高阶谱元区域分解算法求解定常方腔驱动流

主要利用Jacobian-free的Newton-Krylov方法求解定常不可压缩Navier-Stokes方程,将基于高阶谱元法的区域分解Stokes算法的非定常时间推进步作为Newton迭代的预处理,回避了传统Newton方法Jacobian矩阵的显式装配,节省了程序内存,同时降低了Newton迭代线性系统的条件数,且没有非线性对流项的隐式求解,大大加快了收敛速度。对有分析解的Kovasznay流动的计算结果表明,本高阶谱元法在空间上有指数收敛的谱精度,且对定常解的Newton迭代是二次收敛的。本文模拟了二维方腔顶盖一致速度驱动流,同基准解符合得很好,表明本文方法是准确可靠的。本文还考虑了Re=800时方腔顶盖正弦速度驱动流,除得到已知的一个稳定对称解和一对稳定非对称解外,还获得了一对新的不稳定的非对称解。     

Efficient quadrature for a boundary element method to compute three-dimensional Stokes flow

A collocation-type boundary element method based on bilinear B-splines is used for the numerical solution of the Stokes Dirichlet problem in bounded domains D ? R³. The computation of the influence matrix requires the numerical evaluation of weakly singular integrals on the domain boundary if the usual double-layer potential ansatz is chosen. Here mostly standard methods with disjoint grids for collocation and integration are used. We develop a special integration scheme based on triangular co-ordinates near the singularity and show its efficiency compared with the method mentioned above.     

Complete general solution of Stokes equations for plane boundaries

The completeness of a representation for a solution of Stokes equations is proved which is suitable for solving problems which involve plane boundaries. We discuss two theorems for Stokes flow past a plane boundary with different boundary conditions and illustrate them with examples.     

Numerical solution of axisymmetric cavity flows using the boundary element method

This paper presents a formulation of the boundary element method (BEM) for solution of axisymmetric cavity flow problems. The governing equation is written in terms of Stokes' stream function, requiring a new fundamental solution to be found. The iterative procedure for adjusting the free-surface position is similar to that used for planar cavity flows. Numerical results are compared with finite difference and finite element solutions, showing the robustness of the BEM model.     

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。     

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.     

A boundary element formulation using the simple method

A new boundary element method is described for calculation of the steady incompressible laminar flows. The method is based on the well-known SIMPLE algorithm. The new boundary element method allows one to find the fields of the pressure and velocity corrections without inner iterations, thus reducing the computational time drastically. This makes it different from the method developed by Patankar and Spalding.³² However, the new method demands a much larger computer strorage. The boundary integral equations are discretized with the help of constant boundary elements and constant cells. The values of the integrals along the boundary elements and the cells for the two-dimensional domain are found analytically. To preserve the stability in the iteration process, under-relaxation for the convection terms is used. This paper gives the results of calculations of the flows between two plane parallel plates at Re = 20 and Re = 200, the flows in a square cavity with a moving upper lid at Re = 1 and Re = 100 and the flow in a plane channel with sudden symmetric expansion at Re =46·6.     

Numerical solution of a free surface problem by a boundary element method

This paper describes a method for the numerical solution of a Riabouchinsky cavity flow. Application of a boundary element method leads to a system of non-linear equations. The mild singularity appearing at the separation point is treated with the introduction of a curved boundary element, which satisfies the exact behaviour of the free boundary in that neighbourhood.     

On the solution of Stokes equations for plane boundaries

In a recent paper(Li et al., Acta Mech. Sin. 31,32–44, 2015), the authors claimed that the general solution of steady Stokes flows can be compactly expressed using only two harmonic functions. They present two cases of a flat plate translating through a viscous fluid. The present paper shows that such a two-harmonic solution does not describe the rotation of a circular plate in an unbounded fluid and thus confirms that at least three independent harmonics are required to express the general solution of Stokes equations.     

Dynamic simulation of the hydrodynamic interaction among immersed particles in Stokes flow

A numerical method for the dynamic simulation of the hydrodynamic interaction among particles in Stokes flow is developed. The method couples the quasi-static Stokes equations for the fluid with the equilibrium equations for the particles. The boundary element method is used to represent the velocity at a general field point in terms of surface velocities and stresses. However, neither the stresses nor the velocities are assumed to be known on the surface of the particles. Kinematic equations relating the linear and angular velocities at the centroids of the particles to the surface velocities are combined with the discretized boundary element equations and the equilibrium equations to generate a system of linear equations. The associated coefficient matrix is correspondent to the grand resistance matrix which relates the velocities of the particles to a given geometry.     

自由面势流问题边界元法的数值色散误差

以连续及离散Fourier分析研究自由面势流问题边界元法的数值色散误差,并从理论上探讨有关计算中数值色散误差的改善问题.研究表明:对于该问题的数值色散误差而言,重要的在于以问题相应的离散算子考察计及各种数值手段后的总体色散误差,而非仅考虑该数值手段自身的数值色散误差大小.高阶面元、自由面域外奇点或适当的耦合方法是降低有关问题算子总体色散误差的较好选择.     

Stokes flow before plane boundary with mixed stick-slip boundary conditions

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.     

自由面势流问题的域外奇点边界元法及其数值误差分析

讨论了域外奇点边界元法在自由面势流问题计算中的作用，并以连续及离散Fourier分析对该方法(就m阶面元的一般情况)进行数值误差分析，导出了计及面元阶数、奇点至自由面垂向距离、配置点移动、差分格式等因素影响的数值误差一般表达式。从理论上证明了自由面势流问题计算中采用域外奇点法可改善离散产生的数值色散误差并能结合配置点前移(向上游)等方法以数值满足辐射条件。     

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 total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

A theorem for a shear stress-free plane boundary in Stokes flow

A theorem for non-axisymmetrical Stokes flow about a shear stress-free plane boundary is established by using a representation for the velocity and pressure fields for the same flow in terms of biharmonic and harmonic functions. A corollary of the theorem is derived which gives the axisymmetrical Stokes flow in terms of the Stokes function about the same boundary. The formulae for drag and torque on the boundary are also given. A few illustrative examples are presented.     

Analysis of convective diffusion problem with first-order chemical reaction by boundary element method

A boundary element method is presented for a steady state convective diffusion problem with a first-order chemical reaction. In the chemical reaction system the usual conservation law for calculation of the diagonal components of the matrix H cannot be satisfied. Thus in the boundary element formulation the emphasis is laid on how to compute these diagonal components. The main aim of this paper is to present the method of numerical quadrature to obtain the principal values, using the splitting technique to treat the singular integration.