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NASA NTRS1978

Atmospheric-wake vortex interactions

The interactions of a vortex wake with a turbulent stratified atmosphere are investigated with the computer code WAKE. It is shown that atmospheric shear, turbulence, and stratification can provide the dominant mechanisms by which vortex wakes decay. Computations included the interaction of a vortex wake with a viscous ground plane. The observed phenomenon of vortex bounce is explained in terms of secondary vorticity produced on the ground. This vorticity is swept off the ground and advected about the vortex pair, thereby altering the classic hyperbolic trajectory. The phenomenon of the solitary vortex is explained as an interaction of a vortex with crosswind shear. Here, the vortex having the sign opposite that of the sign of the vorticity in the shear is dispersed by a convective instability. This instability results in the rapid production of turbulence which in turn disperses the smoke marking the vortex.

Bilanin, A. J.↗

NASA NTRS1978

A high order accurate finite element algorithm for high Reynolds number flow prediction

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Baker, A. J.↗

NASA NTRS1978

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.↗

NASA NTRS1978

Effects of Moisture Profiles and Laminate Configuration on the Hygro Stresses in Advanced Composites

An integrated hygrothermo-mechanical theory was used to predict the effects of three moisture profiles on the ply hygro stresses in angleplied laminates. The moisture profiles were linear, parabolic and hyperbolic. Moisture content varied from 1 percent in the exposed ply to zero in the protected ply. The angleplied laminates were of two generic configurations. The results obtained are summarized graphically to illustrate the effects of both moisture profile and laminate configuration. The results indicate that ply transverse tensile hygro stresses may reach sufficiently high magnitudes to cause transply cracking.

Chamis, C. C.↗

NASA NTRS1978

An old integration scheme for compressible flows revisited, refurbished and put to work

A scheme for integrating the Euler equations of compressible flow in any hyperbolic case is presented. The scheme relies on the concept of characteristics but is strictly a finite difference scheme. Improvements in accuracy and physical consistence due to the scheme are discussed and results of its application to complex flows are shown.

Moretti, G.↗

NASA NTRS1977

Supercritical cascade design

The use of the method of complex extension to achieve better aerodynamic designs for supercritical cascades applicable to transonic turbomachinery is discussed. The method permits the computation of analytical solutions to elliptic, hyperbolic or mixed second-order partial equations in two dimensions. Boundary value problems formulated to develop an airfoil shape having a prescribed speed distribution for subsonic flow and a nearby speed distribution in the transonic case are also considered. Computing times necessary to run the blade design program are described as acceptably short.

Korn, D. G.↗

NASA NTRS1977

A balanced diagnostic system compatible with a barotropic prognostic model

A system of diagnostic equations for the velocity field, or wind laws, for a barotropic primitive-equation model of large-scale atmospheric flow is derived. Attention is given to the classical balance equation and its ellipticity condition. Numerical solutions of the diagnostic system are presented, including examples of cases of the mixed elliptic-hyperbolic type and cases with non-zero divergence. Procedures for implementing such a system are outlined, along with a review of factors in using the technique for operational numerical weather prediction.

Ghil, M.↗

NASA NTRS1977

An upper limit for the flux of interstellar dust grains in the solar system

Data from the LEAM (Lunar Ejecta and Meteorite experiment, a micrometeorite detector at the Apollo 17 landing site) have been examined for evidence of interstellar (IS) dust grains traversing the solar system. The analysis technique considers IS grains approaching the solar system from the local solar apex. A model calculates the grains' hyperbolic orbits into the solar system and predicts the impact directions on the moon. The observations are then compared with the predicted impact directions to measure the IS dust flux. No evidence has been found (at the 97.5% confidence level) for a flux greater than about 6 hundred-thousandths per sq m/sec for particles of at least 2 by 10 to the -14th power g.

Tomandl, D. A.↗

NASA NTRS1977

Initial energy and perihelion distributions of Oort-cloud comets

A Monte Carlo model of stellar perturbations of the Oort cloud is used to study the distributions in energy and perihelion of comets entering the planetary region for the first time. The model is run for a variety of initial states and a range of velocity perturbations. In all cases the resulting orbits are uniformly distributed in perihelion distance in the planetary region, q less than 20 AU. Most orbits are confined to a fairly narrow range in 1/a and hyperbolic orbits are rare.

Weissman, P. R.↗

NASA NTRS1978

Halley comet rendezvous with a SEPS vehicle

An analysis of the performance of a Solar Electric Propulsion System (SEPS) vehicle rendezvousing with Halley's comet just prior to its Frebruary 1986 perihelion is described. A calculus of variations mathematical formulation is used to maximize Halley arrival mass while giving effect to the influence of solar array size, launch date, arrival date, and insertion hyperbolic excess velocity. Numerical sensitivity relief, thrust system modeling, trajectory characteristics and ion engine operating conditions are discussed and illustrated. Results indicate a rendezvous is feasible with a minimal advance in solar cell and ion engine technology.

