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NASA NTRS1981

Theory of dense hydrogen

Equations of state for molecular and metallic hydrogen are calculated in an accurate and comparable manner by using the density-functional method applied to static lattices. By isolating the electronic energies in this way, attention can be drawn to the relative importance of the protonic degrees of freedom. From the equation-of-state results it can be concluded that a remnant molecular pairing is preferred in a band-overlap metallic state, complete dissociation occurring only at very high densities. An accurate determination of the pressure needed to achieve complete dissociation is shown to require a self-consistent treatment of electron and proton degrees of freedom.

Chakravarty, S.↗

NASA NTRS1993

Trajectory optimization based on differential inclusion

A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a 1-D rocket ascent problem ('Goddard Problem') in presence of a dynamic pressure constraint.

Seywald, Hans↗

NASA NTRS1993

Trajectory optimization based on differential inclusion

A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a I-D rocket ascent problem ('Goddard Problem') in presence of a dynamic pressure constraint.

Seywald, Hans↗

NASA NTRS1993

Trajectory optimization based on differential inclusion

A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a 1-D rocket ascent problem (`Goddard Problem') in presence of a dynamic pressure constraint.

Seywald, Hans↗

NASA NTRS1983

A theory of nondilute spray evaporation based upon multiple drop interactions

Drop interactions, changes in drop temperature, and the evolution of the ambient conditions due to heating and evaporation were taken into account in a theory of drop evaporation. A sphere of influence is defined for each drop, with all droplets having equal separation distances. Global conservation and state equations for a droplet spray are formulated, together with equations for single drop evaporation in finite surroundings. A criterion is defined for complete evaporation and the dynamics of spray evaporation are discussed. The theory is noted to be applicable for both dilute and nondilute spray evaporation effects.

Bellan, J.↗

NASA NTRS1990

Closed-form solution of decomposable stochastic models

Markov and semi-Markov processes are increasingly being used in the modeling of complex reconfigurable systems (fault tolerant computers). The estimation of the reliability (or some measure of performance) of the system reduces to solving the process for its state probabilities. Such a model may exhibit numerous states and complicated transition distributions, contributing to an expensive and numerically delicate solution procedure. Thus, when a system exhibits a decomposition property, either structurally (autonomous subsystems), or behaviorally (component failure versus reconfiguration), it is desirable to exploit this decomposition in the reliability calculation. In interesting cases there can be failure states which arise from non-failure states of the subsystems. Equations are presented which allow the computation of failure probabilities of the total (combined) model without requiring a complete solution of the combined model. This material is presented within the context of closed-form functional representation of probabilities as utilized in the Symbolic Hierarchical Automated Reliability and Performance Evaluator (SHARPE) tool. The techniques adopted enable one to compute such probability functions for a much wider class of systems at a reduced computational cost. Several examples show how the method is used, especially in enhancing the versatility of the SHARPE tool.

Sjogren, Jon A.↗

NASA NTRS1984

Use of the Burton-prim-slichter Equation at High Growth Rates

The following must be assumed in order to derive the classic Burton-Prim-Slichter equation for segregation during crystal growth: isobaric, isothermal, isopotential or uncharged species, binary, planar interface, steady state, constant diffusion coefficient in fluid, no diffusion in crystal, no lateral convection within fluid film at interface with complete mixing beyond (stagnant film model), and either density, total concentration or partial molar volumes constant in the fluid phase. In addition, the effective distribution coefficient and the interfacial distribution coefficient must be defined properly. The velocity in the equation is the growth rate times a factor correcting for the difference in volumetric properties between crystal and fluid. The stagnant film thickness is found to be a function of freezing rate, with the precise functionality depending on the type of stirring occurring in the growth fluid.

Wilcox, W. R.↗

NASA NTRS1983

Space Shuttle propulsion parameter estimation using optimal estimation techniques, volume 1

The mathematical developments and their computer program implementation for the Space Shuttle propulsion parameter estimation project are summarized. The estimation approach chosen is the extended Kalman filtering with a modified Bryson-Frazier smoother. Its use here is motivated by the objective of obtaining better estimates than those available from filtering and to eliminate the lag associated with filtering. The estimation technique uses as the dynamical process the six degree equations-of-motion resulting in twelve state vector elements. In addition to these are mass and solid propellant burn depth as the ""system'' state elements. The ""parameter'' state elements can include aerodynamic coefficient, inertia, center-of-gravity, atmospheric wind, etc. deviations from referenced values. Propulsion parameter state elements have been included not as options just discussed but as the main parameter states to be estimated. The mathematical developments were completed for all these parameters. Since the systems dynamics and measurement processes are non-linear functions of the states, the mathematical developments are taken up almost entirely by the linearization of these equations as required by the estimation algorithms.

