Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-ε turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.     

Finite difference method of transinet nonlinear free surface wave problems

A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.     

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.     

Higher-order theory of internal solitary waves with a free surface in two-layer fluid system of finite depth

A higher-order approximation theory of internal solitary waves with a free surface is presented. Using the method of strained co-ordinates, the third-order approximation evolution equation of interface has been found. An analytic expression of the wave velocity is given. The evolution equation has been solved numerically. It is found that the effects of free surface on the shape and wave velocity of solitary wave are O(ε²), and the third-order numerical solutions are closer to experimental data than the first-and second-order solutions.     

用二维欧拉方法模拟滑移爆轰作用下金属圆筒的收敛运动

本文用二维流体-弹塑性欧拉编码对金属圆筒在滑移爆轰作用下的收敛运动作了数值模拟。文中考虑了起爆端对金属圆筒运动的影响,给出了圆筒自由面不同时刻的运动姿态和自由面速度随时间变化规律。计算结果与实验结果符合较好。     

A fully hydrodynamic model for three‐dimensional,free‐surface flows

The hydrostatic pressure assumption has been widely used in studying water movements in rivers, lakes, estuaries, and oceans. While this assumption is valid in many cases and has been successfully used in numerous studies, there are many cases where this assumption is questionable. This paper presents a three‐dimensional, hydrodynamic model for free‐surface flows without using the hydrostatic pressure assumption. The model includes two predictor–corrector steps. In the first predictor–corrector step, the model uses hydrostatic pressure at the previous time step as an initial estimate of the total pressure field at the new time step. Based on the estimated pressure field, an intermediate velocity field is calculated, which is then corrected by adding the non‐hydrostatic component of the pressure to the estimated pressure field. A Poisson equation for non‐hydrostatic pressure is solved before the second intermediate velocity field is calculated. The final velocity field is found after the free surface at the new time step is computed by solving a free‐surface correction equation. The numerical method was validated with several analytical solutions and laboratory experiments. Model results agree reasonably well with analytical solutions and laboratory results. Model simulations suggest that the numerical method presented is suitable for fully hydrodynamic simulations of three‐dimensional, free‐surface flows. Copyright © 2003 John Wiley & Sons, Ltd.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Planning and control of 3-D nano-manipulation

The use of atomic force microscope (AFM) as a nanomanipulator has been evolving for various kinds of nano-manipulation tasks. Due to the bow effect of the piezo scanner of the AFM, the AFM space is different from the Cartesian space. Traditional nanomanipulation based on AFM is only a 2-D operation and does not consider the bow effect of the piezotube. In this paper, different 3-D nanomanipulation tasks using AFM such as nanolithography, pushing and cutting have been discussed. 3-D path planning is performed directly in the AFM space and the 3-D paths are generated based on the 3-D topography information of the surface represented in the AFM space. This approach can avoid the mappings between the AFM space and Cartesian space in planning. By following the generated motion paths, the tip can either follow the topography of the surface or move across the surface by avoiding collision with bumps. Nanomanipulation using this method can be considered as the “true“ 3-D operations since the cantilever tip can be controlled to follow any desired 3-D trajectory within the range of AFM space. The experimental study shows the effectiveness of the planning and control scheme.     

On the derivation of an admissibility condition for phase boundary propagation in an SMA bar based on a 3-D formulation

There is some considerable difficulty in determining the solution uniquely for a propagating phase boundary in shape memory alloy (SMA) bar. In this paper, we establish an admissibility condition starting from a three-dimensional (3-D) internal-variable formulation to resolve this issue. We adopt a 3-D formulation in literature which is based on a constitutive model with specific forms of the Helmholtz free energy and dissipation rate. Then the 3-D dynamical equations are reduced to the 1-D rod equations for three phase regions (coupled with the radial effect and surface condition) by using two small parameters. Connection conditions at the phase interfaces are determined. By considering the traveling-wave solution for the rod system, we eventually derive three conditions across a sharp phase boundary corresponding to the 1-D sharp-interface model, including the two usual jump conditions and an additional condition. The third condition is then used to supplement the 1-D sharp-interface model to study an impact problem. The unique solution is constructed analytically for all possible impact velocity, including three kinds of wave patterns according to different levels of the impact velocity. The results are compared with those obtained by the maximal dissipation rate criterion.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

用Level Set方法求解具有自由面的流动问题

为采用Ｌｅｖｅｌ Ｓｅｔ方法来计算有自由的流动问题提出了一种方案,把自由水面视为水和空气的交界面,两种介质用统一的Ｎ－Ｓ方法求解,在自由面两侧分别采用各自的密度和粘性,并在自由面上给以适当的光滑;采用边界元法求解双调和方程来确定距离函数;Ｎ－Ｓ方程用投影法求解,文中给出了二维水池水面振荡和瞬时溃坝问题的算例,可以看出用ＬｅｖｅｌＳｅｔ方法求解有自由面流动问题是有效的。     

三维非定常可压跨声速有旋流动的统一变域变分理论

本文利用Ｃｌｅｂｓｃｈ变量,应用半反推法首次建立了三维非定常可压跨声有旋流体的广义变分原理及变分原理族,应用变域变分工具,几乎所有界面条件（包括激波、尾涡面条件等）都转化成了自然边界条件,这给有限元计算带来了很大的方便。     

Oblique water entry of a cone by a fully three-dimensional nonlinear method

The hydrodynamic problem of a cone entering the water surface obliquely has been analyzed by the three-dimensional (3-D) incompressible velocity potential theory with the fully nonlinear boundary conditions on the moving free surface and body surface boundary. The time stepping method is used in the stretched coordinate system defined as the ratio of the physical system to the distance that the cone has travelled into water. The boundary element method is used to solve the potential at each time step. Both triangular element mesh and quadrilateral element mesh have been used. Discretisation of the body surface and the free surface is applied regularly during the simulation to account for their change and deformation, and data from the old mesh is transferred into the new one through interpolation. Both the dynamic and kinematic free surface boundary conditions are satisfied through the Eulerian form. In particular the free surface elevation and potential variation are traced at a given azimuth of the cylindrical coordinate system, in the direction parallel to the body surface or perpendicular to the free surface to avoid multi-valued function. Detailed convergence study with respect to time step and element size has been undertaken and high accuracy has been achieved. Results for the cone in vertical entry are compared with those obtained from the 2-D axisymmetric method and good agreement is found. Simulations are made for cones of various deadrise angles and different oblique entries and detailed results are provided.     

