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NASA NTRS1994

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

Kenny, Sean P.↗

NASA NTRS1994

A hydrogen-oxygen rocket engine coolant passage design program (RECOP) for fluid-cooled thrust chambers and nozzles

The design of coolant passages in regeneratively cooled thrust chambers is critical to the operation and safety of a rocket engine system. Designing a coolant passage is a complex thermal and hydraulic problem requiring an accurate understanding of the heat transfer between the combustion gas and the coolant. Every major rocket engine company has invested in the development of thrust chamber computer design and analysis tools; two examples are Rocketdyne's REGEN code and Aerojet's ELES program. In an effort to augment current design capabilities for government and industry, the NASA Lewis Research Center is developing a computer model to design coolant passages for advanced regeneratively cooled thrust chambers. The RECOP code incorporates state-of-the-art correlations, numerical techniques and design methods, certainly minimum requirements for generating optimum designs of future space chemical engines. A preliminary version of the RECOP model was recently completed and code validation work is in progress. This paper introduces major features of RECOP and compares the analysis to design points for the first test case engine; the Pratt & Whitney RL10A-3-3A thrust chamber.

Tomsik, Thomas M.↗

NASA NTRS1994

Study of multiple cracks in airplane fuselage by micromechanics and complex variables

Innovative numerical techniques for two dimensional elastic and elastic-plastic multiple crack problems are presented using micromechanics concepts and complex variables. The simplicity and the accuracy of the proposed method will enable us to carry out the multiple-site fatigue crack propagation analyses for airplane fuselage by incorporating such features as the curvilinear crack path, plastic deformation, coalescence of cracks, etc.

Denda, Mitsunori↗

NASA NTRS1994

Atmospheric effects on the risetime and waveshape of sonic booms

Accurate prediction of human response to sonic booms from proposed HSCT aircraft depends on a knowledge of the waveshape and risetime of the boom at the ground. In previous work, we have developed a numerical technique to predict the combined effects of molecular absorption and finite wave distortion on the sonic boom as it propagates from the aircraft to the top of the turbulent boundary layer. We have more recently developed a scattering center based model to calculate the effects of turbulence on the sonic boom waveform as it propagates through this boundary layer. Calculations have been performed using single scales of turbulence and compared to measurements at Edwards AFB in the late 1960's. A model of the atmosphere involving two scales each for convective and mechanical turbulence has been developed and fit to meteorological data collected during JAPE 2. Scattering calculations employing this model underpredict the number of unperturbed waveforms. In order to develop a more realistic model of the atmosphere, the JAPE 2 meteorological data has been fit to a von Karman spectrum. Results of scattering using this multi-scale model will be presented. The combination of finite wave effects with turbulent scattering predictions includes the principal effects of the atmosphere on the sonic boom from the HSCT.

Raspet, Richard↗

NASA NTRS1994

Numerical simulation of supersonic boundary layer transition

The present contribution reviews some of the recent progress obtained at our group in the direct numerical simulation (DNS) of compressible boundary layer transition. Elements of the different simulation approaches and numerical techniques employed are surveyed. Temporal and spatial simulations, as well as comparisons with results obtained from Parabolized Stability Equations, are discussed. DNS results are given for flat plate boundary layers in the Mach number range 1.6 to 4.5. A temporal DNS at Mach 4.5 has been continued through breakdown all the way to the turbulent stage. In addition results obtained with a recently developed extended temporal DNS approach are presented, which takes into account some nonparallel effects of a growing boundary layer. Results from this approach are quite close to those of spatial DNS, while preserving the efficiency of the temporal DNS.

