格子法理论及其在计算流体力学中的应用研究
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摘要
作为经典的流体力学、数值计算方法和计算机技术三者的有机结合体，计算流体力学异军突起，得到不断发展和壮大，成为科学研究和工程设计的有效手段。理论分析、实验研究和数值模拟三者相互渗透，不仅推动了流体力学理论的新发展，而且加强了流体力学的工程应用。格子法就是在此背景下孕育发展起来的一种极具魅力的数值计算方法。
本文在广泛调研，追踪国内外研究进展及现状的基础上，针对目前格子法在工程应用中所面临的主要问题，开展了以下几方面的研究工作：
首先运用Chapman-Enskog展开及多尺度分析技术，揭示了格子法与传统数值计算方法间的内在联系，阐述了格子模型的思想本质及其建模过程。
其次，按照物理问题数学建模的原则，对格子法的误差进行了分析，给出了格子BGK方程再现Navier-Stokes方程时的压缩误差项，并数值验证了格子模型的声速及粘性系数等相关参数的精度，表明格子模型尽管具有时空二阶精度，能满足工程计算的要求，但随着Mach数增大，压缩误差逐渐成为主要误差，必须予以消除。
本文第四章建立了以压力、速度为独立变量的不可压格子BGK模型。其中密度保持为常数，使得从格子BGK方程能够导出宏观不可压NS方程，理论上消除了因密度变化导致的压缩误差。通过对定常、非定常问题的数值结果的比较，证明了本文不可压模型能有效消除压缩误差。进而对方腔流的速度场和压力场进行了计算，绘制了不同雷诺数下的流线图及压力等高线图，得到的回流涡的位置和流函数的值和现有的数据十分吻合，表明本文不可压格子模型是可靠的。
第五章考虑了数值方法另外一个方面的问题，在均匀流和剪切流两种背景流场下，运用Von Neumann线性分析法，针对d2q7、d2q9及d3q15格子模型，分析了质量分布参数、波数、松弛时间和平均流速等决定模型稳定性的主要参数对模型稳定性的影响，得到了对流场数值计算具有指导意义的一般性结论与线性稳定性标准N≥R_e~(0.58)。
本文第六章从另一种途径改善模型的稳定性，同时结合现有格子模型受网格严重限制的实际，提出了格子模型的贴体坐标系算法。直接从离散的速
武汉理工大学博士学位论文
度Boltzmann出发，物理域离散为保持物体几何外形的贴体曲线网格，然后
运用坐标变换，将定解问题变换到计算域中求解。首先计算了存在理论解的
圆柱Couette流动，并针对不同数目的离散网格、不同格子模型以及不同坐标
系算法，对结果进行了比较，不仅再次印证了本文第四章给出的不可压格子
模型具有满意的计算精度，而且表明本章的贴体坐标算法是行之有效的。然
后计算了圆柱绕流。得到的低雷诺数下的回流长度、分离角、阻力系数、压
力系数和现有的试验或数值结果相比，都能良好吻合。显示出来的最高雷诺
数为9500的流线分布，准确地描述了流动随时间的变化过程，捕捉了可能存
在的运动模式。
最后对本文进行了总结，并指出了今后进一步研究的主要方向。
A new force suddenly rises, the computational fluid dynamics, as the organic union of classic fluid mechanics, numerical computation methods and computer science, develops to become the efficient means of both science research and engineering design. The interaction among theoretical analysis, experimental research and numerical simulation promotes both the theoretic renovation and engineering applications of fluid mechanics. In this background, the lattice Boltzmann method comes forth and progresses rapidly as an attractive numerical computation method.
Aiming at the key issues of lattice Boltzmann method applications in engineering design, the main work, based on the extensive inquisition and pursuing the evolution and status in quo, is presented as following:
Firstly, the intrinsic relationship between lattice Boltzmann method and other traditional numerical methods is interpreted through the Chapman-Enskog expansion and multi-scale analytic technique. And the essential idea and foundation of lattice Boltzmann model are set forth.
