Perturbations of a Spacecraft Orbit During an Hyperbolic Flyby
A first-order perturbation theory exists for the flyby problem with the Born approximation as the reference orbit.
perturbations Hyperbolic flyby Celestrial Mechanic↗
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An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case
Two methods for developing high order single step explicit algorithms on symmetric stencils with data on only one time level are presented. Examples are given for the convection and linearized Euler equations with up to the eighth order accuracy in both space and time in one space dimension, and up to the sixth in two space dimensions. The method of characteristics is generalized to nondiagonalizable hyperbolic systems by using exact local polynominal solutions of the system, and the resulting exact propagator methods automatically incorporate the correct multidimensional wave propagation dynamics. Multivariate Taylor or Cauchy-Kowaleskaya expansions are also used to develop algorithms. Both of these methods can be applied to obtain algorithms of arbitrarily high order for hyperbolic systems in multiple space dimensions. Cross derivatives are included in the local approximations used to develop the algorithms in this paper in order to obtain high order accuracy, and improved isotropy and stability. Efficiency in meeting global error bounds is an important criterion for evaluating algorithms, and the higher order algorithms are shown to be up to several orders of magnitude more efficient even though they are more complex. Stable high order boundary conditions for the linearized Euler equations are developed in one space dimension, and demonstrated in two space dimensions.
Goodrich, John W.↗
High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs
This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.
Nielsen, Tanner B.↗
High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes
The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.
Barth, Timothy↗
Development and application of unified algorithms for problems in computational science
A framework is presented for developing computationally unified numerical algorithms for solving nonlinear equations that arise in modeling various problems in mathematical physics. The concept of computational unification is an attempt to encompass efficient solution procedures for computing various nonlinear phenomena that may occur in a given problem. For example, in Computational Fluid Dynamics (CFD), a unified algorithm will be one that allows for solutions to subsonic (elliptic), transonic (mixed elliptic-hyperbolic), and supersonic (hyperbolic) flows for both steady and unsteady problems. The objectives are: development of superior unified algorithms emphasizing accuracy and efficiency aspects; development of codes based on selected algorithms leading to validation; application of mature codes to realistic problems; and extension/application of CFD-based algorithms to problems in other areas of mathematical physics. The ultimate objective is to achieve integration of multidisciplinary technologies to enhance synergism in the design process through computational simulation. Specific unified algorithms for a hierarchy of gas dynamics equations and their applications to two other areas: electromagnetic scattering, and laser-materials interaction accounting for melting.
Shankar, Vijaya↗
Dynamics of micrometeoroids
The spatial and orbital distributions of meteoroids of mass less than 10 to the -8 g are discussed. Pioneer 8 and 9, HEOS-2 and Helios 1 observations of the smallest meteoroids leaving the solar system in hyperbolic orbits due to radiation pressure (beta particles), intermediate mass particles arriving from the heliocentric spacecraft apex direction (alpha meteoroids) and the largest micrometeoroids observed with no preferred directionality (sporadic meteoroids) are reviewed, and evidence for the three groups of dynamically different micrometeoroids in lunar microcrater statistics is indicated. The collisional model of the origin of dust particles from the dispersion of sporadic meteoroids of cometary origin due to the Poynting-Robertson effect and collisions is then applied to account for the dynamics of the beta and alpha micrometeoroids.
Gruen, G.↗
A probability of encounter with interstellar comets and the likelihood of their existence
A theory of the probability of encounter of the sun with an interstellar comet at a distance comparable to the earth-sun distance is formulated, and a general expression is derived establishing the relationship among the influx rate of interstellar comets, the perihelion distance, the space density of the comets, the Maxwellian distribution of comet velocities in the interstellar cloud, and the cloud's systematic velocity relative to the sun. An upper limit of .0006 solar mass per cubic parsec is established for the space density of interstellar comets. The theoretical distribution of semimajor axes of interstellar comets shows that a strong hyperbolic excess must be present in the orbits of a majority of interstellar comets regardless of the dynamical characteristics of the comet cloud, except when the cloud is moving along with the sun and the distribution of individual velocities has a very low dispersion. This case, however, implies a possibility of capture by the sun and thus becomes a problem of an Oort-type cloud.
