Improved Boundary Conditions Research Articles

High-order splitting methods for the incompressible Navier-Stokes equations

A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular, improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods. A new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules. These schemes exhibit much broader stability regions as compared to Adams-family schemes, typically used in splitting methods. Their stability properties remain almost constant as the accuracy of the integration increases, so that robust third- or higher-order time-accurate schemes can readily be constructed that remain stable at relatively large CFL number. The new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation. A model Stokes problem is studied in detail, and several examples of Navier-Stokes solutions of flows in complex geometries are reported. Comparison is made with the previously used first-order in time spectral element splitting and non-splitting (e.g., Uzawa) schemes. High-order splitting/spectral element methods combine accuracy in space and time, and flexibility in geometry, and thus can be very efficient in direct simulations of turbulent flows in complex geometries.

Nonlinear Orographically Forced Planetary Waves

Abstract Traditionally, stationary wave models have been linearized about a zonal-mean flow and the response calculated to various fixed orographic and thermal forcings. In this paper it is shown that the inclusion of nonlinear interactions can significantly modify the solution for orographic forcing. This arises primarily from the improved boundary condition that allows the flow to be deflected around the mountain as well as over it. In this context a useful conceptual model is obtained by linearization based on the smallness of the latitudinal extent of a mountain. The nonlinear model is considerably less sensitive to the zonal-mean surface flow and, in some instances, the perturbation amplitude decreases with increasing surface flow. The nonlinear, hemispheric solutions for full orography and wintertime basic state are shown for both the Northern and Southern hemispheres. They suggest that the direct effect of orographic forcing alone accounts for less than one-half of the observed time mean asymmetries.

Generalized impedance boundary conditions in scattering

The authors present generalized surface impedance conditions that improve medium characterization. One of the most common conditions that is often used to simulate a dielectric coating is the standard impedance boundary condition, and the improved boundary condition discussed can be viewed as a generalization of this, distinguished by the presence of higher-order derivatives of the field. By virtue of the derivatives, the conditions are less local in character, and the additional degrees of freedom can be used to better simulate the material properties of a surface. Following a review of the standard impedance condition, the new conditions are introduced. Several forms are presented, each with coefficients which are functions of the geometry and material properties of the surface. The applications of these to a variety of scattering problems are then discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved prospects for estimating insolation for calculating regional evapotranspiration from remotely sensed data

Insolation is a key parameter in most formulations for estimating regional evapotranspiration and/or primary productivity. Many attempts have been made to replace several parameters in these formulations with remotely sensed data. Insolation, however, has been obtained from conventional observations. The objective of this paper is to demonstrate that satellite methods for deriving insolation can be improved on the regional scale beyond what is achievable on large scales. We have developed a realistic surface albedo model, which was used as an improved boundary condition in an insolation model, and also, to specify clear/cloudy thresholds. It was demonstrated that as a result, the mean bias error (MBE) was reduced by as much as 50%, to about 6 W m −2 for a monthly mean. Such improvements and accuracies have implications for local evapotranspiration estimation methods.

Improved boundary conditions for the time-dependent Schrödinger equation

An improved boundary condition treatment for the time-dependent Schrödinger equation applied to resonant-tunneling diode simulation has been developed. This treatment is half-implicit, or centered in time, and allows larger time steps than the previous explicit treatment. The method does not complicate the time-dependent calculation since the resulting matrix is still tridiagonal.

Consistent Boundary Conditions for Reduced Navier-Stokes (RNS) Scheme Applied to Three-Dimensional Viscous Flows

A consistent and efficient set of boundary conditions are developed for the multi-sweep space marching pressure-elliptic Reduced Navier-Stokes (RNS) scheme as applied for three-dimensional internal viscous flow problems. No-slip boundary conditions are directly imposed on the solid walls. There is no iteration procedure required in the cross plane to ensure mass conservation across each marching plane. The finite difference equations forming the coefficient matrix are ordered such that the surface normal velocity is specified on all the solid walls; unlike external flows, a pressure boundary condition in the cross plane is not required. Since continuity is directly satisfied at all points in the flow domain, the first order momentum equations can be solved directly for the pressure without the need for a Poisson pressure correction equation. The procedure developed herein can also be applied with periodic boundary conditions. The analysis is given for general compressible flows. Incompressible flow solutions are obtained, for straight and curved ducts of square cross section, to validate the procedure. The solutions of these test cases are used to demonstrate the applicability of the RNS scheme, with the improved boundary conditions, for internal flows with strong interaction as would be encountered in ducts and turbomachinery geometries.

