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NASA NTRS1981

Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Implicit, noniterative, finite difference schemes were recently developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture was that unconditional stability requires implicit numerical boundary conditions. An apparent counter example was the space time extrapolation considered by Gustafsson, Kreiss, and Sunstrom. Spatial (implicit) and space time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals are examined. The results indicate that for unconditional stability with a finite number of spatial mesh intervals, the numerical boundary conditions must be implicit.

Beam, R. M.↗

NASA NTRS1981

Numerical model of the moon's rotation

The differential equations for the Euler angles that define the moon's orientation with respect to an inertial coordinate frame have been integrated numerically. This numerical model of the moon's rotation, when used in conjunction with current lunar-orbit and earth-rotation models, fits lunar laser ranging observations over a five-year period within 28 cm (rms). The present model is also compared with the numerical model of Williams (1975) and Eckhardt's (1981) semianalytical model.

Cappalo, R. J.↗

NASA NTRS1981

A numerical study of two-dimensional shock vortex interaction

The numerical analysis of shock vortex interaction in a two-dimensional channel is discussed in this paper. The Euler equations and the two-step MacCormack scheme are employed in this numerical analysis. The comparison of the numerical results with the theoretical analyses and experimental results in the literature has provided unique insights into the dynamics of the transient interaction process and the needs for further theoretical developments. The results are useful in applications to problems in aeroacoustics and the transient aerodynamics of high performance aircraft.

Pao, S. P.↗

NASA NTRS1981

Manufacturing in space: Fluid dynamics numerical analysis

Natural convection in a spherical container with cooling at the center was numerically simulated using the Lockheed-developed General Interpolants Method (GIM) numerical fluid dynamic computer program. The numerical analysis was simplified by assuming axisymmetric flow in the spherical container, with the symmetry axis being a sphere diagonal parallel to the gravity vector. This axisymmetric spherical geometry was intended as an idealization of the proposed Lal/Kroes growing experiments to be performed on board Spacelab. Results were obtained for a range of Rayleigh numbers from 25 to 10,000. For a temperature difference of 10 C from the cooling sting at the center to the container surface, and a gravitional loading of 0.000001 g a computed maximum fluid velocity of about 2.4 x 0.00001 cm/sec was reached after about 250 sec. The computed velocities were found to be approximately proportional to the Rayleigh number over the range of Rayleigh numbers investigated.

Robertson, S. J.↗

NASA NTRS1982

The numerical analysis of the rotational theory for the formation of lunar globules

The morphology of lunar globules is studied through the application of a numerical analysis of their rotation in space during cooling. It is assumed that molten rock is shot from the surface of the moon, solidifies in space above the moon and then falls back to the surface. The rotational theory studied makes the following assumptions: the volume of the molten rock does not change during cooling; the angular momentum is conserved; there are no internal motions because of the high viscosity of the molten rock, i.e., in equilibrium the globule is rotating as a rigid body; finally, the kinetic reaction of the globule to the forces is fast relative to the rate of cooling, i.e., the globule reaches equilibrium at constant energy. These assumptions are subjected to numerical analysis yielding good agreement between the actual globule shapes and the numerical results, but leaving some doubt as to the validity of the rotational theory due to the failure to establish the existence of true local minima and an incomplete understanding of the thermokentics.

Ross, J.↗

NASA NTRS1982

Engine dynamic analysis with general nonlinear finite element codes. Part 2: Bearing element implementation overall numerical characteristics and benchmaking

Finite element codes are used in modelling rotor-bearing-stator structure common to the turbine industry. Engine dynamic simulation is used by developing strategies which enable the use of available finite element codes. benchmarking the elements developed are benchmarked by incorporation into a general purpose code (ADINA); the numerical characteristics of finite element type rotor-bearing-stator simulations are evaluated through the use of various types of explicit/implicit numerical integration operators. Improving the overall numerical efficiency of the procedure is improved.

