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Green function in solving linear and nonlinear second order ordinary differential equations including examples in finding rendezvous and periodic orbits of restricted three body system
Shapes of gear teeth produced by gear cutters of specified shape simulated computationally, according to approach based on principles of differential geometry. Results of computer simulation displayed as computer graphics and/or used in analyses of design, manufacturing, and performance of gears. Applicable to both standard and non-standard gear-tooth forms. Accelerates and facilitates analysis of alternative designs of gears and cutters. Simulation extended to study generation of surfaces other than gears. Applied to cams, bearings, and surfaces of arbitrary rolling elements as well as to gears. Possible to develop analogous procedures for simulating manufacture of skin surfaces like automobile fenders, airfoils, and ship hulls.
This paper presents a transient model of catalytic combustion suitable for isolated channels and monolith reactors. The model is a lumped two-phase (gas and solid) model where the gas phase is quasi-steady relative to the transient solid. Axial diffusion is neglected in the gas phase; lateral diffusion, however, is accounted for using transfer coefficients. The solid phase includes axial heat conduction and external heat loss due to convection and radiation. The combustion process utilizes detailed gas and surface reaction models. The gas-phase model becomes a system of stiff ordinary differential equations while the solid phase reduces, after discretization, into a system of stiff ordinary differential-algebraic equations. The time evolution of the system came from alternating integrations of the quasi-steady gas and transient solid. This work outlines the numerical model and presents some sensitivity studies on important parameters including internal transfer coefficients, catalytic surface site density, and external heat-loss (if applicable). The model is compared to two experiments using CO fuel: (1) steady-state conversion through an isothermal platinum (Pt) tube and (2) transient propagation of a catalytic reaction inside a small Pt tube. The model requires internal mass-transfer resistance to match the experiments at lower residence times. Under mass-transport limited conditions, the model reasonably predicted exit conversion using global mass-transfer coefficients. Near light-off, the model results did not match the experiment precisely even after adjustment of mass-transfer coefficients. Agreement improved for the first case after adjusting the surface kinetics such that the net rate of CO adsorption increased compared to O2. The CO / O2 surface mechanism came from a sub-set of reactions in a popular CH4 / O2 mechanism. For the second case, predictions improved for lean conditions with increased external heat loss or adjustment of the kinetics as in the first case. Finally, the results show that different initial surface-species distribution leads to different steady-states under certain conditions. These results demonstrate the utility of a lumped two-phase model of a transient catalytic combustor with detailed chemistry.
Work conducted by Norstrud (1971) has been extended to lifting flows with the inclusion of embedded continuous supercritical regions. The approach taken follows some fundamental steps proposed for two-dimensional flows by Oswatitsch (1950). The governing integral equation is replaced by a system of nonlinear algebraic equations. The method of parametric differentiation is applied to the solution of this system of equations. The analytical analysis is discussed together with a numerical analysis in the case of a wing configuration with arbitrary thickness distribution.
Theoretical analysis shows that the amount of slow growth occurring prior to fracture in a plane stress tension specimen subjected to a subcritical stress intensity level is affected by the ductility and rheological sensitivity of the material, the rate of loading, the initial crack size, and the geometrical configuration of the test. These parameters enter the governing nonlinear differential equation which describes the quasi-static crack extension in a rate-sensitive Tresca solid. Examples of numerical integration are given and employed to predict the maximum fracture toughness attained in a plane stress tension test.
The Chandrasekhar equations arising in optimal control problems for linear distributed parameter systems are considered. The equations are derived via approximation theory. This approach is used to obtain existence, uniqueness, and strong differentiability of the solutions and provides the basis for a convergent computation scheme for approximating feedback gain operators. A numerical example is presented to illustrate these ideas.
The Navier-Stokes equations represent an extremely good model of the physical phenomena encountered in most aeronautical problems. However, the computational resource needed to solve the Navier-Stokes equations are so large that even with today's supercomputers, it is necessary to make use of simpler models. A large number of external aerodynamic problems can be accurately described by a simpler model. This model consists of an outer inviscid flow plus a boundary-layer thickness correction for the vehicle shape. The outer inviscid model may be represented by the potential equation or by the Euler equation. The present paper provides the foundations for the numerical solution of the Euler equations. The governing equations are considered, taking into account conservation laws, the medium, the differential form of the conservation laws, generalized solutions, shock-fitting, and characteristics. Attention is also given to initial and boundary conditions, existence and uniqueness, and rotational phenomena.
The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.
Using the generalized Rayleigh Gans approximation along with the radiative transfer method, a bistatic backscattering model for a layer of randomly oriented, elliptic-shaped leaves is formulated. Following a similar procedure the bistatic scattering model for a layer of needle-shaped leaves is also developed to simulate coniferous vegetation. The differences between the scattering characteristics of the deciduous and coniferous leaves are illustrated numerically for different orientation and incidence angles. It is found that both like and cross polarizations are needed to differentiate the difference in scattering due to the shapes of the scatterers. The calculated backscattering coefficients are compared with measured values from artificial canopies with circular-shaped leaves.
Different numerical techniques have been examined for simulating the underexpanded plumes in supersonic flows. The results revealed that the monotone differentiable limiter is superior to the TVD (total variation diminishing) minmod limiter when the MUSCL (monotonic upstream schemes for conservation laws) approach is used for Roe's upwind scheme. Roe's upwind scheme is found to be superior to the central difference scheme not only for capturing shock waves, but also for capturing shear flows.