Source record↗

NASA NTRS1990

Synoptic/planetary-scale interactions and blocking over the North Atlantic Ocean

Work was completed on the height tendency diagnoses of two extratropical cyclones that occurred upstream from the blocking event studied previously. One developed explosively over water 60 to 36 hours before the block first appeared, while the second developed explosively over the southeastern United States during the time of block formation. In both cases, both vorticity and temperature advection were consistently important forcing mechanisms. This is in contrast to the block itself, in which vorticity advection was easily the dominant forcing mechanism. Latent heat release was also significant, accounting for about 50 percent of the total height falls in the cyclone below 850 mb. Estimates of latent heat release were greatly enhanced by coupling parameterized estimates with values derived from GOES IR data using an algorithm developed by Marshall's F. R. Robertson. Among the difficulties encountered in this work was the identification of an appropriate lower boundary condition for the solution of the height tendency equation. The zero value currently used tends to yield underestimates of the lower troposphere height tendencies. To address this problem a new diagnostic technique was developed in cooperation with Dr. Peter Zwack of the University of Quebec at Montreal. Based on an equation Dr. Zwack had previously developed (the Zwack-Okossi development equation), researchers now have a relationship that is completely consistent with the height tendency equation and provides estimates of lower boundary geostrophic vorticity or height tendencies. Finally, comparison of the SAT (satellite data) and NOSAT (no satellite data) analyses is progressing well. The present focus is on both the new diagnostic technique and the SAT/NOSAT comparisons. The former is being tested on the southeastern United States cyclone case previously mentioned and compared with the height tendency diagnoses already completed. The latter are being examined for the blocking case described in the publications cited in this summary. In addition to obtaining statistics that will allow general comparison of the two analyses, it will be possible to determine whether conclusions about the dynamics of the block development are influenced by the analysis set used.

Smith, Phillip J.↗

NASA NTRS1996

Far-Infrared Water Emissions from Magnetohydrodynamic Shock Waves

Nondissociative, magnetohydrodynamic, C-type shock waves are expected to be a prodigious source of far-infrared water emissions in dense interstellar regions. We have constructed a model to calculate the farinfrared H20 line spectra that emerge from such shocks. Using the best estimates currently available for the radiative cooling rate and the degree of ion-neutral coupling within the shocked gas, we modeled the temperature structure of MHD shocks using standard methods in which the charged and neutral particles are treated separately as two weakly coupled, interpenetrating fluids. Then we solved the equations of statistical equilibrium to find the populations of the lowest 179 and 170 rotational states of ortho- and para-H2O We have completed an extensive parameter study to determine the emergent H2O line luminosities as a function of preshock density in the range n(H2) equals 10(exp 4) - 10(sup 6.5)/cc and shock velocity in the range upsilon(sub s) = 5 - 40 km/ s. We find that numerous rotational transitions of water are potentially observable using the Infrared Space Observatory and the Submillimeter Wave Astronomy Satellite and may be used as diagnostics of the shocked gas. We have also computed the rotational and ro-vibrational emissions expected from H2, CO, and OH, and we discuss how complementary observations of such emissions may be used to further constrain the shock conditions. In common with previous studies, we come close to matching the observed H2, and high-J CO emissions from the Orion-KL star-forming region on the basis of a single shock model. We present our predictions for the strengths of H2O line emission from the Orion shock, and we show how our results may be scaled to other regions where molecular shocks are likely to be present.

Kaufman, Michael J.↗

NASA NTRS1981

The Numerical Studies Program for the Atmospheric General Circulation Experiment (AGCE) for Spacelab Flights

The atmospheric general circulation experiment (AGCE) numerical design for Spacelab flights was studied. A spherical baroclinic flow experiment which models the large scale circulations of the Earth's atmosphere was proposed. Gravity is simulated by a radial dielectric body force. The major objective of the AGCE is to study nonlinear baroclinic wave flows in spherical geometry. Numerical models must be developed which accurately predict the basic axisymmetric states and the stability of nonlinear baroclinic wave flows. A three dimensional, fully nonlinear, numerical model and the AGCE based on the complete set of equations is required. Progress in the AGCE numerical design studies program is reported.

