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All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

Influence of boundary approximations and conditions on finite difference solutions

Numerical representations of boundary approximations and conditions for three problems are investigated to determine the resulting global accuracy of the steady state solution. Numerical accuracy with various boundary approximations is determined for quasi-one-dimensional inviscid flow in a duct with the interior grid points evaluated using the MacCormack scheme. When an extrapolation approximation with first order local truncation error is used, the global second order accuracy of the difference scheme can be destroyed. For one dimensional flow in a porous medium, an implicit midpoint difference scheme which is consistent with the boundary conditions is developed without the need of boundary approximations. A dissipative model problem is solved with the boundary conditions discretized with first and second order accuracy. The overall second order accuracy of the difference scheme is destroyed if first order numerical representation of one of the boundary conditions is used. With a boundary approximation, the second order global accuracy of the model problem is retained if either second order extrapolation or first order representation of the governing equation is used.

Blottner, F. G.↗
NASA NTRS1989

New approximate boundary conditions for large eddy simulations of wall-bounded flows

Two new approximate boundary conditions have been applied to the large eddy simulation of channel flow with and without transpiration. These new boundary conditions give more accurate results than those previously in use, and allow significant reduction of the required CPU time over simulations in which no-slip conditions are applied. Mean velocity profiles and turbulence intensities compare well both with experimental data and with the results of resolved simulations. The influence of the approximate boundary conditions remains confined near the point of application and does not affect the turbulence statistics in the core of the flow.

Piomelli, Ugo↗
NASA NTRS1992

Electromagnetic scattering by coated convex surfaces and wedges simulated by approximate boundary conditions

Asymptotic/high-frequency solutions are developed for analyzing the non-specular scattering mechanisms associated with coated convex surfaces and edges simulated by approximate boundary conditions. In particular, the standard impedance boundary conditions (SIBC's) and the second order generalized impedance boundary conditions (GIBC's) are employed for a characterization of the edge diffraction, creeping wave, and surface diffracted wave contributions. To study the creeping wave and surface diffracted wave mechanisms, rigorous UTD (uniform geometrical theory of diffraction) diffraction coefficients are developed for a convex coated cylinder simulated with SIBC's and GIBC's. The ray solutions obtained remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. The diffraction coefficient for a convex cylinder are obtained via a generalization of the corresponding ones of the circular cylinder. To validate the asymptotic/high-frequency solution, integral equations are derived for both E and H-polarization and solved numerically using the method of moments. Results are presented for a single and three layered coated convex cylinder. Some insights are also provided on the accuracy of the employed GIBC's versus SIBC's for application to curved surfaces. To characterize the scattering by impedance wedges illuminated at skew incidence, diffraction coefficients are derived from an approximate solution of the governing functional difference equations. This solution exactly recovers the known ones for an impedance half plane or an arbitrary wedge at normal incidence, and to validate it for other wedge angles, a moment method code was used. Finally, to test the usefulness of the approximate skew incidence impedance wedge diffraction coefficient for three dimensional structures, equivalent currents are derived in the context of the physical theory of diffraction (PTD) for a finite length impedance wedge of arbitrary internal angles. These are incorporated in a standard general purpose PTD code and results are presented for a number of different impedance structures.

Syed, H. H.↗
NASA NTRS1981

Stable boundary approximations for a class of implicit schemes for the one-dimensional inviscid equations of gas dynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations for initial-boundary-value problems of the hyperbolic type. For the numerical experiments, select the one-dimensional inviscid gas-dynamic equations in conservation-law form is selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit versus implicit boundary schemes, unconditionally stable schemes, and underspecification or overspecification of boundary conditions.

Yee, H. C.↗
NASA NTRS1979

Drift boundary approximations in simple magnetospheric convection models

The paper analyzes features of particle behavior in magnetospheric convection using simple analytic forms for the electrostatic potential. It is shown that analytic approximations for the shape and position of surfaces delimiting closed and open particle orbits can be derived. These approximations are good for any pitch angle and at most energies. They break down only for the medium-energy protons that are predicted by such models to penetrate far inside the plasmapause. The derived expressions allow rapid tests of a variety of electric field models when analyzing particle data, and this is illustrated by intercomparing field models with several observations.

Southwood, D. J.↗
NASA NTRS1982

Approximate boundary condition procedure for the two-dimensional numerical solution of vortex wakes

Research on efficient computational methods for general vorticity fields has been conducted in connection with a need for basic research on vortex-dominated flows. The present investigation is concerned with the evolution of vortex wakes behind aircraft wings. An efficient procedure is presented for the calculation of the boundary values used in the numerical solution of the unsteady, incompressible, two-dimensional Navier-Stokes equations for an unbounded flow field. The extent of the computational grid can be reduced compared to methods utilizing standard boundary conditions, without loss of accuracy. The efficiencies realized make it feasible to calculate the vortex wake development for realistic wing configurations, including the merging of multiple vortices, for Reynolds numbers of about 10,000 based on wing chord.

