On the use of approximate analytical solutions in solving optimum trajectory problems
Method for numerical solutions of nonlinear optimum trajectory problems using approximate analytic method
APPROXIMATION METHOD↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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On singlet s-wave electron-hydrogen scattering.
Discussion of various zeroth-order approximations to s-wave scattering of electrons by hydrogen atoms below the first excitation threshold. The formalism previously developed by the author (1967, 1968) is applied to Feshbach operators to derive integro-differential equations, with the optical-potential set equal to zero, for the singlet and triplet cases. Phase shifts of s-wave scattering are computed in the zeroth-order approximation of the Feshbach operator method and in the static-exchange approximation. It is found that the convergence of numerical computations is faster in the former approximation than in the latter.
Madan, R. N.↗
Analytical and experimental study of flow phenomena in noncavitating rocket pump inducers
The flow processes in rocket pump inducers are summarized. The experimental investigations were carried out with air as the test medium. The major characteristics features of the rocket pump inducers are low flow coefficient (0.05 to 0.2) large stagger angle (70 deg to 85 deg) and high solidity blades of little or no camber. The investigations are concerned with the effect of viscosity not the effects of cavitation. Flow visualization, conventional and hot wire probe measurement inside and at the exit of the blade passage, were the analytical methods used. The experiment was carried out using four three and two bladed inducers with cambered blades. Both the passage and the exit flow were measured. The basic research and boundary layer investigation was carried out using a helical flat plate (of some dimensions as the inducer blades tested), and flat plate helical inducer (four bladed). Detailed mean and turbulence flow field inside the passage as well as the exit of the rotor were derived from these measurement. The boundary layer, endwall, and other passage data reveal extremely complex nature of the flow, with major effects of viscosity present across the entire passage. Several analyses were carried out to predict the flow field in inducers. These included an approximate analysis, the shear pumping analysis, and a numerical solution of exact viscous equations with approximate modeling for the viscous terms.
Lakshminarayana, B.↗
Numerical solution of transonic full stream function equations in conservation form
The stream function equation in conservation form is solved iteratively based on the artificial compressibility method. The density is not a unique function of the mass flux. In order to avoid the ambiguity near the sonic line, the density is updated in terms of the velocity, which is obtained through a simple integration of a first order equation step by step in the flow field. Iteration algorithms and finite difference approximations are discussed and numerical results of both conservative and nonconservative calculations are presented.
Hafez, M. M.↗
Modeling the small-scale deposition of snow onto structured Arctic sea ice during a MOSAiC storm using snowBedFoam 1.0.
The remoteness and extreme conditions of the Arctic make it a very difficult environment to investigate. In these polar regions covered by sea ice, the wind is relatively strong due to the absence of obstructions and redistributes a large part of the deposited snow mass, which complicates estimates for precipitation hardly distinguishable from blowing or drifting snow. Moreover, the snow mass balance in the sea ice system is still poorly understood, notably due to the complex structure of its surface. Quantitatively assessing the snow distribution on sea ice and its connection to the sea ice surface features is an important step to remove the snow mass balance uncertainties (i.e., snow transport contribution) in the Arctic environment. In this work we introduce snowBedFoam 1.0., a physics-based snow transport model implemented in the open-source fluid dynamics software OpenFOAM. We combine the numerical simulations with terrestrial laser scan observations of surface dynamics to simulate snow deposition in a MOSAiC (Multidisciplinary Drifting Observatory for the Study of Arctic Climate) sea ice domain with a complicated structure typical for pressure ridges. The results demonstrate that a large fraction of snow accumulates in their vicinity, which compares favorably against scanner measurements. However, the approximations imposed by the numerical framework, together with potential measurement errors (precipitation), give rise to quantitative inaccuracies, which should be addressed in future work. The modeling of snow distribution on sea ice should help to better constrain precipitation estimates and more generally assess and predict snow and ice dynamics in the Arctic.
