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NASA NTRS1963

Resonance Cases and Small Divisors in a Third Integral of Motion

In this paper a general discussion of the resonance cases in an axially-symmetric potential field is presented, when the unperturbed frequencies in the radial and z direction have a rational ratio. The general form of the third integral is not valid in these cases because of the appearance of divisors of the form (m(exp 2)P-n(exp 2)Q), which become zero in the resonance cases. However, a new isolating integral of the unperturbed case is available, and this can be used to construct a third integral in the form of a power series and eliminate all secular terms. Three cases are distinguished, (alpha) m+n>4, (beta) m+n=4, and (gamma) m+n<4. In the first case the orbits are rather similar to those of the general irrational case. In the third case th e orbits show a quite peculiar character, which, however, can be explained rather accurately by a first-order theory of the third integral. Numerical integrations were made for the cases P=16Q, 4P=9Q, and P=4Q. The third integral, given in first- or second-order approximation, is rather well conserved. Case beta and the cases of small divisors, when m(exp 2)P-n(exp 2)Q is near zero but not equal to zero, are discussed in Paper II.

Contopoulos, George↗
NASA NTRS2015

Comparison of High-Order and Low-Order Methods for Large-Eddy Simulation of a Compressible Shear Layer

The objective of this work is to compare a high-order solver with a low-order solver for performing large-eddy simulations (LES) of a compressible mixing layer. The high-order method is the Wave-Resolving LES (WRLES) solver employing a Dispersion Relation Preserving (DRP) scheme. The low-order solver is the Wind-US code, which employs the second-order Roe Physical scheme. Both solvers are used to perform LES of the turbulent mixing between two supersonic streams at a convective Mach number of 0.46. The high-order and low-order methods are evaluated at two different levels of grid resolution. For a fine grid resolution, the low-order method produces a very similar solution to the high-order method. At this fine resolution the effects of numerical scheme, subgrid scale modeling, and filtering were found to be negligible. Both methods predict turbulent stresses that are in reasonable agreement with experimental data. However, when the grid resolution is coarsened, the difference between the two solvers becomes apparent. The low-order method deviates from experimental results when the resolution is no longer adequate. The high-order DRP solution shows minimal grid dependence. The effects of subgrid scale modeling and spatial filtering were found to be negligible at both resolutions. For the high-order solver on the fine mesh, a parametric study of the spanwise width was conducted to determine its effect on solution accuracy. An insufficient spanwise width was found to impose an artificial spanwise mode and limit the resolved spanwise modes. We estimate that the spanwise depth needs to be 2.5 times larger than the largest coherent structures to capture the largest spanwise mode and accurately predict turbulent mixing.

large eddy simulations↗
NASA NTRS2016

Hybrid Differential Dynamic Programming with Stochastic Search

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Aziz, Jonathan↗
NASA NTRS2016

Hybrid Differential Dynamic Programming with Stochastic Search

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASAs Dawn mission. The Dawn trajectory was designed with the DDP-based Static Dynamic Optimal Control algorithm used in the Mystic software. Another recently developed method, Hybrid Differential Dynamic Programming (HDDP) is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Aziz, Jonathan↗
NASA NTRS2015

Time-Accurate Unsteady Pressure Loads Simulated for the Space Launch System at Wind Tunnel Conditions

A transonic flow field about a Space Launch System (SLS) configuration was simulated with the Fully Unstructured Three-Dimensional (FUN3D) computational fluid dynamics (CFD) code at wind tunnel conditions. Unsteady, time-accurate computations were performed using second-order Delayed Detached Eddy Simulation (DDES) for up to 1.5 physical seconds. The surface pressure time history was collected at 619 locations, 169 of which matched locations on a 2.5 percent wind tunnel model that was tested in the 11 ft. x 11 ft. test section of the NASA Ames Research Center's Unitary Plan Wind Tunnel. Comparisons between computation and experiment showed that the peak surface pressure RMS level occurs behind the forward attach hardware, and good agreement for frequency and power was obtained in this region. Computational domain, grid resolution, and time step sensitivity studies were performed. These included an investigation of pseudo-time sub-iteration convergence. Using these sensitivity studies and experimental data comparisons, a set of best practices to date have been established for FUN3D simulations for SLS launch vehicle analysis. To the author's knowledge, this is the first time DDES has been used in a systematic approach and establish simulation time needed, to analyze unsteady pressure loads on a space launch vehicle such as the NASA SLS.