A simple formula for diffusion calculations involving wall reflection and low density

Diffusion theory is often employed to calculate the effects of wall destruction on the local concentration of an active species immersed in a scattering gas. In many situations the spatial dependence of the concentration is given to a good approximation by the fundamental diffusion mode, and the local loss frequency can be calculated using the container’s fundamental mode diffusion length Λ. The additional assumption that the density of the active species may be taken to be zero at the container boundaries gives a value of Λ=Λ0 which depends only on the container dimensions, but use of Λ0 can be seriously in error if the diffusion mean free path λm is comparable to the dimensions, or if the particle reflection coefficient R becomes of significance. An improved boundary condition may be written simply in terms of the linear extrapolation length λ, whose inverse is the logarithmic gradient of the particle density at the boundary. The equation λ=2(1+R)λm/3(1−R) allows the representation of the full range of possible values of the particle reflection coefficient, 0&amp;lt;R&amp;lt;1, and extends the usefulness of the diffusion approximation to low scatterer densities. In the collisionless limit, the predicted particle loss frequency is identical to that predicted from the average chord length. Using this boundary condition, the dependence of Λ on λ/Λ0 has been computed for a range of simple container shapes, by solving the transcendental equations involved. This has allowed the identification of a dimensionless scaling variable, l0λ/Λ20, where l0 is the ratio of the container volume to its surface area. For all cases considered the simple empirical approximation Λ2=(Λ20+l0λ) is accurate when λ is very large or very small compared to Λ0, and disagrees most with the numerical solutions in the region where λ and Λ0 are comparable, with the worst case error being 11%.

Boundary conditions for incompressible flows

A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.

Simulation of the High-Frequency Response of a Heterogeneous Ground through the Finite Difference Technique

To simulate the propagation of an electromagnetic plane wave in an inhomogeneous ground, the finite difference approach can be used. One of the main problems in using this method is imposing the boundary conditions near the ground surface, especially at high frequency. Indeed, for the E polarization, the upper top of the numerical grid must be sufficiently far away from the air-ground interface in order to neglect the field due to the heterogeneities and the discretization of the atmosphere is necessary. For magneto-telluric modeling, improved boundary conditions have already been proposed. This paper deals with a new condition, valid everywhere in air and which can be applied for E and H polarization. Thus even at high frequency, as for radar applications, only one line is added to the grid discretizing the ground.

Resistance calculation of a cylindrical semiconductor with continuously variable conductivity along its axis covered by circular electrodes

In this paper we present a new method of resistance calculation of a cylindrical semiconductor, few bases of which are covered by circular electrodes of different areas. On the semiconductor base provided with the smaller circular electrode an improved boundary condition is applied. The potential variation in the semiconductor is determined by an exact solution of the continuity equation by matching cylindrical periodic functions in the space domain on the semiconductor—electrode boundary supposing that the conductivity of the electrode is finite. Three types of conductivity profiles are considered. The conductivity of the cylindrical semiconductor along its axis is either constant, or varies linearly or exponentially. The results presented in the paper can be applied in the design, measurement and manufacture of semiconductor components having the shape of a cylinder.

Application of Improved Numerical Techniques to the Tsunami Response of Island Systems

A finite-difference numerical model based on the classical linear long-wave equations has been developed to study the tsunami response of multiple-island systems. The first application of this model was to the Hawaiian Islands. Early results were satisfactory. However, one drawback to the model was the use, at the outer open boundaries of the basin, of a radiation condition which allowed only wave energy approaching the boundary at a normal angle to pass out of the basin without reflection. Recently, an improved boundary condition has been employed which takes into account the radial propagation of scattered waves near the boundary. The earlier model tests were repeated using this new condition. The major features of the results remained relatively unchanged. Certain secondary features of the response. which had made analysis of model results quite complicated, were either eliminated or greatly reduced.

