Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.     

Large energy behavior of the velocity distribution for the hard-sphere gas

The initial-value problem for the Boltzmann-Lorentz equation for hard spheres at zero temperature is shown to be ill defined, the general solution depending on an arbitrary function. The uniqueness of the solution can be obtained by imposing the conservation of the number of particles (Carleman's type of condition does not suffice). The linearized Boltzmann equation for hard spheres is then analyzed, as it occurs in Enskog's method for calculating transport coefficients. It is demonstrated that in the case of viscosity and diffusion it is necessary to add supplementary conditions to obtain the uniqueness of the solution. The nonuniform character of Enskog's expansion and violation of positivity in the large velocity region are exhibited.     

Callen-like method for the classical Heisenberg ferromagnet

A study of the d-dimensional classical Heisenberg ferromagnetic model in the presence of a magnetic field is performed within the two-time Green function’s framework in classical statistical physics. We extend the well known quantum Callen method to derive analytically a new formula for magnetization. Although this formula is valid for any dimensionality, we focus on one- and three- dimensional models and compare the predictions with those arising from a different expression suggested many years ago in the context of the classical spectral density method. Both frameworks give results in good agreement with the exact numerical transfer-matrix data for the one-dimensional case and with the exact high-temperature-series results for the three-dimensional one. In particular, for the ferromagnetic chain, the zero-field susceptibility results are found to be consistent with the exact analytical ones obtained by M.E. Fisher. However, the formula derived in the present paper provides more accurate predictions in a wide range of temperatures of experimental and numerical interest.     

Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.     

广义边界条件和概率流密度的连续性

以四参量族－广义边界条件代替有限深方势阱内粒子波函数的对数的一阶导数连续性条件，表明这种替换比用广义边界条件取代一盒子内自由粒子波函数在两壁处为零的边界条件显行更加合理些。我们发现，如对参量ρ和θ取任意值，则会导致概率流密度连续性被破坏。我们找到了约束ρ和θ取值的条件，并得到一组新的解析解。     

Enhancement of single mode operation in coaxial optical waveguide using DB boundary conditions

In this study, a competent numerical strategy to compute the dispersion of optical waveguides is presented and propagation of electromagnetic waves in a coaxial optical waveguide with DB boundary conditions is instigated. For this intend, cylindrical coordinates are here being used to derive the DB boundary conditions and to obtain field components for the modes. The propagation constant for the waveguide to be studied is determined by solving the Bessel and the modified Bessel functions. The cutoff frequencies for various lower order modes have been calculated and their dispersion characteristics are plotted correspondingly. The behavior of the coaxial optical waveguide under DB boundary conditions is shown to be significantly different from that of coaxial optical waveguide and conventional optical waveguide under traditional or tangential boundary conditions. Finally, the effect of waveguide dimensions on the mode cutoff frequencies and fabrication issues are also addressed.     

Exact boundary conditions for the initial value problem of convex conservation laws

The initial value problem of convex conservation laws, which includes the famous Burgers’ (inviscid) equation, plays an important rule not only in theoretical analysis for conservation laws, but also in numerical computations for various numerical methods. For example, the initial value problem of the Burgers’ equation is one of the most popular benchmarks in testing various numerical methods. But in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support, so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations. In this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries, which contain a finite computational domain, by using the Lax’s exact formulas for the convex conservation laws. The well-posedness of the initial-boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described. Numerical tests with the proposed artificial boundary conditions are carried out by using the Lax–Friedrichs monotone difference schemes.     

Two transparent boundary conditions for the electromagnetic scattering from two-dimensional overfilled cavities

We consider electromagnetic scattering from two-dimensional (2D) overfilled cavities embedded in an infinite ground plane. The unbounded computational domain is truncated to a bounded one by using a transparent boundary condition (TBC) proposed on a semi-ellipse. For overfilled rectangular cavities with homogeneous media, another TBC is introduced on the cavity apertures, which produces a smaller computational domain. The existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established. In the exterior domain, the 2D scattering problem is solved in the elliptic coordinate system using the Mathieu functions. In the interior domain, the problem is solved by a finite element method. Numerical experiments show the efficiency and accuracy of the new boundary conditions.     