Sekanina, Z.↗
Pressure Distribution Measurements at High Speed and Oblique Incidence of Flow
The present report contains the results of a series of observations obtained for a wing of symmetrical profile for different angles of yaw. The shock tunnel with 0.4m x 0.4m cross section, of the Institute for Gas Dynamics at L.F.A. - Braunchweig was available for this work. The profile used was a 9 percent, thick hyperbolic profile with maximum thickness 40 percent aft, which was calculated by the method of F. Ringleb (2). Since the conformal transformation of this profile is known, the theoretical pressure distribution could be determined exactly for the case of incompressible flow. Then by the use of the Prandtl rule one may extend the comparison of theory and experiment to the case of higher velocity of incident flow.
Lippisch, A.↗
Where Are the Interstellar Comets?
The existing theories of comet dynamics held that large numbers of comets are ejected to interstellar space and that other stellar systems would do the same; however, no comet on a hyperbolic orbit has ever been observed. Calculations developed by Duncan et al. are used to show that most comets thought to be ejected would actually be captured by the Oort Cloud.
comets interstellar comets Oort Cloud orbital dyna↗
Dynamical capture of field stars by interstellar clouds
Evidence suggesting the existence of old stars in young galactic clusters, has motivated an investigation of a model in which such stars are captured from the field during the gas cloud collapse phase of star formation. A hyperbolic field star cannot be gravitationally captured by a static cloud since the star's total energy is conserved. However, if the cloud's potential is an explicit function of time, the energy of a penetrating star can be decreased. As a first step in modeling the dynamical capture of field stars, capture by freely falling, homogeneous spherical cloud is considered. This simplified model is solved, and a prediction is obtained for number of captured field stars in terms of the cloud parameters and dispersion velocities of the field stars.
Whitman, P. G.↗
Numerical methods for systems of conservation laws of mixed type using flux splitting
The essentially non-oscillatory (ENO) finite difference scheme is applied to systems of conservation laws of mixed hyperbolic-elliptic type. A flux splitting, with the corresponding Jacobi matrices having real and positive/negative eigenvalues, is used. The hyperbolic ENO operator is applied separately. The scheme is numerically tested on the van der Waals equation in fluid dynamics. Convergence was observed with good resolution to weak solutions for various Riemann problems, which are then numerically checked to be admissible as the viscosity-capillarity limits. The interesting phenomena of the shrinking of elliptic regions if they are present in the initial conditions were also observed.
Shu, Chi-Wang↗
A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods
The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.
Yee, H. C.↗
A three-dimensional model of corotating streams in the solar wind. 1: Theoretical foundations
The theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions (3-D) is discussed. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 solar radii where the flow is supersonic) is approximately described by the nonlinear, single fluid, polytropic (magneto-) hydrodynamic equations. Efficient numerical techniques for solving this complex system of coupled, hyperbolic partial differential equations are outlined. The formulation is inviscid and nonmagnetic, but methods allow for the potential inclusion of both features with only modest modifications. One simple, highly idealized, hydrodynamic model stream is examined to illustrate the fundamental processes involved in the 3-D dynamics of stream evolution. Spatial variations in the rotational stream interaction mechanism were found to produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction.
Pizzo, V. J.↗
Dynamics of groups around interacting double ellipticals: Measuring dark matter haloes
Binary galaxies, as binary stars, are important to measure masses, as suggested by Page (1952). Because three orbit parameters are measurable for galaxies at one instant of time, severe uncertainties remain in the orbit and mass determinations. These uncertainties can partly be overcome by statistical studies of selected samples and/or n-body simulations. Close double galaxies (and isolated galaxies) could also be useful to estimate dynamical masses if we can find test particles around them. Interacting elliptical pairs or dumb-bell galaxies are found with a large range, between 0-1200 km s(exp -1), of relative radial velocities. Standard 2-body orbit calculations, highly uncertain due to projection factors, suggest for the largest velocity differences very large galaxy masses, if the systems are bound and stationary. However, recent n-body simulations model these binaries as galaxies captured from hyperbolic orbits, requiring masses of order a few times 10(exp 11) solar maximum (Borne et al. 1988), but producing systems that are short lived. A different picture appears when we study observationally the dynamical mass of interacting double ellipticals using faint satellite galaxies. These satellites contribute little luminosity and, presumably, little mass to the system. The authors present results of two such groups, basically forming systems of test particles, around the dumb-bells NGC 4782/3 and IC 5049. They also briefly discuss the satellite group around the central dumb-bell in the cluster Sersic 40/6. Apparently, they detect large quantities of dark matter in the vicinity of these dumb-bell galaxies, because the system masses of approx. 4.5 times 10(exp 13) solar mass and 8 times 10(exp 13) solar mass for NGC 4782/3 and IC 5049, respectively, are quite high. Likewise, the mass of the Sersic 40/6 inner core is 7 times 10(exp 13) solar mass. The possibility that a common massive dark matter halo increases the merging times of these types of galaxies is suggested. Here, the authors assume H sub o = 100 km s (exp -1) Mpc(exp -1).