According to the mathematic modeling principle of physical problem, the error of lattice Boltzmann model is analyzed in Chapter 3. The nonlinear deviation term from the Navier-Stokes equation is given, and the main model coefficients, such as speed of sound, viscosity and so on, are verified by numerical computation, the results show that the lattice Boltzmann method has second order precision in space and in time which satisfy the engineering application, whereas, the compressible effect can't be neglected along with Mach number increasing, and must be reduced or eliminated.
Chapter 4 proposes an incompressible lattice BGK model, in which the velocity and pressure instead of the mass density that is a constant are the independent dynamic variables. And the incompressible Navier-Stokes equations are exactly derived from this incompressible LBGK model, thus the compressible effect due to the density fluctuation is theoretically eliminated. The results of steady or unsteady flow show that the incompressible lattice BGK model is validate in reducing the compressibility error. And then, the cavity flow is simulated, and the streamline and pressure contour at different Reynolds
number are plotted, the stream function and location of vortex centers are agree well with the previous results, which indicate the incompressible lattice BGK model is reliable.
Numerical stability, the other issue of the lattice Boltzmann method, is discussed in Chapter 5. Corresponding to the uniform and shear background flow, the stability of d2q7 d2q9 and d3ql5 model is analyzed through the Von Neumann linear stability theory, both the conclusion about the mass distribution parameters, the wave number, the relaxation time
and the uniform velocity, and the linear stability criterion N R0.58 are instructive to
numerical simulation of flow.
Chapter 6 presents the lattice Boltzmann algorithm on body-fitted coordinates system, which not only avoids the restriction of uniform regular mesh, but also enhances the stability. After the coordinate transformation, the discrete velocity Boltzmann equation is solved directly in computational domain to preserve the geometry of body. The results comparison of cylindrical couette flow at different size meshes, different lattice BGK models and different algorithms shows that, not only the precision of the incompressible lattice BGK model is satisfactory, but also the curvilinear coordinate algorithm is efficient. Therefore, the circular cylinder flow is simulated. And the results of the wake length, separation angle, and drag or pressure coefficient for low Reynolds number are excellent agreement with previous experimental and numerical results. Furthermore, the streamlines for Reynolds numbers up to 9500 display the evolution with time of flow, and capture the possible flow pattern at different times.
Finally, the whole paper is summarized and the further research interests are put forward in Chapter 7.
本文在广泛调研，追踪国内外研究进展及现状的基础上，针对目前格子法在工程应用中所面临的主要问题，开展了以下几方面的研究工作：
首先运用Chapman-Enskog展开及多尺度分析技术，揭示了格子法与传统数值计算方法间的内在联系，阐述了格子模型的思想本质及其建模过程。
其次，按照物理问题数学建模的原则，对格子法的误差进行了分析，给出了格子BGK方程再现Navier-Stokes方程时的压缩误差项，并数值验证了格子模型的声速及粘性系数等相关参数的精度，表明格子模型尽管具有时空二阶精度，能满足工程计算的要求，但随着Mach数增大，压缩误差逐渐成为主要误差，必须予以消除。
本文第四章建立了以压力、速度为独立变量的不可压格子BGK模型。其中密度保持为常数，使得从格子BGK方程能够导出宏观不可压NS方程，理论上消除了因密度变化导致的压缩误差。通过对定常、非定常问题的数值结果的比较，证明了本文不可压模型能有效消除压缩误差。进而对方腔流的速度场和压力场进行了计算，绘制了不同雷诺数下的流线图及压力等高线图，得到的回流涡的位置和流函数的值和现有的数据十分吻合，表明本文不可压格子模型是可靠的。
第五章考虑了数值方法另外一个方面的问题，在均匀流和剪切流两种背景流场下，运用Von Neumann线性分析法，针对d2q7、d2q9及d3q15格子模型，分析了质量分布参数、波数、松弛时间和平均流速等决定模型稳定性的主要参数对模型稳定性的影响，得到了对流场数值计算具有指导意义的一般性结论与线性稳定性标准N≥R_e~(0.58)。
本文第六章从另一种途径改善模型的稳定性，同时结合现有格子模型受网格严重限制的实际，提出了格子模型的贴体坐标系算法。直接从离散的速
武汉理工大学博士学位论文
度Boltzmann出发，物理域离散为保持物体几何外形的贴体曲线网格，然后
运用坐标变换，将定解问题变换到计算域中求解。首先计算了存在理论解的
圆柱Couette流动，并针对不同数目的离散网格、不同格子模型以及不同坐标
系算法，对结果进行了比较，不仅再次印证了本文第四章给出的不可压格子
模型具有满意的计算精度，而且表明本章的贴体坐标算法是行之有效的。然
后计算了圆柱绕流。得到的低雷诺数下的回流长度、分离角、阻力系数、压
力系数和现有的试验或数值结果相比，都能良好吻合。显示出来的最高雷诺
数为9500的流线分布，准确地描述了流动随时间的变化过程，捕捉了可能存
在的运动模式。
最后对本文进行了总结，并指出了今后进一步研究的主要方向。
