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NASA NTRSDate not supplied

Comparison of OVERFLOW Computational and Experimental Results for a Blunt Mars Entry Vehicle Concept during Supersonic Retropropulsion

Simulations of unsteady supersonic retropropulsion (SRP) flow over a Hypersonic Inflatable Aerodynamic Decelerator (HIAD) blunt-body vehicle were performed using the OVERFLOW Computational Fluid Dynamics (CFD) solver. High-fidelity flow solver techniques, including Detached Eddy Simulation (DES) turbulence modeling and Adaptive Mesh Refinement (AMR), were employed to obtain improved realism in CFD predictions. Simulation conditions and geometry configurations were designed to match specific runs in the Descent System Study (DSS) wind tunnel testing (WTT) campaign. The accuracy of each simulation is assessed by direct comparison to experimental data. Comparisons of computational predictions of the SRP flowfield and bow shock shape to experimental schlieren imaging show reasonable prediction of mean shock shape, with approximately 10% similarity in shock standoff distance for selected conditions, as well as similarity in local, time-varying fluctuations of the shock-plume interaction. Comparisons of discrete measurements of surface pressure coefficient (Cp) indicate CFD accuracy within approximately 10% of the experiment across the majority of the model heatshield, with larger variations at some of the heatshield edge locations with stronger flow unsteadiness. Simulated unsteadiness of these chaotic flows, which were highly dynamic and multi-modal, was shown to be within 20-40% of experimentally-measured pressure standard deviation (SD) for the majority of the sampled locations.

Supersonic Retropropulsion↗

NASA NTRSDate not supplied

Comparison of OVERFLOW Computational and Experimental Results for a Blunt Mars Entry Vehicle Concept during Supersonic Retropropulsion

Supersonic Retropropulsion↗

NASA NTRS2007

Fuel-Optimal Trajectories in a Planet-Moon Environment Using Multiple Gravity Assists

For low energy spacecraft trajectories such as multi-moon orbiters for the Jupiter system, multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In this paper, we outline a procedure to obtain a family of zero-fuel multi-moon orbiter trajectories, using a family of Keplerian maps derived by the first author previously. The maps capture well the dynamics of the full equations of motion; the phase space contains a connected chaotic zone where intersections between unstable resonant orbit manifolds provide the template for lanes of fast migration between orbits of different semimajor axes. Patched three body approach is used and the four body problem is broken down into two three-body problems, and the search space is considerably reduced by the use of properties of the Keplerian maps. We also introduce the notion of Switching Region where the perturbations due to the two perturbing moons are of comparable strength, and which separates the domains of applicability of the corresponding two Keplerian maps.

Ross, Shane D.↗

NASA NTRS1990

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗

NASA NTRS1991

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609 .... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗

NASA NTRSDate not supplied

Ocean-Land Teleconnections and Chaotic Atmospheric Variability in Climate Models, with Implications for Real-World Behavior

This study introduces a unique approach to estimating a fundamental but unmeasurable property of Nature: ρo 2 , the fraction of the variance of a given meteorological variable that is determined by boundary forcing (specifically, SSTs) as opposed to chaotic atmospheric dynamics. Here, we focus on continental monthly-averaged near-surface air temperature, or T2M. Rather than determining the fractions of SST-explained T2M variance found for a specific atmospheric general circulation model (AGCM) and then assuming that these fractions also represent Nature, we perform an analysis that entails: (i) evaluating a specific AGCM’s ability to capture observed variability with an infinite number of ensemble members, (ii) quantifying the degree to which different modeling systems agree on SST-land teleconnections, and (iii) combining these results into our novel estimates of ρo 2 . The derived ρo 2 fields suggest, with caveats, that AGCMs tend to overestimate the fraction of explained T2M variance (the signal-to-total variance ratio) in the Tropics.

Randal D Koster↗

NASA NTRS1990

Are earthquakes an example of deterministic chaos?

A simple mass-spring model is used to systematically examine the dynamical behavior introduced by fault zone heterogeneities. The model consists of two sliding blocks coupled to each other and to a constant velocity driver by elastic springs. The state of this system can be characterized by the positions of the two blocks relative to the driver. A simple static/dynamic friction law is used. When the system is symmetric, cyclic behavior is observed. For an asymmetric system, where the frictional forces for the two blocks are not equal, the solutions exhibit deterministic chaos. Chaotic windows occur repeatedly between regions of limit cycles on bifurcation diagrams. The model behavior is similar to that of the one-dimensional logistic map. The results provide substantial evidence that earthquakes are an example of deterministic chaos.

