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At least 217 records · Page 12

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NASA NTRS1972

An implicit finite-difference solution to the viscous shock layer, including the effects of radiation and strong blowing

An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.

Garrett, L. B.↗

NASA NTRS1973

Simultaneous observation solutions for NASA-MOTS and SPEOPT station positions on the North American datum

Simultaneous observations of the GEOS-I and II flashing lamps by the NASA MOTS and SPEOPT cameras on the North American Datum (NAD) were analyzed using geometrical techniques to provide an adjustment of the station coordinates. Two separate adjustments were obtained. An optical data only solution was computed in which the solution scale was provided by the Rosman-Mojave distance obtained from a dynamic station solution. In a second adjustment, scaling was provided by processing simultaneous laser ranging data from Greenbelt and Wallops Island in a combined optical-laser solution. Comparisons of these results with previous GSFC dynamical solutions indicate an rms agreement on the order of 4 meters or better in each coordinate. Comparison with a detailed gravimetric geoid of North America yields agreement of 3 meters or better for mainland U.S. stations and 7 and 3 meters, respectively, for Bermuda and Puerto Rico.

Reece, J. S.↗

NASA NTRS1973

A global station coordinate solution based upon camera and laser data - GSFC 1973

Results for the geocentric coordinates of 72 globally distributed satellite tracking stations consisting of 58 cameras and 14 lasers are presented. The observational data for this solution consists of over 65,000 optical observations and more than 350 laser passes recorded during the National Geodetic Satellite Program, the 1968 Centre National d'Etudes Spatiales/Smithsonian Astrophysical Observatory (SAO) Program, and International Satellite Geodesy Experiment Program. Dynamic methods were used. The data were analyzed with the GSFC GEM and SAO 1969 Standard Earth Gravity Models. The recent value of GM = 398600.8 cu km/sec square derived at the Jet Propulsion Laboratory (JPL) gave the best results for this combination laser/optical solution. Solutions are made with the deep space solution of JPL (LS-25 solution) including results obtained at GSFC from Mariner-9 Unified B-Band tracking. Datum transformation parameters relating North America, Europe, South America, and Australia are given, enabling the positions of some 200 other tracking stations to be placed in the geocentric system.

Marsh, J. G.↗

NASA NTRS1973

An interative solution of an integral equation for radiative transfer by using variational technique

An effective iterative technique is introduced to solve a nonlinear integral equation frequently associated with radiative transfer problems. The problem is formulated in such a way that each step of an iterative sequence requires the solution of a linear integral equation. The advantage of a previously introduced variational technique which utilizes a stepwise constant trial function is exploited to cope with the nonlinear problem. The method is simple and straightforward. Rapid convergence is obtained by employing a linear interpolation of the iterative solutions. Using absorption coefficients of the Milne-Eddington type, which are applicable to some planetary atmospheric radiation problems. Solutions are found in terms of temperature and radiative flux. These solutions are presented numerically and show excellent agreement with other numerical solutions.

Yoshikawa, K. K.↗

NASA NTRS1973

Backlund transformation for solutions of the Korteweg-de Vries equation

We present a Backlund transformation operating on solutions of the Korteweg-de Vries equation. It provides a unifying approach to the theory of solitary waves (solutions). Recursive application of the transformation to any solution of the Korteweg-de Vries equation generates a hierarchy of solutions for which an algebraic recursion relation is obtained. Every such hierarchy (one of which is the known family of pure multisoliton solutions) thus admits a nonlinear superposition principle.

Wahlquist, H. D.↗

NASA NTRS1974

Solutions for laminar boundary layers with separation and reattachment

Numerical solutions of the laminar, incompressible boundary-layer equations are presented for flows involving separation and reattachment. Regular solutions are obtained with an inverse approach in which either the displacement thickness or the skin friction is specified, and the pressure is deduced from the solution. A stream function/vorticity formulation of the boundary-layer equations is used to eliminate the unknown pressure. Solutions of the resulting finite-difference equations, in which the flow direction is taken into account, are obtained by a successive column iteration scheme. Results are compared with Klineberg and Steger's (1974) separated boundary-layer calculations, and with Briley's (1971) solution of the Navier-Stokes equations for a separated region.

Carter, J. E.↗

NASA NTRS1975

Almost periodic solutions to difference equations

The theory of Massera and Schaeffer relating the existence of unique almost periodic solutions of an inhomogeneous linear equation to an exponential dichotomy for the homogeneous equation was completely extended to discretizations by a strongly stable difference scheme. In addition it is shown that the almost periodic sequence solution will converge to the differential equation solution. The preceding theory was applied to a class of exponentially stable partial differential equations to which one can apply the Hille-Yoshida theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous equation (which can be approximated by almost periodic sequences) which are the solutions to appropriate discretizations. Two methods of discretizations are discussed: the strongly stable scheme and the Lax-Wendroff scheme.

Bayliss, A.↗

NASA NTRS1974

An improved solution of the simplified two-phase plume

An exact solution of the two-phase, gas-particle, solid propellant rocket exhaust plume flowfield is presented. The good agreement between the exact solution and the experimental data proves that the accuracy of the present technique is acceptable for, at least, preliminary design calculations. A good comparison between the plume impingement effects predicted by the exact solution and those by the best computer code available has also been found for various propellant formulations and rocket sizes. This shows that the present exact solution does describe the basic characteristics of the two-phase plume. In case of a parametric analysis for obtaining an optimum design, the present exact solution will be a more convenient tool than any one of the numerical techniques especially when many parameters are involved in the optimization study. Several typical results are shown herein and their physical significance is discussed.

