Second-order Boundary Condition Research Articles

A tutorial on the parabolic equation (PE) model used for long range sound propagation in the atmosphere

The theory of the parabolic equation (PE) method and the manner in which a finite difference numerical solution scheme can be constructed are presented. The treatment deals with both narrow and wide angle cases. The implementation of a second-order accurate ground boundary condition and an upper boundary condition, designed to minimise reflections into the computational region, are also described.

A numerical study of vector absorbing boundary conditions for the finite-element solution of Maxwell's equations

It is reported that in 3-D vector solutions to Maxwell's equations, boundary conditions of the Bayliss-Turkel kind can be used to absorb outgoing radiation. The boundary conditions were used with curvilinear brick finite elements to analyze spherical test problems for which the exact fields are known. Errors due to incomplete absorption decrease as the outer boundary is moved farther away. The second-order boundary condition is appreciably more accurate than the first-order, at the same cost. >

Productivity enhancement in hydrofractured oil reservoirs

Low permeability formations are often hydrofractured to increase the production rate of oil and gas. This process creates a thin, but highly permeable, fracture which provides an easy path for oil and gas to flow through the reservoir to the borehole. Here we examine the payoff of hydrofracturing by determining the increased production rate of a hydrofractured well. We find explicit formulas for the steady production rate in the three regimes of small, intermediate, and large (dimensionless) fracture conductivities. Previously, only the formula for the large fracture conductivity case was known.We assume that Darcy flow pertains throughout the reservoir. Then, the steady fluid flow through the reservoir is governed by Laplace's equation with a second-order boundary condition along the fracture. We first analyze this boundary value problem for the case of small fracture conductivities. An explicit formula for the production rate is obtained for this case, essentially by combining singular perturbation methods with spectral methods in a function space which places the second-order boundary condition on the same footing as Laplace's equation. Next, we re-cast Laplace's equation as a variational principle which has the second-order boundary condition as its natural boundary condition. This allows us to use simple trial functions to derive accurate estimates of the production rate in the intermediate conductivity case. Then, an asymptotic analysis is used to find the production rate for the large fracture conductivity case. Finally, the asymptotic and variationally-derived production rate formulasare compared to exact values of the production rate, which have been obtained numerically.It may be feasible to create more than a single fracture about a borehole. So we also develop similar asymptotic and variational formulas for the production rate of a well with multiple fractures.

Asymptotic boundary conditions for finite element analysis of three-dimensional transmission line discontinuities

The finite element method is used to analyze open three-dimensional transmission line structures in the quasi-TEM regime. Starting from the general solution of the Laplace equation in spherical coordinates, a set of asymptotic boundary conditions are derived for three-dimensional quasi-static problems for a spherical outer boundary. The second-order boundary condition is generalized to a box-shaped outer boundary and implemented in the finite element method to solve the potential problem of a rectangular microstrip patch. Numerical results show that the asymptotic boundary conditions yield more accurate results than those obtainable with a perfectly conducting shield placed at the same location.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The well-posedness of an initial-boundary-value problem for the wave equation with a second-order boundary condition

The well-posedness of an initial-boundary-value problem for the wave equation with a second-order boundary condition

Second-Order Accurate Boundary Conditions for Compressible Flows

The accuracy and the consistency of the methods commonly used for the numerical application of the boundary conditions on the surface of solid bodies immerged in an inviscid compressible flow is examined. A new method is proposed for the application of second-order accurate boundary conditions in the case of isentropic irrotational compressible flows. The method is applied to a circular cylinder in uniform subsonic and low transonic flows. The conservation law equations are solved using the second-order accurate difference scheme of MacCormack. The results indicate a definite improvement over lower-accuracy methods. The method can be easily generalized to apply on three-dimensional solid surfaces.

Numerical computation of the hypersonic leading edge problem using the Burnett equations

The hypersonic rarefied flow near the sharp leading edge of a flat plate is computed for the first time using a finite-difference solution of the complete, unsteady Burnett equations. The computation is advanced in time from the initial conditions until the steady-state solution is reached. The computational region extends from the leading edge to the strong-interaction regime. Schamberg’s second-order wall slip and temperature jump boundary conditions are employed at the wall. The numerical results are compared with experimental data, a Monte Carlo simulation, and a finite-difference solution of the complete Navier–Stokes equations. Based on these comparisons, it is evident that the Burnett equations in conjunction with Schamberg’s second-order boundary conditions give a much less accurate description of the rarefied flow field near the leading edge than do the Navier–Stokes equations.

An approximate second-order wing theory.

T note describes the development of a general but approximate second-order wing theory. In the absence of a numerically tractable, mathematically precise determination of supersonic flow over three-dimensional wings, several methods for improving on linear perturbation theory have been evolved. One method combines the first-order differential equation with an exact boundary condition; comparisons with experiment and theoretical results, where available, show that the method is of value. The present method takes this approach a stage further in that some account is taken of the second-order differential equation via an approximate particular integral, but only the second-order boundary condition is satisfied. The approximate particular integral may be interpreted as an approximation to the source distribution associated with the second-order effect. Comparison with an exact theory and a second-order theory available for the special case of a supersonic edge delta wing shows that, in these circumstances, the error arising from the approximate particular integral is not significant. However, it is possible that erroneous results could arise in other cases; the range of applicability has yet to be determined. In 1950, Sugo proposed a second-order theory for supersonic flow over three-dimensional wings of arbitrary planform. Lee has examined this paper and shown that Sugo's theory incorporated some errors and assumptions. Nevertheless, Sugo's method of developing an approximate particular integral for the nonhomogeneous second-order equation is of interest. Lee has used a similar approximate particular integral, and the results obtained so far from the modified theory are encouraging; although still semiempirical, they are certainly superior to those of the original work. Lee's theory, like Sugo's theory, probably approaches second-order accuracy only near the wing surface. This restriction, which was not altogether clear in Sugo's original work, seems to have been disregarded by Wallace and Clarke (Ref. 3, p. 184) in an application of Sugo's theory to cruciform wings of arbitrary profile; such an application requires a particular integral valid throughout the extended region of flow between the planar wing surfaces.

Laminar slip flow of an electrically conducting incompressible rarefied gas in a channel with a transverse magnetic field

The effects of a constant external magnetic field on the laminar, fully developed flow of an electrically conducting incompressible rarefied gas in a nonconducting parallel-plate channel are studied. Consideration is given to the slip-flow regime, wherein a gas velocity discontinuity occurs at the channel walls. It is found that the magnitude of the slip velocity is unaffected by the magnetic-field strength for a given pressure drop, but that the mean gas velocity and wall friction coefficient are functions of both the velocity slip coefficient and the magnetic-field strength. The effect of a second-order slip-flow boundary condition is briefly discussed.

Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations

Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations

Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations

Received June 21, 1962. This research was supported by the Air Force Office of Scientific Research. In May, 1961 the author submitted a thesis containing the principle part of this paper to the Rice University in partial fulfillment of the requirements for a degree of Master of Arts. * ,p(x, t) E Ca'd(R) if and only if sp is continuously differentiable a times with respect to x and a times with respect to t in the region R. 405