Huang, Jie↗

NASA NTRS2002

The Retrospective Iterated Analysis Scheme for Nonlinear Chaotic Dynamics

Atmospheric data assimilation is the name scientists give to the techniques of blending atmospheric observations with atmospheric model results to obtain an accurate idea of what the atmosphere looks like at any given time. Because two pieces of information are used, observations and model results, the outcomes of data assimilation procedure should be better than what one would get by using one of these two pieces of information alone. There is a number of different mathematical techniques that fall under the data assimilation jargon. In theory most these techniques accomplish about the same thing. In practice, however, slight differences in the approaches amount to faster algorithms in some cases, more economical algorithms in other cases, and even give better overall results in yet some other cases because of practical uncertainties not accounted for by theory. Therefore, the key is to find the most adequate data assimilation procedure for the problem in hand. In our Data Assimilation group we have been doing extensive research to try and find just such data assimilation procedure. One promising possibility is what we call retrospective iterated analysis (RIA) scheme. This procedure has recently been implemented and studied in the context of a very large data assimilation system built to help predict and study weather and climate. Although the results from that study suggest that the RIA scheme produces quite reasonable results, a complete evaluation of the scheme is very difficult due to the complexity of that problem. The present work steps back a little bit and studies the behavior of the RIA scheme in the context of a small problem. The problem is small enough to allow full assessment of the quality of the RIA scheme, but it still has some of the complexity found in nature, namely, its chaotic-type behavior. We find that the RIA performs very well for this small but still complex problem which is a result that seconds the results of our early studies.

Todling, Ricardo↗

NASA NTRS1994

Advanced data assimilation in strongly nonlinear dynamical systems

Advanced data assimilation methods are applied to simple but highly nonlinear problems. The dynamical systems studied here are the stochastically forced double well and the Lorenz model. In both systems, linear approximation of the dynamics about the critical points near which regime transitions occur is not always sufficient to track their occurrence or nonoccurrence. Straightforward application of the extended Kalman filter yields mixed results. The ability of the extended Kalman filter to track transitions of the double-well system from one stable critical point to the other depends on the frequency and accuracy of the observations relative to the mean-square amplitude of the stochastic forcing. The ability of the filter to track the chaotic trajectories of the Lorenz model is limited to short times, as is the ability of strong-constraint variational methods. Examples are given to illustrate the difficulties involved, and qualitative explanations for these difficulties are provided. Three generalizations of the extended Kalman filter are described. The first is based on inspection of the innovation sequence, that is, the successive differences between observations and forecasts; it works very well for the double-well problem. The second, an extension to fourth-order moments, yields excellent results for the Lorenz model but will be unwieldy when applied to models with high-dimensional state spaces. A third, more practical method--based on an empirical statistical model derived from a Monte Carlo simulation--is formulated, and shown to work very well. Weak-constraint methods can be made to perform satisfactorily in the context of these simple models, but such methods do not seem to generalize easily to practical models of the atmosphere and ocean. In particular, it is shown that the equations derived in the weak variational formulation are difficult to solve conveniently for large systems.

Miller, Robert N.↗

NASA NTRS1994

Infinite possibilities: Computational structures technology

Computational Fluid Dynamics (or CFD) methods are very familiar to the research community. Even the general public has had some exposure to CFD images, primarily through the news media. However, very little attention has been paid to CST--Computational Structures Technology. Yet, no important design can be completed without it. During the first half of this century, researchers only dreamed of designing and building structures on a computer. Today their dreams have become practical realities as computational methods are used in all phases of design, fabrication and testing of engineering systems. Increasingly complex structures can now be built in even shorter periods of time. Over the past four decades, computer technology has been developing, and early finite element methods have grown from small in-house programs to numerous commercial software programs. When coupled with advanced computing systems, they help engineers make dramatic leaps in designing and testing concepts. The goals of CST include: predicting how a structure will behave under actual operating conditions; designing and complementing other experiments conducted on a structure; investigating microstructural damage or chaotic, unpredictable behavior; helping material developers in improving material systems; and being a useful tool in design systems optimization and sensitivity techniques. Applying CST to a structure problem requires five steps: (1) observe the specific problem; (2) develop a computational model for numerical simulation; (3) develop and assemble software and hardware for running the codes; (4) post-process and interpret the results; and (5) use the model to analyze and design the actual structure. Researchers in both industry and academia continue to make significant contributions to advance this technology with improvements in software, collaborative computing environments and supercomputing systems. As these environments and systems evolve, computational structures technology will evolve. By using CST in the design and operation of future structures systems, engineers will have a better understanding of how a system responds and lasts, more cost-effective methods of designing and testing models, and improved productivity. For informational and educational purposes, a videotape is being produced using both static and dynamic images from research institutions, software and hardware companies, private individuals, and historical photographs and drawings. The extensive number of CST resources indicates its widespread use. Applications run the gamut from simpler university-simulated problems to those requiring solutions on supercomputers. In some cases, an image or an animation will be mapped onto the actual structure to show the relevance of the computer model to the structure. Transferring the digital files to videotape presents a number of problems related to maintaining the quality of the original image, while still producing a broadcast quality videotape. Since researchers normally do not create a computer image using traditional composition theories or video production requirements, often the image loses some of its original digital quality and impact when transferred to videotape. Although many CST images are currently available, those that are edited into the final project must meet two important criteria: they must complement the narration, and they must be broadcast quality when recorded on videotape.

