谱元法和高阶时间分裂法求解方腔顶盖驱动流

详细推导了谱元方法的具体计算公式和时间分裂法的具体计算过程 ;对一般的时间分裂法进行了改进 ,即对非线性步分别用 3阶 Adams-Bashforth方法和 4阶显式 Runge-Kutta法 ,粘性步采用 3阶隐式 Adams-Moulton形式 ,提高了时间方向的离散精度 ,同时还改进了压力边界条件 ,采用 3阶的压力边界条件 ;利用改进的时间分裂方法分解不可压缩 Navier-Stokes方程 ,并结合谱元法计算了移动顶盖方腔驱动流 ,提高了方法可以计算的 Re数 ,缩短了达到收敛的时间 ,并将结果与基准解进行比较 ;分析了移动顶盖方腔驱动流中 Re数对流场分布的影响。     

方腔双壁反向驱动流涡结构演化的数值模拟

采用微分求积法计算分析了不同雷诺数的二维方腔驱动流涡结构特性.数值模拟着重研究了雷诺数从0.01到1000变化对方腔双壁反向驱动流涡结构演化的影响,给出了涡演化过程的流型图和分叉图.结果表明,当雷诺数接近0时,腔内流动呈现对称的涡结构流型;随着雷诺数增加,子涡的大小和中心位置发生变化,鞍点始终位于方腔的中心,腔内流动形成非对称的斜扭流型;当雷诺数增大到某一临界值后,单一大涡占住整个方腔,大涡的形状变得更圆;如果雷诺数继续增加,方腔左上角和右下角同时出现二级涡,大涡中心始终位于方腔中心不变.     

交错网格上应用有限谱QUICK法计算方腔流动

王健平教授提出的有限谱方法是一种局域化谱方法，具有精度高、无相位差、应用灵活等特点，在以往的实践中取得了很大成功。本文在交错网格上对二维驱动方腔流问题进行计算，求解了二维不可压缩流动的涡流流函数方程。其中微分部分采用有限谱法进行处理，对流项的处理则应用了QUICK格式。本文计算了雷诺数为1000、5000、10000、20000等多种情况，将所得的结果进行分析，并将中线上的速度分别同已有的文献数据进行对比，从而，验证有限谱微分的正确性和其在实际应用中的可行性。     

谱元方法求解二维不可压缩Navier-Stokes方程

利用有限元的思想并结合谱方法的精度提出求解偏微分方程的谱元方法,在元素内插值函数使用伪谱Chebyshev逼近,并将此方法应用于求解不可压Navier-Stokes方程,具体求解了二维方腔顶盖驱动流,与公认基准解对比获得了较好的结果。     

Chebyshev谱元方法结合并行算法求解三维区域的Helmholtz方程

本文探讨了采用Chebyshev谱元方法结合并行计算求解三维区域的Helmholtz方程问题。首先应用变分方法,得到了带有第一类边界条件的三维区域Helmholtz方程的弱形式。然后在三维的标准单元内,采用Chebyshev正交多项式展开函数u和试函数v,并且将其带入弱形式方程,通过积分,得到单元刚度矩阵;通过合成单元刚度矩阵,得到总体矩阵。最后通过基于MPI的并行计算,求解了以总体矩阵为系数的方程组,得到了Helmholtz方程的数值解,和解析解对比表明了数值解的正确性,并且数值解具有8阶精度。在并行求解方程组过程中,充分利用矩阵的对称性和矢量存储来获取上三角元素,这大幅的节约了存储量和计算进程间的通讯量,获得的并行效率可达76.6%。     

Poiseuille-Bénard流的出口边界条件及其谱元法计算

研究二维矩形管道中底部加热的不可压缩Ｐｏｉｓｅｕｉｌｌｅ－Ｂｅｎａｒｄ流的谱元法数值计算问题．讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响．通过干扰区、干扰幅度和计算时间的比较,确定比较理想的出口边界条件．     

同位网格摄动有限体积格式求解浮力驱动方腔流

利用对流扩散方程的摄动有限体积格式,在Rayleigh数从10$^{3}$ 到10$^{8}$的范围内对浮力驱动方腔流动问题作了数值模拟. 对流扩散方程的摄动 有限体积格式具有一阶迎风格式的简洁形式,使用相同的基点,重构近似精度高,特别是两 相邻控制体中心到公共界面的距离相等或不相等,PFV格式公式相同等优点. 在数值模拟中, 无论均匀网格还是非均匀网格均获得与DSC方法、自适应有限元法、多重网格法等Benchmark 解相符较好的数值结果,证明UPFV格式对高Rayleigh数对流传热问题的适用性和有效性.     

基于非线性光流方程的高阶时域DIC算法

针对极端测试环境中采集的低质量测试图像,提出一种解析高精度变形场的高阶时域DIC(Digital Image Correlation)算法.采用模拟散斑方法建立了含高斯白噪声的亚像素级位移图像,通过最小二乘迭代法求解高阶时域DIC算法的位移量,分析了算法的测试精度.研究结果表明,非线性光流方程能够描述变形前后像素点灰度...     

