Orbit prediction accuracy theory
Equations for estimating orbit prediction accuracy theory, with partial derivatives calculated from two-body elliptical orbit theory
ORBIT EQUATION↗
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Pines method for circular and elliptical orbits
The Pines method for trajectory integration is derived from the equations of two-body motion. The concept of the perturbation derivative is explained and a formal definition is presented. An algorithm for applying the Pines method to numerical integration of trajectories is included.
Semar, C.↗
Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite
A mathematical model is developed for studying the effects of gravity gradient torque on the attitude stability of a tumbling triaxial rigid satellite. Poisson equations are used to investigate the rotation of the satellite (which is in elliptical orbit about an attracting point mass) about its center of mass. An averaging method is employed to obtain an intermediate set of differential equations for the nonresonant, secular behavior of the osculating elements which describe the rotational motions of the satellite, and the averaged equations are then integrated to obtain long-term secular solutions for the osculating elements.
Liu, J. J. F.↗
Implicit/explicit analysis of interactive phenomena in: supersonic, chemically-reacting, mixing and boundary layer problems
An efficient implicit/explicit procedure for solving the parabolized Navier-Stokes (PNS) equations to analyze supersonic, viscous, chemically-reacting flows is presented. The approach combines a fully-implicit algorithm for solving the parabolic mixing/boundary layer equations with an explicit algorithm for determining the hyperbolic pressure field. The unified model, SPLITP, contains provisions for treating bounding/embedded subsonic regions using a pressure-splitting procedure which suppresses elliptic (upstream influence) effects without recourse to sublayer approximations. Several two-equation turbulence models with compressibility and curvature-correction terms are incorporated into SPLITP. Applications to a variety of flow problems are presented which include: ducted supersonic mixing and combustion flowfields, free jets and plumes, and, curved boundary layers and wall jets.
Dash, S. M.↗
Elliptic Relaxation of a Tensor Representation for the Redistribution Terms in a Reynolds Stress Turbulence Model
A formulation to include the effects of wall proximity in a second-moment closure model that utilizes a tensor representation for the redistribution terms in the Reynolds stress equations is presented. The wall-proximity effects are modeled through an elliptic relaxation process of the tensor expansion coefficients that properly accounts for both correlation length and time scales as the wall is approached. Direct numerical simulation data and Reynolds stress solutions using a full differential approach are compared for the case of fully developed channel flow.
Carlson, J. R.↗
Vortex breakdown incipience: Theoretical considerations
The sensitivity of the onset and the location of vortex breakdowns in concentrated vortex cores, and the pronounced tendency of the breakdowns to migrate upstream have been characteristic observations of experimental investigations; they have also been features of numerical simulations and led to questions about the validity of these simulations. This behavior seems to be inconsistent with the strong time-like axial evolution of the flow, as expressed explicitly, for example, by the quasi-cylindrical approximate equations for this flow. An order-of-magnitude analysis of the equations of motion near breakdown leads to a modified set of governing equations, analysis of which demonstrates that the interplay between radial inertial, pressure, and viscous forces gives an elliptic character to these concentrated swirling flows. Analytical, asymptotic, and numerical solutions of a simplified non-linear equation are presented; these qualitatively exhibit the features of vortex onset and location noted above.
Berger, Stanley A.↗
Large-scale computation of incompressible viscous flow by least-squares finite element method
The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.
Jiang, Bo-Nan↗
Lateral Migration and Rotational Motion of Elliptic Particles in Planar Poiseuille Flow
Simulations of elliptic particulate suspensions in the planar Poiseuille flow are performed by using the lattice Boltzmann equation. Effects of the multi-particle on the lateral migration and rotational motion of both neutrally and non-neutrally buoyant elliptic particles are investigated. Low and intermediate total particle volume fraction f(sub a) = 13%, 15%, and 40% are considered in this work.
Qi, Dewei↗
Instability of jets of arbitrary geometry
This paper describes a calculation technique for determining the stability of jets of arbitrary cross section. In particular, elliptic and rectangular jets are considered. The numerical procedure involves both a conformal transformation between the computational domain and the physical plane and a solution of the transformed stability equation in the computational domain. Modern, efficient, conformal mappings are used for both simply and doubly connected domains. The numerical solution is based on a finite difference/pseudospectral discretization of the stability equation. The technique is verified by comparison with previous calculations for circular and elliptic jets. Calculations are performed for the stability of elliptic and rectangular jets of aspect ratio 2. Growth rates, phase velocities, and pressure eigenfunctions are calculated.
