亚、跨、超音速及不可压流动的数值分析方法的研究

为了对亚、跨、超音速及不可压无粘流动进行数值模拟,将LU－SGS方法与预处理方法结合,给出了PLU－SGS方法。方程离散基于有限体积法,采用高阶精度AUSMPW格式。方程求解采用了特征边界条件。通过典型算例的数值试验对比分析,表明PLU－SGS方法可以有效地对亚、跨、超音速及不可压流动进行数值模拟,并具有较高的计算精度和收敛速度。     

高黏性流动有限元模拟的迭代稳定分步算法

分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中，已有的显式和半隐式分步算法由于 其显式本质，必须采用小时间步长计算，不但降低了计算效率，同时也常与为使分步算法稳 分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中，已有的显式和半隐式分步算法由于 其显式本质，必须采用小时间步长计算，不但降低了计算效率，同时也常与为使分步算法稳 定必须满足的最小时间步长要求冲突. 本文目的是构造一种含迭代格式的分步算法，它能在 保证精度的前提下大幅度地增大时间步长. 方腔流和平面Poisseuille流数值计算结果证实 了此特点，该方法被有效应用于充填流动过程的数值模拟.     

MHD控制超声速边界层的理论研究和数值分析

对MHD(mechanisms of magnetohy drodynamics)控制超声速平板湍流边界层的机理进行了理论研究和数值模拟. 理论上,采用等离子体低频近似碰撞频率模型,建立等离子体中电子和离子的力平衡方程,得到等离子体速度、极化电场以及边界层速度. 数值上,通过空间HLLE格式、LU--SGS时间推进求解时均磁流体动力学湍流方程,其中湍流模型采用sst--k\omega双方程模型. 研究结果表明:(1)边界层速度的理论结果和数值结果误差在7%范围内;(2)只有磁场而电场为零时,洛仑兹力起到减小摩阻的作用. 施加电场后,洛仑兹力能够加速边界层低速区流体;(3) 在边界层外层,越靠近壁面,作用参数越小;而在边界层近壁区黏性底层,虽然惯性力减小, 但黏性力却迅速增加,因此越靠近壁面,作用参数反而越大,加速低速流的代价增加.      

含高阶余项的非线性动力系统数值计算法

建立了一种求解非线性动力系统高精度数值计算的新方法,重构了等价的非线性动力系统方程,该方程考虑了非线性函数的任意高阶项,并给出了该方程的Duhamel积分表达式,在时间步长内用Newton-Raphson法进行数值迭代求解,该方法能连续满足微分方程而不只是在离散的步长端点满足方程,从而打破了传统的Euler型有限差分法。计算实例表明,该方法计算精度高于传统的Runge-Kutta,Newmark-β和Wilson-θ等方法。     

配点法解非线性结构的增量动力平衡方程

本文提出了一个用配点法求解增量动力平衡方程的方法,在本法中取三次 B 样条函数作试函数,计算公式简单明了,计算工作量较它法为少.在分析非线性振动问题中,时间步长的改变会引起计算工作的大量增加,本文引入一个“动力平衡迭代法”,采用此法时,时间步长可不再改变,计算工作量可进一步大量减少。     

高雷诺数湍流风场大涡模拟的并行直接求解方法

在环境流体力学中,风场是风沙流、风雪流等自然环境特性问题研究的动力源和基础.通常采用壁湍流模型进行风场大涡模拟(large eddy simulation, LES)计算,但受到计算规模的限制使得高雷诺数风场的模拟计算难以实现.并行计算技术是解决大规模高雷诺数风场大涡模拟的关键技术之一.在不可压湍流风场的LES模拟中,压力泊松方程的并行计算技术是进行规模并行计算的困难点.根据风场流动模拟计算的特点,采用水平网格等距而垂直于地面网格非等距,在解决规模并行计算中求解压力泊松方程的难点问题时,利用FFT解耦三维泊松方程使其变为垂向的一维三对角方程,并利用可并行的三对角方程PDD求解技术,可建立三维泊松方程的直接并行求解技术.结合其它容易并行的动量方程计算,本文建立风场LES模拟的并行直接求解方法 (parallel direct method-LES, PDM-LES).在超级计算机上对新方法进行并行效率测试,并行计算效率达到90%.新的方法可用于进行湍流风场大涡模拟的大规模并行计算.计算结果表明,湍流风场瞬时速度分布近壁面存在条带状的拟序结构,平均场的速度分布符合速度对数律特性,风场湍流特性基本合理.     

