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All sourcesNASA NTRSDOE OSTI
NASA NTRS2004

Rayleigh-Sommerfield Diffraction vs Fresnel-Kirchhoff, Fourier Propagation and Poisson's Spot

The boundary conditions imposed on the diffraction problem in order to obtain the Fresnel-Kirchhoff (FK) solution are well-known to be mathematically inconsistent and to be violated by the solution when the observation point is close to the diffracting screen 1-3. These problems are absent in the Rayleigh-Sommerfeld (RS) solution. The difference between RS and FK is in the inclination factor and is usually immaterial because the inclination factor is approximated by unity. But when this approximation is not valid, FK can lead to unacceptable answers. Calculating the on-axis intensity of Poisson s spot provides a critical test, a test passed by RS and failed by FK. FK fails because (a) convergence of the integral depends on how it is evaluated and (b) when the convergence problem is xed, the predicted amplitude at points near the obscuring disk is not consistent with the assumed boundary conditions.

RS (RAYLEIGH SOMMERFELD)↗
NASA NTRS1986

An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

Young, D. P.↗
NASA NTRS2004

Further Improvement in 3DGRAPE

3DGRAPE/AL:V2 denotes version 2 of the Three-Dimensional Grids About Anything by Poisson's Equation with Upgrades from Ames and Langley computer program. The preceding version, 3DGRAPE/AL, was described in Improved 3DGRAPE (ARC-14069) NASA Tech Briefs, Vol. 21, No. 5 (May 1997), page 66. These programs are so named because they generate volume grids by iteratively solving Poisson's Equation in three dimensions. The grids generated by the various versions of 3DGRAPE have been used in computational fluid dynamics (CFD). The main novel feature of 3DGRAPE/AL:V2 is the incorporation of an optional scheme in which anisotropic Lagrange-based trans-finite interpolation (ALBTFI) is coupled with exponential decay functions to compute and blend interior source terms. In the input to 3DGRAPE/AL:V2 the user can specify whether or not to invoke ALBTFI in combination with exponential-decay controls, angles, and cell size for controlling the character of grid lines. Of the known programs that solve elliptic partial differential equations for generating grids, 3DGRAPE/AL:V2 is the only code that offers a combination of speed and versatility with most options for controlling the densities and other characteristics of grids for CFD.

Alter, Stephen↗
NASA NTRS1985

Computation of transonic vortex flows past delta wings Integral equation approach

The steady full-potential equation is written in the form of Poisson's equation, and the solution of the velocity field is expressed in terms of an integral equation. The solution consists of a surface integral of vorticity distribution on the wing and its free-vortex sheets and a volume integral of source distribution within a volume around the wing and its free-vortex sheets. The solution is obtained through successive iteration cycles. The source distribution is computed by using a mixed finite-difference scheme of the Murman-Cole type. The method is applied to delta wings. Numerical examples show that a conical shock is captured on the suction side of the wing. It is attached to the lower surface of the leading-edge vortex but does not necessarily reach to the wing surface.

Kandil, O. A.↗
NASA NTRS1988

Computational technique for compressible vortex flows using the integral equation solution

The steady full-potential equation is written in the form of Poisson's equation, and the solution for the velocity field is expressed in terms of an integral equation. The integral solution consists of two surface integrals and one volume integral. The solution is obtained through successive iteration cycles. Each cycle of iteration consists of two sub-cycles, an inner cycle for wake relaxation and an out cycle for the strength of the source distribution integrals representing the flow compressibility. The density gradients in the source distribution is computed by using a type-differencing scheme of the Murman-Cole type. The method is applied to delta wings and the numerical examples show that a curved shock is captured on the wing suction side beneath the leading edge vortex sheet. Recently, a modified version of the scheme was applied to rectangular wings. In this modified scheme, the surface integral terms were computed by using a bilinear distribution of vorticity on triangular vortex panels which represent the wing and its wake. The results were compared with the available experimental data and they are in good agreement.

Kandil, Osama A.↗
NASA NTRS1989

Oxygen acoustic solitary waves in a magnetized plasma

Ion-acoustic solitary waves in a magnetized plasma containing an arbitrary mixture of H(+) and O(+) ions are studied. A nonlinear wave equation has been derived from the Poisson-Vlasov equations, including a uniform magnetic field and dissipation due to reflected electrons. When dissipation is ignored, the equation has soliton solutions associated with both oxygen and hydrogen acoustic modes, which can be either rarefactive or compressive depending on the ion concentrations and the electron/ion temperature ratio and, more weakly, on the bulk drifts of the species. If electron reflection is included, the solitary wave can be intensified. Under somewhat restrictive conditions the oxygen solitary wave is rarefactive and propagates with a velocity comparable to that observed by the Viking satellite. The three-dimensional solitons obey a relation of scales parallel to the magnetic field and in the transverse direction. Computer simulations of one-dimensional versions of the nonlinear wave equation are presented.