Fowlis, W. W.↗

NASA NTRS1991

Optimal guidance of aero-assisted transfer vehicles based on matched asymptotic expansions

This paper addresses a number of issues related to the matched asymptotic expansions analysis of skip trajectories, or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions. The procedure for matching inner and outer solutions, and using the composite solution to satisfy boundary conditions is rigorously followed in developing a complete algebraic solution to the problem of inclination change with minimum energy loss. Repeated solution of these algebraic equations along the trajectory, treating each current state as an initial state, constitute a feedback guidance algorithm.

Calise, A. J.↗

NASA NTRS1949

Dislocation Theory of the Fatigue of Metals

A dislocation theory of fatigue failure for annealed solid solutions is presented. On the basis of this theory, an equation giving the dependence of the number of cycles for failure on the stress, the temperature, the material parameters, and the frequency is derived for uniformly stressed specimens. The equation is in quantitative agreement with the data. Inasmuch as one material parameter is indicated to be temperature-dependent and its temperature dependence is unknown, it is impossible to predict the temperature dependence of the number of cycles for failure. A predicted quantitative correlation between fatigue and creep was found to exist, which suggests the practical possibility of obtaining fatigue data for annealed solid solutions and elements from steady-state creep-rate data for these materials. As a result of this investigation, a modification of the equation for the steady-state creep rate previously developed on the basis of the dislocation theory is suggested. Additional data are required to verify completely the dislocation theory of fatigue.

Machlin, E S↗

NASA NTRS2019

Permeation Rate Equations for Hydrogen and Deuterium in a Palladium-Silver Alloy

Mass transfer of a gas through a selective, solid membrane is an effective method for separation of desired species. This selective permeability is evident in the flow of hydrogen and the isotope deuterium through palladium-silver metal alloy media. In this study, based upon Sieverts’s law and the Arrhenius diffusion equation, an empirical correlation was developed to determine the steady-state permeation rate R, dependence on media temperature T, gas supply pressure p(sub S), and gas backpressure p(sub B) on the lower pressure side. Because of an extensive range of experimental conditions and complete reporting of raw data, the research by Ackerman and Koskinas was used as a source for data allowing empirical equation fitting. Unfortunately, those authors reported best-fit equations that poorly represented their own results. To improve the modeling of the original data and demonstrate the quality of the measurements, the current study develops improved hydrogen and deuterium permeation rate equations: P = 4.22×10(exp −6) A[exp(−704/T)](sq. root p(sub S) − sq. root p(sub B))/t for hydrogen P = 2.12×10(exp −6) A[exp(−468/T )](sq. root p(sub S) − sq. root p(sub B))/t for deuterium for values of cross-sectional area A (sq.cm), medium thickness t (cm), pressure p (psia), and temperature T (Kelvin), giving a permeation rate in mole/minute. These equations model permeation rate data more closely than do several other existing literature sources.

Smith, Phillip J.↗

NASA NTRS2000

Steady-State Computation of Constant Rotational Rate Dynamic Stability Derivatives

Dynamic stability derivatives are essential to predicting the open and closed loop performance, stability, and controllability of aircraft. Computational determination of constant-rate dynamic stability derivatives (derivatives of aircraft forces and moments with respect to constant rotational rates) is currently performed indirectly with finite differencing of multiple time-accurate computational fluid dynamics solutions. Typical time-accurate solutions require excessive amounts of computational time to complete. Formulating Navier-Stokes (N-S) equations in a rotating noninertial reference frame and applying an automatic differentiation tool to the modified code has the potential for directly computing these derivatives with a single, much faster steady-state calculation. The ability to rapidly determine static and dynamic stability derivatives by computational methods can benefit multidisciplinary design methodologies and reduce dependency on wind tunnel measurements. The CFL3D thin-layer N-S computational fluid dynamics code was modified for this study to allow calculations on complex three-dimensional configurations with constant rotation rate components in all three axes. These CFL3D modifications also have direct application to rotorcraft and turbomachinery analyses. The modified CFL3D steady-state calculation is a new capability that showed excellent agreement with results calculated by a similar formulation. The application of automatic differentiation to CFL3D allows the static stability and body-axis rate derivatives to be calculated quickly and exactly.