Padovan, J.↗

NASA NTRS1981

Simulation of a numerical filter for enhancing earth radiation budget measurements

The Earth Radiation Budget Experiment has the objective to collect the radiation budget data which are needed to determine the radiation budget at the top of the atmosphere (TOA) on a regional scale. A second objective is to determine the accuracy of the results. Three satellites will carry wide and medium field of view radiometers which measure the longwave and shortwave components of radiation. Scanning radiometers will be included to detect small spatial features. A proposal has been made to employ for the nonscanning radiometers a one-dimensional numerical filter which reduces satellite measurements to TOA radiant excitances. The numerical filter was initially formulated by House (1980). It enhances the resolution of the radiation budget along the satellite groundtrack. The accuracy of the numerical filter estimate is studied by simulating the data gathering and measurement inversion process. The results of the study are discussed, taking into account two error sources.

Green, R. N.↗

NASA NTRS1982

Numerical simulation of viscous-inviscid interactions on indented nose tips

An implicit numerical algorithm to solve the unsteady thin-layer Navier-Stokes equations in a strong conservative form has been used to compute the viscous flow over indented nose tips placed in a supersonic free stream. Numerical solutions are presented for axisymmetric and three-dimensional indented configurations for laminar flow conditions. Results demonstrate the capability of the present numerical procedure to predict flow fields that contain strong viscous-inviscid interactions, including boundary-layer separation, reattachment, and embedded discontinuities. Comparisons with available experimental data for the shock shape and surface pressure are also given.

Rizk, Y. M.↗

NASA NTRS1981

International Conference on Numerical Methods in Fluid Dynamics, 7th, Stanford University, Stanford and Moffett Field, CA, June 23-27, 1980, Proceedings

Topics discussed include polygon transformations in fluid mechanics, computation of three-dimensional horseshoe vortex flow using the Navier-Stokes equations, an improved surface velocity method for transonic finite-volume solutions, transonic flow calculations with higher order finite elements, the numerical calculation of transonic axial turbomachinery flows, and the simultaneous solutions of inviscid flow and boundary layer at transonic speeds. Also considered are analytical solutions for the reflection of unsteady shock waves and relevant numerical tests, reformulation of the method of characteristics for multidimensional flows, direct numerical simulations of turbulent shear flows, the stability and separation of freely interacting boundary layers, computational models of convective motions at fluid interfaces, viscous transonic flow over airfoils, and mixed spectral/finite difference approximations for slightly viscous flows.

Reynolds, W. C.↗

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 NTRS1982

Numerical simulation of magnetohydrodynamic shock propagation in the corona

Recent developments in the field of numerical simulation models for the study of shock wave propagation in the corona are presented. These models are based on gasdynamic (GD) and ideal (that is, dissipationless, except at shocks) magnetohydrodynamic (MHD) theories. The characteristics and physical interpretations of the results derived from these models are discussed in some detail. The most significant physical results obtained to date are provided by the two-dimensional non-planar, time-dependent, MHD numerical simulation model. In this model, the non-linear interaction among the three essential MHD waves, i.e., fast-, slow-, and Alfven waves are demonstrated. Finally, the physical relevance of these numerical simulation models in relation to observed solar activity is presented.

Wu, S. T.↗

NASA NTRS1982

On the energy dependence of the radial diffusion coefficient and spectra of inner radiation belt particles - Analytic solutions and comparison with numerical results

A theoretical method by which the energy dependence of the radial diffusion coefficient may be deduced from spectral observations of the particle population at the inner edge of the earth's radiation belts is presented. This region has previously been analyzed with numerical techniques; in this report an analytical treatment that illustrates characteristic limiting cases in the L shell range where the time scale of Coulomb losses is substantially shorter than that of radial diffusion (L approximately 1-2) is given. It is demonstrated both analytically and numerically that the particle spectra there are shaped by the energy dependence of the radial diffusion coefficient regardless of the spectral shapes of the particle populations diffusing inward from the outer radiation zone, so that from observed spectra the energy dependence of the diffusion coefficient can be determined. To insure realistic simulations, inner zone data obtained from experiments on the DIAL, AZUR, and ESRO 2 spacecraft have been used as boundary conditions. Excellent agreement between analytic and numerical results is reported.

