Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

Numerical Solution for Polymer Transient Flows in a Circle Bounded Composite Formation

A new numerical model for transient flows of polymer solution in a circular bounded composite formation is presented in this paper. Typical curves of the wellbore transient pressure are yielded by FEM. The effects of non-Newtonian power-law index, mobility and boundary distance have been considered. It is found that for the mobility ratio larger than 1, which is favorable for the polymer flooding, the pressure derivative curve in log-log form rises up without any hollow. On the other hand, if the pressure derivative curve has a hollow and then is raised up, we say that the polymer flooding fails. Finally, the new model has been extended to more complicated boundary case.

A HYBRID METHOD OF FRACTIONAL STEPS WITH PREDICTOR-CORRECTOR DIFFERENCE-PSEUDOSPECTRUM FOR NUMERICAL-SOLUTION OF THE CONVECTION-DOMINATED FLOW PROBLEMS

A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L(2)-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathematically for a nonlinear convection-dominated flow. The error estimation is given and the superiority of this method is verified by numerical test.

A Numerical Method for Calculating Transmission Coefficients Across Arbitrary Potential Barriers with High Accuracy

We report a new method for calculating transmission coefficients across arbitrary potential barriers based on the Runge-Kutta method. A numerical solution of the Schrodinger equation is calculated using the Runge-Kutta method,and a new model is established to analyze the numerical results to find the transmission coefficient. This technique is applied to various cases, such as parabolic potential barrier and double-barrier structures. Transmission probability with high precision is obtained and discussed. The tunnelling current density through a MOS structure is also explored and the result coincides with the Fowler-Nordheim model,which indicates the applicability of our method.

A Study of Composite Beam with Shape Memory Alloy Arbitrarily Embedded under Thermal and Mechanical Loadings

The constitutive relations and kinematic assumptions on the composite beam with shape memory alloy (SMA) arbitrarily embedded are discussed and the results related to the different kinematic assumptions are compared. As the approach of mechanics of materials is to study the composite beam with the SMA layer embedded, the kinematic assumption is vital. In this paper, we systematically study the kinematic assumptions influence on the composite beam deflection and vibration characteristics. Based on the different kinematic assumptions, the equations of equilibrium/motion are different. Here three widely used kinematic assumptions are presented and the equations of equilibrium/motion are derived accordingly. As the three kinematic assumptions change from the simple to the complex one, the governing equations evolve from the linear to the nonlinear ones. For the nonlinear equations of equilibrium, the numerical solution is obtained by using Galerkin discretization method and Newton-Rhapson iteration method. The analysis on the numerical difficulty of using Galerkin method on the post-buckling analysis is presented. For the post-buckling analysis, finite element method is applied to avoid the difficulty due to the singularity occurred in Galerkin method. The natural frequencies of the composite beam with the nonlinear governing equation, which are obtained by directly linearizing the equations and locally linearizing the equations around each equilibrium, are compared. The influences of the SMA layer thickness and the shift from neutral axis on the deflection, buckling and post-buckling are also investigated. This paper presents a very general way to treat thermo-mechanical properties of the composite beam with SMA arbitrarily embedded. The governing equations for each kinematic assumption consist of a third order and a fourth order differential equation with a total of seven boundary conditions. Some previous studies on the SMA layer either ignore the thermal constraint effect or implicitly assume that the SMA is symmetrically embedded. The composite beam with the SMA layer asymmetrically embedded is studied here, in which symmetric embedding is a special case. Based on the different kinematic assumptions, the results are different depending on the deflection magnitude because of the nonlinear hardening effect due to the (large) deflection. And this difference is systematically compared for both the deflection and the natural frequencies. For simple kinematic assumption, the governing equations are linear and analytical solution is available. But as the deflection increases to the large magnitude, the simple kinematic assumption does not really reflect the structural deflection and the complex one must be used. During the systematic comparison of computational results due to the different kinematic assumptions, the application range of the simple kinematic assumption is also evaluated. Besides the equilibrium study of the composite laminate with SMA embedded, the buckling, post-buckling, free and forced vibrations of the composite beam with the different configurations are also studied and compared.