喷水推进器推力的CFD计算方法研究

简要介绍获取喷水推进器推力的理论、试验及数值计算(CFD)方法,重点研究采用动量流量法和壁面积分法计算喷水推进器推力的CFD方法。采用多块网格技术,用六面体结构化网格和四面体非结构化网格相结合的混合网格离散计算区域,采用稳态多参考系方法求解RANS方程,对喷水推进器进水流道、叶轮、导叶体和喷口所组成的整个流场进行数值计算。计算中采用了k-ε湍流模型和标准壁面函数,对用动量流量法计算推力方法中所需的假想流管分界面和进口面的求取做了分析,将两种方法计算的推力与厂商提供的推力特性曲线进行了对比。结果表明,采用CFD计算和分析方法来研究喷水推进系统推力性能是可行、可信的。     

多孔介质壁面剪切湍流速度时空关联的研究

作为一个基础统计量,时空关联函数在湍流问题的研究中有着广泛的应用,是研究湍流噪声、湍流中物质扩散和大涡模拟亚格子模型等问题的重要参考.本文通过建立三维多孔结构壁面剪切湍流模型,采用含Darcy-Brinkman-Forchheimer作用力项的格子Boltzmann方程对无穷大多孔介质平行板之间壁湍流进行了数值模拟,进而研究其速度脉动时空关联函数的统计特性.一方面,根据计算得到的流场数据,对比分析了常规槽道湍流与多孔介质壁面槽道湍流的时间关联函数.另一方面,计算并讨论了不同孔隙率和渗透率的多孔介质壁面对速度脉动时空关联性的影响.通过研究表明:多孔结构壁面剪切湍流的时空关联函数等值线与椭圆理论相符;在研究参数范围内,多孔介质壁面的速度时空关联系数随着孔隙率增大而增大,随着渗透率增大而减小.同时发现在槽道壁面的近壁区、过渡区、对数律区和中心区等不同位置处,速度时空关联呈现较大差异性:越远离壁面位置(对数律区和中心区),其时空关联函数所呈现的关联等值线椭圆越细长,高值相关等值线越集中.多孔介质主要改变速度时空关联椭圆图像的椭圆率,说明多孔介质壁面主要影响湍流横扫速度.     

高雷诺数湍流风场大涡模拟的并行直接求解方法

在环境流体力学中,风场是风沙流、风雪流等自然环境特性问题研究的动力源和基础. 通常采用壁湍流模型进行风场大涡模拟(large eddy simulation, LES)计算,但受到计算规模的限制使得 高雷诺数风场的模拟计算难以实现. 并行计算技术是解决大规模高雷诺数风场大涡模拟的关键技术之一. 在不可压湍流风场的LES模拟中,压力泊松方程的并行计算技术是进行规模并行计算的困难点. 根据风场流动模拟计算的特点,采用水平网格等距而垂直于地面网格非等距,在解决规模并行计算中求解压力泊松方程的难点问题时,利用FFT解耦三维泊松方程使其变为垂向的一维三对角方程, 并利用可并行的三对角方程PDD求解技术,可建立三维泊松方程的直接并行求解技术. 结合其它容易并行的动量方程计算,本文建立风场LES模拟的并行直接求解方法(parallel direct method-LES, PDM-LES). 在超级计算机上对新方法进行并行效率测试,并行计算效率达到90${\%}$. 新的方法可用于进行湍流风场大涡模拟的大规模并行计算. 计算结果表明,湍流风场瞬时速度分布近壁面存在条带状的拟序结构,平均场的速度分布符合速度对数律特性,风场湍流特性基本合理.      