Beam, Sherilee F.↗

NASA NTRS1995

Steady-state dynamic behavior of an auxiliary bearing supported rotor system

This paper investigates the steady-state responses of a rotor system supported by auxiliary bearings in which there is a clearance between the rotor and the inner race of the bearing. A simulation model based upon the rotor of a production jet engine is developed and its steady-state behavior is explored over a wide range of operating conditions for various parametric configurations. Specifically, the influence of rotor imbalance, support stiffness, and damping is studied. It is found that imbalance may change the rotor responses dramatically in terms of frequency contents at certain operating speeds. Subharmonic responses of 2nd order through 10th order are all observed except the 9th order. Chaotic phenomenon is also observed. Jump phenomena (or double-valued responses) of both hard-spring type and soft-spring type are shown to occur at low operating speeds for systems with low auxiliary bearing damping or large clearance even with relatively small imbalance. The effect of friction between the shaft and the inner race of the bearing is also discussed.

Xie, Huajun↗

NASA NTRS1990

The relation between ferroelasticity and superconductivity

The high-temperature superconductivity is explained widely by the layered crystal structure. The one- and two-dimensional subsystems and their interaction are investigated here. It is assumed that the high-T(sub c) superconductivity takes place in the two-dimensional subsystem and the increase of the phase transition temperature from 60 K up to 90 K is the consequence of turning on the influence of one-dimensional chains. The interaction between the two subsystems is transferred along the c axis by the phonons of breathing mode, which causes the hybridization of the electronic bonds between these subsystems. The experimental works indicate that the existence of both the chains Cu(1)-O and their interaction with the superconducting plane of Cu(2)-O modify the temperature of the transition to the superconducting state. It is seen from the neutron scattering data that the rates of the interatomic distance dependencies on temperature are changed around 140 K and 90 K. The 'zig-zag' order in Cu(1)-O chains has been postulated but, on the other hand, the vibrations with a large amplitude only were reported. The bi-stabilized situation of the oxygen ions can be caused by the change of distance between these ions and the Ba ions. It leads to the appearance of a two-well potential. Its parameters depend on temperature and the dynamics of the oxygen ions' movement. They can induce the antipolar order, which can be, however, more or less chaotic. The investigation of the ferroelastic properties of Y-Ba-Cu-O samples lead to the conclusion that they are related to jumps of ions inside the given chain and not to a diffusion between different sites in the ab plane. Researchers deduce thus that the fluctuating oxygen ions from these chains create dipoles in the ab plane. They can be described with the pseudo-spin formalism/ - Pauli matrices/. The system can be described with the Ising model. The pseudo-spins interact with phonons and influence the superconductivity in the second subsystem.

Molak, A.↗

NASA NTRS1991

The relation between ferroelasticity and superconductivity

The high-temperature superconductivity is explained widely by the layered crystal structure. The one- and two-dimensional subsystems and their interaction are investigated here. It is assumed that the high-T(sub c) superconductivity takes place in the two-dimensional subsystem and the increase of the phase transition temperature from 60 K up to 90 K is the consequence of turning on the influence of one-dimensional chains. The interaction between the two subsystems is transferred along the c axis by the phonons of breathing mode, which causes the hybridization of the electronic bonds between these subsystems. The experimental works indicate that the existence of both the chains Cu(1)-O and their interaction with the superconducting plane of Cu(2)-O modify the temperature of the transition to the superconducting state. It is seen from the neutron scattering data that the rates of the interatomic distance dependencies on temperature are changed around 240 K and 90 K. The 'zig-zag' order in Cu(1)-O chains has been postulated but, on the other hand, the vibrations with a large amplitude only were reported. The bi-stabilized situation of the oxygen ions can be caused by the change of distance between these ions and the Ba ions. It leads to the appearance of a two-well potential. Its parameters depend on temperature and the dynamics of the oxygen ions' movement. They can induce the antipolar order, which can be, however, more or less chaotic. The investigation of the ferroelastic properties of Y-Ba-Cu-O samples lead to the conclusion that they are related to jumps of ions inside the given chain and not to a diffusion between different sites in the ab plane. Researchers deduce, thus, that the fluctuating oxygen ions from these chains create dipoles in the ab plane. They can be described with the pseudo-spin formalism (- Pauli matrices). The system can be described with the Ising model. The pseudo-spins interact with phonons and influence the superconductivity in the second subsystem.