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.

Influence of Surface Stress on Frequency of Microcantilever-Based Biosensors (vol 10, pg 307, 2004)

Microcantilever-based biosensors have been found increasing applications in physical, chemical, and biological fields in recent years. When biosensors are used in those fields, surface stress and mass variations due to bio-molecular binding can cause the microcantilever deform or the shift of frequency. These simple biosensors allow biologists to study surface biochemistry on a micro or nano scale and offer new opportunities in developing microscopic biomedical analysis with unique characteristics. To compare and illustrate the influence of the surface stress on the frequency and avoid unnecessary and complicated numerical solution of the resonance frequency, some dimensionless numbers are derived in this paper by making governing equations dimensionless. Meanwhile, in order to analyze the influence of the general surface stress on the frequency, a new model is put forward, and the frequency of the microcantilever is calculated by using the subspace iteration method and the Rayleigh method. The sensitivity of microcantilever is also discussed. (19 refs.)

Shock-induced two-phase flows in an aligned baffle system filled with suspended particles

This paper describes the shock propagation through a dilute gas-particle suspension in an aligned baffle system. Numerical solution to two-phase flows induced by a planar shock wave is given based on the two-continuum model with interphase coupling. The governing equations are numerically solved by using high-resolution schemes. The computational results show the shock reflection and diffraction patterns, and the shock-induced flow fields in the 4-baffle system filled with the dusty gas.

The mechanism of energy storage by shearing the solar magnetic field

In this paper, a complete set of MHD equations have been solved by numerical calculations in an attempt to study the dynamical evolutionary processes of the initial equilibrium configuration and to discuss the energy storage mechanism of the solar atmosphere by shearing the magnetic field. The initial equilibrium configuration with an arch bipolar potential field obtained from the numerical solution is similar to the configuration in the vicinity of typical solar flare before its eruption. From the magnetic induction equation in the set of MHD equations and dealing with the non-linear coupling effects between the flow field and magnetic field, the quantitative relationship has been derived for their dynamical evolution. Results show that plasma shear motion at the bottom of the solar atmosphere causes the magnetic field to shear; meanwhile the magnetic field energy is stored in local regions. With the increase of time the local magnetic energy increases and it may reach an order of 4×10^25 J during a day. Thus the local storage of magnetic energy is large enough to trigger a big solar flare and can be considered as the energy source of solar flares. The energy storage mechanism by shearing the magnetic field can well explain the slow changes in solar active regions.

The wave model of mesothermal plasma near wakes and Korteweg-de Vries equation

The stationary two-dimensional (x, z) near wakes behind a flat-based projectile which moves at a constant mesothermal speed (V∞) along a z-axis in a rarefied, fully ionized, plasma is studied using the wave model previously proposed by one of the authors (VCL). One-fluid theory is used to depict the free expansion of ambient plasma into the vacuum produced behind a fast-moving projectile. This nonstationary, one-dimensional (x, t) flow which is approximated by the K-dV equation can be transformed, through substitution, t=z/V∞, into a stationary two-dimensional (x, z) near wake flow seen by an observer moving with the body velocity (V∞). The initial value problem of the K-dV equation in (x, t) variables is solved by a specially devised numerical method. Comparisons of the present numerical solution for the asymptotically small and large times with available analytical solutions are made and found in satisfactory agreements.

