Results for “phase portrait”

DOE OSTI2020

Analysis of Allison Scanner Phase Portraits Using Action-Phase Coordinates

Allison scanners provide detailed information on the beam transverse phase space. An effective way for analyzing the beam distribution from these measurements is to use action-phase coordinates, where beam propagation in a linear lattice is reduced to advancing the phase. This report presents such analysis for measurements performed with a 2.1 MeV, 5 mA H⁻ beam in the MEBT of the PIP2IT test accelerator at Fermilab. In part, with the choice of calculating the Twiss parameters over the high intensity portion of the beam, the beam core is found to be phase-independent with intensity decreasing exponentially with action, while the beam tails exhibit a clear phase dependence that is stable over the beam line.

43 PARTICLE ACCELERATORS↗

NASA NTRS1991

Shot model parameters for Cygnus X-1 through phase portrait fitting

Shot models for systems having about 1/f power density spectrum are developed by utilizing a distribution of shot durations. Parameters of the distribution are determined by fitting the power spectrum either with analytic forms for the spectrum of a shot model with a given shot profile, or with the spectrum derived from numerical realizations of trial shot models. The shot fraction is specified by fitting the phase portrait, which is a plot of intensity at a given time versus intensity at a delayed time and in principle is sensitive to different shot profiles. These techniques have been extensively applied to the X-ray variability of Cygnus X-1, using HEAO 1 A-2 and an Exosat ME observation. The power spectra suggest models having characteristic shot durations lasting from milliseconds to a few seconds, while the phase portrait fits give shot fractions of about 50 percent. Best fits to the portraits are obtained if the amplitude of the shot is a power-law function of the duration of the shot. These fits prefer shots having a symmetric exponential rise and decay. Results are interpreted in terms of a distribution of magnetic flares in the accretion disk.

Lochner, James C.↗

NASA NTRSDate not supplied

Fluid System Stability Analysis Techniques

Fluid systems, or networks, consist of multiple components that work together to achieve some desired thermofluid state. For any generic application, this fluid state can be a combination of the fluid pressure, flow rate, enthalpy, or species concentration. Fluid system components, such as pumps and valves, are often governed by nonlinear differential equations, resulting in complex component-to-component interactions. System-level fluid network stability occurs when the flow through the system can maintain a steady-state solution in the presence of small perturbations, which depends on these component interactions. System instability, however, can go undetected until issues arise during integrated system testing. This presentation explores a method for system designers to think of the fluid network as an assembly of components, each with their own thermofluid surfaces of partial stability, called nullclines. The intersections of all nullclines yields system-level solutions, called equilibrium points. When designers define operating points, they are tuning system parameters so that these equilibrium points move to the desired location in the thermofluid state plane. However, linearization theory shows us that the dynamic behavior around these points can be unstable. The local stability of these equilibrium points can be assessed analytically with eigen-analysis, or numerically by propagating state-plane samples to construct a phase portrait. Investigating a phase portrait can help designers gain a qualitative understanding of a system’s dynamic performance. This understanding can then help inform requirement definitions, component selection, and operational procedures. This presentation includes an example of the phase portrait technique on a system featuring a centrifugal pump and a back-pressure regulator (BPR). Numerical modeling of this system suggests that equilibrium points on the left-hand side of the pump curve are dynamically unstable.

Fluid System↗

NASA NTRSDate not supplied

Fluid System Stability Analysis Techniques

Fluid systems, or networks, consist of multiple components that work together to achieve some desired thermofluid state. For any generic application, this fluid state can be a combination of the fluid pressure, flow rate, enthalpy, or species concentration. Fluid system components, such as pumps and valves, are often governed by nonlinear differential equations, resulting in complex component-to-component interactions. System-level fluid network stability occurs when the flow through the system can maintain a steady-state solution in the presence of small perturbations, which depends on these component interactions. System instability, however, can go undetected until issues arise during integrated system testing. This presentation explores a method for system designers to think of the fluid network as an assembly of components, each with their own thermofluid surfaces of partial stability, called nullclines. The intersections of all nullclines yields system-level solutions, called equilibrium points. When designers define operating points, they are tuning system parameters so that these equilibrium points move to the desired location in the thermofluid state plane. However, linearization theory shows us that the dynamic behavior around these points can be unstable. The local stability of these equilibrium points can be assessed analytically with eigen-analysis, or numerically by propagating state-plane samples to construct a phase portrait. Investigating a phase portrait can help designers gain a qualitative understanding of a system’s dynamic performance. This understanding can then help inform requirement definitions, component selection, and operational procedures. This presentation includes an example of the phase portrait technique on a system featuring a centrifugal pump and a back-pressure regulator (BPR). Numerical modeling of this system suggests that equilibrium points on the left-hand side of the pump curve are dynamically unstable.