54 ENVIRONMENTAL SCIENCES↗
A mathematical analysis of the theory of interplanetary scintillation in the weak scattering approximation
A simplified analytical technique is presented for modeling the interplanetary scintillation of radio sources of finite angular size with a power-law electron-density-fluctuation power spectrum. The simplification results from the representation of the scintillation spectrum in confluent hypergeometric functions. The approximations presented allow fast numerical evaluation of a spectrum for a weakly scattering but extended medium with less than 10% error over the entire spectrum. Parameters describing anisotropic electron irregularities as well as anisotropic source structure are included, and the dependence of the spectrum normalization on the scales of the medium is derived explicitly. The parametric description of the domains of convergence of the approximate expansions also provides a simple conceptualization of the relative contributions of the scattered radiation along the line of sight to the observed spectrum. This is particularly useful for sources of finite angular size. This technique is applied to previously published observations.
Mitchell, D. G.↗
Quantum Control Landscapes Beyond the Dipole Approximation: Controllability, Singular Controls, and Resources
We investigate the control landscapes of closed n-level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. The latter term is quadratic in the control field. Theoretical analysis of singular controls is presented, which are candidates for producing landscape traps. The results for considering the presence of singular controls are compared to their counterparts in the dipole approximation (i.e., without polarizability). A numerical analysis of the existence of traps in control landscapes for generating unitary transformations beyond the dipole approximation is made upon including the polarizability term. An extensive exploration of these control landscapes is achieved by creating many random Hamiltonians which include terms linear and quadratic in a single control field. The discovered singular controls are all found not to be local optima. This result extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the fluence of the control resulting from optimization. It is also shown that including a polarizability term in an otherwise uncontrollable dipole coupled system removes traps from the corresponding control landscape by restoring controllability. We numerically assess the effect of a polarizability term on a known example of a particular three-level Λ-system with a second order trap in its control landscape. It is found that the addition of the polarizability removes the trap from the landscape. The general practical control implications of these simulations are discussed.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Inelastic Triatom-Atom Quantum Close-Coupling Dynamics in Full Dimensionality: All Rovibrational Mode Quenching of Water Due to the H Impact on a Six-Dimensional Potential Energy Surface
The rovibrational level populations, and subsequent emission in various astrophysical environments, are driven by inelastic collision processes. The available rovibrational rate coefficients for water have been calculated using a number of approximations. We present a numerically exact calculation for the rovibrational quenching for all water vibrational modes due to collisions with atomic hydrogen. The scattering theory implements a quantum close-coupling (CC) method on a high level ab initio six-dimensional (6D) potential energy surface (PES). Total rovibrational quenching cross sections for excited bending levels were compared with earlier results on a 4D PES with the rigid-bender closecoupling (RBCC) approximation. General agreement between 6D-CC and 4DRBCC calculations are found, but differences are evident including the energy and amplitude of low-energy orbiting resonances. Quenching cross sections from the symmetric and asymmetric stretch modes are provided for the first time. The current 6D-CC calculation provides accurate inelastic data needed for astrophysical modeling.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Numerical Study of g-Jitter Induced Double-Diffusive Convection
A finite element study is presented of double-diffusive convection driven by g-jitter in a microgravity environment. Mathematical formulations are presented and extensive simulations are carried out for g-jitter induced fluid flow, temperature distribution, and solutal transport in an alloy system under consideration for space flights. Computations include the use of idealized single-frequency and multi-frequency g-jitter as well as the real g-jitter data taken during an actual Space Shuttle fight. Little correlation is seen between these velocity components for the g-jitter components studied. The temperature field is basically undisturbed by convection because of a small Pr number for the fluid. The disturbance of the concentration field, however, is pronounced, and the local variation of the concentration follows the velocity oscillation in time. It is found that although the concentration field varies in both position and time, the local concentration gradient remains approximately constant in time. Numerical study further indicates that with an increase in g-jitter force (or amplitude), the nonlinear convective effects become much more obvious, which in turn drastically change the concentration fields. The simulated results computed using the g-jitter data taken during space flights show that both the velocity and concentration become random, following approximately the same pattern as the g-jitter perturbations.
