基于SPH的流体仿真数值算法及工程应用研究
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摘要
计算机数值仿真逐渐成为解决现代工程和科学问题的一条重要途径。数值仿真能为理论提供测试和检验,有助于对复杂的物理问题加深认识,甚至还能帮助解释和发现新现象。
基于网格的数值方法虽然已经有广泛的应用,但是在很多方面仍存在不足之处,比如在计算流体动力学中的大变形、运动物质交界面、自由表面等问题时,由于网格产生畸变导致计算误差过大或无法进行,从而使其在许多问题的应用上受到限制。
近年来,无网格法倍受关注,这种方法在许多应用中都优于传统的基于网格的有限元法、有限差分法以及有限体积法等数值方法。本文依托山东大学虚拟工程研究中心和新加坡国立大学ACES实验室,系统地研究了新一代无网格方法—光滑粒子流体动力学方法(SPH)在应用及扩展过程中的相关关键技术,实现了SPH方法的两相耦合应用和三维应用,在此基础上对三维管道气力输送过程进行了数值仿真,扩展了SPH方法的工程应用领域,为SPH方法真正辅助试验创造了条件,具有重要的理论和应用价值。
论文的主要内容如下：
系统总结了基于网格的数值方法,指出其在很多方面存在不足之处,由于网格畸变而使其在许多问题的应用上受到限制,在此基础上,详细分析了新一代无网格法—光滑粒子动力学方法的基本思想和求解问题的过程,由于问题域粒子之间没有任何连接,运算中使用的粒子取决于当前局部分布的粒子,得出了SPH方法的无网格属性和自适应性。系统分析了拉格朗日型的Navier-Stokes方程,应用SPH粒子近似法推导出了Navier-Stokes的密度方程、动量方程和能量方程的SPH表达式。
研究了流体动力学SPH数值计算中的关键技术,指出了为了使算法适合模拟不同的流体特性问题,必须对算法进行特别地处理。在SPH方法中有两种方式对密度进行展开：一种是密度求和法,另一种是连续密度法。虽然密度求和法所需要的计算量大,但因为密度求和法体现了SPH近似法的本质,所以使用得较为广泛。连续性密度法主要用于仿真具有强间断的问题。在SPH方法中,用核函数来近似狄拉克δ函数,分析了在实际应用中常用的两种核函数：高斯核函数和三次样条核函数。
在SPH的应用中,边界条件的处理既是该方法的优点,也是目前的薄弱环节。研究了处理固定边界条件的两种类型的虚粒子,第一种类型的虚粒子设置在固定边界上,第二种类型的虚粒子分布在边界的外部,通常在边界条件不断变化的场合下使用。第二种类型的虚粒子按以下的方式构造,即给定一个实粒子i,则在边界外与实粒子对称处分布一个虚粒子,这些虚粒子具有与相对应实粒子相同的压力和密度,但速度方向相反。为了避免数值震荡,提高算法的稳定性,防止粒子间相互接近时的非物理穿透,在SPH方法的动量方程中引入了人工粘度来进行修正。
本文使用跳蛙法进行时间积分,跳蛙法的优点是计算时所需要的存储量低,而且在每一次计算中只需要进行一次优化估值。给出了SPH算法的程序结构,SPH的基本方法和SPH中其它数值方面的伴随算法使得SPH程序具有一些特殊性质。这些特殊性质都包含在时间积分过程的主循环中。对腔内剪切流动问题、冲击管问题相关算例进行了研究,测试了SPH方法在不同的流体动力学问题中的实用性,对于以上问题,SPH方法均可得到令人满意的结果。
在分析SPH方法单相流基本原理和离散思想的基础上,提出了SPH的两相耦合流动仿真方法。由于SPH方法不需要背景网格,是一种纯无网格方法,在对问题进行粒子离散化的过程中,布置的粒子本身具有物理属性,因此,只要能够正确处理具有相同或者不同属性的粒子之间的相互作用,通过整体粒子的运动分布就能够辨析出不同物质之间的交界面,这样也就能够描述具有不同属性的两相流体的运动情况。在SPH两相耦合流动仿真中,可以通过将固体相或液体相离散为与气体相一样的SPH粒子,但其上携带的是固体相或液体相的物理属性来实现。
研究了保证SPH两相耦合流动数值仿真过程正常进行需要做的技术改进。将密度正则化引入SPH两相耦合流动数值仿真方程,解决了两相流体交界面附近由于密度差异过大导致的边界效应,提高了密度不同或不连续的两相流体交界面处的精度。研究了SPH两相耦合流动中的人工状态方程,通过在两相耦合流动中密度小的物质的状态方程中添加气体内聚力项,解决了密度较小的粒子的逃逸问题。研究了SPH两相耦合流动中的速度修正,引入XSPH中的速度修正方法,防止了粒子间的相互穿透,使得到的两相耦合流动交界面更加清晰和光滑。综合使用SPH两相耦合流动方法,实现了气泡上浮和溃坝两相流动的数值仿真,得到的结果验证了本文提出的SPH两相耦合流动方法的正确性和可行性。
