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Results for “Finite-difference approximation”

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 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1956

Extrapolation techniques applied to matrix methods in neutron diffusion problems

A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.

Mccready, Robert R↗
NASA NTRS1995

Analysis of developing laminar flows in circular pipes using a higher-order finite-difference technique

A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.

Gladden, Herbert J.↗
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 NTRS1974

Numerical solution of the steady-state Navier-Stokes equations for hypersonic flow about blunt axisymmetric bodies

The steady-state Navier-Stokes equations are solved for hypersonic flow about blunt axisymmetric bodies. The equations of motion are solved by successive approximations using an implicit finite-difference scheme. The results are compared with viscous shock-layer theory, experimental data, and time-dependent solutions of the Navier-Stokes equations. It is demonstrated that viscous shock-layer theory is sufficiently accurate for the range of flight conditions normally encountered by entry vehicles.

Anderson, E. C.↗
NASA NTRS1973

An economical method of analyzing transient motion of gas-lubricated rotor-bearing systems.

A method of economically evaluating the hydrodynamic forces generated in a gas-lubricated tilting-pad bearing is presented. The numerical method consists of solving the case of the infinite width bearing and then converting this solution to the case of the finite bearing by accounting for end leakage. The approximate method is compared to the finite-difference solution of Reynolds equation and yields acceptable accuracy while running about one-hundred times faster. A mathematical model of a gas-lubricated tilting-pad vertical rotor systems is developed. The model is capable of analyzing a two-bearing-rotor system in which the rotor center of mass is not at midspan by accounting for gyroscopic moments. The numerical results from the model are compared to actual test data as well as analytical results of other investigators.

Falkenhagen, G. L.↗
NASA NTRS1973

FORTRAN program for calculating velocities and streamlines on the hub-shroud mid-channel flow surface of an axial- or mixed-flow turbomachine. 1: User's manual

A FORTRAN 4 computer program has been developed that obtains a subsonic or shock-free transonic flow solution on the hub-shroud mid-channel flow surface of a turbomachine. The blade row may be fixed or rotating, and may be twisted and leaned. Flow may be axial or mixed, up to 45 deg from axial. Upstream and downstream flow variables may vary from hub to shroud, and provision is made to correct for loss of stagnation pressure. The results include velocities, streamlines, and flow angles on the flow surface; and approximate blade surface velocities. Subsonic solutions are obtained by a finite-difference stream-function solution. Transonic solutions are obtained by a velocity-gradient method, using information from a finite-difference stream-function solution at a reduced mass flow.

Katsanis, T.↗
NASA NTRS1972

Prediction of far flow field in trailing vortices.

A finite-difference machine code is brought to bear on the wake-vortex problem in the quasi-cylindrical boundary-layer approximation. A turbulent-energy model containing new features is developed. Parameters of the model are evaluated by comparison of calculated velocities and turbulent intensities with measurements in an axisymmetric wake. Comparisons are made with a previous calculation of the decay of an isolated vortex and with wind tunnel and flight measurements in trailing vortices. A self-similar solution develops at large axial distance that decays with the square root of distance. A slower decay occurs in the preceding transition region.

Baldwin, B. S.↗
NASA NTRS1973

Apollo 15 measurement of lunar surface brightness temperatures - Thermal conductivity of the upper 1.5 meters of regolith.

In situ measurements of lunar surface brightness temperatures made as a part of the Apollo Lunar Surface Experiments Package at the Apollo 15 Hadley Rille landing site are reported. Data derived from five thermocouples of the Heat Flow Experiment, which are lying on or just above the surface, are used to examine the thermal properties of the upper 15 cm of the lunar regolith using eclipse and nighttime cool-down temperatures. Application of finite-difference techniques in modeling the lunar soil shows that the thermocouple data are best fit by a model consisting of a low-density and low-thermal conductivity surface layer approximately 2 cm thick overlying a region increasing in conductivity and density with depth. Conductivities on the order of 0.00001 W per cm per deg K are postulated for the upper layer, with conductivity increasing to the order of 0.0001 W per cm per deg K at depths exceeding 20 cm. An increase in mean temperature with depth indicates that the ratio of radiative to conductive transfer at 350 K is 2.7 for at least the upper few centimeters of lunar soil; this value is nearly twice that measured for returned lunar fines.

Keihm, S. J.↗
NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗
NASA NTRS1974

Viscous shock-layer flows for the space shuttle windward plane of symmetry

Predictions are presented for hypersonic, nonequilibrium viscous flows over nonanalytic blunt bodies at reentry conditions. Emphasis is placed on predictions of the viscous flowfield for the windward plane of symmetry of the space shuttle and other axisymmetric bodies which approximate the space shuttle geometry. The equations which govern the viscous shock-layer flow are presented, and the method by which the equations are solved is discussed. The predictions of the present finite-difference method are compared with predictions of other numerical methods as well as with experimental data.

