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Results for “Finite-Difference Methods”

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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
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

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

Development of solution techniques for nonlinear structural analysis

Nonlinear structural solution methods in the current research literature are classified according to order of the solution scheme, and it is shown that the analytical tools for these methods are uniformly derivable by perturbation techniques. A new perturbation formulation is developed for treating an arbitrary nonlinear material, in terms of a finite-difference generated stress-strain expansion. Nonlinear geometric effects are included in an explicit manner by appropriate definition of an applicable strain tensor. A new finite-element pilot computer program PANES (Program for Analysis of Nonlinear Equilibrium and Stability) is presented for treatment of problems involving material and geometric nonlinearities, as well as certain forms on nonconservative loading.

Vos, R. G.↗
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 NTRS1973

The transonic aerofoil problem with embedded shocks.

The integral equation approach to the mixed flow problem of infinite wings at high subsonic speeds is adopted for non-circulatory and circulatory (lifting) flows. The solutions are determined from a system of non-linear algebraic equations and, to ensure always unique solutions, the method of differentiation with respect to a parameter has been applied. The resulting Cauchy problem is then solved with the linearised flow solution as the initial value vector. For the case of embedded shocks in the flow field, the method of steepest descent has been added to the calculation scheme. Results for subcritical and supercritical flows past aerofoils are given and compared with solutions obtained by finite-difference techniques.

Norstrud, 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 NTRS1972

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Scott, R. K.↗
NASA NTRS1974

Numerical calculations of two dimensional, unsteady transonic flows with circulation

The feasibility of obtaining two-dimensional, unsteady transonic aerodynamic data by numerically integrating the Euler equations is investigated. An explicit, third-order-accurate, noncentered, finite-difference scheme is used to compute unsteady flows about airfoils. Solutions for lifting and nonlifting airfoils are presented and compared with subsonic linear theory. The applicability and efficiency of the numerical indicial function method are outlined. Numerically computed subsonic and transonic oscillatory aerodynamic coefficients are presented and compared with those obtained from subsonic linear theory and transonic wind-tunnel data.

Beam, R. M.↗
NASA NTRS1950

Direct method of design and stress analysis of rotating disks with temperature gradient

A method is presented for the determination of the contour of disks, typified by those of aircraft gas turbines, to incorporate arbitrary elastic-stress distributions resulting from either centrifugal or combined centrifugal and thermal effects. The specified stress may be radial, tangential, or any combination of the two. Use is made of the finite-difference approach in solving the stress equations, the amount of computation necessary in the evolution of a design being greatly reduced by the judicious selection of point stations by the aid of a design chart. Use of the charts and of a preselected schedule of point stations is also applied to the direct problem of finding the elastic and plastic stress distribution in disks of a given design, thereby effecting a great reduction in the amount of calculation. Illustrative examples are presented to show computational procedures in the determination of a new design and in analyzing an existing design for elastic stress and for stresses resulting from plastic flow.

Manson, S S↗
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 NTRS1973

Nonlinear transient responses of structures by the spatial finite-element method.

Based upon the Principle of Virtual Work and D'Alembert's Principle, the assumed-displacement version of the spatial finite-element method is developed to predict the large deflection transient responses of structures including elastic-plastic, strain hardening, and strain-rate material behavior. The formulations are developed in detail for curved beamlike structures undergoing planar (1) Bernouilli-Euler-type or (2) Timoshenko-type deformation behavior. The resulting equations of motion are solved timewise by a finite-difference numerical procedure. The present predictions are evaluated via several beam and ring examples for which experimental measurements and independent finite-difference predictions in both space and time are available; very good agreement is noted. The consequences of employing several types of timewise finite-difference operators are examined. Also, some comparisons between finite-element predictions and finite-difference predictions are shown to illustrate 'typical comparisons' of efficiency for a given prediction accuracy.

Wu, R. W.-H.↗
NASA NTRS1972

Numerical solutions of reactive fluid flows during postignition transients in hybrid rocket systems.

A computational method has been developed for the study of the post-ignition transients in hybrid rocket systems. The particular system chosen consisted of a gaseous oxidizer flowing within a tube of solid fuel, resulting in heterogeneous combustion. With the appropriate assumptions, two-dimensional, time-dependent conservation equations were derived for the reacting gas phase, and for the solid phase, in a cylindrical coordinate system. These were then programmed for numerical computation, using two implicit finite-difference schemes, the Lax-Wendroff scheme for the gas phase, and the Crank-Nicolson scheme for the solid phase. Appropriate initial and boundary conditions were represented, including heat and mass conservation at the interface between gas and solid. Initially, no attempt was made to relate the recession rate at the surface to the surface temperature, or to include heat transfer by radiation. A simple case was selected for preliminary calculations, with aluminum and oxygen as fuel and oxidizer, and aluminum oxide as the product.

Hung, W. S. Y.↗
NASA NTRS1974

The numerical calculation of laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for several separated flows and the accuracy of the method is verified. A computer program listing and complete solution case are included.

Klineberg, J. M.↗
NASA NTRS1974

On laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for a number of separated flows and the accuracy of the method is verified.

Klineberg, J. M.↗
NASA NTRS1972

Finite element analysis of large transient elastic-plastic deformations of simple structures, with application to the engine rotor fragment containment/deflection problem

Assumed-displacement versions of the finite-element method are developed to predict large-deformation elastic-plastic transient deformations of structures. Both the conventional and a new improved finite-element variational formulation are derived. These formulations are then developed in detail for straight-beam and curved-beam elements undergoing (1) Bernoulli-Euler-Kirchhoff or (2) Timoshenko deformation behavior, in one plane. For each of these categories, several types of assumed-displacement finite elements are developed, and transient response predictions are compared with available exact solutions for small-deflection, linear-elastic transient responses. The present finite-element predictions for large-deflection elastic-plastic transient responses are evaluated via several beam and ring examples for which experimental measurements of transient strains and large transient deformations and independent finite-difference predictions are available.

Wu, R. W.↗