不可压多松弛格子Boltzmann方法的研究及其应用
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摘要
无论是燃烧还是气固两相流的计算,都有一个共同的研究领域——流体力学,对流体力学的研究是研究复杂物理现象的基础。格子Boltzmann方法(LBM)是上世纪80年代末从格子气自动机(LGCA)发展而来的一种新的计算流体数值方法。与传统数值方法的研究视角不同,LBM是从微观粒子运动的层面来对流体进行数值模拟的。LBM的描述对象是单一粒子的分布函数,分布函数的控制方程为经典Boltzmann方程。而LB方程则是Boltzmann方程在相空间的离散形式,这种离散包括粒子速度空间、时间空间的离散。通过Chapman-Enskog展开,利用物理量的守恒关系,在满足小Knudsen数和小Mach数条件下,可以将LB方程还原到描述流体运动的宏观流体力学方程。从而,我们可以通过数值模拟粒子的分布来达到描述宏观流体运动的目的。
格子Boltzmann方法是与现代计算机匹配的高效新算法,它具有天然并行性、结构简单、易于编程的优点,在流体力学等领域得到了广泛的应用,已经成为研究非线性现象和复杂系统的重要方法之一。
但是,格子Boltzmann方法至今还有不完善之处,比如外力的处理,不可压多松弛模型的研究。该方法虽然已经在多相流、多孔介质流、悬浮粒子流、磁流体力学等领域取得了很大的成功,但是目前在微重力流体力学中的应用研究甚少。另外,尽管LB方法程序简洁,但是随着模拟问题的复杂性和人们对模拟结果精度要求的提高,使得计算量剧增,因此对程序的优化至关重要,它直接影响着LB方法在工程实际中的应用。因此,本文就以上提出的几个方面做了有益的尝试,为相关工作的深入展开奠定了必要的基础。
首先,我们构造了一种求解含外力项的Navier-Stokes方程的格子Boltzmann模型。不同于已有的模型,将外力的空间导数加到演化方程中,通过Chapman-Enskog(C-E)展开,不需要多余的假设可以恢复到宏观方程。详细讨论了三种离散格式,可以证明,现有的一些模型是本文提出模型的特例。数值实验结果表明,我们提出的方法具有二阶数值精度和较好的数值稳定性。
其次,提出了二维九速和八速不可压多松弛模型,该模型基于Guo提出的LBGK模型。通过Gram-Schimidt正交化过程,构造了八速模型的线性变换矩阵,该矩阵满足Ginzburg给出的通用的格式。通过多尺度展开,两类模型都可以恢复到不可压的宏观方程,该模型消除了已有模型中存在的可压缩效应。对各种问题的数值实验结果表明,模型的数值稳定性很好。为模拟不可压流动提供一种数值稳定性较好的方法。
第三,对于微重力流体力学中一类重要的流动——热毛细对流,我们构造了双分布的LB模型。采用非平衡态外推格式使得边界处理变得极其简单可行。对二维上表面是自由边界的矩形容器内的熔体做数值实验,验证了该模型的正确性。因此,LB在微流体力学中的应用是可行的,为研究微重力环境下的各种流体提供了一种新的介观方法。
第四,以经典算例-方腔流为例,对格子Boltzmann方法的核心代码进行了优化,主要做了时间和空间上的优化,优化的程序计算效率提高数倍。在并行的框架下,核心演化的代码换为优化后的程序,计算效率仍然有大幅度的提高。
总之,本文提出两个多松弛不可压LB模型,改善了单松弛LB模型的数值稳定性;进而提出一种外力处理的通用格式,并给出了严格的数学推导;研究了微重力环境下表面张力驱动的热毛细对流问题,为LB方法在该领域的应用提供很好的基础;设计的高效LB算法提升了其实际应用的空间。
Whatever computation of combusion or gas-partile flow, there is a basic research field - fluid mechanics. Lattice Boltzmann method (LBM) is a newly developed computational fluid numerical method, which is originated from lattice gas cellular automata (LGCA) at the end of 1980's last centrury. LBM simulates the fluid movement at the microscopic particle level that is absolutely different from the conventional numerical methods. Single particle distribution satisfying classic Boltzmann equation is described by LBM. The so-called LB equation is a special discrete form of continuum Boltzmann equation in discrete particle velocity, discrete spatial and temporal spaces. By using the Chapman-Enskog multi-scale analysis technique and the conservations of physical variables, the LB equation can be recovered to the fluid dynamical equation at macroscopic level with the low Knudsen number and low Mach number assumptions. As a result, we can obtain the macroscopic fluid movement through calculating the particle distribution numerically.
