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NASA NTRS1976

Three-dimensional potential flow over hills and oval mounds

An analysis was made of the potential flow behavior for an initially uniform flow passing over a single axisymmetric hill, an oval mound, and a combination of two hills. Small perturbation theory was used, and the resulting Laplace equation for the perturbation velocity potential was solved by using either a product solution or a Green's function. The three dimensional solution is of interest in calculating the pressure distribution around obstacles, the flow of pollutants carried by the wind, and the augmentation of wind velocity for windmill siting. The augmentation in velocity at the top of a hill was found to be proportional to the hill height relative to a characteristic width dimension of the hill. An axisymmetric hill produced about 20 percent less velocity increase than a two dimensional ridge having the same cross-sectional profile.

Siegel, R.↗

NASA NTRS1980

Boundary conditions for the numerical solution of elliptic equations in exterior regions

Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.

Bayliss, A.↗

NASA NTRS1981

A title-gap flow model for use in aerodynamic loads assessment of space shuttle thermal protection system: Parallel gap faces

The problem of predicting aerodynamic loads on the insulating tiles of the space shuttle thermal protection system (TPS) is discussed and seen to require a method for predicting pressure and mass flux in the gaps between tiles. A mathematical model of the tile-gap flow is developed, based upon a slow viscous (Stokes) flow analysis, and is verified against experimental data. The tile-gap pressure field is derived from a solution of the two-dimensional Laplace equation; the mass-flux vector is then calculated from the pressure gradient. The means for incorporating this model into a lumped-parameter network analogy for porous-media flow is given. The means for incorporating this model into a lumped-parameter network analogy for porous-media flow is given. The flow model shows tile-gap mass flux to be very sensitive to the gap width indicating a need for coupling the TPS flow and tile displacement calculation. Analytical and experimental work to improve TPS flow predictions and a possible shuttle TPS hardware modification are recommended.

Dwoyer, D. L.↗

NASA NTRS1982

A numerical method for the dynamics of non-spherical cavitation bubbles

A boundary integral numerical method for the dynamics of nonspherical cavitation bubbles in inviscid incompressible liquids is described. Only surface values of the velocity potential and its first derivatives are involved. The problem of solving the Laplace equation in the entire domain occupied by the liquid is thus avoided. The collapse of a bubble in the vicinity of a solid wall and the collapse of three bubbles with collinear centers are considered.

Lucca, G.↗

NASA NTRS1982

Capillary forces exerted by liquid drops caught between crossed cylinders. A 3-D meniscus problem with free contact line

The Young-Laplace equation is solved for three-dimensional menisci between crossed cylinders, with either the contact line fixed or the contact angle prescribed, by means of the Galerkin/finite element method. Shapes are computed, and with them the practically important quantities: drop volume, wetted area, capillary pressure force, surface tension force, and the total force exerted by the drop on each cylinder. The results show that total capillary force between cylinders increases with decreasing contact angle, i.e. with better wetting. Capillary force is also increases with decreasing drop volume, approaching an asymptotic limit. However, the wetted area on each cylinder decreases with decreasing drop volume, which raises the question of the optimum drop volume to strive for, when permanent bonding is sought from solidified liquid. For then the strength of the bond is likely to depend upon the area of contact, which is the wetted area when the bonding agent was introduced in liquid form.

Patzek, T. W.↗

NASA NTRS1982

Flow through the tile gaps in the Space Shuttle Thermal Protection System

The problem of predicting aerodynamic loads on the insulating tiles of the Space Shuttle Thermal Protection System (TPS) is discussed and seen to require a method for predicting pressure and mass flux in the gaps between tiles. A mathematical model of the tile-gap flow is developed based upon a slow viscous (Stokes) flow analysis and is verified against available experimental data. This model derives the tile-gap pressure field from a solution of the two-dimensional Laplace equation; the mass flux vector is then calculated from the pressure gradient. The means for incorporating this model into a lumped-parameter network analogy for porous-media flow is also given. The flow model shows tile-gap mass flux to be very sensitive to the gap width indicating a need for coupling the TPS flow and tile displacement calculations. Finally recommendations are made concerning additional analytical and experimental work to improve TPS flow predictions.

Dwoyer, D. L.↗

NASA NTRS1982

Wind-tunnel wall interference corrections for three-dimensional flows

A procedure for the evaluation of wall interference corrections for three-dimensional aircraft configurations is presented. The Mach number and angle-of-attack corrections are obtained by numerically solving the Laplace equation in a parallelepiped with boundary conditions supplied mainly from experimental pressure measurements. A portion of these measurements and other wind-tunnel data required by the procedure may be replaced by theoretical estimates if not available from experiments. The accuracy of the correction results will then depend on the accuracy of these estimates. The correction procedure is applied to an isolated wing and to a wing-tail configuration in a solid-wall wind tunnel. It is found that neglecting twist and camber corrections for the wing effectively increases the tail angle-of-attack correction. Two different Mach number corrections can be calculated for the wing and tail. However, since only one Mach number correction is allowed for both the wing and the tail, and since the wing surface area is larger than the tail surface area, the final correction tends to be closer to the required wing correction. This is a source of error for the tail results.

