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Thomson theory of arbitrarily intense elliptically polarized plane electromagnetic wave scattering by free electrons, solving electron equations of motion
The present work describes a mathematical model and numerical scheme for the computer-aided calculation of two- and three-dimensional transonic flow over an isolated yawed wing with oblique shock waves and a trailing vortex sheet. The flow is modeled by the potential equation for irrotational flow which is hyperbolic at supersonic points and elliptic at subsonic points. A coordinate-invariant difference scheme is used in which retarded difference formulas are constructed to conform with the local flow direction. The resulting 'rotated' difference scheme allows complete flexibility in the choice of a coordinate system. Shock waves are located automatically in the form of compression bands spread over a few mesh widths. The scheme has proven to be stable and convergent throughout the transonic range. Calculations have been performed for Mach number up to 1.2 and yaw angles up to 60 deg, the likely operating range of a yawed-wing transport designed to fly at supersonic speeds. Calculations becomes less accurate towards upper end of range, because the difference scheme is first-order accurate in the supersonic range.
The numerical determination of combustor flowfields is of great value to the combustor designer. An a priori knowledge of the flow behavior can speed the combustor design process and reduce the number of experimental test rigs required to arrive at an optimal design. Even 2-D steady incompressible isothermal flow predictions are of use; many codes of this kind are available, each employing different techniques to surmount the difficulties arising from the nonlinearity of the governing equations and from typically irregular combustor geometries. Mapping techniques (algebraic and elliptic PDE), and adaptive grid methods (both multi-grid and grid embedding) as applied to axisymmetric combustors are discussed.
The NCOREL full-potential method with an entropy correction is presently applied to supersonic missile flowfield problems. After defining the salient characteristics of the method, a combination of linear theory with NCOREL and experimental data is used to isolate the nonlinear features of the supersonic flow so that the influence of geometry and flow conditions on the development of such flow nonlinearities can be appreciated. Comparisons of experimental longitudinal force and moment data with NCOREL and various linear theory predictions are presented for several generic missile airframe configurations of circular and elliptic cross section. The NCOREL code solves the nonconservative full potential equation in a spherical coordinate system; exact boundary conditions are defined on the missile surface.
SAGE is a user-friendly, highly efficient, two-dimensional self-adaptive grid code based on Nakahashi and Deiwert's variational principles method. Grid points are redistributed into regions of high flowfield gradients while maintaining smoothness and orthogonality of the grid. Efficiency is obtained by splitting the adaption into 2 directions and applying one-sided torsion control, thus producing a 1-D elliptic system that can be solved as a set of tridiagonal equations.
An investigation of the linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is presented. The effects of the entropy gradient are investigated, and for the particular entropy profile chosen, the positive average entropy in the vortex core is destabilizing, while the opposite is true for the negative average entropy gradient. Finite-difference methods are used to study the linear initial value problem. These methods are found to be capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behavior is found to correspond to the most unstable normal mode in all cases. The numerical solution of the Euler equations is used to study the nonlinear development of an elliptic-mode perturbation.
An investigation of the damping characteristics of dashpots was carried out combining theory and experiment. Laminar flow was assumed and three equations for the steady velocity of a piston moving in a cylinder filled with liquid were derived. In the first equation, the piston was assumed coaxial in the cylinder and, in the second equation, the piston was assumed eccentric in the cylinder with an element of the piston in contact with the cylinder wall. The third equation is for a piston of circular cross section in an elliptical cylinder. Experiments showed that the piston is normally eccentric in the cylinder. The pistons tested were 1.25 and 2 inches in diameter 0.062 to 1.00 inch long, and the clearances varied from 1.36 to 5.16 x 10 to the 3rd power inch. The difference in pressure on the two sides of the piston varied from about 2 to 55 pounds per square inch. The piston velocities for each assembly were measured with damping liquids of three different viscosities. At high piston velocities, when turbulent flow exists, the observed velocities were much lower than the velocities calculated on the basis of laminar flow. Results for a wide range of Reynolds numbers are presented in graphical form.
TDIGG is fast and versatile computer program for generating two-dimensional computational grids for use in programs solving equations of flow by finite-difference methods. Both algebraic and elliptic grid-generation systems included. Enables user to view results of each iteration. Written in FORTRAN 77.
Direct numerical simulations (DNS) of the acoustic receptivity process on a semi-infinite flat plate with a modified-super-elliptic (MSE) leading edge are performed. The incompressible Navier-Stokes equations are solved in stream-function/vorticity form in a general curvilinear coordinate system. The steady basic-state solution is found by solving the governing equations using an alternating direction implicit (ADI) procedure which takes advantage of the parallelism present in line-splitting techniques. Time-harmonic oscillations of the farfield velocity are applied as unsteady boundary conditions to the unsteady disturbance equations. An efficient time-harmonic scheme is used to produce the disturbance solutions. Buffer-zone techniques have been applied to eliminate wave reflection from the outflow boundary. The spatial evolution of Tollmien-Schlichting (T-S) waves is analyzed and compared with experiment and theory. The effects of nose-radius, frequency, Reynolds number, angle of attack, and amplitude of the acoustic wave are investigated. This work is being performed in conjunction with the experiments at the Arizona State University Unsteady Wind Tunnel under the direction of Professor William Saric. The simulations are of the same configuration and parameters used in the wind-tunnel experiments.
A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Karman-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.
