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NASA NTRS1975

Relaxation methods in fluid mechanics

The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

Lomax, H.↗

NASA NTRS1975

Accurate integration of geostationary orbits with Burdet's focal elements

The theory of Burdet's focal elements (1969) is outlined. The differential equations are presented, and the initial value problem is described together with the transformation to rectangular coordinates and classical elements. The focal elements are well defined for zero eccentricity and inclination and can be adopted for the computation of elliptic, parabolic, and hyperbolic motion. For the numerical integration of near-geostationary orbits, a comparison of the efficiency is made between focal elements, Kustaanheimo-Stiefel (1971) theory, and rectangular coordinates. For this class of orbits, a higher accuracy has been obtained by integrating elements than integrating rectangular coordinates.

Flury, W.↗

NASA NTRS1975

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

Martin, E. D.↗

NASA NTRS1975

The mini-O, a digital superhet, or a truly low-cost Omega navigation receiver

A quartz tuning fork filter circuit and some unique CMOS clock logic methods provide a very simple OMEGA-VLF receiver with true hyperbolic station pair phase difference outputs. An experimental system was implemented on a single battery-operated circuit board requiring only an external antenna preamplifier, and LOP output recorder. A bench evaluation and preliminary navigation tests indicate the technique is viable and can provide very low-cost OMEGA measurement systems. The method is promising for marine use with small boats in the present form, but might be implemented in conjunction with digital microprocessors for airborne navigation aids.

Burhans, R. W.↗

NASA NTRS1976

Conformal coordinates associated with uniformly accelerated motion

Specific problems in the theory of relativity are often simplified by an appropriate choice of the coordinate system. Restricted conformal coordinates provide an especially simple analysis of motion with uniform acceleration, known as hyperbolic motion. Conformal coordinates x', t' may be obtained from Cartesian coordinates x, t by the transformation x'+ct'=F(x+ct) and x'-ct'=G(x-ct), where c is the velocity of light. A variable motion of the x' system is determined by the choice of the functions F and G.

Jones, R. T.↗

NASA NTRS1976

The evolution of comet orbits

The origin of comets and the evolution of their orbits are discussed. Factors considered include: the law of survival of comets against ejection on hyperbolic orbits; short-period comets are not created by single close encounters of near-parabolic comets with Jupiter; observable long-period comets do not evolve into observable short-period comets; unobservable long-period comets with perihelia near Jupiter can evolve into observable short-period comets; long-period comets cannot have been formed or created within the planetary region of the solar system (excluding the effects of stellar perturbations); it is possible that some of the short-period comets could have been formed inside the orbit of Neptune; circularly-restricted three-body problem, and its associated Jacobi integral, are not valid approximations to use in studying origin and evolution of comets.

Everhart, E.↗

NASA NTRS1974

Investigation of a quadrupole ultra-high vacuum ion pump

The new nonmagnetic ion pump resembles the quadrupole ionization gage. The dimensions are larger, and hyperbolically shaped electrodes replace the four rods. Their surfaces follow y sq. = 36 + x sq. (x, y in centimeters). The electrodes, 55 cm long, are positioned lengthwise in a tube. At one end a cathode emits electrons; at the other end a narrowly wound flat spiral of tungsten clad with titanium on cathode potential can be heated for titanium evaporation. Electrons accelerated by a dc potential of the surface electrodes oscillate between the ends on rotational trajectories, if a high frequency potential superimposed on the dc potential is properly adjusted. Pumping speeds (4-100 liter/sec) for different gases at different peak voltages (1000-3000V) at corresponding frequencies (57-100 MHz), and at different pressures 0.00001 to the minus 9 power Torr were observed. The lowest pressure reached was below 10 to the minus 10 power Torr.

Schwarz, H. J.↗

NASA NTRS1975

Diagonalization and simultaneous symmetrization of the gas-dynamic matrices

The hyperbolic nature of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation for diagonalizing an arbitrary linear combination of coefficient matrices. It is shown that the individual matrices are simultaneously symmetrized by the similarity transformation. The transformations and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.

Warming, R. F.↗

NASA NTRS1975

The powered swingby and its application to Jupiter orbit mission design

This paper is a description of recent studies of constrained optimal transfer between given hyperbolic asymptotes with an emphasis on the practical aspects of the concept for powered Galilean satellite swingby Jupiter orbit insertion. The effects of orbital inclination on performance gain are discussed and the targeting problem for establishing the optimal swingby is solved. Comparisons of some candidate mission profiles are presented and the advantages and disadvantages of the powered swingby orbit insertion are discussed in the context of recent interactions with the scientific community on the relative merits of several mission profiles now under consideration.