Fluid System↗

NASA NTRS1988

The determination of the topological structure of skin friction lines on a rectangular wing-body combination

A short tutorial in the application of topological ideas to the intepretation of oil flow patterns is presented. Topological concepts such as critical points, phase portraits, topological stability, and indexing are discussed. These concepts are used in an ordered procedure to construct phase portraits of skin friction lines with oil flow patterns for a wing-body combination and two angles of attack. The relationship between the skin friction phase portrait and planar cuts of the velocity field is also discussed.

Yates, Leslie A.↗

NASA NTRS1986

Viscous starting jets

The transient motion which is produced when a viscous incompressible fluid is forced from an initial state of rest is studied. The equations for unsteady particle paths, written in terms of similarity variables, are analyzed as a quasi-autonomous system with the Reynolds number treated as a parameter. By finding and classifying critical points in the system's phase portrait, the flow structure is examined. It is shown that: (1) bifurcations in the phase portrait occur at specific values of the Reynolds number of the flow in question, and (2) the exact solutions of the Stokes equations for the low-Reynolds-number limit contain two critical Reynolds numbers and three distinct states of motion which culminate in the onset of a vortex roll-up.

Cantwell, Brian J.↗

NASA NTRS1984

Transition and mixing in impulsively started jets and vortex rings

A new method is used to analyze transition in impulsively started jets and vortex rings. The method uses similarity variables to reduce the equations for unsteady particle paths to a quasi-autonomous system with the Reynolds number as a parameter. The flow pattern is represented in terms of the phase portrait of this system and the structure of the flow is defined by its critical points. Flow transition is examined in terms of bifurcations in the topology of the phase portrait which occur as the Reynolds number is increased. There is no assumption that the flow is parallel or that it is subjected to small disturbances. The method of analysis, through straightforward, is new and lies outside the usual small disturbance theory used to determine the stability of various profile shapes.

Cantwell, B. J.↗

DOE OSTI2020

Measurements of a 2.1 MeV H− beam with an Allison scanner

Transverse 2D phase space distributions of a 2.1 MeV, 5 mA H− beam are measured at the Proton Improvement Plan II Injector Test accelerator at Fermilab with an Allison scanner. This paper describes the design, calibration, and performance of the scanner along with the main results from beam measurements. Analyses of the recorded phase portraits are performed primarily in action-phase coordinates. The stability of the action under linear optics makes it easier to compare measurements taken under different beamline conditions. The amplitude of a single measured point (“pixel”) is proportional to the phase density in the corresponding portion of the beam. When the Twiss parameters are calculated using only the high-phase density part of the beam, the pixel amplitude in the beam core is found to be decreasing approximately exponentially with action and to be phase-independent. Outside of the core, the amplitudes decrease with action at a slower rate than in the core. This “tail” comprises 10%–30% of the beam, with 0.1% of the total measured intensity extending beyond action 10–20 times larger than the rms emittance. The transition from the core to the tail is accompanied by the appearance of two “branches” that are separated in phase and extend beyond the core. A set of selected measurements shows that there is no measurable emittance dilution along the beamline; the beam parameters are practically constant over a 0.5 ms pulse; and scraping in various parts of the beamline is an effective way to decrease the transverse tails by removing the branches.

Richard, C. (ORCID:000000029891135X)↗

NASA NTRS1993

On the global dynamics of adaptive systems - A study of an elementary example

The inherent nonlinear character of adaptive systems poses serious theoretical problems for the analysis of their dynamics. On the other hand, the importance of their dynamic behavior is directly related to the practical interest in predicting such undesirable phenomena as nonlinear oscillations, abrupt transients, intermittence or a high sensitivity with respect to initial conditions. A geometrical/qualitative description of the phase portrait of a discrete-time adaptive system with unmodeled disturbances is given. For this, the motions in the phase space are referred to normally hyperbolic (structurally stable) locally invariant sets. The study is complemented with a local stability analysis of the equilibrium point and periodic solutions. The critical character of adaptive systems under rather usual working conditions is discussed. Special emphasis is put on the causes leading to intermittence. A geometric interpretation of the effects of some commonly used palliatives to this problem is given. The 'dead-zone' approach is studied in more detail. The predicted dynamics are compared with simulation results.