Westphalen, H.↗

NASA NTRS1983

Numerical computation of transonic flow governed by the full-potential equation

Numerical solution techniques for solving transonic flow fields governed by the full potential equation are discussed. In a general sense relaxation schemes suitable for the numerical solution of elliptic partial differential equations are presented and discussed with emphasis on transonic flow applications. The presentation can be divided into two general categories: An introductory treatment of the basic concepts associated with the numerical solution of elliptic partial differential equations and a more advanced treatment of current procedures used to solve the full potential equation for transonic flow fields. The introductory material is presented for completeness and includes a brief introduction (Chapter 1), governing equations (Chapter 2), classical relaxation schemes (Chapter 3), and early concepts regarding transonic full potential equation algorithms (Chapter 4).

Holst, T. L.↗

NASA NTRS1983

A numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution

A numerical method is presented for studying one-dimensional electron plasma evolution under typical interplanetary conditions. The method applies the Fourier-Fourier transform approach to a plasma model that is a generalization of the electrostatic Vlasov-Poisson system of equations. Conservation laws that are modified to include the plasma model generalization and also the boundary effects of nonperiodic solutions are given. A new conservation law for entropy in the transformed space is then introduced. These conservation laws are used to verify the numerical solutions. A discretization error analysis is presented. Two numerical instabilities and the methods used for their suppression are treated. It is shown that in interplanetary plasma conditions, the bump-on-tail instability produces significant excitation of plasma oscillations at the Bohm-Gross frequency and its second harmonic. An explanation of the second harmonic excitation is given in terms of wave-wave coupling during the growth phase of the instability.

Klimas, A. J.↗

NASA NTRS1983

A numerical model of gravity wave breaking and stress in the mesosphere

The goal of the study is to calculate numerically the deceleration and heating caused by breaking gravity waves. The effect of the radiative dissipation of the wave is included as vertical-wavelength-dependent Newtonian cooling. The parameterization for zonal deceleration is extended by breaking gravity waves (Lindzen, 1981) to include the turbulent diffusion of heat and momentum. After describing the numerical model, the numerical results are presented and compared with the parameterizations in a noninteractive model of the mean zonal wind. Attention is then given to the transport of constituents by gravity waves and the attendant turbulent zone. It is noted that if gravity wave breaking were not an intermittent process, gravity wave stresses would produce an adiabatic mesosphere with a zonal mean velocity close to the phase speed of the breaking wave.

Schoeberl, M. R.↗

NASA NTRS1983

Numerical investigation of unsteady flow development in a nozzle-duct configuration

A computational approach has been used to resolve the transient phenomena caused by a rocket engine starting up and exhausting into a short duct. The results are obtained from the finite-difference Navier-Stokes, continuity, and energy equations for a single component gas. Using a recently developed numerical technique to account for the time-varying boundary conditions at the nozzle throat and the duct openings, the numerical simulation of the idealized ignition flow model has yielded some insight into the complex interactions between the wave patterns and the vortical flow. The present findings verify the hypothesis that a lateral expansion of the jetstream outside the nozzle will generate an over-pressure pulse on the duct wall if the jetstream is restricted within the nozzle by flow separation. The numerical simulations of this complex flow problem should be useful for improving simple correlation techniques used for engineering purposes.

Li, C. P.↗

NASA NTRS1982

Numerical generation of composite three dimensional grids by quasilinear elliptic systems

A three-dimensional grid generation technique designed to numerically construct a boundary-conforming grid within a three-dimensional region bounded by a closed surface is described. The boundary values are generated numerically by a modified elliptic system and are used to compute grid control parameters that are contained in the elliptic systems. The interior grid distribution is governed by the distribution of points on the boundary as well as by the boundary's geometric shape. The composite three-dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions. The details of the method and its implementation are presented, displaying numerical results for three-dimensional composite grid for a wing-body combination and surface grids.

Thomas, P. D.↗

NASA NTRS1983

Hybrid experimental-numerical stress analysis

The hybrid experimental-numerical stress-analysis technique, which saw limited applications during the 1950's, has been resurrected with the vastly improved numerical techniques of the 1970's. By inputing the experimental results as initial and boundary conditions, modern computer codes are executed in its generation and application modes to yield results which are unobtainable when only one of the two techniques is used. The hybrid technique thus exemplifies the complementary role of the experimental and numerical techniques.

Kobayashi, A. S.↗