实现了SPH方法在三维流体流动中的技术应用。分析了全配对搜索法和链表搜索法,将Bucket搜索优化算法和树形搜索优化算法引入到SPH方法中,并给出了程序实现,有效地提高了光滑粒子法的计算效率。深入研究了粒子对的相互作用,由于粒子间的相互作用是基于点对点方式的,因此可在SPH仿真分析中应用成对相互作用法来提高计算效率和节省存储空间。成对相互作用法是通过应用最近相邻粒子搜索法来实现的,并为之后进行的SPH求和法存储必须的数据。研究了提高计算结果精度的几种方法：在对核近似连续性理论分析的基础上,提出了初始粒子的配置方法,须尽量将粒子均匀分布,并且使所有粒子质量相同或者质量呈连续变化；采用一种简单高效的基于密度变换的可变光滑长度技术；对光滑长度进行优化和松弛更新,给出了优化和松弛过程的步骤和参数,有效解决了现有SPH方法在处理三维流体流动过程中计算误差大的问题。
应用SPH方法实现了三维管道气力输送过程的数值仿真,分析了影响管道气力输送过程的主要因素,推导了粒子速度和输送气体速度的关系,提出了拟合方法。研究了气力输送过程中管道边界的实现方法,给出了使用应变率及应力的迭代近似法求解粘性力的方法,并分析了管道中输送物料的分布及压力情况。提出了管道边界层流效应的仿真方法：通过设置边界虚粒子和实粒子的粘度参数,而在管道中部的粒子并不设置粘度参数,按无粘处理,能够模拟边界层流效应。输送过程中气流碰到刚性壁面后将产生壁面热量,研究了壁面热量能量转换,给出了考虑人工热量条件下的SPH能量方程。合理简化影响要素,综合运用前面的相关技术,实现了管道输送问题的SPH程序原型。
研究了管道气力输送SPH数值仿真方法的VC++实现技术,由于Fortran具有接近数学公式的自然描述,并且计算精度高,在计算机里具有很高的执行效率,是目前流行较广的一种面向过程的适用于科学计算的高级语言,因此整个SPH程序实现选用Fortran来编写,但是它在实现人机对话、交互处理方面不很理想,界面也不够美观,图形处理功能也比较欠缺。VC++是目前个人计算机上深受欢迎的面向对象的程序设计环境。VC++具有强大的功能,但在科学计算方面实现起来却比较复杂。因此,用VC++和Fortran进行混合编程可以编制出具有友好界面和良好计算功能的程序。本文通过综合运用SPH基本原理,SPH两相耦合流动方法,SPH三维仿真技术,实现了气力输送过程的SPH数值仿真,在保持计算效率的基础上,通过VC++调用Fortran生成的动态链接库(.dll)文件,实现了SPH程序的交互功能。
综上所述,本文提出的SPH两相耦合流动仿真方法,将SPH方法延伸到了两相耦合流动中,完善了现有的光滑粒子流体动力学应用理论。本文实现了SPH方法在三维仿真流动中的应用,将树形搜索优化算法引入到SPH方法中,有效地提高了光滑粒子法的计算效率,特别是在粒子数量较多的时候,计算效率优势更加明显。本文给出了提高计算结果精度的几种方法。首次将SPH方法应用在三维管道气力输送仿真中,扩展了SPH方法的工程应用领域,给出了考虑粘性和壁面热量转换的计算方法,分析了管道中输送物料的分布及压力情况,提出了管道边界层流效应的仿真方法。在保持计算效率的基础上,通过VC++调用Fortran生成的动态链接库文件,实现了SPH程序的交互功能,为SPH方法真正辅助试验创造了条件。这些工作具有重要的理论和工程应用价值。
本课题得到了国家建设高水平大学公派研究生项目(留金出[2007]3020)的支持。
Numerical simulation using computers has increasingly become a very important approach for solving problems in engineering and science. It plays a valuable role in providing tests and examinations for theories, offering insights to complex physics, and assisting in the interpretation and even the discovery of new phenomena.
