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.     

A boundary element method for a non-linear free surface problem

In this paper a numerical method to compute the wave resistance of a body submerged in a free stream of finite and infinite depth is presented. Non-linear effects on the free surface are taken into account by an iterative procedure; the solution is in the form of a single-layer potential. For the 2D problem, results are shown for both the cases of finite and infinite depth of the fluid domain, with special emphasis on the supercritical flow in which the consistency of the scheme is pointed out. The method is also extended to the 3D case of a spheroid submerged in deep water. All the results presented are compared with experimental data and analytical solutions available in the literature.     

Second-order boundary element method calculations of hydrodynamic interactions between particles in close proximity

A second-order boundary element technique was developed to simulate the 3D hydrodynamic interactions between multiple particles of arbitrary shape. This paper reports the results of an extensive validation procedure aimed at demonstrating the convergence characteristics of the technique, especially in cases where the particles are in close proximity. The quadratic elements are superior to the lower-order elements in terms of accuracy, computer storage and CPU time required, thus resulting in a significant improvement in the overall computational efficiency. Superparametric discretization improves the accuracy over isoparametric discretization but lowers the convergence rate of the method. When the interparticle gap becomes very small (less than 1% of the particle radius), the numerical solution diverges owing to inaccurate determination of the element contributions in the gap region. An adaptive subdomain integration scheme was developed that dramatically improved the integration accuracy and provided convergent solutions for problems of very small gaps down to 0–01% of the particle diameter.     

A boundary element method for the simulation of non-spherical bubbles and their interactions near a free surface

The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix are eliminated using coordinate transformation and so-called 4 π rule. The solid angle for the open surface is treated in direct method based on its definition. Several kinds of configurations for the bubbles and free surface have been investigated. The pressure contours during the evolution of bubbles are obtained in our model and can better illuminate the mechanism underlying the motions of bubbles and free surface. The bubble dynamics and their interactions have close relation with the standoff distances, buoyancy parameters and initial sizes of bubbles. Completely different bubble shapes, free surface motions, jetting patterns and pressure distributions under different parameters can be observed in our model, as demonstrated in our calculation results.     

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.     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

A net inflow method for incompressible viscous flow with moving free surface

A new finite element procedure called the net inflow method has been developed to simulate time-dependent incompressible viscous flow including moving free surfaces and inertial effects. As a fixed mesh approach with triangular element, the net inflow method can be used to analyse the free surface flow in both regular and irregular domains. Most of the empty elements are excluded from the computational domain, which is adjusted successively to cover the entire region occupied by the liquid. The volume of liquid in a control volume is updated by integrating the net inflow of liquid during each iteration. No additional kinetic equation or material marker needs to be considered. The pressure on the free surface and in the liquid region can be solved explicitly with the continuity equation or implicitly by using the penalty function method. The radial planar free surface flow near a 2D point source and the dam-breaking problem on either a dry bed or a still liquid have been analysed and presented in this paper. The predictions agree very well with available analytical solutions, experimental measurements and/or other numerical results.     

An efficient three-dimensional semi-implicit finite element scheme for simulation of free surface flows

An efficient semi-implicit finite element model is proposed for the simulation of three-dimensional flows in stratified seas. The body of water is divided into a number of layers and the two horizontal momentum equations for each layer of water are first integrated vertically. Nine-node Lagrangian quadratic isoparametric elements are employed for spatial discretization in the horizontal domain. The time derivatives are approximated using a second-order-accurate semi-implicit time-stepping scheme. The distinguishing feature of the proposed numerical scheme is that only nodal values on the same vertical line are coupled. Two test cases for which analytic solutions are available are employed to test the proposed scheme. The test results show that the scheme is efficient and stable. A numerical experiment is also included to compare the proposed scheme with a finite difference scheme.     

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.     

A boundary element method for anisotropic coupled thermoelasticity

Summary A boundary element formulation is presented for the solution of the equations of fully coupled thermoelasticity for materials of arbitrary degree of anisotropy. By employing the fundamental solutions of anisotropic elastostatics and stationary heat conduction, a system of equations with time-independent matrices is obtained. Since the fundamental solutions are uncoupled and time-independent, a domain integral remains in the representation formula which contains the time-dependence as well as the thermoelastic coupling. This domain integral is transformed to the boundary by means of the dual reciprocity method. By taking this approach, the use of dynamic fundamental solutions is avoided, which enables an efficient calculation of system matrices. In addition, the solution of transient processes as well as, free and forced vibration analysis becomes straightforward and can be carried out with standard time-stepping schemes and eigensystem solvers. Another important advantage of the present formulation is its versatility, since it includes a number of simplified thermoelastic theories, viz. the theory of thermal stresses, coupled and uncoupled quasi-static thermoelasticity, and stationary thermoelasticity. The accuracy of the new thermoelastic boundary element method is demonstrated by a number of example problems. Support by the Deutsche Forschungsgemeinschaft (DFG) of the Graduate Collegium Modelling and discretization methods for continua and fluids (GKKS) at the University of Stuttgart is gratefully acknowledged.     