A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions

Combining order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and convergence. The global convergence order in maximum norm is O(τ^2−α + h²). The accuracy and efficiency of the scheme are checked by two numerical tests.     

Effect of boundary conditions on the invariant density of noisy maps at fully-developed chaos

The invariant density of one-dimensional maps in the regime of fully-developed chaos with uncorrelated additive noise is considered. Boundary conditions are shown to play a significant role in determining the precise form of the invariant density, via the manner in which they handle the spill-over, caused by the noise, of orbits beyond the interval. The known case of periodic boundary conditions is briefly recapitulated. Analytic solutions for the invariant density that are possible under certain conditions are presented with applications to specific well-known maps. The case of ‘sticky’ boundaries is generalized to ‘re-injection at the nearest boundary’, and the exact functional equations determining the invariant density are derived. Interesting boundary layer effects are shown to occur, that lead to significant modifications of the invariant density corresponding to the unperturbed (noise-free) case, even when the latter is a constant — as illustrated by an application of the formalism to the noisy tent map. All our results are non-perturbative, and hold good for any noise amplitude in the interval.     

POD-based wall boundary conditions for the numerical simulation of turbulent channel flows

Simulation of turbulent wall-bounded flows requires a high spatial resolution in the wall region, which limits the range of Reynolds numbers which can be effectively reached. In previous work, we proposed proper orthogonal decomposition (POD) based wall boundary conditions to bypass the simulation of the inner wall region. Tests were carried out for direct numerical simulation at a low Reynolds number Re_τ = 180. The boundary condition is based on the POD spatial eigenfunctions which are determined a priori in the full channel. It consists of a three-component velocity field on the plane y⁺ = 50 which is reconstructed at each instant from a combination of selected eigenfunctions. The coefficients of the combination are estimated from the simulation in the reduced domain using the threshold-based reconstruction method described in Podvin et al. The study is now extended to large-eddy simulation at higher Reynolds numbers Re_τ = 295 and Re_τ = 590. Two versions of the reconstruction method are considered. In the first version, both the phases and the moduli of the coefficients are allowed to vary. In the second version, only the phases are adjusted. We find that the latter method is associated with improved statistics and is relatively robust with respect to the reconstruction threshold. However, it is sensitive to the details of the numerical simulation, unlike the former method, which is associated with less accurate statistics and is more dependent on the reconstruction threshold.     

Application of the thermodynamics of curved boundary layers to the scaled particle theory of hard-sphere fluids

The thermodynamics of curved boundary layers, with the assumption that the distance between the surface of a fluid cavity and its surface of tension is a quadratic function of the cavity radius, is applied to the exact thermo-dynamic expression forG, the central function of scaled particle theory. The coefficients in the quadratic representation are determined so as to satisfyall five of the known exact conditions onG valid for cavity radii between one-half and one molecular diameter. The results of the calculation are displayed as the hard-sphere equation of state, the boundary tension associated with the surface of tension, and the distance between the cavity surface and the surface of tension. Although the hard-sphere equation of state obtained by this method using all five conditions onG is more accurate than in the case where only two or three conditions are used, the original scaled particle theory, in whichG itself was represented simply by a quadratic function of inverse powers of cavity radius, still yields the more accurate equation of state. Nevertheless, the present approach limits approximations to the distance between the cavity surface and the surface of tension, a small quantity in itself. The path to a still more improved theory remains well defined, contingent upon the discovery of additional exact conditions, and does not depend, as in the original scaled particle theory, on a form forG arrived at in a somewhat ad hoc manner.Research supported under NSF Grant #GP-12408.     

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.     

The Bethe ansatz for the six-vertex model with rotated boundary conditions

A method is suggested for derivation of the Bethe ansatz equations for the six-vertex model on a square lattice rotated at an arbitrary angle with respect to the coordinate axes. The method is based on the random walk representation for configurations of the model. The equations for the ice model on the rotated lattice are derived and some numerical results are obtained.     

