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Numerical solutions of Navier-Stokes equations are presented for the supersonic laminar flow over a two-dimensional compression corner. A well-known time-dependent method has been used wherein the asymptotic steady solutions of the unsteady Navier-Stokes equations are obtained with the Brailovskaia (1965) finite-difference scheme.
The heat flow in a confined perfect gas in low gravity is investigated, including the effects of conduction and thermal convection. Buoyancy-driven flow is neglected, due to the low-gravity environment, but the effect of thermoacoustic motion due to fluid compressibility is included. One-dimensional mathematical models are constructed from the conservation equations for a compressible, viscous, heat-conducting fluid. A conservative, time-dependent finite-difference method is used to generate numerical solutions on a digital computer. Problems for flat plates and cylindrical segments are solved for specified thermal boundary conditions. Numerical results are given which indicate that thermoacoustic convection can significantly increase the transient heat flow over conduction model predictions for cases where a confined gas is rapidly heated.
We present a systematic perturbation theory, extendable in principle to all orders of magnitude, for the solution of the equations of motion of an ideal Josephson element shunted by a resistance and driven by a dc current source and a small time-dependent source. We present second-order results for the case in which the time dependence is that of a single sinusoid, and these results are compared with other numerical and analytical calculations. Near, but not on, the first constant voltage step where the perturbation theory appears divergent, the current-voltage characteristic is calculated by means of a nonperturbative adiabatic procedure. The impedance and responsivity agree with earlier results.
Based on a detailed study of time-dependent or viscous remanence (VRM), thermoremanence (TRM) and magnetic granulometry of soil breccia 14313, single-domain particles of iron 100 to 200 A in size are proposed as the major carriers of natural remanence (NRM) in this rock. The VRM of 14313 is unusually intense and exhibits a logarithmic time decrease of VRM which ceases fairly abruptly after a time about equal to the original exposure to the field. The partial TRM spectrum reveals both a high-blocking-temperature fraction, scarcely affected by AF demagnetization to 1000 Oe, and an unusual concentration of blocking temperatures just above room temperature. The former fraction would contribute a very hard and stable component to any NRM of lunar origin, but the latter fraction, which accounts for the pronounced VRM of 14313 and undoubtedly has imparted a large viscous NRM component in the earth's field, is also surprisingly hard. A substantial portion (20 to 40%) is not demagnetized by an 800-Oe field.
Recent innovations in digital computer technology have enabled engineers to analyze shell structures of complex configurations without unduly restrictive approximations. An attempt is made to compare the various programs now generally available from the point of view of the advantages of the relative technique utilized, as well as the programmed state of the art. Many of the comparisons are based on the sample problems solved by the STARS-2 system of programs. These examples indicate both the structural detail which can be analyzed by, and the analytical capabilities available in, the numerical shell-of-revolution programs. All advantages and differences are demonstrated by use of solutions for realistic shell problems in the areas of statics, stability, vibrations, and dynamic response of shells subjected to time-dependent loadings.
Study of the dynamic stability of plates which are constructed of several alternating soft and hard layers and are subjected to time-dependent periodic in-plane loads. A theory that accounts for all of these effects is presented for multilayer sandwich plates. The plate thickness consists of (k - 1) orthotropic soft core layers sandwiched between k hard isotropic membrane layers, each of which may have a different thickness and elastic properties. It is assumed that the core layers carry only the transverse shear stresses, while the hard membrane layers carry the in-plane normal and shear stresses. The complementary variational principle for dynamics is used to derive the governing differential equations and the necessary boundary conditions for the dynamic stability of the sandwich plate. The equations governing the vibration of the plate and those governing its static stability are deduced from the more general equations for dynamic stability.
The time-dependent continuity equations, including diffusion, were solved for the first six energy levels of molecular nitrogen for conditions in the thermosphere corresponding to stable auroral red (SAR) arcs. The results show that molecular nitrogen is excited vibrationally to the degree that the rate constant for the ionospheric loss process, O(+) + N2 yields NO(+) + N, is increased by as much as a factor of 7.6 at F2 region altitudes. It was found that deviations from the energetically equivalent Boltzmann distribution were large, causing the rate constant to be as much as 1.6 times the rate constant calculated for the Boltzmann distribution. These results indicate that SAR arc intensities as small as 58 R can produce noticeable increases in the ionosphere ion-atom interchange reaction rate and hence in the rate of loss of ionospheric electrons. It is suggested that the observed decrease of electron density in the F2 region in SAR arcs can probably be explained by enhanced reaction rates for ion-atom interchange between O(+) and N2 caused by vibrational excitation of molecular nitrogen by electron impact.
Measurements of the light ion thermal plasma distribution in the magnetosphere frequently show apparent isolated patches of enhanced plasma density in the trough region beyond the main plasmasphere. These patches of light ions viewed along a satellite orbit appear detached from the main plasmasphere. By using a simple time-dependent convection model to determine the length of time a magnetic flux tube has been closed and in daylight (a rough indicator of the expected equatorial plasma density variation), the most prominent 'detached' regions measured by the mass spectrometer on Ogo 5 in the noon-dusk quadrant are seen on a global scale to be readily interpreted as filamentary extensions of the plasmasphere, called plasma tails. Hence on a global scale the pronounced detached regions may be attached to the main plasmasphere.
A review of developments in a new model for boundary-layer separation is presented. The separation model encompasses timevarying as well as steady-flow conditions. Engineering criteria in terms of calculable pressure gradients are obtained for both laminar and turbulent boundary-layer separation. These criteria, together with limited information on time-dependent separations, should be of value in predicting flows in turbomachinery. The reattachment of boundary layers is also shown to fit within the separation model.
