Search NASA⌕ Search

SEARCH · Search NASA

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
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 NTRS1975

Hypersonic ionizing air viscous shock-layer flows over nonanalytic blunt bodies

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. The principal emphasis is placed on predictions of the viscous flowfield for the windward plane of symmetry of the space shuttle orbiter and other axisymmetric bodies which approximate the shuttle orbiter geometry. Experimental data on two slender sphere-cones at hypersonic conditions are also considered. The present predictions agreed well with experimental data and with the past predictions. Substantial differences were found between present predictions and more approximate methods.

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 NTRS1976

Combustion of hydrogen injected into a supersonic airstream (a guide to the HISS computer program)

A computer program based on a finite-difference, implicit numerical integration scheme is described for the prediction of hydrogen injected into a supersonic airstream at an angle ranging from normal to parallel to the airstream main flow direction. Results of calculations for flow and thermal property distributions were compared with 'cold flow data' taken by NASA/Langley and show excellent correlation. Typical results for equilibrium combustion are presented and exhibit qualitatively plausible behavior. Computer time required for a given case is approximately one minute on a CDC 7600. A discussion of the assumption of parabolic flow in the injection region is given which demonstrates that improvement in calculation in this region could be obtained by a partially-parabolic procedure which has been developed. It is concluded that the technique described provides an efficient and reliable means for analyzing hydrogen injection into supersonic airstreams and the subsequent combustion.

Dyer, D. F.↗
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 NTRS1975

A cubic spline approximation for problems in fluid mechanics

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1975

Viscous flow solutions with a cubic spline approximation

A cubic spline approximation is used for the solution of several problems in fluid mechanics. This procedure provides a high degree of accuracy even with a nonuniform mesh, and leads to a more accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several typical integration schemes are presented. For two-dimensional flows a spline-alternating-direction-implicit (SADI) method is evaluated. The spline procedure is assessed and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
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.↗
NASA NTRS1974

A comparison of numerical solutions of the advective equation

Second- and third-order finite-difference methods recently applied to problems in high-speed fluid flow are applied to the model advection equation cast in conservative form. The differencing methods considered use forward time differencing with both centered and preferential space differences employed in the predictor-corrector sequences. The free parameter required for stability in the third-order method is adjusted to cause the solution to be either minimum dispersive or minimum dissipative in nature. Results indicate that the third-order method using minimum dispersion is the most accurate method tested. Computer time requirements are approximately twice those needed for second-order techniques.

Anderson, D.↗
NASA NTRS1975

An investigation of several numerical procedures for time-asymptotic compressible Navier-Stokes solutions

The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.

Rudy, D. H.↗
NASA NTRS1976

Implicit finite-difference procedures for the computation of vortex wakes

Implicit finite-difference procedures for the primitive form of the incompressible Navier-Stokes and the compressible Euler equations are used to compute vortex wake flows. The partial differential equations in strong conservation-law form are transformed to cluster grid points in regions with large changes in vorticity. In addition to clustering, fourth-order accurate, spatial difference operators are used to help resolve the flow-field gradients. The use of implicit time-differencing permits large time steps to be taken since temporal variations are typically small. Computational efficiency is achieved by approximate factorization. Both two-dimensional and preliminary three-dimensional calculations are described and qualitatively compared with existing experimental data.

Steger, J. L.↗