Espana, Martin D.↗

DOE OSTI2024

Evolution of highly multimodal Rayleigh–Taylor instabilities

Rayleigh–Taylor (RT) instabilities are important fluid instabilities that arise in inertial confinement fusion (ICF) capsule implosions, and many other contexts. Multi-mode coupling is observed in experiments and plays a substantial role in material mix from RT instabilities. In this work, we study the evolution of highly multimodal perturbations (power law distribution) that approximate those found at manufactured material interfaces. We use simulations of over 2000 different perturbations in the LANL code xRAGE to identify distinct phases in the processes of bubble growth and bubble merger which can be visualized in a 2D phase portrait with clear regimes of mode growth and decay. Our results show that the dynamic evolution of the instability strongly depends on the mode of the perturbations and mode interactions. The merger process accelerates bubble growth. A non-Markovian region and a transition of the instability from: (1) initial exponential growth to (2) linear growth and to (3) quadratic growth and asymptotic behavior, are clearly captured in the phase space. We have developed a quantitative model of bubble growth that reproduces the dynamic behavior of ensembles of perturbations. Implications for ICF capsules designed for robustness against instabilities are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Modal dynamics of high-frequency transverse combustion instabilities

This paper investigates the modal dynamics of self-excited high frequency transverse instabilities in a multi-nozzle, swirl-stabilized can combustor. It utilizes results from multiple pressure sensors to decompose the disturbance into clockwise (CW) and counter-clockwise (CCW) waves, enabling a reconstruction of the pressure amplitude, phase, anti-nodal line location, and spin ratio. While these types of decompositions have been demonstrated on small scale combustors, particularly annular combustors, this is the first study to analyze these results for a high power, multi-nozzle can combustor. Its particular focus is in characterizing the system evolution in phase space and the nature of the system attractors under linearly unstable conditions; results are presented in spin ratio-phase difference space, so that limit cycle oscillations appear as fixed points. Results are shown for conditions where three different attractors dominate (standing, CW, and CCW) but are taken for very long time periods (~50,000 cycles) so that the system's interaction with these attractors can be observed. For the standing mode-dominated case, the phase portrait shows spiral trajectories converging to a single fixed point. For the spinning mode dominated cases, however, multiple fixed points and saddle points were observed, and the trajectories showed a more complicated structure. It was also shown that even though the CW mode is dominant, this mode intermittently switches to standing or CCW modes for short period. These results suggest that the strength of each attractor depends on test conditions and that random noise stochastically perturbs the system so that large regions of the phase space are explored. Lastly, one interesting observation is that the angular velocity of the anti-nodal line linearly varies with the spin ratio under standing-mode dominant conditions.

42 ENGINEERING↗

DOE OSTI2023

Vibro-impact analysis and characterization of pipeline conveying fluids with multi-segmented motion-limiting constraints

Previous studies of the cantilevered pipeline conveying fluid system have included motion-limiting constraints in the form of trilinear springs. While this is desirable in experimental scenarios, it may not be representative of real-world applications. Therefore, here, this study focuses on multi-segmented motion-limiting constraints. As this type of motion-limiting constraint has not been investigated with a cantilevered pipeline system, a wide variety of outer and inner constraint stiffness and constraint gap sizes are investigated in this study to gain a comprehensive understanding of how the multi-segmented constraints affect the dynamics of the cantilevered pipeline. In this effort, bifurcation diagrams, phase portraits, Poincare maps, time histories, and power spectra are used to investigate the dynamics of the system, and the fluid flow speeds where dynamic characteristics are considered. In general, it is found that critical flow speeds like when the pipe sticks in the constraints are reduced as the constraint stiffnesses are increased. Additionally, the sticking flow speed occurred at lower flow speeds as the gap sizes of the inner and outer constraints decrease, and a larger constraint offset results in a smaller inner gap size leading to critical behaviors occurring at earlier flow speeds.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

The oscillating Fischer-Tropsch reaction

The mechanistic steps that underlie the formation of higher hydrocarbons in catalytic carbon monoxide (CO) hydrogenation at atmospheric pressure over cobalt-based catalysts (Fischer-Tropsch synthesis) have remained poorly understood. Here, we reveal nonisothermal rate-and-selectivity oscillations that are self-sustained over extended periods of time (>24 hours) for a cobalt/cerium oxide catalyst with an atomic ratio of cobalt to cerium of 2:1 (Co 2 Ce 1 ) at 220°C and equal partial pressures of the reactants. A microkinetic mechanism was used to generate rate-and-selectivity oscillations through forced temperature oscillations. Experimental and theoretical oscillations were in good agreement over an extended range of reactant pressure ratios. Additionally, phase portraits for hydrocarbon production were constructed that support the thermokinetic origin of our rate-and-selectivity oscillations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1989

On the Nature of Navier-stokes Turbulence

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.