Alter, Stephen J.↗
NASA NTRS2016

Mapping the Gas Turbulence in the Coma Cluster: Predictions for Astro-H

Astro-H will be able for the first time to map gas velocities and detect turbulence in galaxy clusters. One of the best targets for turbulence studies is the Coma cluster, due to its proximity, absence of a cool core, and lack of a central active galactic nucleus. To determine what constraints Astro-H will be able to place on the Coma velocity field, we construct simulated maps of the projected gas velocity and compute the second-order structure function, an analog of the velocity power spectrum. We vary the injection scale, dissipation scale, slope, and normalization of the turbulent power spectrum, and apply measurement errors and finite sampling to the velocity field. We find that even with sparse coverage of the cluster, Astro-H will be able to measure the Mach number and the injection scale of the turbulent power spectrum-the quantities determining the energy flux down the turbulent cascade and the diffusion rate for everything that is advected by the gas (metals, cosmic rays, etc.). Astro-H will not be sensitive to the dissipation scale or the slope of the power spectrum in its inertial range, unless they are outside physically motivated intervals. We give the expected confidence intervals for the injection scale and the normalization of the power spectrum for a number of possible pointing configurations, combining the structure function and velocity dispersion data. Importantly, we also determine that measurement errors on the line shift will bias the velocity structure function upward, and show how to correct this bias.

clusters: intracluster medium↗
NASA NTRS2017

A Smoothed Eclipse Model for Solar Electric Propulsion Trajectory Optimization

Solar electric propulsion (SEP) is the dominant design option for employing low-thrust propulsion on a space mission. Spacecraft solar arrays power the SEP system but are subject to blackout periods during solar eclipse conditions. Discontinuity in power available to the spacecraft must be accounted for in trajectory optimization, but gradient-based methods require a differentiable power model. This work presents a power model that smooths the eclipse transition from total eclipse to total sunlight with a logistic function. Example trajectories are computed with differential dynamic programming, a second-order gradient-based method.

trajectory design↗
NASA NTRS2017

Low Pressure Flame Blowoff from the Forward Stagnation Region of a Blunt-Nosed Cast PMMA Cylinder in Axial Mixed Convective Flow

Low-pressure blowoff experiments were conducted with a stagnation flame stabilized on the forward tip of cast PMMA rods in a vertical wind tunnel. Pressure, forced flow velocity, gravity, and ambient oxygen concentration were varied. Stagnation flame blowoff is determined from a time-stamped video recording of the test. The blowoff pressure is determined from test section pressure transducer data that is synchronized with the time stamp. The forced flow velocity is also determined from the choked flow orifice pressure. Most of the tests were performed in normal gravity, but a handful of microgravity tests were also conducted to determine the influence of buoyant flow velocity on the blowoff limits. The blowoff limits are found to have a linear dependence between the partial pressure of oxygen and the total pressure, regardless of forced flow velocity and gravity level. The flow velocity (forced and/or buoyant) affects the blowoff pressure through the critical Damkohler number residence time, which dictates the partial pressure of oxygen at blowoff. This is because the critical stretch rate increases linearly with increasing pressure at low pressure (sub-atmospheric pressures) since a second-order overall reaction rate with two-body reactions dominates in this pressure range.

flammability↗
NASA NTRS2017

A Smoothed Eclipse Model for Solar Electric Propulsion Trajectory Optimization

Solar electric propulsion (SEP) is the dominant design option for employing low-thrust propulsion on a space mission. Spacecraft solar arrays power the SEP system but are subject to blackout periods during solar eclipse conditions. Discontinuity in power available to the spacecraft must be accounted for in trajectory optimization, but gradient-based methods require a differentiable power model. This work presents a power model that smooths the eclipse transition from total eclipse to total sunlight with a logistic function. Example trajectories are computed with differential dynamic programming, a second-order gradient-based method.

optimization↗
NASA NTRS2016

Production of Peroxy Nitrates in Boreal Biomass Burning Plumes over Canada During the BORTAS Campaign