Miner, E. W.↗
NASA NTRS1973

Decay of Far-Flowfield in Trailing Vortices

Methods for reduction of velocities in trailing vortices of large aircraft are of current interest for the purpose of shortening the waiting time between landings at central airports. We have made finite-difference calculations of the flow in turbulent wake vortices as an aid to interpretation of wind-tunnel and flight experiments directed toward that end. Finite-difference solutions are capable of adding flexibility to such investigations if they are based on an adequate model of turbulence. Interesting developments have been taking place in the knowledge of turbulence that may lead to a complete theory in the future. In the meantime, approximate methods that yield reasonable agreement with experiment are appropriate. The simplified turbulence model we have selected contains features that account for the major effects disclosed by more sophisticated models in which the parameters are not yet established. Several puzzles are thereby resolved that arose in previous theoretical investigations of wake vortices.

Baldwin, B. S.↗
NASA NTRS1973

On the numerical simulation of three-dimensional transonic flow with application to the C-141 wing

Results computed by a finite-difference, relaxation algorithm are presented for the supercritical flow (M = 0.825) about the C-141 airplane wing, which has sweep, taper, and twist. Comparisons with both wind-tunnel and flight data indicate that computed solutions of the classical transonic small disturbance equation can accurately simulate high Reynolds number flows when the shock sweep angle is small. It is also shown that this equation poorly approximates the complete potential equation when embedded shock waves are swept at angles greater than about 15 deg. Hence, a more consistent small disturbance equation is derived for use in more general cases.

Lomax, H.↗
NASA NTRS1973

Preliminary study of a wall stabilized constructed arc

An iterative, implicit, finite-difference numerical technique is described which is suitable for obtaining solutions to the governing equations for a gas flowing in an axially symmetric constricted-arc heater. The method is shown to provide adequate solutions for three cases of simple pipe flows found in the literature, and for flow in a constricted-arc heater by direct comparison with experimental data. The comparison with arc-heater data includes static pressure, arc voltage, and wall heat flux, all as a function of axial location, and a radial temperature profile at one axial station. The arc-heater data were taken with air as the test gas at a heater inlet pressure approximately 0.40 atm and two flow rates of 2.2 and 4.8 g/sec. The arc currents investigated were between 377 and 584 amperes.

Graves, R. A.↗
NASA NTRS1974

Computer program for calculating laminar, transitional, and turbulent boundary layers for a compressible axisymmetric flow

A finite-difference program is described for calculating the viscous compressible boundary layer flow over either planar or axisymmetric surfaces. The flow may be initially laminar and progress through a transitional zone to fully turbulent flow, or it may remain laminar, depending on the imposed boundary conditions, laws of viscosity, and numerical solution of the momentum and energy equations. The flow may also be forced into a turbulent flow at a chosen spot by the data input. The input may contain the factors of arbitrary Reynolds number, free-stream Mach number, free-stream turbulence, wall heating or cooling, longitudinal wall curvature, wall suction or blowing, and wall roughness. The solution may start from an initial Falkner-Skan similarity profile, an approximate equilibrium turbulent profile, or an initial arbitrary input profile.

Albers, J. A.↗
NASA NTRS1973

Approximate analysis of containment/deflection ring responses to engine rotor fragment impact.

The transient responses of containment and/or deflection rings to impact from an engine rotor-blade fragment are analyzed. Energy and momentum considerations are employed in an approximate analysis to predict the collision-induced velocities which are imparted to the fragment and to the affected ring segment. This collision analysis is combined with the spatial finite-element representation of the ring and a temporal finite-difference solution procedure to predict the resulting large transient elastic-plastic deformations of containment/deflection rings. Some comparisons with experimental data are given.

Wu, R. W.-H.↗
NASA NTRS1973

Nonlinear vibrations of rectangular plates.

A finite-difference method is developed to determine the large amplitude dynamic responses of thin elastic plates subjected to uniform pressure pulse-type loads. Four different sets of boundary conditions are considered. Some specific problems are solved. The results are compared with approximate solutions obtained by Yamaki (1961). The numerical method presented provides an accurate and efficient approximate solution to the problem, and should be useful as a check on other approximate methods. The grid-size and the time-step necessary for obtaining numerical stability depend on the particular problem. For many cases the method converges rapidly and a rather large grid-size and time-step is adequate.

Bayles, D. J.↗
NASA NTRS1973

Comparison of finite-difference schemes for analysis of shells of revolution

Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.

Noor, A. K.↗
NASA NTRS1974

Hypersonic chemically reacting viscous shock layers over sphere-cones and cylinder-wedges

Hypersonic, nonequilibrium viscous flow over nonanalytic blunt bodies is considered. The equations which govern the viscous shock-layer flow are presented and the method by which the equations are solved is discussed. The predictions of the present finite-difference method are compared with other numerical predictions as well as with experimental data. Three flow conditions are considered; the experimental, wind tunnel conditions of Pappas and Lee for a 7.5 deg sphere-cone at Mach 13 and two cases considered by Kang and Dunn, a 9 deg sphere-cone at 233,000 ft and a 20 deg sphere-cone at 280 and 310 Kft. The predictions of the present method agreed well with the experimental heat-transfer data, but substantial differences were found between the present predictions and the more approximate predictions of Kang and Dunn for heat-transfer distributions and temperature profiles.

Miner, E. W.↗