Molak, A.↗

NASA NTRS1998

On the Origin of Chaos in the Asteroid Belt

We consider the effect of gravitational perturbations from Jupiter on the dynamics of asteroids, when Jupiter is itself perturbed by Saturn. The presence of Saturn introduces a number of additional frequencies into Jupiters orbit. These frequencies in turn produce chaos in narrow regions on either side of the chaotic zones associated with the mean motion resonances between the asteroids and Jupiter. The resonant arguments of these three-body resonances contain the longitudes of Jupiter and the asteroid together with either the secular frequency 9-6, or the longitude of Saturn. Resonances involving the longitude of Saturn are analogs of the Laplace resonance in the Jovian satellite system. We show that many three-body resonances involving the longitude of Saturn are chaotic. We give simple expressions for the width of the chaotic region and the associated Lyapunov time. In some cases the chaos can produce a diffusive growth in the 4 eccentricity of the asteroid that leads to ejection of the asteroid on times shorter than the age of the solar system. We give simple estimates for the diffusion time. Finally, we present the results of numerical integrations testing the theory.

Murray, N.↗

NASA NTRS1992

Symplectic maps for the n-body problem - Stability analysis

The stability of new symplectic n-body maps is examined from the point of view of nonlinear dynamics. The resonances responsible for the principal artifacts are identified. These are resonances between the stepsize and the difference of mean motions between pairs of planets. For larger stepsizes resonant perturbations are evident in the variation of the energy of the system corresponding to these stepsize resonances. It is shown that the principal instability of the method can be predicted and corresponds to the overlap of the stepsize resonances. It is noted that the analysis suggests that other artifacts will occur. For example, the overlap of a stepsize resonance with a resonance of the actual system may also give a region of chaotic behavior that is an artifact. It is pointed out that the fact that the principal artifacts corresponds to a particular set of stepsize resonances suggests that it may be possible to perturbatively remove the effect when the stepsize resonances are nonoverlapping.

Wisdom, Jack↗

NASA NTRS1989

Evolution of comets into asteroids

This paper presents observational evidence, together with recent theoretical developments, supporting the hypothesis that at least some asteroids might be extinct or dormant cometary nuclei. The observations include the discovery of a number of apparent asteroids in chaotic Jupiter-crossing orbits; the IRAS discovery of 1983 TB, an asteroid in the same orbit as the Geminid meteor shower; the apparent low activity levels determined for several short-period comet nuclei including Comet Halley; and observations of possible cometary activity in some earth-crossing asteroids. Theoretical developments include explorations of dynamical mechanisms capable of delivering main-belt asteroids into earth-crossing orbits, and an understanding of possible processes which may affect comets during their long residence in the Oort cloud and lead to the formation of nonvolatile crusts before and after they enter the planetary system.

Weissman, Paul R.↗

NASA NTRS2006

A Unique Perspective from Space on our Planet: Science, Technologies and Applications

The study of Planet earth is a very complex problem. It has many non-linear and chaotic systems operating in parallel and have interdependencies. In reality, these systems/phenomena s are not well understood or mathematically modeled because of our lack of knowledge of such intricate processes. However, in order to further the subject of Earth as an integrated system, space provides excellent vantage points to look at these processes in multidimensional framework. For example, we can make strives to understand the global water cycle, carbon cycle, atmospheric chemistry, biomass changes, oceans and solid Earth variations by making multitude of global measurements such as soil moisture, precipitation, tropospheric and stratospheric gases, aerosols, tropospheric winds, ocean salinity, ocean color, vegetation cover, crustal dynamics and many more. Such suites of measurements derive the coupled models so we may predict the changes due to natural and anthropogenic forcing. NASA along with other international space agencies have made tremendous investments in recent years in developing and flying remote sensing space borne sensors to enable these measurements. These science measurements and products are further used to address pressing issues such as coastal zone erosion, air quality, severe weather, water availability and quality, public health, fires, earthquakes, land slides and others for societal benefits. This presentation provides a comprehensive overview of NASA s science investigations, related technologies and satellites/sensors and applications.

Habib, Shaid↗

NASA NTRS1988

The dynamics of coorbital satellite systems

The dynamical behavior of N coorbital satellites moving with the same average mean motion around a primary has been studied both analytically and by numerical integrations for N from 2 to 9 satellites. Simplified dynamical equations are used to determine the different stationary configurations and their local stability against infinitesimal perturbations. The ring of equally spaced identical satellites is found to be locally unstable for N of six or less, while for N from two to eight there exists another stable compact stationary configuration with separations of 60 deg or less between adjacent satellites. For N of seven or more the equally spaced configuration becomes locally stable, and for N of nine or more it is the only stationary configuration. The motion becomes chaotic for large-amplitude perturbations. The chaotic motion fills a restricted region of phase space whose outer boundary is determined by the maximum velocity curve.

Salo, H.↗