Despite grid-based numerical method has been widely used in many respects, there are still shortcomings, such as in large deformation of computational fluid dynamics, interfaces of movement material, free surface and so on, which limit their applications in many complex problems due to mesh distortion leading to too large calculation error or can not be carried out.
In recent years, meshless method has been paid great attention, this method in many applications are superior to the traditional grid-based finite element method, finite difference method and the finite volume method and so on.This paper relies on Virtual Engineering Research Center of Shandong University and Centre for Advanced Computations in Engineering Science of NUS, makes a systematic study on corresponding technologies of a new generation meshless method-smooth particle hydrodynamics (SPH) method in the application process. This paper achieves the applications of SPH two-phase coupling and three-dimensional, and a numerical simulation of duct pneumatic conveying process, which extends engineering application fields of SPH method, creating the conditions for a real-assisted testing of SPH, has important theoretical and application value.
The main contents in this thesis are as follows:
A detailed analysis of basic ideas and solving problems process of the Smoothed Particle Hydrodynamics method is developed. Because there is no connection between particles of domain, the particles computed depend on the current local distribution of particles, so SPH method is meshless with adaptability attributes. In the thesis a systematical analysis of the Lagrange Navier-Stokes equations is studied, and the SPH expression with Navier-Stokes equations of the density, the momentum equations and the energy equations using SPH particle approximation is derived.
The key technologies of fluid dynamics SPH numerical calculation are investigated. The algorithm should be handled specially in order to be suitable for simulating flow problem with different character. There are two approaches to evolve density in the conventional SPH method. The first approach is the summation density. Another approach of particle approximation for density is the continuity density. Although the density summation approach needs more computational effort, the summation density approach seems more popular in practical applications of SPH, partly because it well represents the essence of the SPH approximation. For simulating events with strong discontinuity, the continuity density approach is preferred. In the SPH method, the Dirac function is approximated with the smoothing function, two kinds of smoothing functions in the commonly used:Gaussian kernel function and the cubic spline kernel function are analysed.
In SPH applications, the boundary condition is both advantage and weakness of this method nowadays. Two types of virtual particles used to deal with boundary condition are studied. The virtual particles of the first type are located right on the solid boundary. The virtual particles of the second type fill in the boundary region, which are usually used in the situation of boundary condition always changed. The virtual particles of the type II are constructed in the following way. For a certain real particle i, if it is located within the distance of kh, from the boundary, a virtual particle is placed symmetrically on the outside of the boundary. These virtual particles have the same density and pressure as the corresponding real particles but opposite velocity. In order to avoid the shock value and improve the stability of the algorithm to prevent the particles from non-physical penetration when they are close to each other, artificial viscosity is introduced to SPH momentum equation to amend.
The discrete SPH equations are integrated with Leap-Frog algorithm in this thesis. The advantage of the Leap-Frog algorithm is low memory storage required in the computation and the efficiency for one force evaluation per step. Program structure of SPH algorithm is given. The basic SPH method and the accompanied other various algorithms of SPH are result in some special features in the SPH coding. These special features are generally involved under the main loop of time integration process. Shock tube problem and shear driven cavity problem are studied to test usefulness of SPH method in different fluid dynamics problems, SPH method can get satisfactory results in the above questions.
Based on analyzing the basic principles of single-phase flow and discrete thought, SPH two-phase coupling simulation method is proposed. SPH method is a pure meshless method without background grid. The particles are of physical properties in the discretization process, so long as they can properly handle the interaction between particles with the same or different properties, the interface between different materials can be discriminated by the distribution of the whole movement of particles, so the movement of two-phase coupling flow can be described with different properties. In the SPH two-phase coupled flow simulation, solid phase or liquid phase is discreted the same SPH particles as gas phase particles, carrying the properties of solid phase or liquid phase.