Guo, Y.↗

NASA NTRS1994

Accelerated panel methods using the fast multipole method

Panel methods are commonly used in computational fluid dynamics for the solution of potential flow problems. The methods are a numerical technique based on the surface distribution of singularity elements. The solution is the process of finding the strength of the singularity elements distributed over the body's surface. This process involves the solution of the matrix problem Pq = p' for a set of unknowns q. The Fast Multipole Method is used to directly compute q without using matrix solvers. The algorithm works in O(N) time for N points, a great improvement over standard matrix solvers. In panel methods, the surface of a body is divided into a series of quadrilateral panels. The methods involve the computation of the influence of all other panels on each individual panel. The influence is based on the surface distribution, though this can be approximated by the area for distant panels. An alternative approximation, though with arbitrary accuracy, is to develop a multipole expansion about the center of the panel to describe the effect of a given panel on distant points in space. The expansion is based on the moments of the panel, thus allow the use of various surface distributions without changing the basic algorithm, just the computation of the various moments. The expansions are then manipulated in a tree walk to develop Taylor series expansions about a point in space which describe the effect of all distant panels on any point within a volume of convergence. The effect of near panels then needs to be computed directly, but the effect of all distant panels can be computed by simply evaluating the resulting expansion. The Fast Multipole Method has been applied to panel methods for the solution of source and doublet distributions. A major feature of the algorithm is that the algorithm does not change to derive the potential and velocity for sources and doublets. The same expansions can be used for both sources and doublets. Since the velocity is related to the potential, and the doublet potential is related to the z-component of the source velocity, all values can be derived from the same expansion by taking a series of partial derivatives. This requires more expansion terms to be kept since terms are lost in the process of taking partial derivatives. Thus to maintain accuracy for the doublet computation, more terms are required than if just evaluating for sources. The resulting Fast Multipole code should then parallelize better than classical panel methods due to the locality of data dependencies found in the Fast Multipole Method. Theoretically the parallelized code should execute in O(log N) time with O(N) processors, though this is not practical. Ongoing work includes implementing the parallel accelerated panel method, including methods to improve the load balancing of the problem by taking advantage of the known geometry of panels, and to encorporate sensitivity analysis into the algorithm.

Leathrum, James F., Jr.↗

NASA NTRS1995

Calculated coupling efficiency between an elliptical-core optical fiber and an optical waveguide over temperature

To determine the feasibility of coupling the output of a single-mode optical fiber into a single-mode rib waveguide in a temperature varying environment, a theoretical calculation of the coupling efficiency between the two was investigated. Due to the complex geometry of the rib guide, there is no analytical solution to the wave equation for the guided modes, thus, approximation and/or numerical techniques must be utilized to determine the field patterns of the guide. In this study, three solution methods were used for both the fiber and guide fields; the effective-index method (EIM), Marcatili's approximation, and a Fourier method. These methods were utilized independently to calculate the electric field profile of each component at two temperatures, 20 C and 300 C, representing a nominal and high temperature. Using the electric field profile calculated from each method, the theoretical coupling efficiency between an elliptical-core optical fiber and a rib waveguide was calculated using the overlap integral and the results were compared. It was determined that a high coupling efficiency can be achieved when the two components are aligned. The coupling efficiency was more sensitive to alignment offsets in the y direction than the x, due to the elliptical modal field profile of both components. Changes in the coupling efficiency over temperature were found to be minimal.

Tuma, Margaret L.↗

NASA NTRS1994

Strong gravitational lensing statistics as a test of cosmogonic scenarios

Gravitational lensing statistics can provide a direct and powerful test of cosmic structure formation theories. Since lensing tests, directly, the magnitude of the nonlinear mass density fluctuations on lines of sight to distant objects, no issues of 'bias' (of mass fluctuations with respect to galaxy density fluctuations) exist here, although lensing observations provide their own ambiguities of interpretation. We develop numerical techniques for generating model density distributions with the very large spatial dynamic range required by lensing considerations and for identifying regions of the simulations capable of multiple image lensing in a conservative and computationally efficient way that should be accurate for splittings significantly larger than 3 seconds. Applying these techniques to existing standard Cold dark matter (CDM) (Omega = 1) and Primeval Baryon Isocurvature (PBI) (Omega = 0.2) simulations (normalized to the Cosmic Background Explorer Satellite (COBE) amplitude), we find that the CDM model predicts large splitting (greater than 8 seconds) lensing events roughly an order-of-magnitude more frequently than the PBI model. Under the reasonable but idealized assumption that lensing structrues can be modeled as singular isothermal spheres (SIS), the predictions can be directly compared to observations of lensing events in quasar samples. Several large splitting (Delta Theta is greater than 8 seconds) cases are predicted in the standard CDM model (the exact number being dependent on the treatment of amplification bias), whereas none is observed. In a formal sense, the comparison excludes the CDM model at high confidence (essentially for the same reason that CDM predicts excessive small-scale cosmic velocity dispersions.) A very rough assessment of low-density but flat CDM model (Omega = 0.3, Lambda/3H(sup 2 sub 0) = 0.7) indicates a far lower and probably acceptable level of lensing. The PBI model is consistent with, but not strongly tested by, the available lensing data, and other open models would presumably do as well as PBI. These preliminary conclusions and the assumptions on which they are based can be tested and the analysis can be applied to other cosmogonic models by straightforward extension of the work presented here.

