Transformation of a potential function under coordinate rotations
Potential function transformation under coordinate rotations, deriving coefficients for Laplace equation series expansion
Levie, S. L., Jr.↗
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Analysis of heat transfer for compressible flow in two-dimensional porous media
Compressible gas flow in two dimensional porous wall, calculating heat transfer and mass flow by conformal mapping for Laplace equation solution
Goldstein, M. E.↗
A crack problem with four distinct harmonic functions
The problem of an elastic solid containing a semi-infinite plane crack subjected to concentrated shears parallel to the edge of the crack is considered. A closed form solution using four distinct harmonic functions (none of which can be taken arbitrarily) is found to satisfy the finite displacement and inverse square root stress singularity at the edge of the crack. Explicit expressions in terms of elementary functions are given for the distribution of stress and displacement in the solid. These are obtained by employing Fourier and Kontorovich-Lebedev integral transforms and certain singular solutions of Laplace equations in three dimensions. The variations of the intensity of the local stress field along the crack border are shown graphically. An example is presented, which is in contrast with the conclusion established in the literature that one of the four Papkovich-Neuber functions in three-dimensional elasticity may be arbitrarily set to zero.
Sih, G. C.↗
Application of Papkovich-Neuber potentials to a crack problem.
The problem of an elastic solid containing a semi-infinite plane crack subjected to concentrated shears parallel to the edge of the crack is considered in this paper. A closed form solution using four harmonic functions is found to satisfy the finite displacement and inverse square root stress singularity at the edge of the crack. Explicit expressions in terms of elementary functions are given for the distribution of stress and displacement in the solid. These are obtained by employing Fourier and Kontorovich-Lebedev integral transforms and certain singular solutions of Laplace equations in three dimensions. The variations of the intensity of the local stress field along the crack border are shown graphically.
Kassir, M. K.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