Rizk, M. H.↗

NASA NTRS1983

Geomagnetism of earth's core

Instrumentation, analytical methods, and research goals for understanding the behavior and source of geophysical magnetism are reviewed. Magsat, launched in 1979, collected global magnetometer data and identified the main terrestrial magnetic fields. The data has been treated by representing the curl-free field in terms of a scalar potential which is decomposed into a truncated series of spherical harmonics. Solutions to the Laplace equation then extend the field upward or downward from the measurement level through intervening spaces with no source. Further research is necessary on the interaction between harmonics of various spatial scales. Attempts are also being made to analytically model the main field and its secular variation at the core-mantle boundary. Work is also being done on characterizing the core structure, composition, thermodynamics, energetics, and formation, as well as designing a new Magsat or a tethered satellite to be flown on the Shuttle.

Benton, E. R.↗

NASA NTRS1982

The dependence of the shape and stability of captive rotating drops on multiple parameters

Asymptotic and numerical techniques in bifurcation theory are applied to the Young-Laplace equation governing meniscus shape in order to analyze the dependence of the shape and stability of rigidly rotating drops held captive between corotating solid faces on multiple parameters. Asymptotic analysis of the evolution of drop shape from the cylindrical as a function of distance between the solid faces, drop volume, rotational Bond number and gravitational Bond number shows that some shape bifurcations from cylinders to wavy, axisymmetric menisci are ruptured by small changes in drop volume or gravity. Computer calculations of axisymmetric drop shapes based on a finite element representation of the interface and numerical algorithms for tracking shape families and singular points are then used to map drop stability for the four-dimensional parameter space. The results of the asymptotic and numerical analyses are shown to agree well within the limited range of parameters where the asymptotic analysis is valid.

Ungar, L. H.↗

NASA NTRS1985

Slender body theory and Space Shuttle transonic aerodynamics

A computational implementation of transonic slender body theory and the equivalence rule has been utilized to study transonic flow field around the Space Shuttle Orbiter. The far field is described by a nonlinear axisymmetric Karman-Guderley model and the near field by a cross flow Laplace equation boundary value problem. The latter is treated using a source panel method. Preliminary comparisons with experiments give encouraging indications that the model can be useful for quick turnaround estimates. Areas of refinement to obtain more accurate predictions are discussed.

Malmuth, N. D.↗

NASA NTRS1985

Surface Tensions and Their Variations with Temperature and Impurities

The surface tensions in this work were determined using the sessile drop technique. This method is based on a comparison of the profile of a liquid drop with the profile calculated by solving the Young-Laplace equation. The comparison can be made in several ways; the traditional Bashforth-Adams procedure was used in conjunction with recently calculated drop shape tables which virtually eliminate interpolation errors. Although previous study has found little difference in measurements with pure and oxygen doped silicon, there is other evidence suggesting that oxygen in dilute concentrations severely depresses the surface tension of silicon. The surface tension of liquid silicon in purified argon atmospheres was measured. A temperature coefficient near -0.28 mJ/square meters K was found. The experiments show a high sensitivity of the surface tension to what is believed are low concentrations of oxygen. Thus one cannot rule out some effect of low levels of oxygen in the results. However, the highest surface tension values obtained in conditions which minimized the residual oxygen pressure are in good agreement with a previous measurement in pure hydrogen. Therefore, depression of the surface tension by oxygen is insignificant in these measurements.

Hardy, S. C.↗

NASA NTRS1986

Aperture plane potential control for thermal ion measurements

The effects produced by the addition of an aperture plane to control the bias around an instrument for low-energy ion measurements on satellites collecting data in the plasmasphere and over the polar cap are analyzed. The analysis is based on the design and functions of the retarding ion mass spectrometer (RIMS) on the DE 1 satellite. The NASA Charging Analyzer Program, which treats the spacecraft as a cylinder, was used to generate one set of predictions. A second model involves solution of the Laplace equation with the spacecraft treated as a sphere. Both models were used to predict the barrier height expected at the aperture in response to the bias induced to offset the potential. Comparisons with plasma data show that in the thin sheath regime of the plasmasphere the detectors act as if the potential is shifted, thereby lowering the energy resolution of the instrument. Outside the plasmasphere the barrier height dominates the spin curve variations more than the potential. Partial potential control is available from induced bias apertures if control is active and guided by electron detectors with sensitivities in the 0.5-50.0 eV energy range.