Quintana, H.↗
A three-dimensional model of corotating streams in the solar wind. I - Theoretical foundations
The paper is concerned with the development of the theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 Rs where the flow is supersonic) is approximately described by the nonlinear, single-fluid, polytropic magnetohydrodynamic or hydrodynamic equations. Efficient numerical techniques are outlined for solving this complex system of coupled, hyperbolic partial differential equations. The present formulation is inviscid and nonmagnetic, but the methods used allow for the potential inclusion of both features with only modest modifications. A simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution. It is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction. Comparison with simpler models demonstrates the essential nonlinear, multidimensional nature of the interplanetary dynamics.
Pizzo, V.↗
Hypervelocity impact microfoil perforations in the LEO space environment (LDEF, MAP AO-023 experiment)
The Microabrasion Foil Experiment comprises arrays of frames, each supporting two layers of closely spaced metallic foils and a back-stop plate. The arrays, deploying aluminum and brass foil ranging from 1.5 to some 30 microns were exposed for 5.78 years on NASA's LDEF at a mean altitude of 458 km. They were deployed on the North, South, East, West, and Space pointing faces; results presented comprise the perforation rates for each location as a function of foil thickness. Initial results refer primarily to aluminum of 5 microns thickness or greater. This penetration distribution, comprising 2,342 perforations in total, shows significantly differing characteristics for each detector face. The anisotropy confirms, incorporating the dynamics of particulate orbital mechanics, the dominance of incorporating extraterrestrial particulates penetrating thicknesses greater than 20 microns in Al foil, yielding fluxes compatible with hyperbolic geocentric velocities. For thinner foils, a disproportionate increase in flux of particles on the East, North, and South faces shows the presence of orbital particulates which exceed the extraterrestrial component perforation rate at 5 micron foil thickness by a factor of approx. 4.
Mcdonnell, J. A. M.↗
Spiral Lamber's Problem with Generalized Logarithmic Spirals
Lambert’s problem subject to a continuous acceleration is solved using the family of generalized logarithmic spirals. Thanks to the existence of two first integrals related to the energy and angular momentum surprising analogies with the Keplerian case are found. A minimum-energy spiral transfer exists. Increasing the value of the constant of the generalized energy yields pairs of conjugate spiral trajectories. The properties of such spirals are strongly connected with the properties of conjugate Keplerian orbits. When the generalized constant of the energy reaches a critical value the two solutions degenerate into a pair of parabolic spirals, one of which connects the two vectors through infinity. From that point the spiral transfers become hyperbolic. Generalized logarithmic spirals admit closed-form solutions to all the required magnitudes including the time of flight, providing a deep insight into the dynamics of the problem. In addition, the maximum acceleration along the transfer is found analytically so the solutions that violate the design constraints on the maximum thrust acceleration can be rejected without any further computations. When the time of flight is fixed there is still a degree of freedom in the solution, related to a control parameter. Resonant transfers appear naturally thanks to the symmetry properties of the generalized logarithmic spirals. The problem of designing a low-thrust transfer between two bodies can be reduced to solving the corresponding spiral Lambert’s problem. In order to show the versatility of the method it is applied to the design of an asteroid tour and to explore launch opportunities to Mars.\
Roa, Javier↗
On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations
The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.
Deshpande, S. M.↗