The observations collected during the BOReal forest fires on Tropospheric oxidants over the Atlantic using Aircraft and Satellites (BORTAS) campaign in summer 2011 over Canada are analysed to study the impact of forest fire emissions on the formation of ozone (O3 and total peroxy nitrates (sigma)PNs, (sigma)ROONO2. The suite of measurements on board the BAe-146 aircraft, deployed in this campaign, allows us to calculate the production of O3 and of (sigma)PNs, a long-lived NOx reservoir whose concentration is supposed to be impacted by biomass burning emissions.In fire plumes, profiles of carbon monoxide (CO), which is a well-established tracer of pyrogenic emission, show concentration enhancements that are in strong correspondence with a significant increase of concentrations of (sigma)PNs, where as minimal increase of the concentrations of O3 and NO2 is observed. The (sigma)PN and O3 productions have been calculated using the rate constants of the first- and second-order react Pions of volatile organic compound (VOC) oxidation. The (sigma)PN and O3 productions have also been quantified by 0-D model simulation based on the Master Chemical Mechanism. Both methods show that in fire plumes the average production of (sigma)PNs and O3 are greater than in the background plumes, but the increase of (sigma)PN production is more pronounced than the O3 production. The average (sigma)PN production in fire plumes is from 7 to 12 times greater than in the background, whereas the average O3 production in fire plumes is from 2 to 5 times greater than in the background. These results suggest that, at least for boreal forest fires and for the measurements recorded during the BORTAS campaign,fire emissions impact both the oxidized NOy and O3;but (1)(sigma)PN production is amplified significantly more thanO3 production and (2) in the forest fire plumes the ratio between the O3 production and the (sigma)PN production is lower than the ratio evaluated in the background air masses, thus confirming that the role played by the (sigma)PNs produced during biomass burning is significant in the O3 budget. The implication of these observations is that fire emissions in some cases, for example boreal forest fires and in the conditions reported here, may influence more long-lived precursors of O3than short-lived pollutants, which in turn can be transported and eventually diluted in a wide area.

biomass burning↗
NASA NTRS2017

Simulation Study of Landau Damping Near the Persisting to Arrested Transition

A one-dimensional electrostatic filtered Vlasov-Poisson simulation study is discussed. The transition from persisting to arrested Landau damping that is produced by increasing the strength of a sinusoidal perturbation on a background Vlasov-Poisson equilibrium is explored. Emphasis is placed on observed features of the electron phase-space distribution when the perturbation strength is near the transition value. A single ubiquitous waveform is found perturbing the space-averaged phase-space distribution at almost any time in all of the simulations; the sole exception is the saturation stage that can occur at the end of the arrested damping scenario. This waveform contains relatively strong, very narrow structures in velocity bracketing (plus or minus velocity (sub res)) - the velocities at which electrons must move to traverse the dominant field mode wavelength in one of its oscillation periods - and propagating with (plus or minus velocity (sub res)) respectively. Local streams of electrons are found in these structures crossing the resonant velocities from low speed to high speed during Landau damping and from high speed to low speed during Landau growth. At the arrest time, when the field strength is briefly constant, these streams vanish. It is conjectured that the expected transfer of energy between electrons and field during Landau growth or damping has been visualized for the first time. No evidence is found in the phase-space distribution to support recent well-established discoveries of a second-order phase transition in the electric field evolution. While trapping is known to play a role for larger perturbation strengths, it is shown that trapping plays no role at any time in any of the simulations near the transition perturbation strength.

plasma simulation↗
NASA NTRS2017

The Impact Trajectory of Asteroid 2008 TC3

Asteroid 2008 TC3 was the rst asteroid ever discovered before reaching Earth. By using the almost 900 astrometric observations acquired prior to impact we estimate the trajectory of 2008 TC3 and the ground-track of the impact location as a function of altitude. For a reference altitude of 100 km the impact location 3- formal uncertainty is a 1.4 km 0.15 km ellipse with a semimajor axis azimuth of 105. We analyze the contribution of modeling errors and nd that the second-order zonal harmonics of the Earth gravity eld moves the ground-track by more than 1 km and the location along the ground-track by more than 2 km. Non-zonal and higher order harmonics only change the impact prediction by less than 20 m. The contribution of the atmospheric drag to the trajectory of 2008 TC3 is at the numerical integration error level, a few meters, down to an altitude of 50 km. Integrating forward to lower altitudes and ignoring the break-up of 2008 TC3, the atmospheric drag causes an along-track error that can be as large as a few kilometers at sea level. The locations of the recovered meteorites is consistent with the computed ground-track.