A new force suddenly rises, the computational fluid dynamics, as the organic union of classic fluid mechanics, numerical computation methods and computer science, develops to become the efficient means of both science research and engineering design. The interaction among theoretical analysis, experimental research and numerical simulation promotes both the theoretic renovation and engineering applications of fluid mechanics. In this background, the lattice Boltzmann method comes forth and progresses rapidly as an attractive numerical computation method.
Aiming at the key issues of lattice Boltzmann method applications in engineering design, the main work, based on the extensive inquisition and pursuing the evolution and status in quo, is presented as following:
Firstly, the intrinsic relationship between lattice Boltzmann method and other traditional numerical methods is interpreted through the Chapman-Enskog expansion and multi-scale analytic technique. And the essential idea and foundation of lattice Boltzmann model are set forth.
According to the mathematic modeling principle of physical problem, the error of lattice Boltzmann model is analyzed in Chapter 3. The nonlinear deviation term from the Navier-Stokes equation is given, and the main model coefficients, such as speed of sound, viscosity and so on, are verified by numerical computation, the results show that the lattice Boltzmann method has second order precision in space and in time which satisfy the engineering application, whereas, the compressible effect can't be neglected along with Mach number increasing, and must be reduced or eliminated.
Chapter 4 proposes an incompressible lattice BGK model, in which the velocity and pressure instead of the mass density that is a constant are the independent dynamic variables. And the incompressible Navier-Stokes equations are exactly derived from this incompressible LBGK model, thus the compressible effect due to the density fluctuation is theoretically eliminated. The results of steady or unsteady flow show that the incompressible lattice BGK model is validate in reducing the compressibility error. And then, the cavity flow is simulated, and the streamline and pressure contour at different Reynolds
number are plotted, the stream function and location of vortex centers are agree well with the previous results, which indicate the incompressible lattice BGK model is reliable.
Numerical stability, the other issue of the lattice Boltzmann method, is discussed in Chapter 5. Corresponding to the uniform and shear background flow, the stability of d2q7 d2q9 and d3ql5 model is analyzed through the Von Neumann linear stability theory, both the conclusion about the mass distribution parameters, the wave number, the relaxation time
and the uniform velocity, and the linear stability criterion N R0.58 are instructive to
numerical simulation of flow.
Chapter 6 presents the lattice Boltzmann algorithm on body-fitted coordinates system, which not only avoids the restriction of uniform regular mesh, but also enhances the stability. After the coordinate transformation, the discrete velocity Boltzmann equation is solved directly in computational domain to preserve the geometry of body. The results comparison of cylindrical couette flow at different size meshes, different lattice BGK models and different algorithms shows that, not only the precision of the incompressible lattice BGK model is satisfactory, but also the curvilinear coordinate algorithm is efficient. Therefore, the circular cylinder flow is simulated. And the results of the wake length, separation angle, and drag or pressure coefficient for low Reynolds number are excellent agreement with previous experimental and numerical results. Furthermore, the streamlines for Reynolds numbers up to 9500 display the evolution with time of flow, and capture the possible flow pattern at different times.
Finally, the whole paper is summarized and the further research interests are put forward in Chapter 7.
引文
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