems, the general formulation is that the approaching flow may be rotational. Particular attention was given to the phenomenon of choking under nonuniform flow conditions. Numerical integration of the governing ordinary differential equations was also investigated. This scheme of analysis was also applied to study the interacting flow field within an ejector system.

Howlett, L. D.↗