Qian, S.↗
NASA NTRS1991

Generating Three-Dimensional Grids About Anything

Three-Dimensional Grids About Anything by Poisson's Equation (3DGRAPE) computer program designed to make computational grids in or about almost any shape. Generated by solution of Poisson's differential equations in three dimensions. Program automatically finds its own values for inhomogeneous terms giving near-orthogonality and controlled grid-cell height at boundaries. Grids generated applied to both viscous and inviscid aerodynamic problems, and to problems in other areas of fluid dynamics. Written in 100 percent FORTRAN 77.

Sorenson, Reese L.↗
NASA NTRS1994

Research on Orbital Plasma-Electrodynamics (ROPE)

Since the development of probe theory by Langmuir and Blodgett, the problem of current collection by a charged spherically or cylindrically symmetric body has been investigated by a number of authors. This paper overviews the development of a fully three-dimensional particle simulation code which can be used to understand the physics of current collection in three dimensions and can be used to analyze data resulting from the future tethered satellite system (TSS). According to the TSS configurations, two types of particle simulation models were constructed: a simple particle simulation (SIPS) and a super particle simulation (SUPS). The models study the electron transient response and its asymptotic behavior around a three dimensional, highly biased satellite. The potential distribution surrounding the satellite is determined by solving Laplace's equation in the SIPS model and by solving Poisson's equation in the SUPS model. Thus, the potential distribution in space is independent of the density distribution of the particles in the SUPS model but it does depend on the density distribution of the particles in the SUPS model. The evolution of the potential distribution in the SUPS model is described. When the spherical satellite is charged to a highly positive potential and immersed in a plasma with a uniform magnetic field, the formation of an electron torus in the equatorial plane (the plane in perpendicular to the magnetic field) and elongation of the torus along the magnetic field are found in both the SIPS and the SUPS models but the shape of the torus is different. The areas of high potential that exist in the polar regions in the SUPS model exaggerate the elongation of the electron torus along the magnetic field. The current collected by the satellite for different magentic field strengths is investigated in both models. Due to the nonlinear effects present in SUPS, the oscillating phenomenon of the current collection curve during the first 10 plasma periods can be seen (this does not appear in SIPS). From the parametric studies, it appears that the oscillating phenomenon of the current collection curve occurs only when the magnetic field strength is less than 0.2 gauss for the present model.

Wu, S. T.↗
NASA NTRS1975

Advances in Application of Fast Semidirect Computational Methods in Transonic Flow

This paper is intended as a review and summary of the advances made in a recently developed approach for rapid numerical solution of the equations of inviscid transonic aerodynamics. The investigation has been limited to two-dimensional, steady, inviscid flow over airfoils in a subsonic free stream, with emphasis on development of a rapid computational technique, rather than on generality of application. The approach uses finite-difference algorithms called "fast direct elliptic solvers" within an iteration scheme. "Direct" means that the entire computation field is solved at once, rather than in successive traverses over the field as in a point- or line-relaxation method. Such an iterative method is referred to as "semidirect." The iterative convergence can be faster than in other relaxation methods because changes are felt simultaneously at all points in each succeeding iteration. Direct elliptic solvers and semidirect methods have restrictions, but these are gradually being removed. Direct solvers were first developed for solving Poisson's equation on a rectangle without interior boundaries. A method to treat first-order systems, a direct Cauchy-Riemann solver has also been developed. Numerical treatment of part of a system of nonlinear equations by a Poisson solver has been reported. Also Poisson solvers in semidirect methods were used for nonseparable elliptic equations. The semidirect method was extended to the solution of a problem of mixed type, where the improved Murman-Cole transonic small-disturbance difference equations were solved. A slightly supercritical flow over a biconvex airfoil was treated successfully, but the iterations did not converge for more strongly supercritical conditions In another work the addition of terms ot both sides of the difference equations stabilized the iteration for supercritical conditions with large supersonic zones. For this, the Cauchy-Riemann solver was revised to incl,ude the needed terms. Most recently, the evaluation of parameters for rapid convergence and comparisons, with Murman's line-relaxation method was described. The method was extended to full second order accuracy in a fully conservative formulation in another work.