Thermoelastic constitutive equations are derived for a material undergoing solidification or hardening as the result of a chemical reaction. The derivation is based upon a two component model whose composition is determined by the degree of hardening, and makes use of strain-energy considerations. Constitutive equations take the form of stress rate-strain rate relations, in which the coefficients are time-dependent functions of the composition. Specific results are developed for the case of a material of constant bulk modulus which undergoes a transition from an initial liquidlike state into an isotropic elastic solid. Potential applications are discussed.
This paper discusses the time and temperature dependence of the thermoelectric properties of n- and p-type 80 at. % Si-20 at. % Ge alloy. It is shown that the time-temperature behavior of the properties of both polarity types of this alloy can be described by means of a diffusion-limited dopant precipitation model due to Lifshitz and Slyozov (1961). An experimental program for the determination of the long-term behavior of the properties of the alloy is described in terms of the Lifshitz-Slyozov model, and the thermoelectric properties of the alloy are given for temperatures in the range from room temperature to 1000 C as a function of time up to and including twelve years. The data given for the p-type alloy represent the first time-dependent thermoelectric property data ever published for the p-type silicon-germanium alloys.
Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the time-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.
Time-dependent solutions of a one-fluid model of the interplanetary medium are investigated. This set of unsteady hydrodynamic equations has been written in conservation form in order to apply the Lax-Wendroff method for the solution of this problem. The initial condition is specified by a pulse at 1 solar radius. The equilibrium condition is chosen to be the steady solution of a quiet solar wind. The specified solar disturbances in this calculation are allowed to be both sub- and supersonic by the present theoretical formulation. The results are presented in terms of density, velocity, and temperature profiles of the interplanetary gas flow at heliocentric distances up to about 10 AU at any particular time. The trajectories of disturbances for various initial pulses are shown. Some 1972 solar-flare observational data are compared with these theoretical calculations. From these calculations, the effects on the interplanetary environment, due to the propagation of solar disturbances, can be determined.
A series of publications on the source-distribution methods for evaluating the aerodynamics of thin wings at supersonic speeds is summarized, extended, and unified. Included in the first part are the deviations of: (a) the linearized partial-differential equation for unsteady flow at a substantially constant Mach number. b) The source-distribution solution for the perturbation-velocity potential that satisfies the boundary conditions of tangential flow at the surface and in the plane of the wing; and (c) the integral equation for determining the strength and the location of sources to describe the interaction effects (as represented by upwash) of the bottom and top wing surfaces through the region between the finite wing boundary and the foremost Mach wave. The second part deals with steady-state thin-wing problems. The third part of the report approximates the integral equation for unsteady upwash and includes a solution of approximate equation. Expressions are then derived to evaluate the load distributions for time-dependent finite-wing motions.
A method is presented for determining the time-dependent flow over a rectangular wing moving with a supersonic forward speed and undergoing small vertical distortions expressible as polynomials involving spanwise and chordwise distances. The solution for the velocity potential is presented in a form analogous to that for steady supersonic flow having the familiar "reflected area" concept discovered by Evvard. Particular attention is paid to indicial-type motions and results are expressed in terms of generalized indicial forces. Numerical results for Mach numbers equal to 1.1 and 1.2 are given for polynomials of the first and fifth degree in the chordwise and spanwise directions, respectively, on a wing having an aspect ratio of 4.
An algorithm has been developed for time-dependent forced convective diffusion-reaction having convection by a recirculating flow field within the drop that is hydrodynamically coupled at the interface with a convective external flow field that at infinity becomes a uniform free-streaming flow. The concentration field inside the droplet is likewise coupled with that outside by boundary conditions at the interface. A chemical reaction can take place either inside or outside the droplet, or reactions can take place in both phases. The algorithm has been implemented, and for comparison results are shown here for the case of no reaction in either phase and for the case of an external first order reaction, both for unsteady behavior. For pure interphase mass transfer, concentration isocontours, local and average Sherwood numbers, and average droplet concentrations have been obtained as a function of the physical properties and external flow field. For mass transfer enhanced by an external reaction, in addition to the above forms of results, we present the enhancement factor, with the results now also depending upon the (dimensionless) rate of reaction.
Observations of the Echo I balloon satellite have been compared with a theory including the following perturbing effects: (1) solar radiation pressure; (2) lunar and solar gravitation; (3) second, third, and fourth harmonics of the earth's gravitational potential; and (4) atmospheric drag. With a set of orbital elements at the 26th day of the lifetime of the satellite, it was possible to match the observational data to 180 days with root mean square residuals as follows: Delta-a = 17.9 km, Delta-e = 0.0021, Delta-i = 0.0177 deg., Delta-omega = 1.1231 deg., Delta-Omega = 0.4821 deg., Delta-perigee height = 7.50 km. No differential correction has been applied as yet. Values of atmospheric density between 1500 and 930 km, assuming neutral drag effects only, have been inferred from the orbital data. The connection between solar activity and drag is also examined. As the Echo I perigee height continues to oscillate between 900 and 1500 km, more valuable orbital data will be obtained and atmospheric properties will be deduced. Further refinements in the mathematical model, especially in a time-dependent model atmosphere, should bring a substantial reduction in the residuals of the observations.
A method for the theoretical determination of the lifetime of com- pressed plates at elevated temperatures is presented. In this approach, linearized equations are used throughout with the assumption that the plate material is a standard linear solid. The critical time (lifetime) is determined by reducing the time-dependent behavior to the time- independent response of purely elastic buckling. Theoretically predicted lifetimes of 2024-T3 (formerly 24S-T3) aluminum-alloy plates at 450 F are compared with experimental values obtained in previous work.