LB method is a high performance computing method with notable advantages, such as fully parallelism, easy implementation and simple codes. Until now, it has been widely applied in computational fluid dynamics and has become one of the most important tools for nonlinear science and complex system.
However, there still exist some disadvantages in LBM, such as the treatment of the force term and multi-relaxation-time LB model. The applications of LBM have achieved great success in multiphase flow, porous flow, suspending particle flow, magnetohydrodynamics, etc. But less research on the microgravity fluid is found. On the other hand, although the code of LBM is simple, the computation may be increased dramatically as the more complex problem and the needed better numerical accuracy. It is important to do some optimization in order to apply the LBM in the engineering. Therefore, we make some valuable relevant research work to remedy the gap.
Firstly, a novel scheme of the lattice BGK (LBGK) model with a force term has been proposed. Unlike the existing models, an appropriate term was added in the evolutionary equation. Through the Chapman-Enskog (C-E) procedure the Navier-Stokes (N-S) equations with a force term can be recovered with the kinetic viscosity. Three discrete methods of the added term have been discussed in detail. It can be proved that some existing models are the special cases of the model. Numerical simulation results show that the scheme is of second numerical accuracy and better stability.
Second, two-dimensional nine-velocity and eight-velocity lattice Boltzmann models with multi-relaxation-time are proposed for incompressible flows, in which the equilibria in the momentum space are derived from an earlier incompressible lattice Boltzmann model with single relaxation time by Guo. Via the Gram-Schimidt orthogonalization procedure the eight-by-eight transformation matrix can be constructed which satisfies the general form by Ginzburg. Through the Chapman-Enskog expansion, the incompressible Navier-Stokes equations are recovered from the two models which eliminate the compressible effect in the existed models. The results of numerical tests exhibits much better numerical stability than the single relaxation time model. These two models afford a new method for incompressible flow.
In the third part, we constructed a two distribution function LBGK model for the thermocapillary flow in microgravity. The boundary conditions for the ther-malcapillary flow are treated using the non-equilibrium extrapolation scheme. The model is validated by simulating the thermocapillary flow in a two-dimensional square cavity with a single free surface and differentially heated side walls. It is feasible to apply the LBM to the microgravity fluid dynamics.
Finally, We take the classical problem-cavity flow as an example and optimize the kernel codes of the LBM. The optimization include two aspects: time and space. The efficiency of the optimized code increased much more. In the parallel frame, the efficiency also increased if the kernel code is taken the optimized code.
In conclusion, this thesis proposes two incompressible LB models which improve numerical stablity of LBGK model, and a general scheme for force term proved. Furthermore the thermocapillary flow in microgravity has been considered and this work has made many valuable efforts to accelerate the applications of LBM. At last, the high performance algorithm can broaden the applications of LBM in our subject.
格子Boltzmann方法是与现代计算机匹配的高效新算法,它具有天然并行性、结构简单、易于编程的优点,在流体力学等领域得到了广泛的应用,已经成为研究非线性现象和复杂系统的重要方法之一。
但是,格子Boltzmann方法至今还有不完善之处,比如外力的处理,不可压多松弛模型的研究。该方法虽然已经在多相流、多孔介质流、悬浮粒子流、磁流体力学等领域取得了很大的成功,但是目前在微重力流体力学中的应用研究甚少。另外,尽管LB方法程序简洁,但是随着模拟问题的复杂性和人们对模拟结果精度要求的提高,使得计算量剧增,因此对程序的优化至关重要,它直接影响着LB方法在工程实际中的应用。因此,本文就以上提出的几个方面做了有益的尝试,为相关工作的深入展开奠定了必要的基础。
首先,我们构造了一种求解含外力项的Navier-Stokes方程的格子Boltzmann模型。不同于已有的模型,将外力的空间导数加到演化方程中,通过Chapman-Enskog(C-E)展开,不需要多余的假设可以恢复到宏观方程。详细讨论了三种离散格式,可以证明,现有的一些模型是本文提出模型的特例。数值实验结果表明,我们提出的方法具有二阶数值精度和较好的数值稳定性。
其次,提出了二维九速和八速不可压多松弛模型,该模型基于Guo提出的LBGK模型。通过Gram-Schimidt正交化过程,构造了八速模型的线性变换矩阵,该矩阵满足Ginzburg给出的通用的格式。通过多尺度展开,两类模型都可以恢复到不可压的宏观方程,该模型消除了已有模型中存在的可压缩效应。对各种问题的数值实验结果表明,模型的数值稳定性很好。为模拟不可压流动提供一种数值稳定性较好的方法。
第三,对于微重力流体力学中一类重要的流动——热毛细对流,我们构造了双分布的LB模型。采用非平衡态外推格式使得边界处理变得极其简单可行。对二维上表面是自由边界的矩形容器内的熔体做数值实验,验证了该模型的正确性。因此,LB在微流体力学中的应用是可行的,为研究微重力环境下的各种流体提供了一种新的介观方法。
第四,以经典算例-方腔流为例,对格子Boltzmann方法的核心代码进行了优化,主要做了时间和空间上的优化,优化的程序计算效率提高数倍。在并行的框架下,核心演化的代码换为优化后的程序,计算效率仍然有大幅度的提高。
总之,本文提出两个多松弛不可压LB模型,改善了单松弛LB模型的数值稳定性;进而提出一种外力处理的通用格式,并给出了严格的数学推导;研究了微重力环境下表面张力驱动的热毛细对流问题,为LB方法在该领域的应用提供很好的基础;设计的高效LB算法提升了其实际应用的空间。