Cen, Renyue↗

NASA NTRS1994

Axisymmetric modeling of cometary mass loading on an adaptively refined grid: MHD results

The first results of an axisymmetric magnetohydrodynamic (MHD) model of the interaction of an expanding cometary atmosphere with the solar wind are presented. The model assumes that far upstream the plasma flow lines are parallel to the magnetic field vector. The effects of mass loading and ion-neutral friction are taken into account by the governing equations, whcih are solved on an adaptively refined unstructured grid using a Monotone Upstream Centered Schemes for Conservative Laws (MUSCL)-type numerical technique. The combination of the adaptive refinement with the MUSCL-scheme allows the entire cometary atmosphere to be modeled, while still resolving both the shock and the near nucleus of the comet. The main findingsare the following: (1) A shock is formed approximately = 0.45 Mkm upstream of the comet (its location is controlled by the sonic and Alfvenic Mach numbers of the ambient solar wind flow and by the cometary mass addition rate). (2) A contact surface is formed approximately = 5,600 km upstream of the nucleus separating an outward expanding cometary ionosphere from the nearly stagnating solar wind flow. The location of the contact surface is controlled by the upstream flow conditions, the mass loading rate and the ion-neutral drag. The contact surface is also the boundary of the diamagnetic cavity. (3) A closed inner shock terminates the supersonic expansion of the cometary ionosphere. This inner shock is closer to the nucleus on dayside than on the nightside.

Gombosi, Tamas I.↗

NASA NTRS1993

The interaction of plume heads with compositional discontinuities in the Earth's mantle

The effects of compositional discontinuities of density and viscosity in the Earth's mantle on the ascent of mantle plume heads is studied using a boundary integral numerical technique. Three specific problems are considered: (1) a plume head rising away from a deformable interface, (2) a plume head passing through an interface, and (3) a plume head approaching the surface of the Earth. For the case of a plume attached to a free-surface, the calculated time-dependent plume shapesare compared with experimental results. Two principle modes of plume head deformation are observed: plume head elingation or the formation of a cavity inside the plume head. The inferred structure of mantle plumes, namely, a large plume head with a long tail, is characteristic of plumes attached to their source region, and also of buoyant material moving away from an interface and of buoyant material moving through an interface from a high- to low-viscosity region. As a rising plume head approaches the upper mantle, most of the lower mantle will quickly drain from the gap between the plume head and the upper mantle if the plume head enters the upper mantle. If the plume head moves from a high- to low-viscosity region, the plume head becomes significantly elongated and, for the viscosity contrasts thought to exist in the Earth, could extend from the 670 km discontinuity to the surface. Plume heads that are extended owing to a viscosity decrease in the upper mantle have a cylindrical geometry. The dynamic surface topography induced by plume heads is bell-shaped when the top of the plume head is at depths greater than about 0.1 plume head radii. As the plume head approaches the surface and spreads, the dynamic topography becomes plateau-shaped. The largest stresses are produced in the early stages of plume spreading when the plume head is still nearly spherical, and the surface expression of these stresses is likely to be dominated by radial extension. As the plume spreads, compressional stresses on the surface are produced beyond the edges of the plume; consequently, extensional features will be produced above the plume head and may be surrounded by a ring of compressional features.

Manga, Michael↗

NASA NTRS1994

Analysis of the early spectra and light curve of SN 1987A

Numerical modeling of supernova spectra, light curves, and hydrodynamics requires physical inputs, numerical techniques, approximations, and assumptions which must be thoroughly understood in order to study the details of supernova explosions. Here, we discuss some of these in the context of the early evolution of supernova 1987A. Gray radiation-hydrodynamics is used to calculate the bolometric light curve and the hydrodynamic evolution of the supernova. Synthetic spectra are then obtained for the resulting density and velocity structure. The spectrum calculations are performed using a special-relativistic treatment of the radiative transfer equation in the comoving frame, line blanketing by about 10(exp 5) spectral lines, and departures from local thermodynamic equilibrium (LTE) for H I, He I, Mg II, and Ca II. We find that we are able to simultaneously fit the early light curve and spectra reasonably well, using a progenitor model from Arnett (1991a), without fine-tuning the free parameters. Temperature structures and radiative equilibrium, non-LTE effects, homologous expansion, and mean opacities are discussed.