3-D analysis of temperature distribution in the material during pulsed laser and material interaction

Recently, lasers are being increasingly used in the industry owing to their precision and low cost. Material is heated and evaporated during laser and material interaction due to the absorption of laser beams by the material. In this study, a 3-D Laser heating model including evaporation has been solved using the electron- kinetic theory approach. The basis in examining the problem using the kinetic theory approach is to describe the heat conduction through electron-phonon and molecule-phonon collisions. The problem is solved by using the electron-kinetic theory approach in such a way that heat conduction is taken into account until the material is heated to its melting temperature and non-conduction limited heat transfer is considered after the melting temperature is reached. Non-conduction limited heat transfer through the phase change process is resulted from vacancy-molecule collisions. A numerical scheme is introduced to solve the governing equation, owing to the fact that the energy equation resulted is in the form of integro-differential equation. Four different materials, namely iron, nickel, tantalum and titanium are chosen in this study determine the material response to laser pulse heating. For each material, time dependent temperature distribution through the depth of the material and on the surface of the material is computed and analyzed for four different materials.     

A numerical method to solve the m‐terms of a submerged body with forward speed

To model mathematically the problem of a rigid body moving below the free surface, a control surface surrounding the body is introduced. The linear free surface condition of the steady waves created by the moving body is satisfied. To describe the fluid flow outside this surface a potential integral equation is constructed using the Kelvin wave Green function whereas inside the surface, a source integral equation is developed adopting a simple Green function. Source strengths are determined by matching the two integral equations through continuity conditions applied to velocity potential and its normal derivatives along the control surface. After solving for the induced fluid velocity on the body surface and the control surface, an integral equation is derived involving a mixed distribution of sources and dipoles using a simple Green function and one component of the fluid velocity. The normal derivatives of the fluid velocity on the body surface, namely the m‐terms, are then solved by this matching integral equation method (MIEM). Numerical results are presented for two elliptical sections moving at a prescribed Froude number and submerged depth and a sensitivity analysis undertaken to assess the influence of these parameters. Furthermore, comparisons are performed to analyse the impact of different assumptions adopted in the derivation of the m‐terms. It is found that the present method is easy to use in a panel method with satisfactory numerical precision. Copyright © 2002 John Wiley & Sons, Ltd.     

有限深水中骑行波的显式Hamilton描述

小尺度波（扰动波）迭加在大尺度波（未受扰动波）上形成的波动一般之为“骑行波”。研究了有限可变深度的理想不可压缩流体中的骑行波的显式Hamliltn表示，考虑了自由面上流体与空气之间的表面张力。采用自由面高度和自由面上速度势构成的Hamilton正则变量表示骑行波的动能密度，并在未受扰动波的自由面上作一阶展开。运用复变函数论方法处理了二维流动。先用保角变换将物理平面上的流动区域变换到复势平面上的无限长带形区域，然后在复势平面上用Fourier变换解出Laplace方程，最后经Fourier逆变换求出了扰动波速度热所满足的积分方程。作为特例考虑了平坦底部的流动，导出了动能密度的显式表达式。这里给出的积分方程可以替代相当难解的Hamilton正则方程。通过求解积分方程可得出agrange密度的显式表达式。本文提出的方法约研究骑行波的Hamilton描述以及波的相互作用问题提供了新的途径，有助于了解海面的小尺度波的精细结构。     

Perturbation solution to 3-D nonlinear supercavitating flow problems

A 3-D nonlinear problem of supercavitating flow past an axisymmetric body at a small angle of attack is investigated by means of the perturbation method and Fourier-cosine-expansion method. The first three order perturbation equations are derived in detail and solved numerically using the boundary integral equation method and iterative techniques. Computational results of the hydrodynamic characteristics and cavity shapes of each order are presented for nonaxisymmetric supercavitating flow past cones with various apex-angles at different cavitation numbers. The numerical results are found in good agreement with experimental data. The project supported by the National Natural Science Foundation of China     

求解三维裂纹前缘SIF分布时间历程的通用权函数法和有限变分法

将三维热权函数法扩展为适用于表面力、体积力和温度载荷的通用权函数法(UWF).推导出以变分型积分方程表达的UWF法基本方程,从变分的角度,将求解三维热权函数法基本方程的多虚拟裂纹扩展法(MVCE)改造为可以适用于一般的变分型积分方程的一类新型数值方法--有限变分法(FVM).在FVM中可以引入无穷多种线性无关的局部变分模式,可以根据计算要求在求解域中插入任意多个计算节点,单一型裂纹问题FVM所得到的最终方程组的系数矩阵总是一个对称的窄带矩阵,而且对角元总是大数,具有良好的数值计算性能.FVM对于SIF沿裂纹前缘急剧变化的复杂情况具有较好的数值模拟能力和较高的计算精度,利用自身一致性,可以求得三维裂纹前缘SIF的高精度解.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.