Wide Range Of Initial Conditions Research Articles

System optimization using quasilinearization and sensitivity analysis†

A method for solution of two-point-boundary value problems based on quasilinearization and sensitivity analysis is proposed. The combination of these two methods yields an iterative predictor-corrector method which possesses the rapid quadratic convergence features of quasilinearization and which converges over a much wider range of initial conditions and initial trajectories than the basic quasilinearization technique.

Nuclear rocket experimental engine test results

An extensive series of tests of a nuclear rocket experimental engine, XE-Prime, was completed in 1969. The main emphasis of the series was upon characteristics of starting from various conditions and upon operation of the test facility under nuclear firing conditions; however, engine performance at full power and throttled pressure were also investigated. This paper compares selected startup test results with predictions made by computer simulation of the test system, and it briefly describes full-power and high-specific-impulse operation. Representative startup data are presented for automatic temperature startups performed with initial levels of 0.1 wand 2.7 Mw, initial core temperatures of approximately 250° and 1250°R, and drum exponential set points of +11° and —8.5° from critical. The test results showed that startup can be controlled over a wide range of initial conditions.

Non-Equilibrium Processes in the Early Universe

In a recent paper Misner (1968) has suggested that neutrino viscosity is a highly efficient process for removing shear anisotropy during the early stages of the universe, so that the remarkable degree of isotropy observed in the microwave background would be expected to occur whatever the initial conditions. In this paper we suggest that the early universe may not have been in near thermal equilibrium as Misner assumed, so that under a wide range of initial conditions, arbitrarily large anisotropy could occur at any epoch. Therefore, the observed anisotropy does place some restrictions on early conditions in the universe.

An experimental study of transverse ionizing MHD shock waves

The jump conditions across a transverse ionizing MHD shock wave, where the magnetic field is in the plane of the shock, are examined. The conservation laws, in conjunction with Maxwell's laws and the equation of state, yield three jump equations in four unknowns. To uniquely describe jumps across an ionizing wave requires an additional descriptive relationship. The theory of Kulikovskii & Lyubimov and, later, Chu, in which the internal structure of the shock itself supplies the missing relationship, is examined. In particular, Kulikovskii & Lyubimov show, for appropriate ratios of thermal to magnetic diffusivities, that for low-speed waves the magnetic field compression across the shock is unity and the jump equations reduce to the ordinary Rankine—Hugoniot relations. For high-speed shock waves, the magnetic field compression, B2/B1, equals the gas compression across the wave, p2/p1, and the jump equations become the magnetohydrodynamic shock jump relations. Furthermore, intermediate speed shocks induce magnetic field compressions between 1 and p2/p1. An experiment was performed in an inverse pinch where E behind the shock, the shock and piston velocities, and the magnetic field compression across the shock, were measured over a wide range of initial conditions and shock velocities in hydrogen. The jump relations were written with B2/B1 as a parameter and programmed into a digital computer. The program was written for real, equilibrium hydrogen. The program provided easy access to a unique solution of the jump equations for any B2/B1. The experiment tends to confirm the Kulikovskii—Lyubimov—Chu theory. Ordinary shock waves were observed at low speeds and near-MilD shocks were observed at high speeds. Further, the relation was verified for the plasma behind the shock for low-speed shock waves, and was verified to within experimental accuracy for the intermediate class of shock waves.

An experimental study of spherical combustion waves in a hemispherical chamber

Experimental results are presented on spherical deflagration and detonation waves in stoichiometric hydrogen-oxygen mixtures diluted with helium or hydrogen obtained in a hemispherical combustion chamber. Piezopressure gauges, thin-film heat-transfer gauges and glow discharge gauges were used to obtain the data in a range of initial pressures of 75 ⩽ p 1 ⩽ 1000 lb/in 2 and up to 15 moles of helium and 10 of hydrogen dilution. It is shown that the present facility has a great advantage in studying the characteristics of spherical combustion wave processes throughout their interior rather than on their periphery, which is usually the case, over a wide range of initial conditions.