Technical improvement is studied to ensure the normal two-phase coupling flow numerical simulation. The density is normalized to solve boundary effect induced by too large difference near the interface between two phase fluids, to increase accuracy of interface between two phase fluids. The artificial state equation of SPH two phase coupling flow is studied. The problem of escape of smaller particles is solved by adding gas cohesion items in the state equation of particles with smaller density. Velocity correction of SPH two phase coupling is studied. The velocity correction method is introduced to prevent inter-penetration of the particles, so the interface of two-phase coupling is more clear and smooth. Bubbles numerical simulation problem and dam break numerical simulation problem are solved by using two phase coupling SPH method. The result is obtained and is verify that the proposed two-phase coupling SPH method is correct and feasible.
Technical application of SPH in three-dimensional fluid flow is implemented. All-pair search algorithm, linked-list search algorithm are analyzed. And tree search optimization algorithm is introduced to the SPH method. With the program implemented, the computational efficiency is improved effectively. Because the pairwise interaction is based on the way of point to point, computational efficiency is improved and storage space is saved by using pairwise interaction in the SPH simulation and analysis. Thus the pairwise interaction is researched deeply, it is implemented by using the method of nearest neighboring particle searching, which store the necessary data for the coming SPH summation. Several methods for improving computational efficiency are researched. Based on the theoretical analysis of consistency of kernel approximation, the initial configuration method of particles is presented. Particles should be distributed as uniformly as possible, and all the qualities of the particles are the same or changing continuously. It is an effective solution to the problem of large errors of existing three dimensional SPH method in dealing with process of fluid flow. Using a simple and efficient variable smoothing length technology based on density transformation, the smoothing length is optimized and relaxed, and the steps and parameters of optimization and relaxation are given in this thesis.
SPH method is used in duct pneumatic conveying simulation. Duct boundary implement method of pneumatic conveying process is researched. The method for solving viscous force is given using iterative approximation of strain rate and stress. The distribution of transportation materials and pressure in the duct is analyzed. Simulation method for duct boundary layer flow is proposed. The effect of boundary layer flow can be simulated by setting the viscosity parameter of boundary virtual particles and real particles, while central particles in the duct do not have a viscosity parameter, under the no-stick handling, the effect of boundary layer flow can be simulated. Wall heat will be produced when air encounters rigid wall during conveying process, wall heat energy conversion is researched, and SPH energy equation under the condition of artificial heat is given. SPH program prototype for duct conveying problem is implemented by simplifying impact factor reasonably, using related technologies mentioned above.
VC++implementation technology for SPH numerical simulation method of duct pneumatic conveying is researched. Since Fortran is close to a natural description of mathematical formulas, is high calculation precision and implementation efficiency, is a popular process-oriented language, so Fortran is used to achieve the whole SPH process. But it is far from ideal in human-computer dialogue, interactive processing, and its interface is not enough beautiful, graphics processing functions are quite lacking. VC++is a popular object-oriented programming language in personal computer. VC++has powerful features, but it is more complicated in achieving scientific computing. So programming mix VC++and Fortran may get a good computing program with a friendly interface. In the thesis, SPH numerical simulation of pneumatic conveying process is implemented by using SPH basic principle and SPH two-phase coupling method. Based on maintaining computational efficiency, by adoption of VC++calling Fortran generates a dynamic link library (.D11) files, the interactive features of the SPH program are implemented.
In summary, SPH two-phase coupling simulation method presented in the thesis extends SPH method, and completes existing smooth particle hydrodynamics applied theory. SPH method is employed in three-dimensional fluid flow, tree-search optimization algorithm is introduced to the SPH method, effectively improving the computational efficiency of smooth particle method. Especially in a large number of particles, the advantage of computational efficiency is more obvious. Several methods to improve the accuracy of calculated results are mentioned. SPH method is applied to the duct conveying simulation in the first time and extends the engineering applications, and calculation method is given in the condition of viscosity and wall heat conversion. The distribution of transportation materials and pressure in the duct is analyzed. Simulation method for duct boundary layer flow is proposed. Based on maintaining computational efficiency, by adoption of VC++calling Fortran generates a dynamic link library (.dll) files, the interactive features of the SPH program are implemented, creating the conditions for SPH really supporting test. These efforts have important theoretical and engineering application value.