Wang, S. Y.↗

NASA NTRS1974

The influence of the computational mesh on accuracy for initial value problems with discontinuous or nonunique solutions

Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the time-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.

Maccormack, R. W.↗

NASA NTRS1975

Use of numerically generated body-fitted coordinate systems for solution of the Navier-Stokes equations

A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

Thompson, J. F.↗

NASA NTRS1975

Inverse solutions for laminar boundary-layer flows with separation and reattachment

Numerical solutions of the laminar, incompressible boundary layer equations are presented for flows involving separation and reattachment. Regular solutions are obtained with an inverse approach in which either the displacement thickness or the skin friction is specified; the pressure is deduced from the solution. A vorticity-stream-function formulation of the boundary layer equations is used to eliminate the unknown pressure. Solutions of the resulting finite difference equations, in which the flow direction is taken into account, are obtained by several global iteration schemes which are stable and have unconditional diagonal dominance. Results are compared with Klineberg and Steger's separated boundary layer calculations, and with Briley's solution of Navier-Stokes equations for a separated region. In addition, an approximate technique is presented in which the streamwise convection of vorticity is set equal to zero in the reversed flow region; such a technique results in a quick forward marching procedure for separated flows.

Carter, J. E.↗

NASA NTRS1976

Supersonic flow of chemically reacting gas-particle mixtures. Volume 1: A theoretical analysis and development of the numerical solution

A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.

Penny, M. M.↗

NASA NTRS1976

Solution of linear systems by a singular perturbation technique

An approximate solution is obtained for a singularly perturbed system of initial valued, time invariant, linear differential equations with multiple boundary layers. Conditions are stated under which the approximate solution converges uniformly to the exact solution as the perturbation parameter tends to zero. The solution is obtained by the method of matched asymptotic expansions. Use of the results for obtaining approximate solutions of general linear systems is discussed. An example is considered to illustrate the method and it is shown that the formulas derived give a readily computed uniform approximation.

Ardema, M. D.↗

NASA NTRS1976

Turbulent viscous-shock-layer solutions with strong vorticity interaction

Numerical solutions of the viscous-shock-layer equations governing laminar and turbulent flows of a perfect gas and radiating and nonradiating mixtures of perfect gases in chemical equilibrium are presented for hypersonic flow over spherically blunted cones and hyperboloids. The results are compared with boundary-layer and inviscid flow-field solutions. The agreement with the inviscid flow field data is satisfactory. The agreement with boundary-layer solutions is good except in regions of strong vorticity interaction. In these flow regions, the viscous-shock-layer solutions appear to be more satisfactory than the boundary-layer solutions.

Anderson, E. C.↗

NASA NTRS1976

On a numerical solution of the supersonic panel flutter eigenproblem

An automated digital computer procedure is presented in this paper which enables efficient solution of the eigenvalue problem associated with the supersonic panel flutter phenomena. The step-by-step incremental solution procedure is based on an inverse iteration technique which effectively utilizes solution results from the previous step in determining such results during the current solution step. Also, the computations are limited to the determination of a few specific roots only, which are expected to contain the flutter mode, and this is achieved at each step without having to compute any other root. The structural discretization achieved by the finite-element method yields highly banded stiffness, mass, and aerodynamic matrices; the aerodynamic matrix evaluated by the linearized piston theory is real but unsymmetric in nature. The solution algorithm presented in this paper fully exploits the banded form of the associated matrices, and the resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most efficient and economical when compared to existing procedures of such analysis. Numerical results are presented for a two-dimensional panel flutter problem.

Gupta, K. K.↗

NASA NTRS1964

Analytical Solutions for Non-Linear Differential Equations with the Help of a Digital Computer

A technique was developed with the help of a digital computer for analytic (algebraic) solutions of autonomous and nonautonomous equations. Two operational transform techniques have been programmed for the solution of these equations. Only relatively simple nonlinear differential equations have been considered. In the cases considered it has been possible to assimilate the secular terms into the solutions. For cases where f(t) is not a bounded function, a direct series solution is developed which can be shown to be an analytic function. All solutions have been checked against results obtained by numerical integration for given initial conditions and constants. It is evident that certain nonlinear differential equations can be solved with the help of a digital computer.

Cromwell, P. C.↗

NASA NTRS1976

Computer simulation of concentrated solid solution strengthening

The interaction forces between a straight edge dislocation moving through a three-dimensional block containing a random array of solute atoms were determined. The yield stress at 0 K was obtained by determining the average maximum solute-dislocation interaction force that is encountered by edge dislocation, and an expression relating the yield stress to the length of the dislocation and the solute concentration is provided. The magnitude of the solid solution strengthening due to solute atoms can be determined directly from the numerical results, provided the dislocation line length that moves as a unit is specified.

Kuo, C. T. K.↗

NASA NTRS1977

Computer simulation of the motion of a straight dislocation line in concentrated solid solutions. II

An investigation was undertaken to determine if the size and modulus interaction of a solute atom with a screw dislocation and the modulus interaction with an edge dislocation contributed to strengthening, in addition to the size interaction with an edge dislocation. The results indicate that the size interaction between solute atom and an edge dislocation accounts for most of the solid solution strengthening in f.c.c. alloys. The contribution to the yield stress from the modulus interaction with an edge dislocation is less than 15%. The interaction between a solute atom and a screw dislocation is much less than that between a solute atom and an edge dislocation.

Kuo, C. T. K.↗