Hauschildt, Peter H.↗

NASA NTRS1995

Nozzle flow with vibrational nonequilibrium

This research concerns the modeling and numerical solutions of the coupled system of compressible Navier-Stokes equations in cylindrical coordinates under conditions of equilibrium and nonequilibrium thermodynamics. The problem considered was the modeling of a high temperature diatomic gas N2 flowing through a converging-diverging high expansion nozzle. The problem was modeled in two ways. The first model uses a single temperature with variable specific heats as functions of this temperature. For the second model we assume that the various degrees of freedom all have a Boltzmann distribution and that there is a continuous redistribution of energy among the various degrees of freedom as the gas passes through the nozzle. Each degree of freedom is assumed to have its own temperature and, consequently, each system state can be characterized by these temperatures. This suggests that formulation of a second model with a vibrational degree of freedom along with a rotational-translation degree of freedom, each degree of freedom having its own temperature. Initially the vibrational degree of freedom is excited by heating the gas to a high temperature. As the high temperature gas passes through the nozzle throat there is a sudden drop in temperature along with a relaxation time for the vibrational degree of freedom to achieve equilibrium with the rotational-translation degree of freedom. That is, we assume that the temperature change upon passing through the throat is so great that the changes in the vibrational degree of freedom occur at a much slower pace and consequently lags behind the rotational-translational energy changes. This lag results in a finite relaxation time. In this context the term nonequilibrium is used to denote the fact that the energy content of the various degrees of freedom are characterized by two temperatures. We neglect any chemical reactions which could also add nonequilibrium effects. We develop the energy equations for the nonequilibrium model from first principles. The resulting equations, which model the nozzle flow, can be expressed in various forms. In most forms the resulting equations are coupled systems of nonlinear partial differential equations subject to certain boundary conditions. To solve the resulting coupled system of nonlinear partial differential equations, several numerical techniques were investigated: (1) the explicit MacCormack method, (2) the explicit-implicit MacCormack method, (3) the method of operator splitting, (4) factorization schemes, and (5) the Steger-Warming scheme.

Heinbockel, J. H.↗

NASA NTRS1995

Feasibility of Coupling Between a Single-Mode Elliptical-Core Fiber and a Single Mode Rib Waveguide Over Temperature

To determine the feasibility of coupling the output of an optical fiber to a rib waveguide in a temperature environment ranging from 20 C to 300 C, a theoretical calculation of the coupling efficiency between the two was investigated. This is a significant problem which needs to be addressed to determine whether an integrated optic device can function in a harsh temperature environment. Because the behavior of the integrated-optic device is polarization sensitive, a polarization-preserving optic fiber, via its elliptical core, was used to couple light with a known polarization into the device. To couple light energy efficiently from an optical fiber into a channel waveguide, the design of both components should provide for well-matched electric field profiles. The rib waveguide analyzed was the light input channel of an integrated-optic pressure sensor. Due to the complex geometry of the rib waveguide, there is no analytical solution to the wave equation for the guided modes. Approximation or numerical techniques must be utilized to determine the propagation constants and field patterns of the guide. In this study, three solution methods were used to determine the field profiles of both the fiber and guide: the effective-index method (EIM), Marcatili's approximation, and a Fourier method. These methods were utilized independently to calculate the electric field profile of a rib channel waveguide and elliptical fiber at two temperatures, 20 C and 300 C. These temperatures were chosen to represent a nominal and a high temperature that the device would experience. Using the electric field profile calculated from each method, the theoretical coupling efficiency between the single-mode optical fiber and rib waveguide was calculated using the overlap integral and results of the techniques compared. Initially, perfect alignment was assumed and the coupling efficiency calculated. Then, the coupling efficiency calculation was repeated for a range of transverse offsets at both temperatures. Results of the calculation indicate a high coupling efficiency can be achieved when the two components were properly aligned. The coupling efficiency was more sensitive to alignment offsets in the y direction than the x, due to the elliptical modal profile of both components. Changes in the coupling efficiency over temperature were found to be minimal.

Tuma, Margaret L.↗

NASA NTRS1995

An Analytic Approximation to Very High Specific Impulse and Specific Power Interplanetary Space Mission Analysis

A simple, analytic approximation is derived to calculate trip time and performance for propulsion systems of very high specific impulse (50,000 to 200,000 seconds) and very high specific power (10 to 1000 kW/kg) for human interplanetary space missions. The approach assumed field-free space, constant thrust/constant specific power, and near straight line (radial) trajectories between the planets. Closed form, one dimensional equations of motion for two-burn rendezvous and four-burn round trip missions are derived as a function of specific impulse, specific power, and propellant mass ratio. The equations are coupled to an optimizing parameter that maximizes performance and minimizes trip time. Data generated for hypothetical one-way and round trip human missions to Jupiter were found to be within 1% and 6% accuracy of integrated solutions respectively, verifying that for these systems, credible analysis does not require computationally intensive numerical techniques.