Perturbed Morse Oscillator Approximation in Reactive Collisions. II. A Repulsive Potential

The H+Br2→HBr+Br reaction has been considered to form a linear collision intermediate and detailed reactive scattering cross sections have been computed quantum mechanically using a perturbed Morse oscillator approximation. These computations employed a diatomics-in molecules potential-energy surface distorted to make it moderately repulsive, i.e., the potential energy of the collision intermediate was made to be closer to that of the reactants than to that of the products. The resulting cross sections are peaked about the lower allowed vibrational quantum numbers of product HBr for a wide range of initial conditions corresponding to temperatures from 300° to 1000°K. This appearance of the heat of reaction as relative translation rather than as internal excitation is in agreement with both experiment and semiclassical calculations. Specific rate coefficients and pre-exponential factors calculated from the computed cross sections are also in satisfactory agreement with experiment.

A Note on the Dynamical Interaction of Small Interplanetary Particles with the Earth's Upper Atmosphere

Abstract A standard transformation of variables in the non-linear, ordinary differential equation of motion of a small spherical particle moving with an initial cosmic velocity in a viscous medium of non-uniform density, yields a simpler linear differential equation. This equation is further simplified by neglecting the gravitational force, an assumption fully justified on physical grounds. Certain characteristics of the solution are then obtainable with the aid of a single table, for a wide range of initial conditions and particle parameters.

Lunar landing and long-range earth re-entry guidance by application of perturbation theory

A guidance scheme based on linear perturbation theory has been investigated. An improved capability has been achieved by the proper choice of independent variable and by appropriate weighting of the guidance gains computed by linear theory. The capability of this scheme applied to the descent-to-hover phase of lunar landing is demonstrated for two different types of nominal trajectory: a constant-thrus t gravity turn maneuver, and a constant-thrust, constant-pitch-rate maneuver. To demonstrate the performance of this type of guidance scheme for atmosphere entry, it has been applied to the guidance of a vehicle entering the earth's atmosphere at parabolic velocity. Its capability is evaluated for entries from abort conditions, as well as for entries within the normal entry corridor, and effects of variations in life-drag ratio and atmospheric density are investigated. It is shown that for both lunar landing and atmosphere entry this guidance system, which uses a single nominal trajectory, and therefore requires minimum storage capacity, permits guidance to a selected landing site from a wide range of initial conditions.

Hertz impact between a surface and a mass-spring-mass system

Summary According to the Hertz theory, impact between an elastic surface and an elastic sphere is characterized by a non-linear restoring force. In this paper, another mass is connected to the sphere by means of a massless, linear spring. One dimensional impact of this mass-spring-mass system is analyzed. Energy considerations lead to maximum attainable values of the impact force and rebound velocity. Digital solutions of the governing non-linear differential equations are used to obtain typical force-time histories during impact. The digital results are summarized in a non-dimensional form suitable for predicting maximum impact force, duration of contact and rebound velocity for a wide range of initial conditions and system parameters.

Calculations on the thermal history of the Earth

The possible thermal history of a spherically symmetric earth is studied by comparing numerical calculations of the development of temperature within a number of model earths with observations on the present thermal state of the earth. The pertinent observations are the following: (1) The average surface heat flow is 50 ergs/cm2 sec. (2) The mantle of the earth is solid. (3) The electrical conductivity increases rapidly in the outer few hundred kilometers of the earth. It is shown that models which incorporate a wide range of initial conditions and distributions of radioactivity reproduce in a qualitative fashion the electrical conductivity of the earth. Models which have a radioactive content equal to an earth composed of chondritic meteorites reproduce the present surface heat flow to within a factor of 2. This is a consequence of the near coincidence of the present rate of heat production in a chondritic earth and the heat flux of the actual earth. The variability in heat flow among models is due to varying contributions of ‘initial heat,’ higher rate of radioactive heat production in the past, and the depth of burial of heat sources. The solid nature of the mantle provides a further qualitative restraint on possible earth models. However, a lack of experimental data on melting relations at high pressures precludes detailed conclusions. The general features of temperature-depth curves are similar for many models in which energy is transmitted by radiation as well as by ordinary lattice conduction. The gradient of temperature is high near the surface but decreases within the earth as the effective conductivity increases with increasing temperature. The melting temperature is most closely approached or exceeded in the outer few hundred kilometers of the earth. The variation of surface heat flow with time is examined. It is shown that the surface heat flow is constant in time for a wide range of models, provided that heat is transmitted solely by radiation and conduction.