A well integrated experimental/analytical investigation was conducted to provide benchmark quality relevant to a prefilming type airblast fuel nozzle and its interaction with the combustor dome air swirler. The experimental investigation included a systematic study of both single-phase flows that involved single and twin co-axial jets with and without swirl. A two-component Phase Doppler Particle Analyzer (PDPA) was used to document the interaction of single and co-axial air jets with glass beads that simulate nonevaporating spray and simultaneously avoid the complexities associated with fuel atomization processes and attendant issues about the specification of relevant boundary conditions. The interaction of jets with methanol spray produced by practical airblast nozzle was also documented in the spatial domain of practical interest. Model assessment activities included the use of three turbulence models (k-epsilon, algebraic second moment (ASM), and differential second moment (DSM)) for the carrier phase, deterministic or stochastic Lagrangian treatment of the dispersed phase, and advanced numerical schemes. Although qualitatively good comparison with data was obtained for most of the cases investigated, the model deficiencies in regard to modeled dissipation rate transport equation, single length scale, pressure-strain correlation, and other critical closure issues need to be resolved before one can achieve the degree of accuracy required to analytically design combustion systems.
A well integrated experimental/analytical investigation was conducted to provide benchmark quality data relevant to prefilming type airblast fuel nozzle and its interaction with combustor dome air swirler. The experimental investigation included a systematic study of both single-phase flows that involved single and twin co-axial jets with and without swirl. A two-component Phase Doppler Particle Analyzer (PDPA) equipment was used to document the interaction of single and co-axial air jets with glass beads that simulate nonevaporating spray and simultaneously avoid the complexities associated with fuel atomization processes and attendant issues about the specification of relevant boundary conditions. The interaction of jets with methanol spray produced by practical airblast nozzle was also documented in the spatial domain of practical interest. Model assessment activities included the use of three turbulence models (k-epsilon, algebraic second moment (ASM) and differential second moment (DSM)) for the carrier phase, deterministic or stochastic Lagrangian treatment of the dispersed phase, and advanced numerical schemes. Although qualitatively good comparison with data was obtained for most of the cases investigated, the model deficiencies in regard to modeled dissipation rate transport equation, single length scale, pressure-strain correlation, and other critical closure issues need to be resolved before one can achieve the degree of accuracy required to analytically design combustion systems.
The plasma properties and magnetic field structure of the solar corona were determined using coordinated observations obtained with NASA/GSFC's Solar EUV Rocket Telescope and Spectrograph (SERTS), the Very Large Array (VLA), and Kitt Peak photospheric longitudinal magnetograms. A problem was identified with the SERTS calibration as determined from laboratory measurements. A revised calibration curve was derived by requiring that the numerous available measured line intensity ratios agreed with their respective theoretical values. Densities were derived from line intensity ratios, and active region densities were found to typically exceed quiet Sun densities by factors of only about 2. The active region density was found to remain constant across the SERTS slit, despite the fact that the emission line intensities vary significantly. This indicates that the product of the path length and the volume filling factor must vary significantly from the active region outskirts to the central core. Filling factors were derived and found to range from much less than one to nearly unity. Wavelength shifts were examined along the SERTS slit in the spatially resolved spectra, but no evidence was found for significant Doppler shifts in active region 7563 or in the quiet Sun. The numerical procedure developed by Monsignori-Fossi and Landini was used to derive the active region and quiet sun differential emission measure (DEM) from the spatially averaged spectra. A DEM was estimated for each spatial pixel in the two dimensional active region images by scaling the averaged active region DEM based upon corresponding pixel intensities of SERTS Mg IX, Fe XV, and Fe XVI images. These results, along with density measurements, were used in an IDL computer code which calculated the temperature dependence of the coronal magnetic field in each spatial pixel by minimizing the difference between the observed and calculated 20 and 6 cm microwave brightness temperatures.
We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to incur no bias from the choice of the prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model and verify our derivations and bias-free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations in which knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.
High precision and close proximity formation flight is an enabling technology for future space missions and requires an on-board relative navigation capability that is accurate to the mm-level and robust to formation parameters. Common estimation techniques linearize the entire formation about one spacecraft’s position, resulting in degraded accuracy due to linearization errors when separation distances become large. Additionally, methods which decouple absolute and relative state estimates usually require ad-hoc methods to incorporate the estimates together. This work discusses an alternative ”incremental” formulation of the relative navigation problem which is in variant to formation size, robust to coupling between absolute and relative dynamics, and can undergo similarity transformations to smoothly incorporate either absolute or relative information without numerical issues. A specific example of this architecture is presented in the context of formation navigation using Carrier-Differential Global Positioning System measurements, and is compared to a traditional leader-linearized filter. In the presence of accurate measurements, the incremental architecture is shown to reduce linearization errors and mean estimate errors by up to three orders of magnitude without the incorporation of new sensor information.
The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = 0 where f(u) is, say, strictly convex double dot f dot a sub asterisk 0 is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to liner problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitive insight into the large-time behavior of the numerical computation.
The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = O where f(u) is, say, strictly double dot f dot a sub asterisk O is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitative insight into the large-time behavior of the numerical computation.
Many biological systems may profitably be studied as surface phenomena. A model consisting of isotropic growth of a curved surface from a flat sheet is assumed. With such a model, the Gaussian curvature of the final surface determines whether growth rate of the surface is subharmonic or superharmonic. These properties correspond to notions of convexity and concavity, and thus to local excess growth and local deficiency of growth. In biological models where the major factors controlling surface growth are intrinsic to the surface, researchers thus gained from geometrical study information on the differential growth undergone by the surface. These ideas were applied to an analysis of the folding of the cerebral cortex, a geometrically rather complex surface growth. A numerical surface curvature technique based on an approximation to the Dupin indicatrix of the surface was developed. A metric for comparing curvature estimates is introduced, and considerable numerical testing indicated the reliability of this technique.