镜像对称顶盖驱动方腔内流过渡流临界特性研究

流场过渡流临界特性是指流场因流动状态改变而引起的流场物理特性变化. 如流动从定常演化为非定常周期性时, 流动处于过渡状态的物理性质. 它从根本上决定了流动演化模式和流场特性等物理规律, 对认清流动现象的形成机理有重要意义. 本文在之前腔体内流流场过渡流临界特性研究的基础上, 针对镜像对称顶盖驱动方腔内流开展数值模拟和流场稳定性分析研究, 捕捉各流动分岔点, 如Hopf流动分岔点和Neimark-Sacker流动分岔点等, 并揭示其对流场特性的影响; 分析流场演化模式, 随着雷诺数增大从定常状态依次演化为非定常周期性流动、准周期性流动和湍流; 揭示各种流动现象的形成机理, 如流动滞后、对称性破坏、能量级串等; 分析流场拓扑结构, 阐明流场镜像对称性和流场稳定性的关系. 本文研究成果有助于揭示该流场的物理特性, 进一步完善了内流流场特性的研究. 研究发现, 针对本文镜像对称方腔顶盖驱动内流, 流场稳定性的破坏总是以Hopf流动分岔点的出现而发生并且伴随着流场对称性的破坏; 流场演化模式符合经典的Ruelle-Takens模式; 流动从定常状态演化至非定常周期性流动时存在流动滞后现象.      

不可压缩机翼绕流的有限谱法计算

结合有限谱QUICK格式求解不可压缩粘性流问题。这一格式用于模拟不同攻角下的NACA1200机翼绕流问题。利用体积力,提出了将流场速度从0加速到来流速度的方法。区别于传统的压力梯度为零的边界条件,推导出一个更精确的压力边界条件。为使速度散度保持为零,在泊松方程中给速度散度一个特殊的处理。这一成果说明了有限谱法不但具有很高的精度,而且能灵活地和其他格式一起构造出新的格式,从而成功地应用到复杂流场不可压缩流动的数值计算中。     

Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers

A boundary element method for steady two‐dimensional low‐to‐moderate‐Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier–Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non‐linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd.     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

A domain decomposition method for modelling Stokes flow in porous materials

An algorithm is presented for solving the Stokes equation in large disordered two‐dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off‐block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000‐particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd.     

The numerical solution of Stokes flow in a domain with re-entrant boundaries by the boundary element method

Numerical solutions are presented for two-dimensional low Reynolds number flow in a rotating tank with stationary barriers. The boundary element method is employed, assuming straight panels and quadratic source distribution. The feasibility of repositioning the nodes as a way to minimize the error is explored. A stretching parameter places smaller elements near the re-entrant regions. Elementary error analysis shows uniform improvement in the solution with stretching. The changing eddy pattern for different numbers and sizes of the barriers is compared with experimental results.     

矩形空腔内Stokes流的状态空间有限元法

基于Hellinger-Reissner二类变分原理,从平面Stokes流问题的平衡方程、连续性要求和边界条件出发,得到相应的Hamilton函数,建立Hamilton正则方程后,采用分离变量法对场变量进行离散求解:在x方向采用有限元插值,在y方向采用状态空间法给出控制坐标方向的解析解。计算过程中的指数矩阵均采用精细积分法求解,使得本文算法具有高效率、高精度、对步长不敏感的优点。通过对侧边自由液面边界条件的单板驱动矩形空腔Stokes流问题的求解,得到与文献相同的结果,从而验证了本文方法的有效性。本文旨在将弹性力学状态空间有限元法的思想引入到低雷诺数流体力学中,为Hamilton体系下研究复杂边界Stokes流问题提供新的途径。     

Hamilton体系下环扇形域的Stokes流动问题

基于极坐标下Stokes流的基本方程,将径向坐标模拟为时间坐标,推导了Hamilton体系下Stokes流动问题的对偶方程,采用本征向量展开法对环扇形域Stokes流动问题进行了分析,并给出了相应的实际算例,其结果说明了本文方法的有效性。     

Solving high Reynolds‐number viscous flows by the general BEM and domain decomposition method

In this paper, the domain decomposition method (DDM) and the general boundary element method (GBEM) are applied to solve the laminar viscous flow in a driven square cavity, governed by the exact Navier–Stokes equations. The convergent numerical results at high Reynolds number Re = 7500 are obtained. We find that the DDM can considerably improve the efficiency of the GBEM, and that the combination of the domain decomposition techniques and the parallel computation can further greatly improve the efficiency of the GBEM. This verifies the great potential of the GBEM for strongly non‐linear problems in science and engineering. Copyright © 2004 John Wiley & Sons, Ltd.     

Preconditioned finite element algorithms for 3D Stokes flows

Preconditioned conjugate gradient algorithms for solving 3D Stokes problems by stable piecewise discontinuous pressure finite elements are presented. The emphasis is on the preconditioning schemes and their numerical implementation for use with Hermitian based discontinuous pressure elements. For the piecewise constant discontinuous pressure elements, a variant implementation of the preconditioner proposed by Cahouet and Chabard for the continuous pressure elements is employed. For the piecewise linear discontinuous pressure elements, a new preconditioner is presented. Numerical examples are presented for the cubic lid-driven cavity problem with two representative elements, i.e. the Q2-PO and the Q2-P1 brick elements. Numerical results show that the preconditioning schemes are very effective in reducing the number of pressure iterations at very reasonable costs. It is also shown that they are insensitive to the mesh Reynolds number except for nearly steady flows (Re_m → 0) and are almost independent of mesh sizes. It is demonstrated that the schemes perform reasonably well on non-uniform meshes.     

Global flow instability in a lid‐driven cavity

The stability of flow in a lid‐driven cavity is investigated using an accurate numerical technique based on a hybrid scheme with spectral collocation and high‐order finite differences. A global stability analysis is carried out and critical parameters are identified for various aspect ratios. It is found that while there is reasonable agreement with the literature for the critical parameters leading to loss of stability for the square cavity, there are significant discrepancies for cavities of aspect ratios 1.5 and 2. Simulations of the linearized unsteady equations confirm the results from the global stability analysis for aspect ratios A = 1, 1.5 and A = 2. Copyright © 2009 John Wiley & Sons, Ltd.