Martin, E. Dale↗
NASA NTRS1979

Analytical study of the time dependent spacecraft-plasma interaction

A study of the time dependent interaction of an initially uniform equilibrium plasma with a plane conducting surface was made in order to achieve a more complete understanding of the dynamics of the charging process and of the approach to the floating potential on the surface. Numerical solutions of the cold ion equations of motion in conjunction with equilibrium electrons and Poisson's equation show the formation of an ion-rich sheath near the surface and the coupling of the non-neutral region to the undisturbed plasma through a quasineutral rarefaction. Analytical treatment of the quasineutral region showed excellent agreement (within 1 percent) with the numerical results.

Cipolla, J. W., Jr.↗
NASA NTRS1977

An efficient numerical technique for calculating thermal spreading resistance

An efficient numerical technique for solving the equations resulting from finite difference analyses of fields governed by Poisson's equation is presented. The method is direct (noniterative)and the computer work required varies with the square of the order of the coefficient matrix. The computational work required varies with the cube of this order for standard inversion techniques, e.g., Gaussian elimination, Jordan, Doolittle, etc.

Gale, E. H., Jr.↗
NASA NTRS1996

Multigrid, Fractional-Step Computation Of Flow

Speed of computer code solving three-dimensional Navier-Stokes equations of flow of incompressible fluid by fractional-step method increased significantly by use of multigrid procedures. In method, equations solved on general nonorthogonal curvilinear coordinate grid, using volume fluxes. At each time step, computations performed in two fractional steps. In first step, equations of conservation of momentum solved by use of gradient of pressure from previous time step via explicit approximate-factorization method, yielding approximate flow field that does not satisfy equation of conservation of mass. In second step, discrete Poisson-like equation with Neumann-type boundary conditions, formed by combining equations of conservation of momentum and mass, solved iteratively.

Kwak, Dochan↗
NASA NTRS1988

A solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A solution method based on a fractional step approach is developed for obtaining time-dependent solutions of the three-dimensional, incompressible Navier-Stokes equations in generalized coordinate systems. The governing equations are discretized conservatively by finite volumes using a staggered mesh system. The primitive variable formulation uses the volume fluxes across the faces of each computational cell as dependent variables. This procedure, combined with accurate and consistent approximations of geometric parameters, is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain favorable convergence properties of the Poisson solver. The discretized equations are second-order-accurate in time and space and no smoothing terms are added. An approximate-factorization scheme is implemented in solving the momentum equations. A novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two and three-dimensional solutions are compared with other numerical and experimental results to validate the present method.

Rosenfeld, Moshe↗
NASA NTRS1970

High frequency Hall current instability

The Hall current electrojet instability was studied to determine if it generates VLF hiss in the auroral zone of magnetic substorms. Using the kinetic equations with particle conserving Krook collision operator for electrons and ions, and Poisson's equation, it is shown that in the low frequency region, the range of frequency of unstable modes increases as the electron density increases, and that the high frequency component must be taken into account in calculations of turbulent saturation of the Hall current instability driven beyond threshold. It is concluded that Hall currents can generate waves in the VLF hiss frequency band which are related to local auroral electrojet activity.

Lee, K.↗
NASA NTRS1995

Two-dimensional mesh embedding for Galerkin B-spline methods

A number of advantages result from using B-splines as basis functions in a Galerkin method for solving partial differential equations. Among them are arbitrary order of accuracy and high resolution similar to that of compact schemes but without the aliasing error. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. Both integer and non-integer refinement ratios are allowed. The report begins by developing an algorithm for choosing basis functions that yield the desired mesh resolution. These functions are suitable products of one-dimensional B-splines. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented. The scheme is conservative and has uniformly high order of accuracy throughout the domain.

Shariff, Karim↗
NASA NTRS1988

Dynamics of localized ion-acoustic waves in a magnetized plasma

The evolution of negative potential pulses in a magnetized plasma is studied. A three-dimensional nonlinear ion-acoustic wave equation, including nonstationary effects of reflected electrons, has been derived from the Poisson-Vlasov equations with uniform magnetic field. The studies show that a negative potential pulse can be enhanced by drifting electrons. The growing pulse develops asymmetrically with an oscillatory precursor and a local potential jump resembling the early phase of weak double layer formation.

Qian, S.↗
DOE OSTI2024

Nonlinear Poisson–Boltzmann solutions for charged parallel plates: When opposite charges repel

I present an exact solution of the Poisson–Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, L c,1 < L c,2 . For separations less than L c,1 , the force between plates is repulsive. It switches to attractive at L c,1 , but with the electric potential having the same sign on both plates. For L > L c,2 , the force remains attractive, and the potential at the plates has the same sign as the charge on each plate. I also describe charge regulation, determined by pK a , and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. Finally, the implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