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At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Error propagation in the numerical solutions of the differential equations of orbital mechanics

The relationship between the eigenvalues of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigenvalue when linearized about the initial conditions. The real positive eigenvalue causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigenvalues and is numerically stable.

Bond, V. R.↗
NASA NTRS1979

Applying integrals of motion to the numerical solution of differential equations

A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.

Jezewski, D. J.↗
NASA NTRS1980

Applying integrals of motion to the numerical solution of differential equations

A method is developed for using the integrals of systems of nonlinear, ordinary, differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scalar or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.

Vezewski, D. J.↗
NASA NTRS2011

Numerical Solution of the Radiative Transfer Equation: X-Ray Spectral Formation from Cylindrical Accretion onto a Magnetized Neutron Star

Predicting the emerging X-ray spectra in several astrophysical objects is of great importance, in particular when the observational data are compared with theoretical models. This requires developing numerical routines for the solution of the radiative transfer equation according to the expected physical conditions of the systems under study. Aims. We have developed an algorithm solving the radiative transfer equation in the Fokker-Planck approximation when both thermal and bulk Comptonization take place. The algorithm is essentially a relaxation method, where stable solutions are obtained when the system has reached its steady-state equilibrium. Methods. We obtained the solution of the radiative transfer equation in the two-dimensional domain defined by the photon energy E and optical depth of the system pi using finite-differences for the partial derivatives, and imposing specific boundary conditions for the solutions. We treated the case of cylindrical accretion onto a magnetized neutron star. Results. We considered a blackbody seed spectrum of photons with exponential distribution across the accretion column and for an accretion where the velocity reaches its maximum at the stellar surface and at the top of the accretion column, respectively. In both cases higher values of the electron temperature and of the optical depth pi produce flatter and harder spectra. Other parameters contributing to the spectral formation are the steepness of the vertical velocity profile, the albedo at the star surface, and the radius of the accretion column. The latter parameter modifies the emerging spectra in a specular way for the two assumed accretion profiles. Conclusions. The algorithm has been implemented in the XPEC package for X-ray fitting and is specifically dedicated to the physical framework of accretion at the polar cap of a neutron star with a high magnetic field (approx > 10(exp 12) G). This latter case is expected to be of typical accreting systems such as X-ray pulsars and supergiant fast X ray transients.

Fairnelli, R.↗
NASA NTRS1977

Numerical solution of supersonic laminar flow over a three-dimensional compression corner

A newly developed, rapid numerical scheme is extended to three dimensions to solve the complete Navier-Stokes equations for a supersonic, laminar flow over a compression corner with sidewall effects. The program is coded so that it can solve for a general curved ramp surface geometry such as found in inlets and fuselage-wing-flap junctions. A test case of Mach 3.0 flow is calculated. In regions where three-dimensional effects are small, good agreement is obtained between the present calculation and previous two-dimensional solutions. In other regions, the results show complex three-dimensional flow-field interactions including shock-shock and shock/boundary-layer interactions

Hung, C. M.↗
NASA NTRS1983

Numerical solution of the problem of flame propagation by the use of the random element method

A numerical, grid-free algorithm is presented for one-dimensional reaction-diffusion model of laminar flame propagation in premixed gases. It is based on the random element method we developed for the analysis of diffusional processes. The effect of combustion is taken into account by applying the principle of fractional steps to separate the process of diffusion, modeled by the random walk of computational elements, from the exothermic effects of chemical reaction, monitoring their strength. The validity of the algorithm is demonstrated by application to flame propagation problems for which exact solutions exist. The flame speed evaluated by its use oscillates around the exact value at a relatively small amplitude, while the temperature and species concentration profiles are self-correcting in their convergence to the exact solution. A satisfactory resolution is obtained by the use of quite a small number of computational elements which automatically adjust their distribution of fit sharp gradients.

Ghoniem, A. F.↗
NASA NTRS1939

Some notes on the numerical solution of shear-lag and mathematically related problems

The analysis of box beams with shear deformation of the flanges can be reduced to the solution of a differential equation. The same equation is met in other problems of stress analysis. No analytical solutions of this equation can be given for practical cases, and numerical methods of evaluation must be used. Available methods are briefly discussed. Two numerical examples show the application of the step-by-step method of integration to shear-lag problems.

Kuhn, Paul↗
NASA NTRS1978

Numerical solution of a three-dimensional shock wave and turbulent boundary-layer interaction

A rapid numerical scheme is used to solve the complete mass-averaged Navier-Stokes equations for supersonic turbulent flow over a three-dimensional compression corner. A simple eddy viscosity model is developed, and the interaction of a swept shock wave and a three-dimensional turbulent boundary layer is studied. Good agreement is obtained between the present results and experimental measurements for the case of a wedge with an angle of 6 deg on a flat-plate sidewall. For the case of a 12-deg wedge angle, the computed results do not show the existence of a peak pressure found experimentally. However, the range of interaction, the plateau pressure, and the peak heat transfer are closely predicted for all cases. The high heat transfer near the axial corner is due to the thinning of the boundary layer and inflow of fresh high-momentum fluid. The heat transfer is relieved through pressure reduction and boundary-layer thickening.