粘性可压混合层时间稳定性对称紧致差分求解

基于可压扰动方程组的一阶改型，将高精度对称紧致格式引入边值法数值线性稳定分析。对所获非线性离散特征值问题给出了一个通用形式二阶迭代局部算法，实现了时间模式和空间模式的统一求解，并将扰动特征及其特征函数同时得到。据此分析了可压平面自由混合层时间稳定性，涉及二维/三维扰动波、粘性/无粘扰动波、第一/第二模态、特征函数、伪特征值谱等。研究表明，压缩性效应和粘性效应对最不稳定扰动波数和增长率呈相似的减抑作用；在Mc = 1附近，从高波数段开始，粘性效应可强化二维不稳定扰动波由第一模态向第二模态的过渡。

Statistical Simulation of Thin-Film Bearings

The information preservation (IP) method and the direct simulation Monte Carlo (DSMC) method are used to simulate the gas flows between the write/read head and the platter of the disk drive (the slider bearing problem). The results of both methods are in good agreement with numerical solution of the Reynolds equation in the cases studied. However, the DSMC method owing to the problem of large sample size demand and the difficulty in regulating boundary conditions at the inlet and outlet was able to simulate only short bearings, while IP simulates the bearing of authentic length ~1000 m ? and can provide more detailed flow information.

3D Finite Volume Scheme for Czochralski Single Crystal Growth

Czochralski (Cz) technique, which is used for growing single crystals, has dominated the production of single crystals for electronic applications. The Cz growth process involves multiple phases, moving interface and three-dimensional behavior. Much has been done to study these phenomena by means of numerical methods as well as experimental observations. A three-dimensional curvilinear finite volume based algorithm has been developed to model the Cz process. A body-fitted transformation based approach is adopted in conjunction with a multizone adaptive grid generation (MAGG) technique to accurately handle the three-dimensional problems of phase-change in irregular geometries with free and moving surfaces. The multizone adaptive model is used to perform a three-dimensional simulation of the Cz growth of silicon single crystals.Since the phase change interface are irregular in shape and they move in response to the solution, accurate treatment of these interfaces is important from numerical accuracy point of view. The multizone adaptive grid generation (MAGG) is the appropriate scheme for this purpose. Another challenge encountered is the moving and periodic boundary conditions, which is essential to the numerical solution of the governing equations. Special treatments are implemented to impose the periodic boundary condition in a particular direction and to determine the internal boundary position and shape varying with the combination of ambient physicochemical transport process and interfacial dynamics. As indicated above that the applications and processes characterized by multi-phase, moving interfaces and irregular shape render the associated physical phenomena three-dimensional and unsteady. Therefore a generalized 3D model rather than a 2D simulation, in which the governing equations are solved in a general non-orthogonal coordinate system, is constructed to describe and capture the features of the growth process. All this has been implemented and validated by using it to model the low pressure Cz growth of silicon. Accuracy of this scheme is demonstrated by agreement of simulation data with available experimental data. Using the quasi-steady state approximation, it is shown that the flow and temperature fields in the melt under certain operating conditions become asymmetric and unsteady even in the absence of extrinsic sources of asymmetry. Asymmetry in the flow and temperature fields, caused by high shear initiated phenomena, affects the interface shape in the azimuthal direction thus results in the thermal stress distribution in the vicinity, which has serious implications from crystal quality point of view.

Verification of a scaling law in few-cycle laser pulses

The scaling law of photoionization in few-cycle laser pulses is verified in this paper. By means of numerical solution of time-dependent Schrodinger equation, the photoionization and the asymmetry degree of photoionization of atoms with different binding potential irradiated by various laser pulses are studied. We find that the effect of increasing pulse intensity is compensated by deepening the atomic binding potential. In order to keep the asymmetric photoionization unchanged, if the central frequency of the pulse is enlarged by k times, the atomic binding potential should also be enlarged by k times, and the laser intensity should be enlarged by k(3) times. (c) 2005 Optical Society of America.

A scaling law of photoionization in ultrashort pulses

By means of the numerical solution of time-dependant Schrodinger equation, we verify a scaling law of photoionization in ultrashort pulses. We find that for a given carrier-envelope phase and duration of the pulse, identical photoionizations are obtained provided that when the central frequency of the pulse is enlarged by k times, the atomic binding potential is enlarged by k times, and the laser intensity is enlarged by k(3) times. The scaling law allows us to reach a significant control over direction of photoemission and offers exciting prospects of reaching similar physical processes in different interacting systems which constitutes a novel kind of coherent control.