A novel approach for the solution of inviscid flow problems for subsonic compressible flows is described. The approach is based on canonical forms of the equations, in which subsystems governed by hyperbolic operators are separated from those governed by elliptic ones. The discretizations used as well as the iterative techniques for the different subsystems are inherently different. Hyperbolic parts, which describe, in general, propagation phenomena, are discretized using upwind schemes and are solved by marching techniques. Elliptic parts, which are directionally unbiased, are discretized using h-elliptic central discretizations, and are solved by pointwise relaxations together with coarse grid acceleration. The resulting discretization schemes introduce artificial viscosity only for the hyperbolic parts of the system; thus a smaller total artificial viscosity is used, while the multigrid solvers used are much more efficient. Solutions of the subsonic compressible Euler equations are achieved at the same efficiency as the full potential equation.
It is shown that a previously derived semiempirical equation for describing observed ablation rates of isotropic graphites cannot be applied to low-density flows containing dissociated oxygen. Experimentally determined reaction probabilities of isotropic graphites to molecular and atomic oxygen are used to calculate heat-transfer rates and stagnation-point ablation rates for typical conditions. Integrated mass losses are computed for a group of flight trajectories which start from geosynchronous orbit and enter earth's atmosphere in a skipping motion following near-elliptic decaying orbits. A comparison of the results with those obtained by the equation under question shows excellent agreement for steep trajectories, but large discrepancies for shallow trajectories. The differences are attributed to surface oxidation by atomic oxygen.
This paper is concerned with the numerical solution of the secondary vorticity equations in curved ducts, cascades, and rotors. The classical approach of splitting the flow into primary and secondary flow fields is employed and extended to include effects of viscosity, rotation, and density stratification. All elliptical effects are neglected and the Crank-Nicolson method is used to solve the secondary vorticity equation. The secondary flow field is obtained by solving a Poisson equation using a successive over-relaxation method. The results are compared with theoretical and experimental data from stationary ducts, compressor and turbine cascades. The agreement is good for most of the cases.
The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.
Hill's variational equations are solved analytically for the orbital perturbations of a spacecraft nominally in an elliptic orbit around a non-spherical body. The rotation of the central planet about its spin-axis is not considered in the analysis. The perturbations are restricted to the planetary gravitational harmonics only. An extremely simple algorithm is derived to transform the spherical harmonic potentials to the orbital coordinate system, and the resulting accelerations are shown to be simply trigonometric functions of the true anomaly. With the principal matrix solution for the differential equations of the adjoint system given in closed form, the orthogonality of the trigonometric functions makes it possible to obtain an analytic solution for the non-homogeneous problem, at intervals of 2 pi in true anomaly. The solution for orbital perturbations can be extended over several revolutions by applying well-known results from Floquet's theory. The technique is demonstrated with results presented on the spacecraft periapsis altitude for the forthcoming Venus Radar Mapper Mission.
We summarize observations by the MESSENGER spacecraft of highly coherent waves at frequencies between 0.4 and 5 Hz in Mercury's inner magnetosphere. This survey covers the time period from 24 March to 25 September 2011, or 2.1 Mercury years. These waves typically exhibit banded harmonic structure that drifts in frequency as the spacecraft traverses the magnetic equator. The waves are seen at all magnetic local times, but their observed rate of occurrence is much less on the dayside, at least in part the result of MESSENGER's orbit. On the nightside, on average, wave power is maximum near the equator and decreases with increasing magnetic latitude, consistent with an equatorial source. When the spacecraft traverses the plasma sheet during its equatorial crossings, wave power is a factor of 2 larger than for equatorial crossings that do not cross the plasma sheet. The waves are highly transverse at large magnetic latitudes but are more compressional near the equator. However, at the equator the transverse component of these waves increases relative to the compressional component as the degree of polarization decreases. Also, there is a substantial minority of events that are transverse at all magnetic latitudes, including the equator. A few of these latter events could be interpreted as ion cyclotron waves. In general, the waves tend to be strongly linear and characterized by values of the ellipticity less than 0.3 and wave-normal angles peaked near 90 deg. Their maxima in wave power at the equator coupled with their narrow-band character suggests that these waves might be generated locally in loss cone plasma characterized by high values of the ratio beta of plasma pressure to magnetic pressure. Presumably both electromagnetic ion cyclotron waves and electromagnetic ion Bernstein waves can be generated by ion loss cone distributions. If proton beta decreases with increasing magnetic latitude along a field line, then electromagnetic ion Bernstein waves are predicted to transition from compressional to transverse, a pattern consistent with our observations. We hypothesize that these local instabilities can lead to enhanced ion precipitation and directly feed field-line resonances.
Equations of motion for a model of the human body are developed. Basically, the model consists of an elliptical cylinder representing the torso, together with a system of frustrums of elliptical cones representing the limbs. They are connected to the main body and each other by hinges and ball and socket joints. Vector, tensor, and matrix methods provide a systematic organization of the geometry. The equations of motion are developed from the principles of classical mechanics. The solution of these equations then provide the displacement and rotation of the main body when the external forces and relative limb motions are specified. Three simple example motions are studied to illustrate the method. The first is an analysis and comparison of simple lifting on the earth and the moon. The second is an elementary approach to underwater swimming, including both viscous and inertia effects. The third is an analysis of kicking motion and its effect upon a vertically suspended man such as a parachutist.
Weakly multidirectional, long-wavelength cosmic-ray-modified shocks are studied via multiple scale perturbation techniques. The effects of diffraction are discussed in terms of Green's function solutions of the linearized 1 + 3D Burgers and 1 + 3D KdVB equations, and also in terms of solutions with singular Dirac delta initial distributions. The solutions show a monotonic decrease of the wave-front curvature with increasing time owing to the effects of wave diffraction. The shape of the wave surface is discussed in terms of solutions S to the wave eikonal equation corresponding to singular initial conditions. For the fast magnetosonic wave propagating in the positive x-direction, the wave phase surface S = 0 has elliptic cross sections with the planes x = constant and has a convex paraboloidal shape. Plane-wave solutions of the 1 + 3D KdVB equation are discussed.