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NASA NTRS1991

Elimination of secular terms from the differential equations for the elements of perturbed two-body motion

In 1961, Sperling linearized and regularized the differential equations of motion of the two-body problem by changing the independent variable from time to fictitious time by Sundman's transformation (r = dt/ds) and by embedding the two-body energy integral and the Laplace vector. In 1968, Burdet developed a perturbation theory which was uniformly valid for all types of orbits using a variation of parameters approach on the elements which appeared in Sperling's equations for the two-body solution. In 1973, Bond and Hanssen improved Burdet's set of differential equations by embedding the total energy (which is a constant when the potential function is explicitly dependent upon time.) The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion. In the process, another element which is proportional to a component of the angular momentum was introduced. Recently trajectories computed during numerical studies of atmospheric entry from circular orbits and low thrust beginning in near-circular orbits exhibited numerical instability when solved by the method of Bond and Gottlieb (1989) for long time intervals. It was found that this instability was due to secular terms which appear on the righthand sides of the differential equations of some of the elements. In this paper, this instability is removed by the introduction of another vector integral called the delta integral (which replaces the Laplace Vector) and another scalar integral which removes the secular terms. The introduction of these integrals requires a new derivation of the differential equations for most of the elements. For this rederivation, the Lagrange method of variation of parameters is used, making the development more concise. Numerical examples of this improvement are presented.

Bond, Victor R.↗

NASA NTRS1990

Direct simulation of turbulent combustion

Understanding and modeling of turbulent combustion are key-problems in the computation of numerous practical systems. Because of the lack of analytical theories in this field and of the difficulty of performing precise experiments, direct simulation appears to be one of the most attractive tools to use in addressing this problem. The present work can be split into two parts: (1) Development and validation of a direct simulation method for turbulent combustion; (2) Applications of the method to premixed turbulent combustion problems. The goal of part 1 is to define and to test a numerical method for direct simulation of reacting flows. A high level of confidence should be attached to direct simulation results, and this can only be achieved through extensive validation tests. In part 2, direct simulation is used to address some of the many critical problems related to turbulent combustion. At the present time, I have limited this work to premixed combustion and considered only four basic issues: (1) The effect of pressure waves on flame propagation; (2) The interaction between flame fronts and vortices; (3) The influence of curvature on premixed flame fronts; and (4) The validation of flamelet models for premixed turbulent combustion.

Poinsot, T. J.↗

NASA NTRS1991

A Newton algorithm for complex curve fitting

The problem of synthesizing transfer functions from frequency response measurements is considered. Given a complex vector representing the measured frequency response of a physical system, a transfer function of specified order is determined that minimizes the sum of the magnitude-squared of the frequency response errors. This nonlinear least squares minimization problem is solved by an iterative global descent algorithm of the Newton type which converges quadratically near the minimum. The unknown transfer function is expressed as a sum of second order rational polynomials, a parameterization that facilitates a numerically robust computer implementation. The algorithm is developed for single-input, single-output, causal, stable transfer functions.

Spanos, J. T.↗

NASA NTRS1992

Global transformation of rotation matrices to Euler parameters

A global algorithm for transforming rotation matrices to Euler parameters is presented. Although it has no apparent computational or numerical advantage over the known algorithms, it elucidates the relationship between the rotation matrix form and the Euler parameter form. It employs the singular-value decomposition, a numerically ideal algorithm involving orthogonal transformations. Attitude parameterization is reviewed and an analytical framework is provided.

Paielli, Russell A.↗

NASA NTRS1991

Examination of various turbulence models for application in liquid rocket thrust chambers

There is a large variety of turbulence models available. These models include direct numerical simulation, large eddy simulation, Reynolds stress/flux model, zero equation model, one equation model, two equation k-epsilon model, multiple-scale model, etc. Each turbulence model contains different physical assumptions and requirements. The natures of turbulence are randomness, irregularity, diffusivity and dissipation. The capabilities of the turbulence models, including physical strength, weakness, limitations, as well as numerical and computational considerations, are reviewed. Recommendations are made for the potential application of a turbulence model in thrust chamber and performance prediction programs. The full Reynolds stress model is recommended. In a workshop, specifically called for the assessment of turbulence models for applications in liquid rocket thrust chambers, most of the experts present were also in favor of the recommendation of the Reynolds stress model.

Hung, R. J.↗

NASA NTRS1992

Wake in faint television meteors

The two component dustball model was used in numerical lag computation. Detached grain lag is typically less than 2 km, with expected wakes of a few hundred meters. True wake in television meteors is masked by apparent wake due to the combined effects of image persistence and blooming. To partially circumvent this problem, we modified a dual MCP intensified CID video system by addition of a rotating shutter to reduce the effective exposure time to about 2.0 ms. Preliminary observations showed that only 2 of 27 analyzed meteors displayed statistically significant wake.