Trajectory↗
NASA NTRS2015

The Structure, Anharmonic Vibrational Frequencies, and Intensities of NNHNN+

A semi-global potential energy surface (PES) and quartic force field (QFF) based on fitting high-level electronic structure energies are presented to describe the structures and spectroscopic properties of NNHNN+. The equilibrium structure of NNHNN+ is linear with the proton equidistant between the two nitrogen groups and thus of D(sub ∞h) symmetry. Vibrational second-order perturbation theory (VPT2) calculations based on the QFF fails to describe the proton “rattle” motion, i.e., the antisymmetric proton stretch, due to the very flat nature of PES around the global minimum, but performs properly for other modes with sharper potential wells. Vibrational self-consistent field/virtual state configuration interaction (VSCF/VCI) calculations using a version of MULTIMODE without angular momentum terms successfully describe this motion and predict the fundamental to be at 759 cm(exp -1). This is in good agreement with the value of 746 cm(exp -1) from a fixed-node diffusion Monte Carlo calculation and the experimental Ar-tagged result of 743 cm(exp -1). Other VSCF/VCI energies are in good agreement with other experimentally reported ones. Both double-harmonic intensity and rigorous MULTIMODE intensity calculations show the proton transfer fundamental has a very strong intensity.

Astrochemistry↗
NASA NTRS2015

Numerical Simulations of Shock/Plume Interaction Using Structured Overset Grids

Computational simulations using structured overset grids with the Launch Ascent and Vehicle Aerodynamics (LAVA) solver framework are presented for predicting oblique shock/plume interaction effects to near-field sonic boom signatures. Standard second-order accurate as well as higher-resolution numerical discretizations are utilized and compared in the study. The numerical approach is compared with supersonic wind-tunnel data for three cases. The cases include an empty wind-tunnel at the operating conditions, an isolated shockgenerating diamond wedge within the tunnel, and a nozzle with diamond wedge configuration at five different nozzle pressure ratios. Solution sensitivity to numerical discretization is analyzed. Favorable comparisons between the computational results and experimental data of near-field pressure signatures are obtained. A simple prediction method for plume induced shock deflection is developed and results are compared with the CFD data.

Housman, Jeffrey A.↗
NASA NTRS2015

Higher-Order Methods for Compressible Turbulent Flows Using Entropy Variables

Turbulent flows have a large range of spatial and temporal scales which need to be resolved in order to obtain accurate predictions. Higher-order methods can provide greater efficiency for simulations requiring high spatial and temporal resolution, allowing for solutions with fewer degrees of freedom and lower computational cost than traditional second-order computational fluid dynamics (CFD) methods.1 Higher-order methods have been widely used for turbulent flows. However, the reduced numerical stabilization present in higher-order schemes implies that special care needs to be taken in the development of numerical methods to suppress nonlinear instabilities.2–6 In this work we present the development of a higher-order space-time discontinuous Galerkin method with a focus on the aspects of our numerical scheme required for ensuring nonlinear stability for turbulent simulations at high Reynolds numbers.

Diosady, Laslo T.↗
NASA NTRS1989

An analysis of digital phase-locked loops

This report focuses on second-order digital phase-locked loops (DPLLs) with uniformly sampled input, an amplitude-insenstive phase extractor and a conventional loop filter.

Thomas, J. B.↗
NASA NTRS1993

Higher-Order Ionospheric Effects on the GPS Observables and Means of Modeling Them

Based on realistic modeling of the electron density of the ionosphere and using a dipole moment approximation for the earth magnetic field, we are able to estimate the effect of the ionosphere on the GPS signal for a ground user. Our study implies reduction of the second-order ionospheric error to less than 2 mm on the average and, therefore, potential improvement of orbit determination and baseline solutions.

ionosphere↗
DOE OSTI2024

Resolution requirements for high-speed droplet impact along a rigid wall

The mechanical and thermal loads that are produced on high-speed projectiles from a droplet are key in the projectile development.To accurately predict the pressure produced along the wall surface during impact, it is important to correctly model the air/water interface of the droplet. Novel modeling approaches such as the Phase-Field enable control of the (numerical) interface thickness. However, the interplay between the grid resolution (including potential adaptive mesh refinement) and the interface thickness is unclear. In this work, the resolution requirements for numerical simulations of a two-dimensional cylindrical water droplet impacting a rigid wall at Mach four are computationally investigated. Numerical simulations are performed using a second-order accurate scheme with adaptive mesh refinement and a Phase-Field interface model. The choice of numerical interface thickness and resolution parameters affects the maximum pressure and location of the maximum pressure along the wall. The overall maximum pressure does not converge even at high resolutions. The number of levels and thickness (number of cells across the interface) play a role in obtaining a well behaved solution.

Burrell, E.↗