The numerical computation of the preionization phase of a toroidal pinch discharge

Using the two-dimensional MHD computer code ANIMAL. The authors have computed the plasma dynamics of the preionization phase of a toroidal pinch. The physical model includes electron and ion thermal conduction, electron-ion resistivity, electron-neutral resistivity, ionization, dissociation, and line radiation all based on local thermodynamic equilibrium. The code uses ADI finite difference equations with improved boundary conditions which enable the code to follow the oscillatory motion of the plasma. A coupled electrical circuit calculation is performed. The physical model requires the specification of a minimum ionization fraction fmin to set an upper bound on the resistivity. By suitably choosing fmin, they reproduce many of the features of the preionization discharge. Our calculations have suggested a new electrical diagnostic, the measurement of intervals between zero-crossings of current and flux, which is used to help normalize the computations.

Upward and downward continuations in the geopotential determination

The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsCnm andSnm. The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations. In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingCnm andSnm as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations.

Short weak link with distinct chemical potentials at the boundary

The difference between the chemical potentials for pairs (..mu../sub p/) and quasiparticles (..mu..) at the boundaries of a short superconducting weak link, which is usually disregarded, is shown to be important when solving for the behavior of the link using the time-dependent Ginzburg-Landau equation. Selected computational results are presented to illustrate the effects of the improved boundary conditions on the space (time) dependence of ..mu.., ..mu../sub p/, and normal current J/sub n/.

Improved boundary conditions for the numerical solution of E-polarization problems in geomagnetic induction

Summary. New boundary conditions applicable on the sides and top of numerical grids used in the solution of E-polarization problems in electromagnetic induction are proposed. It is demonstrated that these new boundary conditions improve the accuracy of the numerical solution and also that their use allows such problems to be solved on grids of much smaller overall size than were formerly necessary.

Analysis of stellar occultation data: Effects of photon noise and initial conditions

A new inversion technique for obtaining temperature, pressure, and number density profiles of a planetary atmosphere from an occultation light curve is described. This technique employs an improved boundary condition to begin the numerical inversion and permits the computation of errors in the profiles caused by photon noise in the light curve. We present our assumptions about the atmosphere, optics, and noise and develop the equations for temperature, pressure, and number density and their associated errors. By inverting in equal increments of altitude, Δh, rather than in equal increments of time, Δt, the inversion need not be halted at the first negative point on the light curve as required by previous methods. The importance of the boundary condition is stressed, and a new initial condition is given. Numerical results are presented for the special case of inversion of a noisy isothermal light curve. From these results, simple relations are developed which can be used to predict the noise quality of an occultation. It is found that fractional errors in temperature profiles are comparable to those of pressure and number density profiles. An example of the inversion method is shown. Finally, we discuss the validity of our assumptions. In an appendix we demonstrate that minimum fractional errors in scale height determined from the inversion are comparable to those from an isothermal fit to a noisy isothermal light curve.

Goethert's rule with an improved boundary condition

Goethert's rule with an improved boundary condition

Unsteady, One-Dimensional Flow in Resonance Tube : with Wall Friction, Heat Transfer and Interaction on a Contact Surface

The resonance tube is a cylindrical cavity, closed at the downstream end, which is excited to oscillation by a coaxial air-jet. This paper presents a numerical solution of unsteady, one-dimensional, non-isentropic flow in the resonance tubes by the characteristics method, when the wall friction, the heat transfer and the contact surface are taken into account. The calculations on the flow in tubes are performed numerically under improved boundary condition by the normal mesh method with a digital computer. The calculated results are compared with the experimental ones, and they indicate that the role played by wall friction in the thermal effect of resonance tube should not be neglected in comparison with the shock wave propagating in the tube.

Improved Boundary Conditions for the Feautrier Method

Improved Boundary Conditions for the Feautrier Method

Boundary condition model applie to antinucleon-nucleon scattering.-I

An improved boundary condition plus elastic potential has been developed for calculating the elastic, charge exchange and anihilation cross-sections for the antinucleon-nucleon system in the non-relativistic energy region ((50÷300) MeV in the laboratory system). The boundary condition is taken to maximize the absorption crosssection ineach channel of specified isotopic spin, ordinary spin, and total angular momentum. The elastic potential is constructed by modifying the one-pion and two-pion nucleon-nucleon potentials in the usual manner. The results fit the available data reasonably well for both the differential elastic cross-section and the total annihilation cross-section, using a single core, however best fit is obtained sperately when one chooses to fit the annihilation a larger core than the elastic cross-section. Charge-exchange scattering is found to be quite sensitive to the size of the core region.