Robertson, M. C.↗

NASA NTRS1993

Exact methods for modal transient response analysis including feedback control

This paper presents a modal method for the analysis of controlled structural systems that retains the uncoupled nature of the classical transient response analysis of a structure subjected to a prescribed time-varying load. The control force is expanded as a Taylor series that remains on the right side of the equations, and it does not lead to a computational approach that requires coupling between modes on the left side. Retaining a sufficient number of terms in the series produces a solution to the modal equations that is accurate to machine precision. The approach is particularly attractive for large problems in which standard matrix exponential methods become computationally prohibitive. Numerical results are presented to show the accuracy and efficiency of the proposed approach for dynamic feedback compensation of a truss structure with local member modes in the controller bandwidth.

Anderson, Melvin S.↗

NASA NTRS1992

Analysis of rotationally symmetric arrays of apertures on conducting spherical surfaces

Rotationally symmetric arrays of circular waveguide fed apertures on conducting spherical surfaces have been analyzed. Representation of the outside electromagnetic fields in terms of spherical transmission line modes permit the array analysis to be viewed as a spherical waveguide junction discontinuity problem. The apertures, in general, carry dual polarized circular waveguide TE11 modes. Two independent formulations - the unit cell method and the eigenexcitation method that take rotational symmetry into account - are developed and are checked against each other numerically for computational accuracy. These two formulations are analytically linked as discrete Fourier series pairs. It is seen that mutual coupling along the array decays more rapidly when the polarization of excitation is perpendicular to the array plane than when it is parallel to it. This analysis has been applied in the performance analysis of the Dome Lens phased array radar antenna.

Bondyopadhyay, Probir K.↗

NASA NTRS1992

Three-dimensional analytical model for the mixed layer depth

A three-dimensional (3D) semianalytical mixed-layer thermal inertia model for the mixed layer depth is developed, which includes the horizontal advection and diffusion processes modeled previously by Yan et al. (1991) in a one-dimensional (1D) model. The 3D model, which is based on the 3D thermal energy conservation equation, can be used as a supplement to the 1D model in the area where advection is important and for a finer resolution grid prediction. The model can be easily forced by remotely sensed data and is much simpler to compute than numerical 3D mixed-layer turbulence closure models. The new model was verified using data from the AVHRR/multichannel sea surface temperature data set and the Cooperative Ocean Atmosphere Data Set for the North Pacific Ocean, showing good agreement.

Yan, Xiao-Hai↗

NASA NTRS1992

PNS predictions for supersonic/hypersonic flows over finned missile configurations

Finned missile design entails accurate and computationally fast numerical techniques for predicting viscous flows over complex lifting configurations at small to moderate angles of attack and over Mach 3 to 15; these flows are often characterized by strong embedded shocks, so that numerical algorithms are also required to capture embedded shocks. The recent real-gas Flux Vector Splitting technique is here extended to investigate the Mach 3 flow over a typical finned missile configuration with/without side fin deflections. Elliptic grid-generation techniques for Mach 15 flows are shown to be inadequate for Mach 3 flows over finned configurations and need to be modified. Fin-deflection studies indicate that even small amounts of missile fin deflection can substantially modify vehicle aerodynamics. This 3D parabolized Navier-Stokes scheme is also extended into an efficient embedded algorithm for studying small axially separated flow regions due to strong fin and control surface deflections.

Bhutta, Bilal A.↗

NASA NTRS1993

Numerical study of the transient flow in the driven tube and the nozzle section of a shock tunnel

The initial flow in a shock tunnel was examined numerically using computational fluid dynamics (CFD). A finite-volume total variation diminishing (TVD) scheme was used to calculate the transient flow in a shock tunnel. Both viscous and inviscid, chemically nonreacting flows were studied. The study consisted of two parts, the first dealt with the transient flow in the driven-tube/nozzle interface region (inviscid calculations). The effects of varying the geometry in this region was examined. The second part of the study examined the transient flow in the nozzle (viscous calculations). The results were compared to experimental data.

Tokarcik-Polsky, Susan↗

NASA NTRS1991

Direct simulation of turbulent combustion

Understanding and modeling of turbulent combustion are key problems in the computation of numerous practical systems. Because of the lack of analytical theories in this field and of the difficulty of performing precise experiments, direct simulation appears to be one of the most attractive tools to use in addressing this problem. From Sep. 1989 - Sep. 1990, the following problems were studied using direct simulation but also experiments and theoretical models: (1) the influence of curvature on premixed flame fronts; (2) the construction of a model for the flame stretch based on direct simulation and multifractal analysis; and (3) the simulation of the interaction between random flow field and a flame front. The general objective was to improve knowledge of turbulent combustion but also to use this information for turbulent combustion models. The transfer of direct simulation results towards models is also in progress. A submodel to evaluate the flame stretch was derived from the present study and incorporated in the Coherent Flame model used in France for piston and aircraft engines.