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.     

基于预修正快速傅里叶变换高阶边界元方法的多体水动力分析

高阶边界元方法在求解波浪对海上建筑物作用问题中具有诸多优势,但由于它所形成的矩阵是一个满阵,计算量和存储量均为未知量的平方量级,很难满足如多体水动力分析等大尺度多未知量问题的计算需要。本文采用预修正快速傅里叶变换高阶边界元方法（pFFT-HOBEM）,将计算量和存储量均降低到未知量的线性量级。通过对不同未知量时该方法与传统边界元法的计算量与存储量的对比,以及该方法自身各步骤计算时间的对比,研究了不同pFFT网格方案对计算量和存储量的影响,并提出了基于计算时间最小化原则的pFFT网格优化方法。采用本文方法研究了四柱结构在不同频率波浪作用下的作用力及波面分布,并对近场干涉发生时的物理现象进行了分析和讨论。     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Multiscale simulation of viscoelastic free surface flows

A micro–macro approach based on combining the Brownian configuration fields (BCF) method [M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flow using Brownian configuration fields, J. Non-Newtonian Fluid Mech. 70 (1997) 79–101] with an Arbitrary Lagrangian–Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time-dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on the conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409]. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers, (where λ is the characteristic relaxation time of polymer, and is the characteristic rate of strain) compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field.     

Non-linear free surface flows and an application of the Orlanski boundary condition

The focus of this paper is the analysis of spatially two-dimensional non-linear free surface problems. The critical aspects of the problem concern the treatment of the non-linear free surface, the body boundary condition for large motions and the imposition of suitable radiation conditions. To address such complexities, time domain simulation was chosen as the method of analysis. With the use of a finite domain for simulation, a major concern is with the radiation condition to be applied at the open or truncation boundary. For the two-dimensional problem at hand, no theoretical radiation conditions are known to exist. An extension of the Orlanski open boundary condition, based on phase velocity determination at the free surface, is proposed. Three categories of problems were analysed using numerical simulation-namely, freely moving steep waves, waves over a submerged body and forced body motion. Simulation results have been compared with linear theory and experiments.     

A regular boundary element method for fluid flow

The Boundary Element Method is now well established as a valid numerical technique for the solution of field problems, equal to the Finite Element Method in generality and surpassing it in computational efficiency in some cases.¹ In this paper is presented a 'Regular Boundary Element Method' as applied to inviscid laminar fluid flow problems. It involves the formation of a system of regular integral equations obtained by moving the singularity outside the domain of the given problem. It is also shown that non-conforming elements may be used whereby freedoms are not defined at the geometric nodes under the boundary element discretization. A linear element is developed here; higher order variants could easily be defined. Satisfactory numerical results have been obtained using the proposed regular method with both conventional (continuous across the boundary) and non-conforming boundary elements for two-dimensional inviscid laminar fluid flow problems having regular and singular solutions.     

Numerical simulation method for viscoelastic flows with free surfaces—fringe element generation method

To simulate filling flow in injection moulding for viscoelastic fluids, a numerical method, based on a finite element method and a finite volume method, has been developed for incompressible isothermal viscoelastic flow with moving free surfaces. The advantages of this method are, first, good applicability to arbitrarily shaped mould geometries and, second, accurate treatment for boundary conditions on the free surface. Typical filling flows are simulated, namely filling flow into a 1:4 expansion cavity with and without an obstacle. Numerical results predict the position of weld lines and air-traps. The method also indicates the effects of elongational flow on molecular orientation.     

径向积分边界元法确定非均质土石坝渗流自由面

浸润面位置的确定是无压渗流分析中非常重要的问题,本文将径向积分边界单元法应用到渗流问题中,通过直接将区域积分转化为边界积分的技术,克服了传统边界元解决该类问题的缺陷,编写了迭代计算程序,并用二、三维算例证明了算法的有效性。     

A Lagrangian boundary element approach to transient three‐dimensional free surface flow in thin cavities

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The pressure, which in this case is governed by the Laplace's equation, is determined using the boundary element method. A fully Lagrangian approach is implemented for the tracking of the evolving free surface. The domain of computation is the projection of the physical domain onto the (x, y) plane. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. It is found that the flow behavior is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, and inlet flow pressure. Copyright © 2001 John Wiley & Sons, Ltd.