Park, Michael A.↗

NASA NTRS1988

Transonic stability and control of aircraft using CFD methods

Implementation of a capability to calculate longitudinal short-period response in the CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance) finite-difference code is described. The code, developed recently at the NASA Langley Research Center, is capable of solving steady and unsteady flows about complete aircraft configurations and is used primarily for aeroelastic calculations in the critical transonic speed range. The longitudinal short-period equations of motion in state-space form have been coupled to the time-accurate lift and moment calculated by the program. Transient responses to an elevator pulse for free-flying aircraft demonstrate the new capability. A trim routine is also added to the code to obtain trim automatically during steady-state flow field convergence. Stability and control derivatives are estimated from the calculated transient response by a maximum likelihood estimation program. Results for a fighter configuration and a general aviation configuration are presented to assess the capability.

Vinh, Lam-Son↗

NASA NTRS1995

Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

Using a new, second-order accurate numerical method we present dynamical simulations of oblique MHD cosmic-ray (CR)-modified plane shock evolution. Most of the calculations are done with a two-fluid model for diffusive shock acceleration, but we provide also comparisons between a typical shock computed that way against calculations carried out using the more complete, momentum-dependent, diffusion-advection equation. We also illustrate a test showing that these simulations evolve to dynamical equilibria consistent with previously published steady state analytic calculations for such shocks. In order to improve understanding of the dynamical role of magnetic fields in shocks modified by CR pressure we have explored for time asymptotic states the parameter space of upstream fast mode Mach number, M(sub f), and plasma beta. We compile the results into maps of dynamical steady state CR acceleration efficiency, epsilon(sub c). We have run simulations using constant, and nonisotropic, obliquity (and hence spatially) dependent forms of the diffusion coefficient kappa. Comparison of the results shows that while the final steady states achieved are the same in each case, the history of CR-MHD shocks can be strongly modified by variations in kappa and, therefore, in the acceleration timescale. Also, the coupling of CR and MHD in low beta, oblique shocks substantially influences the transient density spike that forms in strongly CR-modified shocks. We find that inside the density spike a MHD slow mode wave can be generated that eventually steepens into a shock. A strong layer develops within the density spike, driven by MHD stresses. We conjecture that currents in the shear layer could, in nonplanar flows, results in enhanced particle accretion through drift acceleration.

Frank, Adam↗

NASA NTRS2011

Experimental Shock Transformation of Gypsum to Anhydrite: A New Low Pressure Regime Shock Indicator

The shock behavior of gypsum is important in understanding the Cretaceous/Paleogene event and other terrestrial impacts that contain evaporite sediments in their targets (e.g., Mars Exploration Rover Spirit detected sulfate at Gusev crater, [1]). Most interest focuses on issues of devolatilization to quantify the production of SO2 to better understand its role in generating a temporary atmosphere and its effects on climate and biota [2,3]. Kondo and Ahrens [4] measured induced radiation emitted from single crystal gypsum shocked to 30 and 40 GPa. They observed greybody emission spectra corresponding to temperatures in the range of 3,000 to 4,000 K that are a factor of 2 to 10 times greater than calculated pressure-density energy equation of state temperatures (Hugoniot) and are high enough to melt gypsum. Chen et al. [5] reported results of shock experiments on anhydrite, gypsum, and mixtures of these phases with silica. Their observations indicated little or no devolatilization of anhydrite shocked to 42 GPa and that the fraction of sulfur, by mass, that degassed is approx.10(exp -2) of theoretical prediction. In another report of shock experiments on calcite, anhydrite, and gypsum, Badjukov et al. [6] observed only intensive plastic deformation in anhydrite shock loaded at 63 GPa, and gypsum converted to anhydrite when shock loaded at 56 GPa but have not experimentally shocked gypsum in a step-wise manner to constrain possible incipient transformation effects. Schmitt and Hornemann [7] shock loaded anhydrite and quartz to a peak pressure of 60 GPa and report the platy anhydrite grains were completely pseudomorphed by small crystallized anhydrite grains. However, no evidence of interaction between the two phases could be observed and they suggested that recrystallization of anhydrite grains is the result of a solid-state transformation. They concluded that significant decomposition of anhydrite requires shock pressures higher than 60 GPa. Gupta et al. [8] reanalyzed the calcite and anhydrite shock wave experiments of Yang [9] using improved equations of state of porous materials and vaporized products. They determined the pressures for incipient and complete vaporization to be 32.5 and 122 GPa for anhydrite GPa which is a factor of 2 to 3 lower than reported earlier by Yang [9]. These studies are not in agreement regarding the onset of sulfate decomposition and documentation of shock effects in gypsum is incomplete.

Bell, Mary S.↗