基于ALE有限元法的流固耦合强耦合数值模拟

针对不同流固耦合问题,提出一种基于任意拉格朗日-欧拉(ALE)有限元技术的分区强耦合算法.运用半隐式特征线分裂算法求解ALE描述下的不可压缩黏性流体Navier-Stokes方程.分别考虑一般平面运动刚体和几何非线性固体,采用复合隐式时间积分法推进结构运动方程,故可选用较大时间步长;进一步应用单元型光滑有限元法求解几何非线性固体大变形,获得更精确结构解且不影响计算效率.运用子块移动技术结合正交-半扭转弹簧近似法高效更新流体动网格;同时将一质量源项引入压力泊松方程满足几何守恒律,无需复杂构造网格速度差分格式.采用简单高效的固定点法配合Aitken动态松弛技术实现各场耦合,可灵活选择先进单场求解技术,具备较好程序模块性.运用本文算法分别模拟了H型桥梁截面颤振问题和均匀管道流内节气阀涡激振动问题.研究表明,数值结果与已有文献数据吻合,计算精度和求解效率均令人满意.     

层合材料的非傅里叶热传导及热应力

论文建立了一个双层材料层合板受瞬态加热情况下的非傅里叶热传导分析模型,用向后差分法得到了温度场的数值解,并对该差分格式的稳定性进行了讨论.给出了温度场随导热时间、热扩散率、空间与时间步长之比以及弛豫时间的变化趋势.同时,通过已经求得的温度场,求得了层合板内的应力场,给出了层合板内的热应力随时间的变化.     

Staggering error control of a class of inelastic processes in random microheterogeneous solids

In this work a staggering solution strategy for the simulation of the time-dependent inelastic mechanical deformation of a class of solids, possessing irregular heterogeneous microstructure, is developed. The system of coupled equations involved consists of (1) a dynamic equation of momentum balance, where the primary field variable is the displacement, (2) an evolution equation for material degradation, where the primary field variable is a state damage function, and (3) an evolution equation for the inelastic strains in the solid where the primary field variable is a plastic strain field. Clearly, the damage and plasticity variables are implicit functions of the displacement, however, for the staggering scheme strategy, it is convenient to formulate them as individual fields during the solution process. The key concept for the strategy to operate efficiently is to estimate and control the so-called staggering error, i.e. the error due to incompletely resolving the coupling between the field equations in a staggering process. This error is a function of the time step size. However, because the coupling is temporally variable, possibly becoming stronger, weaker, or oscillatory, it is extremely difficult to ascertain a priori the time step size needed for prespecified error control. In the present work, to induce desired staggering rates of convergence within each time step, thus controlling the staggering error, an adaptive strategy is developed whereby the time step size is manipulated, enlarged or reduced, to control the intrinsic contraction mapping constant of the staggering system operator. The overall goal is to deliver accurate solutions where temporal discretization error control dictates the upper limits on the time step size, while the iterative staggering strategy refines the step size further to control the staggering error. Three-dimensional numerical experiments are performed to illustrate the solution strategy.     

Multi‐size‐mesh multi‐time‐step algorithm for noise computation on curvilinear meshes

Aeroacoustic problems are often multi‐scale and a zonal refinement technique is thus desirable to reduce computational effort while preserving low dissipation and low dispersion errors from the numerical scheme. For that purpose, the multi‐size‐mesh multi‐time‐step algorithm of Tam and Kurbatskii [AIAA Journal, 2000, 38 (8), p. 1331–1339] allows changes by a factor of two between adjacent blocks, accompanied by a doubling in the time step. This local time stepping avoids wasting calculation time, which would result from imposing a unique time step dictated by the smallest grid size for explicit time marching. In the present study, the multi‐size‐mesh multi‐time‐step method is extended to general curvilinear grids by using a suitable coordinate transformation and by performing the necessary interpolations directly in the physical space due to multidimensional interpolations combining order constraints and optimization in the wave number space. A particular attention is paid to the properties of the Adams–Bashforth schemes used for time marching. The optimization of the coefficients by minimizing an error in the wave number space rather than satisfying a formal order is shown to be inefficient for Adams–Bashforth schemes. The accuracy of the extended multi‐size‐mesh multi‐time‐step algorithm is first demonstrated for acoustic propagation on a sinusoidal grid and for a computation of laminar trailing edge noise. In the latter test‐case, the mesh doubling is close to the airfoil and the vortical structures are crossing the doubling interface without affecting the quality of the radiated field. The applicability of the algorithm in three dimensions is eventually demonstrated by computing tonal noise from a moderate Reynolds number flow over an airfoil. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical and Numerical Results for an Escape Problem

A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called a “window” or “gate”) in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is discovered and proven for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and a sufficient number of sample paths are compared with the exact solution to the Brownian motion (the limit of zero step size), providing an empirical formula for the difference. Histograms of first hitting times are also generated.     