Shu, Y.↗
On the Routh approximation technique and least squares errors
A new method for calculating the coefficients of the numerator polynomial of the direct Routh approximation method (DRAM) using the least square error criterion is formulated. The necessary conditions have been obtained in terms of algebraic equations. The method is useful for low frequency as well as high frequency reduced-order models.
Aburdene, M. F.↗
Effect of microgravity and hypergravity on deposition of 0.5- to 3-micron-diameter aerosol in the human lung
We measured intrapulmonary deposition of 0. 5-, 1-, 2-, and 3-micron-diameter particles in four subjects on the ground (1 G) and during parabolic flights both in microgravity (microG) and at approximately 1.6 G. Subjects breathed aerosols at a constant flow rate (0.4 l/s) and tidal volume (0.75 liter). At 1 G and approximately 1.6 G, deposition increased with increasing particle size. In microG, differences in deposition as a function of particle size were almost abolished. Deposition was a nearly linear function of the G level for 2- and 3-micron-diameter particles, whereas for 0.5- and 1.0-micron-diameter particles, deposition increased less between microG and 1 G than between 1 G and approximately 1.6 G. Comparison with numerical predictions showed good agreement for 1-, 2-, and 3-micron-diameter particles at 1 and approximately 1.6 G, whereas the model consistently underestimated deposition in microG. The higher deposition observed in microG compared with model predictions might be explained by a larger deposition by diffusion because of a higher alveolar concentration of aerosol in microG and to the nonreversibility of the flow, causing additional mixing of the aerosols.
Flight Experiment↗
Dynamics of spin stabilized spacecraft during deployment of telescoping appendages
The dynamics of a spin stabilized spacecraft with telescoping appendages are treated both analytically and numerically. Two types of telescoping appendages are considered: (a) where an end mass is mounted at the end of an (assumed) massless boom; and (b) where the appendage is assumed to consist of a uniformly distributed homogeneous mass throughout its length. For the telescoping system Eulerian equations of motion are developed. During all deployment sequences it is assumed that the transverse component of angular momentum is much smaller than the component along the major spin axis. Closed form analytical solutions for the time response of the transverse components of angular velocities are obtained when the spacecraft hub has a nearly spherical mass distribution. For the more general case, a series solution is obtained and this solution is limited by its radius of convergence. The comparison of the different approximate analytical methods with numerical integration results are studied and it is observed that the oscillatory nature of the responses of the transverse angular velocity components reduces rapidly with faster extension rates.
Sellappan, R.↗
Aeroelastic stability of periodic systems with application to rotor blade flutter
The dynamics of a helicopter blade in forward flight are described by a system of linear differential equations with periodic coefficients. The stability of this periodic aeroelastic system is determined, using multivariable Floquet-Liapunov theory. The transition matrix at the end of the period is evaluated by: (1) direct numerical integration, and (2) a new, approximate method, which consists in approximating a periodic function by a series of step functions. The numerical accuracy and efficiency of the methods is compared, and the second method is shown to be superior by far. Results illustrating the effect of the periodic coefficients and various blade parameters are presented.
Friedmann, P.↗
The uniform asymptotic swallowtail approximation - Practical methods for oscillating integrals with four coalescing saddle points
Methods that can be used in the numerical implementation of the uniform swallowtail approximation are described. An explicit expression for that approximation is presented to the lowest order, showing that there are three problems which must be overcome in practice before the approximation can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.