Olsen, R. C.↗

NASA NTRS1988

Prediction of the oscillating incompressible aerodynamics of a loaded airfoil cascade by a locally analytical method

A complete first order model is developed to predict the oscillating incompressible aerodynamics of an airfoil cascade, including the effects of steady loading. The cascade steady and unsteady flow fields are analyzed by considering a periodic flow channel. The velocity potential is separated into steady and unsteady harmonic components each of which is described by a Laplace equation. An analytical solution in individual grid elements of a body fitted computational grid is then determined, with the complete solution obtained by assembling these local solutions. The capabilities and validity of this model and solution technique are demonstrated by considering the steady and unsteady aerodynamics of both theoretical and experimental cascade configurations.

Chiang, Hsiao-Wei D.↗

NASA NTRS1988

Prediction of oscillating thick cambered aerofoil aerodynamics by a locally analytic method

The effects of mean-flow incidence, airfoil camber, and airfoil thickness on the incompressible aerodynamics of an oscillating airfoil are investigated theoretically, developing and applying a first-order FEM based on locally analytical solutions (LASs). Laplace equations are used to describe the steady and unsteady harmonic velocity potentials; a body-fitted computational grid is employed; grid-element solutions for both potentials are determined using a numerical LAS method; and the LASs are then assembled to obtain a complete solution. Results for a series of flat-plate and Joukowski airfoils are presented in extensive graphs and discussed in detail.

Chiang, Hsiao-Wei D.↗

NASA NTRS1988

PC-CUBE: A Personal Computer Based Hypercube

PC-CUBE is an ensemble of IBM PCs or close compatibles connected in the hypercube topology with ordinary computer cables. Communication occurs at the rate of 115.2 K-band via the RS-232 serial links. Available for PC-CUBE is the Crystalline Operating System III (CrOS III), Mercury Operating System, CUBIX and PLOTIX which are parallel I/O and graphics libraries. A CrOS performance monitor was developed to facilitate the measurement of communication and computation time of a program and their effects on performance. Also available are CXLISP, a parallel version of the XLISP interpreter; GRAFIX, some graphics routines for the EGA and CGA; and a general execution profiler for determining execution time spent by program subroutines. PC-CUBE provides a programming environment similar to all hypercube systems running CrOS III, Mercury and CUBIX. In addition, every node (personal computer) has its own graphics display monitor and storage devices. These allow data to be displayed or stored at every processor, which has much instructional value and enables easier debugging of applications. Some application programs which are taken from the book Solving Problems on Concurrent Processors (Fox 88) were implemented with graphics enhancement on PC-CUBE. The applications range from solving the Mandelbrot set, Laplace equation, wave equation, long range force interaction, to WaTor, an ecological simulation.

Ho, Alex↗

NASA NTRS1988

A numerical method for computing unsteady 2-D boundary layer flows

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results.

Krainer, Andreas↗

NASA NTRS1990

Cascade aerodynamic gust response including steady loading effects

To predict the unsteady convected gust aerodynamic response of a cascade comprised of arbitrary thick and cambered aerofoils in an incompressible, inviscid, flow field, a complete first-order model is formulated. The flow is analyzed by considering a periodic flow channel. The velocity potential is separated into steady and unsteady harmonic components, each described by a Laplace equation. The strong dependence of the unsteady aerodynamics on the steady effects of aerofoil and cascade geometry and incidence angle is manifested in the coupling of the unsteady and steady flow fields through the unsteady boundary conditions. Analytical solutions in individual grid elements of a body-fitted computational grid are then determined, with the complete solution obtained by assembly of these local solutions. The validity and capabilities of this model and solution technique are then demonstrated by analyzing the steady and unsteady aerodynamics of both theoretical and experimental cascade configurations.

Chiang, Hsiao-Wei D.↗

NASA NTRS1991

Electron distribution functions in electric field environments

The amount of current carried by an electric discharge in its early stages of growth is strongly dependent on its geometrical shape. Discharges with a large number of branches, each funnelling current to a common stem, tend to carry more current than those with fewer branches. The fractal character of typical discharges was simulated using stochastic models based on solutions of the Laplace equation. Extension of these models requires the use of electron distribution functions to describe the behavior of electrons in the undisturbed medium ahead of the discharge. These electrons, interacting with the electric field, determine the propagation of branches in the discharge and the way in which further branching occurs. The first phase in the extension of the referenced models , the calculation of simple electron distribution functions in an air/electric field medium, is discussed. Two techniques are investigated: (1) the solution of the Boltzmann equation in homogeneous, steady state environments, and (2) the use of Monte Carlo simulations. Distribution functions calculated from both techniques are illustrated. Advantages and disadvantages of each technique are discussed.

Rudolph, Terence H.↗