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NASA NTRS1989

The solution of the Elrod algorithm for a dynamically loaded journal bearing using multigrid techniques

A numerical solution to a theoretical model of vapor cavitation in a dynamically loaded journal bearing is developed utilizing a multigrid iteration technique. The method is compared with a noniterative approach in terms of computational time and accuracy. The computational model is based on the Elrod algorithm, a control volume approach to the Reynolds equation which mimics the Jakobsson-Floberg and Olsson cavitation theory. Besides accounting for a moving cavitation boundary and conservation of mass at the boundary, it also conserves mass within the cavitated region via a smeared mass or striated flow extending to both surfaces in the film gap. The mixed nature of the equations (parabolic in the full film zone and hyperbolic in the cavitated zone) coupled with the dynamic aspects of the problem create interesting difficulties for the present solution approach. Emphasis is placed on the methods found to eliminate solution instabilities. Excellent results are obtained for both accuracy and reduction of computational time.

Woods, C. M.↗

NASA NTRS1989

Global properties of pseudospectral methods

The present application of polynomial interpolation methods to function-approximation and numerical solutions for hyperbolic and elliptic PDEs allows the explicit construction of the derivative matrix for a general sequence of collocation points. An evaluation of the effect of several factors on the performance of these methods indicates an inability to interpret global methods in terms of local ones; the accuracy of the approximation will differ when the function's large gradients occur near the center of the region or near the boundary, irrespective of the boundary vicinity's collocation-point density.

Solomonoff, A.↗

NASA NTRS1989

Essentially nonoscillatory spectral Fourier methods for shock wave calculations

An essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations is presented. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution.

Cai, Wei↗

NASA NTRS1989

Nonlinear filters for efficient shock computation

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Engquist, Bjorn↗

NASA NTRS1989

A spline-based parameter estimation technique for static models of elastic structures

The problem of identifying the spatially varying coefficient of elasticity using an observed solution to the forward problem is considered. Under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient. Some continuous dependence results are developed for this problem and a spline-based technique is proposed for approximating the unknown coefficient, based on these results. The convergence of the numerical scheme is established and error estimates obtained.

Dutt, P.↗

NASA NTRS1989

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2.

Hussaini, M. Y.↗

NASA NTRS1989

Computational and experimental evaluation of helicopter rotor tips for high speed forward flight

A computational and experimental method for the evaluation of helicopter rotor tips in high-speed forward flight is presented which uses an unsteady full-potential solver on the advancing side of the rotor disk. Forces and moments measured during angle-of-attack sweeps reveal the soft-stall phenomenon of the double-swept planforms as well as the delayed stall of the single-swept hyperbolic tip. Double-swept planforms are shown to exhibit the most favorable performance, creating counter-rotating vortices that augment the lifting capabilities of the blade at high angles of attack and delay the onset of stall.

Sigl, Dave↗

NASA NTRS1988

Optimization of motion control laws for tether crawler or elevator systems

Based on the proposal of a motion control law by Lorenzini (1987), a method is developed for optimizing motion control laws for tether crawler or elevator systems in terms of the performance measures of travel time, the smoothness of acceleration and deceleration, and the maximum values of velocity and acceleration. The Lorenzini motion control law, based on powers of the hyperbolic tangent function, is modified by the addition of a constant-velocity section, and this modified function is then optimized by parameter selections to minimize the peak acceleration value for a selected travel time or to minimize travel time for the selected peak values of velocity and acceleration. It is shown that the addition of a constant-velocity segment permits further optimization of the motion control law performance.

Swenson, Frank R.↗

NASA NTRS1989

The stability of a compressible stratified shear layer

The stability of a shear layer under the effect of gravity is investigated using the compressible magnetohydrodynamic (MHD) equations, including an effective gravity term to represent the curvature effects of the flow and magnetic field line geometry. A general eigenmode equation is derived for a two-dimensional MHD fluid, and an energy-principle analysis to explain the effect of compressibility on the critical Richardson number is presented. For the case of a hyperbolic tangent shear flow and exponential density profile, it was found that, in the Boussinesq approximation, the compressibility raises the critical Richardson number from 1/4 to as much as 1/2, with the exact value depending on the value of the magnetic field at infinity. Under approximation of a strong asymptotic magnetic field, without invoking the Boussinesq approximation, it is shown both analytically and numerically that the density gradient terms cause the shear instability to be dispersive. The long-wavelength stability boundary for the Richardson number J = 0 is characterized by a normalized phase velocity c =

Wang, Z.↗

NASA NTRS1989

Two high-velocity encounters of elliptical galaxies

This paper describes results obtained on a simulation of two high-velocity encounters of NGC 4782/4783 and NGC 2672/2673 binary elliptical galaxies which differ substantially in mass ratio (about 1 for the first pair, and about 10 for the second). CCD images and velocities obtained from digital spectra were used to constrain simulations of the galaxy collisions. The binary orbital elements, the orientation of the orbit in the sky, the time since pericenter, and the dynamical mass of the pair were derived. Results suggested that the dumb-bell galaxy NGC 4782/4783 is not a supermassive galaxy, as was claimed earlier on the basis of the high relative velocity and high central dispersion, but has a moderate mass to luminosity ratio M/L(B) of about 10. It was concluded that its trajectory changed from hyperbolic to elliptical as a result of energy lost during the collision. It was found that the NGC 2672/2673 also has a moderate M/L(B) of about 7.