Whatever computation of combusion or gas-partile flow, there is a basic research field - fluid mechanics. Lattice Boltzmann method (LBM) is a newly developed computational fluid numerical method, which is originated from lattice gas cellular automata (LGCA) at the end of 1980's last centrury. LBM simulates the fluid movement at the microscopic particle level that is absolutely different from the conventional numerical methods. Single particle distribution satisfying classic Boltzmann equation is described by LBM. The so-called LB equation is a special discrete form of continuum Boltzmann equation in discrete particle velocity, discrete spatial and temporal spaces. By using the Chapman-Enskog multi-scale analysis technique and the conservations of physical variables, the LB equation can be recovered to the fluid dynamical equation at macroscopic level with the low Knudsen number and low Mach number assumptions. As a result, we can obtain the macroscopic fluid movement through calculating the particle distribution numerically.
LB method is a high performance computing method with notable advantages, such as fully parallelism, easy implementation and simple codes. Until now, it has been widely applied in computational fluid dynamics and has become one of the most important tools for nonlinear science and complex system.
However, there still exist some disadvantages in LBM, such as the treatment of the force term and multi-relaxation-time LB model. The applications of LBM have achieved great success in multiphase flow, porous flow, suspending particle flow, magnetohydrodynamics, etc. But less research on the microgravity fluid is found. On the other hand, although the code of LBM is simple, the computation may be increased dramatically as the more complex problem and the needed better numerical accuracy. It is important to do some optimization in order to apply the LBM in the engineering. Therefore, we make some valuable relevant research work to remedy the gap.
Firstly, a novel scheme of the lattice BGK (LBGK) model with a force term has been proposed. Unlike the existing models, an appropriate term was added in the evolutionary equation. Through the Chapman-Enskog (C-E) procedure the Navier-Stokes (N-S) equations with a force term can be recovered with the kinetic viscosity. Three discrete methods of the added term have been discussed in detail. It can be proved that some existing models are the special cases of the model. Numerical simulation results show that the scheme is of second numerical accuracy and better stability.
Second, two-dimensional nine-velocity and eight-velocity lattice Boltzmann models with multi-relaxation-time are proposed for incompressible flows, in which the equilibria in the momentum space are derived from an earlier incompressible lattice Boltzmann model with single relaxation time by Guo. Via the Gram-Schimidt orthogonalization procedure the eight-by-eight transformation matrix can be constructed which satisfies the general form by Ginzburg. Through the Chapman-Enskog expansion, the incompressible Navier-Stokes equations are recovered from the two models which eliminate the compressible effect in the existed models. The results of numerical tests exhibits much better numerical stability than the single relaxation time model. These two models afford a new method for incompressible flow.
In the third part, we constructed a two distribution function LBGK model for the thermocapillary flow in microgravity. The boundary conditions for the ther-malcapillary flow are treated using the non-equilibrium extrapolation scheme. The model is validated by simulating the thermocapillary flow in a two-dimensional square cavity with a single free surface and differentially heated side walls. It is feasible to apply the LBM to the microgravity fluid dynamics.
Finally, We take the classical problem-cavity flow as an example and optimize the kernel codes of the LBM. The optimization include two aspects: time and space. The efficiency of the optimized code increased much more. In the parallel frame, the efficiency also increased if the kernel code is taken the optimized code.
In conclusion, this thesis proposes two incompressible LB models which improve numerical stablity of LBGK model, and a general scheme for force term proved. Furthermore the thermocapillary flow in microgravity has been considered and this work has made many valuable efforts to accelerate the applications of LBM. At last, the high performance algorithm can broaden the applications of LBM in our subject.
引文
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