Williams, Craig Hamilton↗

NASA NTRS1995

Numerical investigations in the backflow region of a vacuum plume

Four tasks were completed in this period and results were published in AIAA papers. First, a Boltzmann-2D code, was developed and applied to compute MSFC-A2 nozzle/plume flow field. It solved the two-dimensional Boltzmann-BGK equation using the Finite Difference Discrete Ordinate (FDDO) numerical technique. The code was validated by experimental data for one-dimensional shock structure predictions, paper 95-2056. Successful results for nozzle/plume flow simulation using the developed Boltzmann-2D code were presented at the 1995 AIAA Aerospace Science Conference, paper 95-0627. Second, a computer code solving two-dimensional Burnett equations was developed and applied to low-density nozzle flow field calculation. Results were also published at the 1994 AIAA Thermophysics Conference, paper 94-2055. Third, the developed two-dimensional Burnett code was extended to compute axisymmetric flow field inside MSFC-A2 nozzle, paper 95-2008. The computed nozzle exit conditions are used as input data for Direct Simulation Monte Carlo (DSMC) plume calculation. Fourth, a DSMC code was modified to compute the exhausted plume near the nozzle exit and in the backflow region.

Liaw, Goang-Shin↗

NASA NTRS1995

Thermocapillary flow with evaporation and condensation at low gravity. Part 2: Deformable surface

The free surface behavior of a volatile wetting liquid at low gravity is studied using scaling and numerical techniques. An open cavity model, which was applied in part 1 to investigate fluid flow and heat transfer in non-deforming pores, is used to evaluate the influence of convection on surface morphology with length scales and subcooling/superheating limits of 1 less than or equal to D less than or equal to 10(exp 2) microns and approximately 1 K, respectively. Results show that the menisci shapes of highly wetting fluids are sensitive to thermocapillary flow and to a lesser extent the recoil force associated with evaporation and condensation. With subcooling, thermocapillarity produces a suction about the pore centerline that promotes loss of mechanical equilibrium, while condensation exerts an opposing force that under some conditions offsets this destabilizing influence. With superheating, thermocapillarity and evaporation act in the same direction and mutually foster surface stability. All of these trends are magnified by high capillary and Biot numbers, and the stronger circulation intensities associated with small contact angles. These phenomena strongly depend on the thermal and interfacial equilibrium between the liquid and vapor, and have important ramifications for systems designed to maintain a pressure differential across a porous surface.

Schmidt, G. R.↗

NASA NTRS1995

Electromagnetic Scattering From a Rectangular Cavity Recessed in a 3-D Conducting Surface

The problem of electromagnetic (EM) scattering from an aperture backed by a rectangular cavity recessed in a three-dimensional conducting body is analyzed using the coupled field integral equation approach. Using the free-space Green's function, EM fields scattered outside the cavity are determined in terms of (1) an equivalent electric surface current density flowing on the three-dimensional conducting surface of the object including the cavity aperture and (2) an equivalent magnetic surface current density flowing over the aperture only. The EM fields inside the cavity are determined using the waveguide modal expansion functions. Making the total tangential electric and magnetic fields across the aperture continuous and subjecting the total tangential electric field on the outer conducting three-dimensional surface of the object to zero, a set of coupled integral equations is obtained. The equivalent electric and magnetic surface currents are then obtained by solving the coupled integral equation using the Method of Moments. The numerical results on scattering from rectangular cavities embedded in various three-dimensional objects are compared with the results obtained by other numerical techniques.

Deshpande, M. D.↗

NASA NTRS1995

Global aspects of the motion of the heliospheric termination shock: A gasdynamic model

The heliospheric termination shock is expected to move in response to variation in upstream solar wind conditions. Using numerical techniques, we extend an earlier strictly one-dimensional analytic gas dynamic model of shock motion to two dimensions, to investigate the qualitative features of global behavior of shock motion, and the consequences of latitudinal variation in dynamic pressure. The boundary conditions of the calculation are given by the solar wind parameters as a function of latitude and time on an inner spherical boundary, and a constant pressure (roughly simulating the effect of the local interstellar medium) on an outer boundary. Density variations, specified at the inner boundary as a function of time, are convected into the termination shock. Immediately after the interaction, the shock moves with speeds given by the earlier analytic model. However, as the termination shock propagates outward (or inward), it begins to slow down. After about 2 to 10 years, depending on details of boundary conditions, the signal from the shock interaction has reached the outer boundary and propagates inward to the position of the termination shock, strongly affecting the behavior of the shock. Assuming no further disturbances in the solar wind, the termination shock will reach its new equilibrium after some tens of years. In reality, large-scale variations in solar wind dynamic pressure occur on time scales short in comparison with the eleven year solar cycle, so that one expects that the termination shock is never in an equilibrium position, but rather oscillates inward and outward; this oscillation will vary with heliographic latitude. The effects of a variety of types of solar wind disturbances are investigated and summarized.

Ratkiewicz, R.↗