Balcells, Marc↗

NASA NTRS1989

Analysis of a second-order-accurate finite-volume method for temporally-growing compressible shear layers

A finite-volume method for solving the compressible laminar Navier-Stokes equation in two dimensions is assessed for use in the simulation of transitional flows. The method is second-order-accurate, uses alternating-direction-implicit time integration, and a total-variation-diminishing smoothing operator in the inviscid flux terms. The test problem was a free shear layer with a forced periodicity in x at a specified wavelength and with the parallel mean flow specified as a hyperbolic-tangent profile. A grid refinement investigation verified this method to be second-order-accurate with less than 4 percent error in the growth rate of isolated modes on a 32 by 64 grid. The interaction between modes of similar amplitude was small, less than 1 percent. However, in cases where there was a dominant mode, numerical phase error pumped energy into all other modes. This error results in nonphysical growth rate for modes whose energy content is several orders of magnitude below the dominant mode. In the case when the dominant mode saturates, the pumping action stops and the growth rate of the next largest growing mode regains physical significance.

Atkins, H. L.↗

NASA NTRS1990

Numerical Methods For Chemically Reacting Flows

Issues related to numerical stability, accuracy, and resolution discussed. Technical memorandum presents issues in numerical solution of hyperbolic conservation laws containing "stiff" (relatively large and rapidly changing) source terms. Such equations often used to represent chemically reacting flows. Usually solved by finite-difference numerical methods. Source terms generally necessitate use of small time and/or space steps to obtain sufficient resolution, especially at discontinuities, where incorrect mathematical modeling results in unphysical solutions.

Leveque, R. J.↗

NASA NTRS1989

The generalized drift flux approach: Identification of the void-drift closure law

The main characteristics and the potential advantages of generalized drift flux models are presented. In particular it is stressed that the issue on the propagation properties and on the mathematical nature (hyperbolic or not) of the model and the problem of closure are easier to tackle than in two fluid models. The problem of identifying the differential void-drift closure law inherent to generalized drift flux models is then addressed. Such a void-drift closure, based on wave properties, is proposed for bubbly flows. It involves a drift relaxation time which is of the order of 0.25 s. It is observed that, although wave properties provide essential closure validity tests, they do not represent an easily usable source of quantitative information on the closure laws.

Boure, J. A.↗

NASA NTRS1989

Reexamination of data from the asteroid/meteoroid detector

The discovery of the existence of cosmoids, a class of meteoroid in near hyperbolic orbits, was made in a reevaluation of the Sisyphus Experiment on Pioneer 10 and 11. This experiment measured the spontaneous jetting of cosmoids and showed that the dispersion and increase in brightness occurs in microseconds and lasts only briefly. Cosmoid jetting caused multiple telescope thresholds to be exceeded simultaneously which explains the earlier inabilty to compute trajectories from the measured times in the fields of view. A new calculation correlated the Sisyphus individual event measurements with the zodiacal light. That the meteoroid population is dominated by cosmoids is demonstrated. Reported telescopic small comets, measured by a similar optical technique, appear consistent with the Sisyphus results. Characteristic jetting times measured by Sisyphus also show that the volatile cosmoids could not survive in short period orbits.

Soberman, Robert K.↗

NASA NTRS1990

Viscous shock profiles and primitive formulations

Weak solutions of hyperbolic systems in primitive (non-conservation) form for which a consistent conservation form exists are considered. It is shown that primitive formulations, shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two. Scheme-dependent high order correction terms are derived that enforce consistent viscous shock profiles. The resulting primitive algorithm is conservative to the order of approximation. One dimensional Euler calculations of flows containing strong shocks clearly show that conservation errors in primitive flow calculations are of comparable quality.

Karni, S.↗

NASA NTRS1990

High order filtering methods for approximating hyberbolic systems of conservation laws

In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed essentially non-oscillatory (ENO) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. A filtering method which is developed uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, the full ENO apparatus is used, maintaining the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. High order of accuracy was obtained in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method.

Lafon, F.↗

NASA NTRS1990

GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 2: User's manual and program listing

An efficient computer program, called GRID2D/3D, was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. The theory and method used in GRID2D/3D is described.

Bailey, R. T.↗

NASA NTRS1990

An approximation theory for the identification of linear thermoelastic systems

An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures.

Rosen, I. G.↗