Poinsot, Thierry J.↗

NASA NTRS1955

Determination of lateral-stability derivatives and transfer-function coefficients from frequency-response data for lateral motions

A method is presented for determining the lateral-stability derivatives, transfer-function coefficients, and the modes for lateral motion from frequency-response data for a rigid aircraft. The method is based on the application of the vector technique to the equations of lateral motion, so that the three equations of lateral motion can be separated into six equations. The method of least squares is then applied to the data for each of these equations to yield the coefficients of the equations of lateral motion from which the lateral-stability derivatives and lateral transfer-function coefficients are computed. Two numerical examples are given to demonstrate the use of the method.

Donegan, James J↗

NASA NTRS1994

A sensitivity equation approach to shape optimization in fluid flows

A sensitivity equation method to shape optimization problems is applied. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.

Borggaard, Jeff↗

NASA NTRS1993

The piecewise-parabolic method in curvilinear coordinates

We derive interpolation formulae for a third-order finite difference method in curvilinear, orthogonal coordinate systems. These formulae serve as a supplement to Colella and Woodward's PPM scheme for problems where the coordinate origin is included in the computational domain. Numerical examples of the improved accuracy of the advection scheme near coordinate singularities are shown.

Blondin, John M.↗

NASA NTRS1994

HRMA calibration handbook: EKC gravity compensated XRCF models

This document, consisting of hardcopy printout of explanatory text, figures, and tables, represents one incarnation of the AXAF high resolution mirror assembly (HRMA) Calibration Handbook. However, as we have envisioned it, the handbook also consists of electronic versions of this hardcopy printout (in the form of postscript files), the individual scripts which produced the various figures and the associated input data, the model raytrace files, and all scripts, parameter files, and input data necessary to generate the raytraces. These data are all available electronically as either ASCII or FITS files. The handbook is intended to be a living document and will be updated as new information and/or fabrication data on the HRMA are obtained, or when the need for additional results are indicated. The SAO Mission Support Team (MST) is developing a high fidelity HRMA model, consisting of analytical and numerical calculations, computer software, and databases of fundamental physical constants, laboratory measurements, configuration data, finite element models, AXAF assembly data, and so on. This model serves as the basis for the simulations presented in the handbook. The 'core' of the model is the raytrace package OSAC, which we have substantially modified and now refer to as SAOsac. One major structural modification to the software has been to utilize the UNIX binary pipe data transport mechanism for passing rays between program modules. This change has made it possible to simulate rays which are distributed randomly over the entrance aperture of the telescope. It has also resulted in a highly efficient system for tracing large numbers of rays. In one application to date (the analysis of VETA-I ring focus data) we have employed 2 x 10(exp 7) rays, a substantial improvement over the limit of 1 x 10(exp 4) rays in the original OSAC module. A second major modification is the manner in which SAOsac incorporates low spatial frequency surface errors into the geometric raytrace. The original OSAC included the ability to use Legendre-Fourier polynomials to describe deviations from the basic optical prescription. To this we have added bicubic splines to address a deficiency in the handling of the sharper deformations in the areas of mirror support pads. SAO has developed software (TRANS-FIT) to translate the most common finite element analysis models into these forms for incorporation into the raytrace program.

Tananbaum, H. D.↗

NASA NTRS1994

Large-eddy simulation of a boundary layer with concave streamwise curvature

Turbulence modeling continues to be one of the most difficult problems in fluid mechanics. Existing prediction methods are well developed for certain classes of simple equilibrium flows, but are still not entirely satisfactory for a large category of complex non-equilibrium flows found in engineering practice. Direct and large-eddy simulation (LES) approaches have long been believed to have great potential for the accurate prediction of difficult turbulent flows, but the associated computational cost has been prohibitive for practical problems. This remains true for direct simulation but is no longer clear for large-eddy simulation. Advances in computer hardware, numerical methods, and subgrid-scale modeling have made it possible to conduct LES for flows or practical interest at Reynolds numbers in the range of laboratory experiments. The objective of this work is to apply ES and the dynamic subgrid-scale model to the flow of a boundary layer over a concave surface.

Lund, Thomas S.↗

NASA NTRS1995

Automation based on knowledge modeling theory and its applications in engine diagnostic systems using Space Shuttle Main Engine vibrational data

Humans can perform many complicated tasks without explicit rules. This inherent and advantageous capability becomes a hurdle when a task is to be automated. Modern computers and numerical calculations require explicit rules and discrete numerical values. In order to bridge the gap between human knowledge and automating tools, a knowledge model is proposed. Knowledge modeling techniques are discussed and utilized to automate a labor and time intensive task of detecting anomalous bearing wear patterns in the Space Shuttle Main Engine (SSME) High Pressure Oxygen Turbopump (HPOTP).

Kim, Jonnathan H.↗