Deissler, Robert G.↗

NASA NTRS1990

Studies in astronomical time series analysis. IV - Modeling chaotic and random processes with linear filters

While chaos arises only in nonlinear systems, standard linear time series models are nevertheless useful for analyzing data from chaotic processes. This paper introduces such a model, the chaotic moving average. This time-domain model is based on the theorem that any chaotic process can be represented as the convolution of a linear filter with an uncorrelated process called the chaotic innovation. A technique, minimum phase-volume deconvolution, is introduced to estimate the filter and innovation. The algorithm measures the quality of a model using the volume covered by the phase-portrait of the innovation process. Experiments on synthetic data demonstrate that the algorithm accurately recovers the parameters of simple chaotic processes. Though tailored for chaos, the algorithm can detect both chaos and randomness, distinguish them from each other, and separate them if both are present. It can also recover nonminimum-delay pulse shapes in non-Gaussian processes, both random and chaotic.

Scargle, Jeffrey D.↗

NASA NTRS1992

Turbulence and deterministic chaos

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low Reynolds number fully developed turbulence are compared. Several flows are noted: fully chaotic, complex periodic, weakly chaotic, simple periodic, and fixed-point. Of these, only fully chaotic is classified as turbulent. Besides the sustained flows, a flow which decays as it becomes turbulent is examined. For the finest grid, 128(exp 3) points, the spatial resolution appears to be quite good. As a final note, the variation of the velocity derivatives skewness of a Navier-Stokes flow as the Reynolds number goes to zero is calculated numerically. The value of the skewness is shown to become small at low Reynolds numbers, in agreement with intuitive arguments that nonlinear terms should be negligible.

Deissler, Robert G.↗

NASA NTRS1991

Telerobotic control of a mobile coordinated robotic server

Results from the Master's Degree Thesis of Mr. Robert Stanley, a graduate student supervised by the principal investigator on this project is reported. The goal of this effort is to develop advanced control methods for flexible space manipulator systems. As such, a fuzzy logic controller has been developed in which model structure as well as parameter constraints are not required for compensation. A general rule base is formulated using quantized linguistic terms; it is then augmented to a traditional integral control. The resulting hybrid fuzzy controller stabilizes the structure over a broad range of uncertainties, including unknown initial conditions. An off-line tuning approach using phase portraits gives further insight into the algorithm. The approach was applied to a three-degree-of-freedom manipulator system - the prototype of the coordinated flexible manipulator system currently being designed and built at North Carolina State University.

Lee, Gordon↗

NASA NTRS1993

Solar flux forecasting using mutual information with an optimal delay

Solar flux F(sub 10.7) directly affects the atmospheric density, thereby changing the lifetime and prediction of satellite orbits. For this reason, accurate forecasting of F(sub 10.7) is crucial for orbit determination of spacecraft. Our attempts to model and forecast F(sub 10.7) uncovered highly entangled dynamics. We concluded that the general lack of predictability in solar activity arises from its nonlinear nature. Nonlinear dynamics allow us to predict F(sub 10.7) more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. The forecast horizon is a function of two dynamical invariants: the attractor dimension and the Lyapunov exponent. In recent years, estimation of the attractor dimension reconstructed from a time series has become an important tool in data analysis. In calculating the invariants of the system, the first necessary step is the reconstruction of the attractor for the system from the time-delayed values of the time series. The choice of the time delay is critical for this reconstruction. For an infinite amount of noise-free data, the time delay can, in principle, be chosen almost arbitrarily. However, the quality of the phase portraits produced using the time-delay technique is determined by the value chosen for the delay time. Fraser and Swinney have shown that a good choice for this time delay is the one suggested by Shaw, which uses the first local minimum of the mutual information rather than the autocorrelation function to determine the time delay. This paper presents a refinement of this criterion and applies the refined technique to solar flux data to produce a forecast of the solar activity.

Ashrafi, S.↗