This paper has been supported by the 2007 China PhD abroad program for building high level university.
基于网格的数值方法虽然已经有广泛的应用,但是在很多方面仍存在不足之处,比如在计算流体动力学中的大变形、运动物质交界面、自由表面等问题时,由于网格产生畸变导致计算误差过大或无法进行,从而使其在许多问题的应用上受到限制。
近年来,无网格法倍受关注,这种方法在许多应用中都优于传统的基于网格的有限元法、有限差分法以及有限体积法等数值方法。本文依托山东大学虚拟工程研究中心和新加坡国立大学ACES实验室,系统地研究了新一代无网格方法—光滑粒子流体动力学方法(SPH)在应用及扩展过程中的相关关键技术,实现了SPH方法的两相耦合应用和三维应用,在此基础上对三维管道气力输送过程进行了数值仿真,扩展了SPH方法的工程应用领域,为SPH方法真正辅助试验创造了条件,具有重要的理论和应用价值。
论文的主要内容如下：
系统总结了基于网格的数值方法,指出其在很多方面存在不足之处,由于网格畸变而使其在许多问题的应用上受到限制,在此基础上,详细分析了新一代无网格法—光滑粒子动力学方法的基本思想和求解问题的过程,由于问题域粒子之间没有任何连接,运算中使用的粒子取决于当前局部分布的粒子,得出了SPH方法的无网格属性和自适应性。系统分析了拉格朗日型的Navier-Stokes方程,应用SPH粒子近似法推导出了Navier-Stokes的密度方程、动量方程和能量方程的SPH表达式。
研究了流体动力学SPH数值计算中的关键技术,指出了为了使算法适合模拟不同的流体特性问题,必须对算法进行特别地处理。在SPH方法中有两种方式对密度进行展开：一种是密度求和法,另一种是连续密度法。虽然密度求和法所需要的计算量大,但因为密度求和法体现了SPH近似法的本质,所以使用得较为广泛。连续性密度法主要用于仿真具有强间断的问题。在SPH方法中,用核函数来近似狄拉克δ函数,分析了在实际应用中常用的两种核函数：高斯核函数和三次样条核函数。
在SPH的应用中,边界条件的处理既是该方法的优点,也是目前的薄弱环节。研究了处理固定边界条件的两种类型的虚粒子,第一种类型的虚粒子设置在固定边界上,第二种类型的虚粒子分布在边界的外部,通常在边界条件不断变化的场合下使用。第二种类型的虚粒子按以下的方式构造,即给定一个实粒子i,则在边界外与实粒子对称处分布一个虚粒子,这些虚粒子具有与相对应实粒子相同的压力和密度,但速度方向相反。为了避免数值震荡,提高算法的稳定性,防止粒子间相互接近时的非物理穿透,在SPH方法的动量方程中引入了人工粘度来进行修正。
本文使用跳蛙法进行时间积分,跳蛙法的优点是计算时所需要的存储量低,而且在每一次计算中只需要进行一次优化估值。给出了SPH算法的程序结构,SPH的基本方法和SPH中其它数值方面的伴随算法使得SPH程序具有一些特殊性质。这些特殊性质都包含在时间积分过程的主循环中。对腔内剪切流动问题、冲击管问题相关算例进行了研究,测试了SPH方法在不同的流体动力学问题中的实用性,对于以上问题,SPH方法均可得到令人满意的结果。
在分析SPH方法单相流基本原理和离散思想的基础上,提出了SPH的两相耦合流动仿真方法。由于SPH方法不需要背景网格,是一种纯无网格方法,在对问题进行粒子离散化的过程中,布置的粒子本身具有物理属性,因此,只要能够正确处理具有相同或者不同属性的粒子之间的相互作用,通过整体粒子的运动分布就能够辨析出不同物质之间的交界面,这样也就能够描述具有不同属性的两相流体的运动情况。在SPH两相耦合流动仿真中,可以通过将固体相或液体相离散为与气体相一样的SPH粒子,但其上携带的是固体相或液体相的物理属性来实现。
研究了保证SPH两相耦合流动数值仿真过程正常进行需要做的技术改进。将密度正则化引入SPH两相耦合流动数值仿真方程,解决了两相流体交界面附近由于密度差异过大导致的边界效应,提高了密度不同或不连续的两相流体交界面处的精度。研究了SPH两相耦合流动中的人工状态方程,通过在两相耦合流动中密度小的物质的状态方程中添加气体内聚力项,解决了密度较小的粒子的逃逸问题。研究了SPH两相耦合流动中的速度修正,引入XSPH中的速度修正方法,防止了粒子间的相互穿透,使得到的两相耦合流动交界面更加清晰和光滑。综合使用SPH两相耦合流动方法,实现了气泡上浮和溃坝两相流动的数值仿真,得到的结果验证了本文提出的SPH两相耦合流动方法的正确性和可行性。