Stability and dispersion analysis of reproducing kernel collocation method for transient dynamics

A reproducing kernel collocation method based on strong formulation is introduced for transient dynamics. To study the stability property of this method, an algorithm based on the von Neumann hypothesis is proposed to predict the critical time step. A numerical test is conducted to validate the algorithm. The numerical critical time step and the predicted critical time step are in good agreement. The results are compared with those obtained based on the radial basis collocation method, and they are in good agreement. Several important conclusions for choosing a proper support size of the reproducing kernel shape function are given to improve the stability condition.     

On stabilized finite element methods for the Stokes problem in the small time step limit

Recent studies indicate that consistently stabilized methods for unsteady incompressible flows, obtained by a method of lines approach may experience difficulty when the time step is small relative to the spatial grid size. Using as a model problem the unsteady Stokes equations, we show that the semi‐discrete pressure operator associated with such methods is not uniformly coercive. We prove that for sufficiently large (relative to the square of the spatial grid size) time steps, implicit time discretizations contribute terms that stabilize this operator. However, we also prove that if the time step is sufficiently small, then the fully discrete problem necessarily leads to unstable pressure approximations. The semi‐discrete pressure operator studied in the paper also arises in pressure‐projection methods, thereby making our results potentially useful in other settings. Copyright © 2006 John Wiley & Sons, Ltd.     

MOLECULAR SIMULATION OF DRIVEN CAVITY FLOWS WITH HIGH REYNOLDS NUMBER

采用扩散信息保存(diffusive information preservation,D-IP)方法计算了雷诺数为10²~10⁴的二维方腔流动. D-IP方法是一种基于扩散运动观点的分子模拟方法, 克服了经典直接模拟蒙特卡罗方法对于时间步长和网格大小的严格限制.在计算中, D-IP方法的时间步长和网格大小分别为分子平均碰撞时间和平均自由程的几十倍乃至几百倍, 所得到的方腔流线分布和旋涡的精细结构, 均与Navier-Stokes方程数值解相符.     

COMPARISON OF ACCURACY AND PERFORMANCE FOR LATTICE BOLTZMANN AND FINITE DIFFERENCE SIMULATIONS OF STEADY VISCOUS FLOW

The lattice Boltzmann method (LBM) is used to simulate flow in an infinite periodic array of octagonal cylinders. Results are compared with those obtained by a finite difference (FD) simulation solved in terms of streamfunction and vorticity using an alternating direction implicit scheme. Computed velocity profiles are compared along lines common to both the lattice Boltzmann and finite difference grids. Along all such slices, both streamwise and transverse velocity predictions agree to within 0ċ5% of the average streamwise velocity. The local shear on the surface of the cylinders also compares well, with the only deviations occurring in the vicinity of the corners of the cylinders, where the slope of the shear is discontinuous. When a constant dimensionless relaxation time is maintained, LBM exhibits the same convergence behaviour as the FD algorithm, with the time step increasing as the square of the grid size. By adjusting the relaxation time such that a constant Mach number is achieved, the time step of LBM varies linearly with the grid size. The efficiency of LBM on the CM-5 parallel computer at the National Center for Supercomputing Applications (NCSA) is evaluated by examining each part of the algorithm. Overall, a speed of 13ċ9 GFLOPS is obtained using 512 processors for a domain size of 2176×2176.     

Frequency domain approach to the critical step size of discrete-time recurrent neural networks

It is usually essential to reveal the relationship between continuous-time systems and discrete-time ones. First, a discrete-time recurrent neural network is presented by the Euler scheme in this paper. Then, the time step size is set to a bifurcation parameter and frequency domain approach is adopted for Hopf bifurcation analysis. Moreover, the periodic solutions are obtained by the harmonic balance method; then the stability conditions are presented. The critical step size is determined with which the discrete-time recurrent neural network can inherit the stable state of the continuous-time one. Finally, one numerical example of the discrete-time recurrent neural network is given to support the theoretical analysis.