Hung, C. M.↗
NASA NTRS1980

Comparison between optical measurements and a numerical solution of the flow field within a transonic axial-flow compressor rotor

A comparison between numerical and experimental results is presented for the flowfield within a transonic axial-flow compressor rotor. The rotor was tested at design speed and a wide open throttle discharge condition. The relative tip Mach number was 1.4. A laser anemometer system was used to measure velocity and flow angle upstream, within, and downstream of the rotor. A holographic interferometer was used to visualize the rotor shock system near the tip. The computational procedure solves the full three-dimensional Euler equations using a time-marching technique. Shock location and shape determined from the two optical systems are compared. Calculated relative Mach number and flow angle contours, shock locations, and shock strength are compared to values measured with the laser anemometer.

Strazisar, A. J.↗
NASA NTRS2009

Numerical Solutions for the CAWAPI Configuration on Structured Grids at NASA LaRC, United States

In this chapter numerical simulations of the flow around F-16XL are performed as a contribution to the Cranked Arrow Wing Aerodynamic Project International (CAWAPI) using the PAB3D CFD code. Two turbulence models are used in the calculations: a standard k-epsilon model, and the Shih-Zhu-Lumley (SZL) algebraic stress model. Seven flight conditions are simulated for the flow around the F-16XL where the free stream Mach number varies from 0.242 to 0.97. The range of angles of attack varies from 0 deg to 20 deg. Computational results, surface static pressure, boundary layer velocity profiles, and skin friction are presented and compared with flight data. Numerical results are generally in good agreement with flight data, considering that only one grid resolution is utilized for the different flight conditions simulated in this study. The Algebraic Stress Model (ASM) results are closer to the flight data than the k-epsilon model results. The ASM predicted a stronger primary vortex, however, the origin of the vortex and footprint is approximately the same as in the k-epsilon predictions.

Elmiligui, Alaa A.↗
NASA NTRS1976

Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations

A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.

Hirsh, R. S.↗
NASA NTRS1979

Numerical solution of the azimuthal-invariant thin-layer Navier-Stokes equations

The paper reports a numerical procedure developed for a two-dimensional azimuthal (or planar) invariant form of the thin-layer Navier-Stokes equations. Generalization of the governing equations is described, and the equations are solved with an implicit approximate factorization finite-difference scheme. Inviscid and viscous results are presented for both external and internal flows and for spinning and nonspinning bodies.

Nietubicz, C. J.↗
NASA NTRS1976

A numerical solution of the flow field over a transonic airfoil including strong-shock-induced flow separation

An approximate numerical method has been developed for computing the entire transonic flow field over an airfoil, including the boundary layer, the shock-wave/boundary-layer interaction, and the shock-induced-separation. The inviscid flow, the turbulent boundary layer in the weak interaction zone, and the strong interaction elliptic region are solved separately. The inviscid flow is represented by the solution of the transonic small-disturbance equation, while the boundary layer is solved by the generalized Galerkin method, and the strong interaction region (including the wake) is computed from the time-averaged Navier-Stokes equations. A second-order interaction is computed iteratively using the inviscid pressure distribution as a boundary condition for the boundary layer and Navier-Stokes codes, and the viscous displacement thickness to modify the airfoil geometry for the inviscid computation. The computed pressure distributions and shock-wave profiles are compared with experimental data. Agreement is very good except that the pressure recovery on the trailing edge is greater than the experimental data.

Seginer, A.↗
NASA NTRS1985

Numerical solutions of 3-dimensional Navier-Stokes equations for closed bluff-bodies

The Navier-Stokes equations are solved numerically. These equations are unsteady, compressible, viscous, and three-dimensional without neglecting any terms. The time dependency of the governing equations allows the solution to progress naturally for an arbitrary initial guess to an asymptotic steady state, if one exists. The equations are transformed from physical coordinates to the computational coordinates, allowing the solution of the governing equations in a rectangular parallelepiped domain. The equations are solved by the MacCormack time-split technique which is vectorized and programmed to run on the CDc VPS 32 computer. The codes are written in 32-bit (half word) FORTRAN, which provides an approximate factor of two decreasing in computational time and doubles the memory size compared to the 54-bit word size.

Abolhassani, J. S.↗
NASA NTRS1980

Comments on numerical solution of boundary value problems of the Laplace equation and calculation of eigenvalues by the grid method

The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.

Lyusternik, L. A.↗
NASA NTRS1976

Numerical Solutions of Supersonic and Hypersonic Laminar Compression Corner Flows

An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. A fine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer. Good agreement is obtained between the present computed results and experimental measurement for a Mach number of 14.1 and a Reynolds number of 1.04 x 10(exp 5) with wedge angles of 15 deg, 18 deg, and 24 deg. The details of the pressure variation across the boundary layer are given, and a correlation between the leading edge shock and the peaks in surface pressure and heat transfer is observed.

Hung, C. M.↗
NASA NTRS1979

Numerical solution of the Navier-Stokes equations for arbitrary two-dimensional multi-element airfoils

The development of a numerical simulation of time dependent, turbulent, compressible flow about two dimensional multi-element airfoils of arbitrary shape is described. The basis of this simulation is a technique of automatic numerical generation of coordinate systems fitted to the multiple bodies regardless of their number or shape. Procedures developed whereby the coordinate lines are automatically concentrated in the boundary layer at any Reynolds number are discussed. The compressible turbulent solution involves an algebraic eddy viscosity turbulence model. The laminar version was run for transonic flow at free stream Mach numbers up to 0.9.

Thompson, J. F.↗