Uphoff, C.↗

NASA NTRS1975

Accretional capture of the moon

The effectiveness of accretional capture is studied with the aid of a simple model involving a small secondary body in a hyperbolic orbit which approaches a large primary body. When the separation is a minimum, the mass of the primary body increases. An investigation is conducted regarding the conditions under which this change in mass will result in an elliptic orbit with capture.

Clark, S. P., Jr.↗

NASA NTRS1975

On the computation of two- and three-dimensional steady transonic flows by relaxation methods

The paper is concerned with the application of the Murman and Cole (1971) relaxation scheme to steady, inviscid transonic flow problems in two and three dimensions. This scheme, which automatically accounts for weak shock waves, uses separate difference operators in elliptic and hyperbolic regions. The details of the scheme are described in terms of the original small disturbance formulation of Murman and Cole. In particular, Murman's recent (1973) introduction of fully conservative difference operators to obtain the correct shock jumps is examined. The extension to treating the exact isentropic equation is then covered with special attention given to Jameson's (to appear) rotated difference scheme for supersonic flow regions. The bulk of the discussion is related to two-dimensional procedures, and some comparisons with experiment are made, with emphasis on the effects of viscosity and wind-tunnel walls. Application of the Murman-Cole scheme is then discussed for small disturbances in three dimensions.

Bailey, F. R.↗

NASA NTRS1976

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.↗

NASA NTRS1976

Multidimensional difference schemes with fourth-order accuracy

An explicit finite-difference algorithm is presented for the solution of quasilinear divergence free multidimensional hyperbolic systems. The method consists of four steps per time level. The resulting scheme is fourth-order accurate in both space and time, though the intermediate steps are only first-order accurate. The family of schemes introduced is dissipative, and hence, suitable for both smooth flows and flows containing shocks. This method is compared, in several numerical examples, with both second-order schemes and others that are fourth order in space, but second order in time.

Turkel, E.↗

NASA NTRS1976

Computation of the inviscid supersonic flow over an external axial corner

A second-order finite-difference procedure is used to evaluate the inviscid supersonic flowfield surrounding an external axial corner composed of swept planar compression surfaces and representing the inlets on existing high-speed aircraft. The governing partial differential equations in conservation-law form are hyperbolic with respect to the axial coordinate and are solved iteratively by means of MacCormack's algorithm. The procedure treats both the peripheral shock wave and vortical singularities as discontinuities. Numerical results are presented for two parametric studies regarding the effects on the flowfield of varying the free-stream Mach number and the leading edge sweep of the horizontal wedge. Results of parametric Mach number study agree with the Mach number independence principle in that as the Mach number increases, such characteristics as shock shape, cross-flow sonic line location, and vortical singularity position approach an asymptote.

Kutler, P.↗

NASA NTRS1976

On the Navier-Stokes equations with constant total temperature

For various applications in fluid dynamics, one can assume that the total temperature is constant. Therefore, the energy equations can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed in this paper. It is shown that the system is strictly hyperbolic and well posed for the initial-value problem. Boundary conditions are described such that the linearized system is well posed. The hopscotch method is investigated and numerical results are presented.

Gottlieb, D.↗

NASA NTRS1976

Characteristics procedure for supersonic flows including consideration of viscous contributions to flow rotationality

A characteristics procedure has been developed which includes viscous and conductive transport terms normal to streamlines. By introducing the transport terms as corrections, the equations retain their hyperbolic character. The solutions have been developed using an inverse grid scheme in a streamline-normal network. Inclusion of the transport terms enables tracing of shock waves through the supersonic portion of the boundary layer so that shock reflections are well-located. Excellent agreement is obtained with the results of other characteristics procedures and with the experimental pressure distributions for the waisted body of Winter, Smith and Rotta.

Anderson, B. H.↗

NASA NTRS1976

A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics

Martin, E. D.↗

NASA NTRS1976

Gridless retarding potential analyzer for use in very-low-energy charged particle detection

The theory of the hyperbolic retarding potential analyzer in the electrostatic mode is developed in detail and verified in the laboratory. A monoenergetic electron beam is used for the laboratory investigation. The analyzer (acronym HARP) has advantages over other conventional electrostatic analyzers; among them are less contact potential influence and high throughput because of the symmetry shape of the analyzer. The most useful application of the HARP is in detecting low-energy charged particles. A sample of low-energy particle data obtained in the earth's ionosphere is given.

Shyn, T. W.↗