实现了SPH方法在三维流体流动中的技术应用。分析了全配对搜索法和链表搜索法,将Bucket搜索优化算法和树形搜索优化算法引入到SPH方法中,并给出了程序实现,有效地提高了光滑粒子法的计算效率。深入研究了粒子对的相互作用,由于粒子间的相互作用是基于点对点方式的,因此可在SPH仿真分析中应用成对相互作用法来提高计算效率和节省存储空间。成对相互作用法是通过应用最近相邻粒子搜索法来实现的,并为之后进行的SPH求和法存储必须的数据。研究了提高计算结果精度的几种方法：在对核近似连续性理论分析的基础上,提出了初始粒子的配置方法,须尽量将粒子均匀分布,并且使所有粒子质量相同或者质量呈连续变化；采用一种简单高效的基于密度变换的可变光滑长度技术；对光滑长度进行优化和松弛更新,给出了优化和松弛过程的步骤和参数,有效解决了现有SPH方法在处理三维流体流动过程中计算误差大的问题。
应用SPH方法实现了三维管道气力输送过程的数值仿真,分析了影响管道气力输送过程的主要因素,推导了粒子速度和输送气体速度的关系,提出了拟合方法。研究了气力输送过程中管道边界的实现方法,给出了使用应变率及应力的迭代近似法求解粘性力的方法,并分析了管道中输送物料的分布及压力情况。提出了管道边界层流效应的仿真方法：通过设置边界虚粒子和实粒子的粘度参数,而在管道中部的粒子并不设置粘度参数,按无粘处理,能够模拟边界层流效应。输送过程中气流碰到刚性壁面后将产生壁面热量,研究了壁面热量能量转换,给出了考虑人工热量条件下的SPH能量方程。合理简化影响要素,综合运用前面的相关技术,实现了管道输送问题的SPH程序原型。
研究了管道气力输送SPH数值仿真方法的VC++实现技术,由于Fortran具有接近数学公式的自然描述,并且计算精度高,在计算机里具有很高的执行效率,是目前流行较广的一种面向过程的适用于科学计算的高级语言,因此整个SPH程序实现选用Fortran来编写,但是它在实现人机对话、交互处理方面不很理想,界面也不够美观,图形处理功能也比较欠缺。VC++是目前个人计算机上深受欢迎的面向对象的程序设计环境。VC++具有强大的功能,但在科学计算方面实现起来却比较复杂。因此,用VC++和Fortran进行混合编程可以编制出具有友好界面和良好计算功能的程序。本文通过综合运用SPH基本原理,SPH两相耦合流动方法,SPH三维仿真技术,实现了气力输送过程的SPH数值仿真,在保持计算效率的基础上,通过VC++调用Fortran生成的动态链接库(.dll)文件,实现了SPH程序的交互功能。
综上所述,本文提出的SPH两相耦合流动仿真方法,将SPH方法延伸到了两相耦合流动中,完善了现有的光滑粒子流体动力学应用理论。本文实现了SPH方法在三维仿真流动中的应用,将树形搜索优化算法引入到SPH方法中,有效地提高了光滑粒子法的计算效率,特别是在粒子数量较多的时候,计算效率优势更加明显。本文给出了提高计算结果精度的几种方法。首次将SPH方法应用在三维管道气力输送仿真中,扩展了SPH方法的工程应用领域,给出了考虑粘性和壁面热量转换的计算方法,分析了管道中输送物料的分布及压力情况,提出了管道边界层流效应的仿真方法。在保持计算效率的基础上,通过VC++调用Fortran生成的动态链接库文件,实现了SPH程序的交互功能,为SPH方法真正辅助试验创造了条件。这些工作具有重要的理论和工程应用价值。
本课题得到了国家建设高水平大学公派研究生项目(留金出[2007]3020)的支持。
Numerical simulation using computers has increasingly become a very important approach for solving problems in engineering and science. It plays a valuable role in providing tests and examinations for theories, offering insights to complex physics, and assisting in the interpretation and even the discovery of new phenomena.