Burrows, R. R.↗

NASA NTRS1978

On acceleration of MacCormack's scheme

An acceleration of MacCormack's scheme due to Desideri and Tannehill is analyzed. It is found that for hyperbolic problems one cannot improve upon the efficiency of MacCormack's method. For parabolic problems the time step can be chosen arbitrarily large without loss of stability by an appropriate choice of the acceleration parameters. When applied to the heat equation this method is equivalent to both the Dufort-Frankel scheme and to MSOR.

Gottlieb, D.↗

NASA NTRS1978

Extrapolation methods for dynamic partial differential equations

Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. Computational results are presented confirming the analytic discussions.

Turkel, E.↗

NASA NTRS1978

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.↗

NASA NTRS1978

The artificial compression method for computation of shocks and contact discontinuities. III - Self-adjusting hybrid schemes

This paper presents a new computational method for the calculation of discontinuous solutions of hyperbolic systems of conservation laws, which deal effectively with both shock and contact discontinuities. The method consists of two stages: in the first stage a standard finite-difference scheme is hybridized with a nonoscillatory first order accurate method to provide for the monotonic variation of the solution near discontinuities, and in the second stage artificial compression is applied to sharpen transitions at discontinuities. This modification of a standard finite-difference method results in a scheme which preserves the order of truncation error of the original method and yet yields a sharp and oscillation free transition for both shocks and contact discontinuities. The modification can be easily implemented in existing computer codes.

Harten, A.↗

NASA NTRS1978

A model for close encounters in the planetary problem

A model is proposed for single close encounters between two small masses, m1 and m2, which orbit a much larger mass, M. The main new feature of the model is the assumption of conic motion of the center of mass of m1 and m2 in the gravitational field of M. Comparisons of the model with the three-body equations of motion indicate that the model is a useful approximation for m1, m2 not exceeding 0.00001 M. The model is therefore applicable for encounters between bodies of the order of an earth mass or smaller in the presence of the sun. Comparisons are also made of outcomes obtained by the model with outcomes of numerical integration for a large variety of close encounters. The above comparisons reveal that for many purposes the model is an adequate approximation for encounters with an eccentricity of the hyperbolic orbit of m1 about m2 not less than 4.

Cox, L. P.↗

NASA NTRS1978

Optimum receiver structures for phase-multiplexed modulations

A MAP (maximum a posteriori) estimation loop is derived for a QPSK signal and, by suitable approximation to the nonlinearity which arises as a consequence of the MAP theory, is reconfigured with practical realizations that are valid for high and low SNRs. In particular, it is shown that by approximating the hyperbolic tangent nonlinearity in the MAP estimation loop by the first two terms in its power series, an interesting practical realization of this loop results which applies at low SNRs. The error signal in this loop is formed by multiplying the error signal and lock detector output signal of a conventional biphase Costas loop. A generalized linear in-phase channel configuration is proposed which allows carrier reconstruction from an unbalanced QPSK signal at all ratios of data rates and powers in the two channels, even in the limit as these ratios simultaneously approach unity, i.e., balanced quadriphase.

Simon, M. K.↗

NASA NTRS1977

Numerical calculations of the cross-section for the accretional capture of the moon by the earth

If a primary body is increasing in mass the hyperbolic orbit of a secondary body can become an elliptic orbit. In order to determine the cross-section for the accretional capture of the moon by the earth, a series of numerical calculations has been carried out. Calculations have been carried out for various orbital ellipticities and separations and for various accretion rates. Accretional capture is favored if significant accretion occurs in a period of less than 100 yr. A window for accretional capture occurs if the minimum initial separation of the earth and moon (in astronomical units) is nearly equal to the initial heliocentric lunar eccentricity.

Nordmann, J. C.↗

NASA NTRS1978

Extension of Gauss' method for the solution of Kepler's equation

Gauss' method for solving Kepler's equation is extended to arbitrary epochs and orbital eccentricities. Although originally developed for near parabolic orbits in the vicinity of pericenter, a generalization of the method leads to a highly efficient algorithm which compares favorably to other methods in current use. A key virtue of the technique is that convergence is obtained by a method of successive substitutions with an initial approximation that is independent of the orbital parameters. The equations of the algorithm are universal, i.e., independent of the nature of the orbit whether elliptic, hyperbolic, parabolic or rectilinear.

Battin, R. H.↗