Baty, Roy S.↗
Certain comments on the application of the method of averaging to the study of the rotational motions of a triaxial rigid body
Averaging technique applied to variational equations describing rotational motions of rigid triaxial body in elliptical orbit
Liu, J. J. F.↗
CONIC program user's manual
A user's manual for the current CONIC computer program is presented. The program was written in FORTRAN for use on the TRW Time Share system. The CONIC program is used to calculate the relative motion of two bodies in any conic-section orbit (circular, elliptical, parabolic, or hyperbolic). This manual describes the basic equations used for trajectory propagation and defines the input and output parameters of the program. Sample runs with corresponding figures and a detailed flow chart of the program are included.
Source record↗
Elastohydrodynamic lubrication of point contacts
A procedure for the numerical solution of the complete, isothermal, elastohydrodynamic lubrication problem for point contacts is given. This procedure calls for the simultaneous solution of the elasticity and Reynolds equations. By using this theory the influence of the ellipticity parameter and the dimensionless speed, load, and material parameters on the minimum and central film thicknesses was investigated. Thirty-four different cases were used in obtaining the fully flooded minimum- and central-film-thickness formulas. Lubricant starvation was also studied. From the results it was possible to express the minimum film thickness for a starved condition in terms of the minimum film thickness for a fully flooded condition, the speed parameter, and the inlet distance. Fifteen additional cases plus three fully flooded cases were used in obtaining this formula. Contour plots of pressure and film thickness in and around the contact have been presented for both fully flooded and starved lubrication conditions.
Hamrock, B. J.↗
On least squares approximations to indefinite problems of the mixed type
A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.
Fix, G. J.↗
Numerical investigation of the perpendicular injector flow field in a hydrogen fueled scramjet
A computer program has been developed which numerically solves the full (elliptic) two-dimensional Navier-Stokes and specie equations near a slotted perpendicular hydrogen fuel injector in a scramjet engine. The program currently predicts the turbulent mixing of injected hydrogen fuel and air without reaction, and allows the study of separated regions of flow immediately preceding and following the injector as well as the complex shock-expansion structure produced by the injector in this region of the engine. Results are presented that describe the size of the separated regions near the injector as well as locations where ignition is likely to occur.
Drummond, J. P.↗
Conical pitch angle distributions of very low-energy ion fluxes observed by ISEE 1
Observations are presented of conical distributions of low-energy ion fluxes from throughout the magnetosphere. The data were provided by the plasma composition experiment (PCE) on ISEE 1. ISEE 1 was launched in October 1977 into a highly elliptical orbit with a 30 deg inclination to the equator and 22.5 earth radii apogee. Particular attention is given to data taken when the instrument was in its thermal plasma mode, sampling ions in the energy per charge range 0-100 eV/e. Attention is given to examples of conical distributions in 0- to 100-eV/e ions, the occurrence of conical distributions of 0- to 100-eV ions in local time-geocentric distance and latitude-geocentric distance coordinates, the cone angles in 0- to 100-eV ion conics, Kp distributions of 0- to 100-eV ion conics, and some compositional aspects of 0- to 100-eV ion conics.
Horwitz, J. L.↗
An approximation theory for the identification of nonlinear distributed parameter systems
An abstract approximation framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original finite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed.
Banks, H. T.↗
Deformation of the core-mantle boundary induced by spherical-shell, compressible convection
Single-mode, mean-field equations, for spherical-shell convection are derived, with variable viscosity and compressibility included. Core-mantle boundary (CMB) topography produced by whole mantle convection is computed for various rheologies and different thermal boundary conditions at CMB. Results for long-wavelength undulations show the extreme sensitivity of these signatures to rheological and equation of state parameters. In order to obtain excess ellipticity of about 1 km, the activation energy and volume in the lower mantle should be low, and the temperatures at CMB should be around 4500 K, in accord with recent experimental findings.
Zhang, Shuxia↗
The computation of radiation from nonequilibrium hypersonic flows
The results of the solution of the equations that describe a hypersonic ionized flow about an elliptically blunted cone are presented. The flow conditions correspond to those of the proposed Aeroassist Flight Experiment (AFE) vehicle at altitudes between the perigee at 78 km and the approximate limit of the continuum regime at 90 km. For the free-stream velocities of interest, about 9 km/sec, the flowfield is out of thermo-chemical equilibrium, electronically excited, ionized and radiating. The gas consists of eight-chemical species including free electrons. The thermal state of the gas is modeled with a translational-rotational temperature, four vibrational temperatures for the diatomic species and an electron-electronic temperature. The electronic excitation of molecules is included. The nonequilibrium air radiation from each fluid element is computed and the radiative heat flux at the body surface is determined. The stagnation point radiative heating result agrees with previous calculations.
Candler, Graham↗