Despite grid-based numerical method has been widely used in many respects, there are still shortcomings, such as in large deformation of computational fluid dynamics, interfaces of movement material, free surface and so on, which limit their applications in many complex problems due to mesh distortion leading to too large calculation error or can not be carried out.
In recent years, meshless method has been paid great attention, this method in many applications are superior to the traditional grid-based finite element method, finite difference method and the finite volume method and so on.This paper relies on Virtual Engineering Research Center of Shandong University and Centre for Advanced Computations in Engineering Science of NUS, makes a systematic study on corresponding technologies of a new generation meshless method-smooth particle hydrodynamics (SPH) method in the application process. This paper achieves the applications of SPH two-phase coupling and three-dimensional, and a numerical simulation of duct pneumatic conveying process, which extends engineering application fields of SPH method, creating the conditions for a real-assisted testing of SPH, has important theoretical and application value.
The main contents in this thesis are as follows:
A detailed analysis of basic ideas and solving problems process of the Smoothed Particle Hydrodynamics method is developed. Because there is no connection between particles of domain, the particles computed depend on the current local distribution of particles, so SPH method is meshless with adaptability attributes. In the thesis a systematical analysis of the Lagrange Navier-Stokes equations is studied, and the SPH expression with Navier-Stokes equations of the density, the momentum equations and the energy equations using SPH particle approximation is derived.
The key technologies of fluid dynamics SPH numerical calculation are investigated. The algorithm should be handled specially in order to be suitable for simulating flow problem with different character. There are two approaches to evolve density in the conventional SPH method. The first approach is the summation density. Another approach of particle approximation for density is the continuity density. Although the density summation approach needs more computational effort, the summation density approach seems more popular in practical applications of SPH, partly because it well represents the essence of the SPH approximation. For simulating events with strong discontinuity, the continuity density approach is preferred. In the SPH method, the Dirac function is approximated with the smoothing function, two kinds of smoothing functions in the commonly used:Gaussian kernel function and the cubic spline kernel function are analysed.
In SPH applications, the boundary condition is both advantage and weakness of this method nowadays. Two types of virtual particles used to deal with boundary condition are studied. The virtual particles of the first type are located right on the solid boundary. The virtual particles of the second type fill in the boundary region, which are usually used in the situation of boundary condition always changed. The virtual particles of the type II are constructed in the following way. For a certain real particle i, if it is located within the distance of kh, from the boundary, a virtual particle is placed symmetrically on the outside of the boundary. These virtual particles have the same density and pressure as the corresponding real particles but opposite velocity. In order to avoid the shock value and improve the stability of the algorithm to prevent the particles from non-physical penetration when they are close to each other, artificial viscosity is introduced to SPH momentum equation to amend.
The discrete SPH equations are integrated with Leap-Frog algorithm in this thesis. The advantage of the Leap-Frog algorithm is low memory storage required in the computation and the efficiency for one force evaluation per step. Program structure of SPH algorithm is given. The basic SPH method and the accompanied other various algorithms of SPH are result in some special features in the SPH coding. These special features are generally involved under the main loop of time integration process. Shock tube problem and shear driven cavity problem are studied to test usefulness of SPH method in different fluid dynamics problems, SPH method can get satisfactory results in the above questions.
Based on analyzing the basic principles of single-phase flow and discrete thought, SPH two-phase coupling simulation method is proposed. SPH method is a pure meshless method without background grid. The particles are of physical properties in the discretization process, so long as they can properly handle the interaction between particles with the same or different properties, the interface between different materials can be discriminated by the distribution of the whole movement of particles, so the movement of two-phase coupling flow can be described with different properties. In the SPH two-phase coupled flow simulation, solid phase or liquid phase is discreted the same SPH particles as gas phase particles, carrying the properties of solid phase or liquid phase.