Connor, J. N. L.↗
Benchmark Problems Used to Assess Computational Aeroacoustics Codes
The field of computational aeroacoustics (CAA) encompasses numerical techniques for calculating all aspects of sound generation and propagation in air directly from fundamental governing equations. Aeroacoustic problems typically involve flow-generated noise, with and without the presence of a solid surface, and the propagation of the sound to a receiver far away from the noise source. It is a challenge to obtain accurate numerical solutions to these problems. The NASA Glenn Research Center has been at the forefront in developing and promoting the development of CAA techniques and methodologies for computing the noise generated by aircraft propulsion systems. To assess the technological advancement of CAA, Glenn, in cooperation with the Ohio Aerospace Institute and the AeroAcoustics Research Consortium, organized and hosted the Fourth CAA Workshop on Benchmark Problems. Participants from industry and academia from both the United States and abroad joined to present and discuss solutions to benchmark problems. These demonstrated technical progress ranging from the basic challenges to accurate CAA calculations to the solution of CAA problems of increasing complexity and difficulty. The results are documented in the proceedings of the workshop. Problems were solved in five categories. In three of the five categories, exact solutions were available for comparison with CAA results. A fourth category of problems representing sound generation from either a single airfoil or a blade row interacting with a gust (i.e., problems relevant to fan noise) had approximate analytical or completely numerical solutions. The fifth category of problems involved sound generation in a viscous flow. In this case, the CAA results were compared with experimental data.
Dahl, Milo D.↗
On the breakup of viscous liquid threads
A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions which yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Further more, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inerticial forces in the momentum equations to derive an evolution system modelling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.
Papageorgiou, Demetrios T.↗
Measuring the Hubble constant and our Virgo-infall velocity independently
A sample of spiral galaxies with B(sub T) less than 14.5 located in two local volumes, one in the direction of, but behind, the Virgo Cluster (behind-Virgo volume (BV)) and the other in the opposite direction (anti-Virgo volume (AV)), were used via a Tully-Fisher (TF) relation to derive the following two parameters: H(sub AB), the mean Hubble ratio between AV and BV, and delta v(sub parallel), the peculiar velocity of the Local Group in the direction of the Virgo Cluster (VC) with respect to a uniformly expanding reference system defined by our AV and BV sub-samples. The two sampled volumes, separated by a velocity interval of 5600 km/s, form an antipodal pair. This particular geometry not only allows us to derive the two parameters independently but also reduces the dynamical effect of the Local Supercluster on H(sub AB) without increasing the Malmquist bias. By limiting our sample to spiral galaxies having large velocity widths W(sub R), we effectively reduce the TF scatter and Malmquist bias in our sample. The TF zero point and dispersion were then determined by further correcting for the small residual Malmquist bias. An additional sample of fainter galaxies was used to test for a non-Gaussian tail to the TF disperison. We found no evidence for such a tail and formally give an upper limit of about 18% for the fractional contribution of an unseen tail. The average intrinsic TF dispersion for the dominant Gaussian component is sigma(sub TF)(sup 0) approximately 0.33 mag for W(sub R) approximately equal to or greater than 180 km/s. Our numerical results are delta v(sub parallel) approximately equals 414 +/- 82 km/s and H(sub AB) approximately equals (84.0 +/- 2.4)(1 + epsilon) km/s Mpc, where (1 + epsilon) accounts for any systematic error between the calibrators and the sample galaxies. Various dynamical models were tested to explore the effect on H(sub AB) of the uncertainties in the local velocity field. Constrained by our observed delta v(sub parallel) as well as other observational quantities, we found that the rms deviation from unity of H(sub AB)/H(sub 0) (where H(sub 0) is the Hubble constant for each model) is 5%, making H(sub AB) a good indicator for H(sub 0). Taking this variation as an additional error, our formal estimate for the Hubble constant is H(sub 0) approximately equals (84 +/- 5)(1 + epsilon) km/s Mpc.
Lu, Nanyao Y.↗
Analytic approximations in the study of the solar modulation of electrons
Numerical solutions to the transport equation of galactic cosmic rays in the interplanetary medium have been used to investigate the applicability of commonly used approximate analytic solutions for electrons (10 MeV to 10 GeV). We find that for a given cosmic-ray diffusion coefficient the force-field approximation is in reasonable agreement with the numerical solution at energies above 200 MeV, but deviates significantly at lower energies, depending on the shape of the interstellar electron spectrum. The diffusion-convection approximation agrees generally with the numerical solution within a factor of 2 over the entire energy range.
Cummings, A. C.↗