Weston, R. P.↗
NASA NTRS1987

Acoustic gravity waves: A computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.

Hariharan, S. I.↗
NASA NTRS1988

Acoustic gravity waves - A computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.

Hariharan, S. I.↗
NASA NTRS1980

Analytic and approximate boundary-fitted coordinate systems for fluid flow simulation

A method for constructing a direct functional relation between the computational and physical domains such that boundaries in the computational domain map into boundaries in the physical domain is presented. This method which is called the 'two-boundary technique' is based on generating functions that represent two disconnected boundaries in the physical domain and an independent function which connects one boundary with another to generate grids. The suggested connecting functions are simple linear and cubic parametric polynomials with parameters based on position and derivatives at the disconnected boundaries. The boundaries can be defined by analytical, approximate, and tension spline functions; the grid density in specified regions is controlled by embedding contraction functions in either the boundary representation or the connecting function.

Smith, R. E.↗
NASA NTRS1981

Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

Yee, H. C.↗
NASA NTRS1988

Interaction and merging of vortex filaments

The asymptotic solutions of Navier-Stokes equations for vortex filaments of finite strength with small effective vortical cores are summarized with special emphasis placed on the physical meaning and the practical limit to the applicability of the asymptotic solution. Finite-difference solutions of Navier-Stokes equations for the marging of the filament(s) are described with a focus on the development of the approximate boundary conditions for the computational domain. An efficiency study employing a model problem is used to assess the advantages of the present approximate boundary condition method over previously used techniques. Applications of the present method are presented for the motion and decay of a 3:1 elliptic vortex ring, and for the merging process of a pair of coaxial vortex rings. A numerical procedure for the problem of local merging of vortex filaments, which requires the asymptotic analysis as well as the numerical Navier-Stokes solver, is also presented.

Liu, C. H.↗
NASA NTRS1978

Irradiation stratigraphy in the Apollo 16 deep drill section 60002

Particle track density frequency distributions, abundance of track rich grains and minimum track densities are reported for the upper 20 cm of the 60002 section of the Apollo 16 deep drill core. The principal stratigraphic feature is a boundary approximately 7 cm from the top of the section. Experimental evidence does not conclusively determine whether this contact is an ancient regolith surface or is simply a depositional boundary. If it is an ancient surface, it has a model exposure age of 3 to 7 million years and a reworking depth of about 0.5 cm. However, because track density frequency distributions indicate the mixing of soils of different maturities, we favor interpreting this contact as a depositional boundary. There may be a second depositional boundary approximately 19 cm below the top of 60002.

Blanford, G. E.↗
NASA NTRS1979

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.↗
NASA NTRS1993

On the accurate long-time solution of the wave equation in exterior domains: Asymptotic expansions and corrected boundary conditions

We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.

Hagstrom, Thomas↗
NASA NTRS1979

Computing region moments from boundary representations

The class of all possible formulas for computing arbitrary moments of a region from the region's boundary is derived. The selection of a particular formula depends on the choice of an independent parameter. Several choices of this parameter are explored for region boundaries approximated by polygons. The parameter choice that minimizes computation time for boundaries represented by chain code is derived. Algorithms are presented for computing arbitrary moments for a region from a polygonal approximation of its boundary and for computing low order moments from chain encoded boundaries.

Wilf, J. M.↗
NASA NTRS1950

Approximate Turbulent Boundary-Layer Development in Plane Compressible Flow Along Thermally Insulated Surfaces with Application to Supersonic-Tunnel Contour Correction

Numerical solutions of the differential equation obtained from the momentum theorem for the development of a turbulent boundary layer along a thermally insulated surface in two-dimensional and in radial shock-free flow are presented in tabular form for a range of Mach numbers from 0.100 to 10. The solution can be used in a step-wise procedure with any given distribution of favorable pressure gradients and for zero pressure gradients. Solutions are also given for use with moderate adverse pressure gradients. The mean velocity in the boundary layer is approximated by a power-law profile. In view of the stepwise integration methods to be used, the exponent designated the profile shape can be varied along the surface between the integral fraction limits 1/5 and 1/11 through interpolation. Agreement obtained between theoretical and experimental boundary-layer development in a supersonic nozzle at a nominal Mach number of 2 indicates the general validity of the approximations used in the analysis - in particular, the method of extrapolating low-speed skin-friction relations to high Mach number flows. The extrapolation method used assumes that the skin-friction coefficient depend primarily on Reynolds number, provided that the density and the kinematic viscosity are evaluated at surface conditions.

BOUNDARY LAYER - WIND TUNNEL WALLS↗