Technical improvement is studied to ensure the normal two-phase coupling flow numerical simulation. The density is normalized to solve boundary effect induced by too large difference near the interface between two phase fluids, to increase accuracy of interface between two phase fluids. The artificial state equation of SPH two phase coupling flow is studied. The problem of escape of smaller particles is solved by adding gas cohesion items in the state equation of particles with smaller density. Velocity correction of SPH two phase coupling is studied. The velocity correction method is introduced to prevent inter-penetration of the particles, so the interface of two-phase coupling is more clear and smooth. Bubbles numerical simulation problem and dam break numerical simulation problem are solved by using two phase coupling SPH method. The result is obtained and is verify that the proposed two-phase coupling SPH method is correct and feasible.
Technical application of SPH in three-dimensional fluid flow is implemented. All-pair search algorithm, linked-list search algorithm are analyzed. And tree search optimization algorithm is introduced to the SPH method. With the program implemented, the computational efficiency is improved effectively. Because the pairwise interaction is based on the way of point to point, computational efficiency is improved and storage space is saved by using pairwise interaction in the SPH simulation and analysis. Thus the pairwise interaction is researched deeply, it is implemented by using the method of nearest neighboring particle searching, which store the necessary data for the coming SPH summation. Several methods for improving computational efficiency are researched. Based on the theoretical analysis of consistency of kernel approximation, the initial configuration method of particles is presented. Particles should be distributed as uniformly as possible, and all the qualities of the particles are the same or changing continuously. It is an effective solution to the problem of large errors of existing three dimensional SPH method in dealing with process of fluid flow. Using a simple and efficient variable smoothing length technology based on density transformation, the smoothing length is optimized and relaxed, and the steps and parameters of optimization and relaxation are given in this thesis.
SPH method is used in duct pneumatic conveying simulation. Duct boundary implement method of pneumatic conveying process is researched. The method for solving viscous force is given using iterative approximation of strain rate and stress. The distribution of transportation materials and pressure in the duct is analyzed. Simulation method for duct boundary layer flow is proposed. The effect of boundary layer flow can be simulated by setting the viscosity parameter of boundary virtual particles and real particles, while central particles in the duct do not have a viscosity parameter, under the no-stick handling, the effect of boundary layer flow can be simulated. Wall heat will be produced when air encounters rigid wall during conveying process, wall heat energy conversion is researched, and SPH energy equation under the condition of artificial heat is given. SPH program prototype for duct conveying problem is implemented by simplifying impact factor reasonably, using related technologies mentioned above.
VC++implementation technology for SPH numerical simulation method of duct pneumatic conveying is researched. Since Fortran is close to a natural description of mathematical formulas, is high calculation precision and implementation efficiency, is a popular process-oriented language, so Fortran is used to achieve the whole SPH process. But it is far from ideal in human-computer dialogue, interactive processing, and its interface is not enough beautiful, graphics processing functions are quite lacking. VC++is a popular object-oriented programming language in personal computer. VC++has powerful features, but it is more complicated in achieving scientific computing. So programming mix VC++and Fortran may get a good computing program with a friendly interface. In the thesis, SPH numerical simulation of pneumatic conveying process is implemented by using SPH basic principle and SPH two-phase coupling method. Based on maintaining computational efficiency, by adoption of VC++calling Fortran generates a dynamic link library (.D11) files, the interactive features of the SPH program are implemented.
In summary, SPH two-phase coupling simulation method presented in the thesis extends SPH method, and completes existing smooth particle hydrodynamics applied theory. SPH method is employed in three-dimensional fluid flow, tree-search optimization algorithm is introduced to the SPH method, effectively improving the computational efficiency of smooth particle method. Especially in a large number of particles, the advantage of computational efficiency is more obvious. Several methods to improve the accuracy of calculated results are mentioned. SPH method is applied to the duct conveying simulation in the first time and extends the engineering applications, and calculation method is given in the condition of viscosity and wall heat conversion. The distribution of transportation materials and pressure in the duct is analyzed. Simulation method for duct boundary layer flow is proposed. Based on maintaining computational efficiency, by adoption of VC++calling Fortran generates a dynamic link library (.dll) files, the interactive features of the SPH program are implemented, creating the conditions for SPH really supporting test. These efforts have important theoretical and engineering application value.
This paper has been supported by the 2007 China PhD abroad program for building high level university.
引文
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