Semi-empirical boundary conditions for the linearized acoustic Euler equations using Pseudo-Spectral Time-Domain methods

Pseudo-Spectral Time-Domain algorithms have emerged as new numerical methods for solving Eulerian problems. These methods, in contrast to more common finite-difference, time-domain approaches, provide isotropic dispersion characteristics. However, the technical literature concerning to this topic presents a serious lack of methods for dealing with partially reflecting boundary conditions in order to simulate surfaces of a specified impedance. In the current paper we present a novel semi-empirical formulation for simulating constant impedance boundary conditions within Pseudo-Spectral techniques based on the Fourier transform. Finally, the validations in one and two dimensions by means of different numerical experiments, show the accuracy of the model.     

电磁场切向边值关系的讨论

针对目前电动力学和电磁场理论教材中电磁场切向边值关系的推导方法不一及存在的问题,介绍两种推导电磁场切向边值关系的方法并作了相应的讨论.     

Inflow conditions for spatial direct numerical simulation of turbulent boundary layers

The inflow conditions for spatial direct numerical simulation (SDNS) of turbulent boundary layers should reflect the characteristics of upstream turbulence, which is a puzzle. In this paper a new method is suggested, in which the flow field obtained by using temporal direct numerical simulation (TDNS) for fully developed turbulent flow (only flow field for a single moment is sufficient) can be used as the inflow of SDNS with a proper transformation. The calculation results confirm that this method is feasible and effective. It is also found that, under a proper time-space transformation, all statistics of the fully developed turbulence obtained by both temporal mode and spatial mode DNS are in excellent agreement with each other, not only qualitatively, but also quantitatively. The normal-wise distributions of mean flow profile, turbulent Mach number and the root mean square (RMS) of the fluctuations of various variables, as well as the Reynolds stresses of the fully developed turbulence obtained by using SDNS, bear similarity in nature. Supported by the National Natural Science Foundation of China (Grant No. 90205021), the China Postdoctoral Science Foundation (Grant No. 20060400707), and the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 200328), and partially supported by Liu-Hui Center of Applied Mathematics, Nankai University and Tianjin University     

Characteristic boundary conditions for direct simulations of turbulent counterflow flames

Improved Navier–Stokes characteristic boundary conditions (NSCBC) are formulated for the direct numerical simulations (DNS) of laminar and turbulent counterflow flame configurations with a compressible flow formulation. The new boundary scheme properly accounts for multi-dimensional flow effects and provides nonreflecting inflow and outflow conditions that maintain the mean imposed velocity and pressure, while substantially eliminating spurious acoustic wave reflections. Applications to various counterflow configurations demonstrate that the proposed boundary conditions yield accurate and robust solutions over a wide range of flow and scalar variables, allowing high fidelity in detailed numerical studies of turbulent counterflow flames.     

Remarks on the independence of the free energy from crystalline boundary conditions in the two-dimensional one-component plasma

We study the two-dimensional one-component plasma. We show that given a bound on the one-particle correlation functions in the thermodynamic limit the canonical free energy is independent or free of the Dobrushin-type boundary conditions obtained by putting outside the vessel a regular configuration of fixed charges.Supported by the Deutsche Forschungsgemeinschaft (D.F.G.), Forschungsprojekt Stochastische Felder.     

Solutions of Laplace's equation with simple boundary conditions,and their applications for capacitors with multiple symmetries

We find solutions of Laplace's equation with special boundary conditions, using a general curvilinear system of coordinates. We call this purely geometrical solutions Basic Harmonic Functions (BHF's). From them we obtain more general solutions with arbitrary constant values on the boundaries. Further, the BHF's are used to obtain the capacitance of many electrostatic configurations of conductors. Applications in complex geometries are given. Finally, expressions for electric fields between two conductors and surface charge densities are obtained in terms of generalized curvilinear coordinates. The present method can be extrapolated to other linear homogeneous differential equations.