多孔介质中分数阶流动的格子Boltzmann方法研究

This research is concerning the lattice Boltzmann model and algorithm for the fractional flows in porous media. The entry point of this subject is: abnormal transport phenomena result in some deviations of theoretical predictions of Darcy's law from experimental results, but the application of fractional derivative can describe some phenomena..In view of the successful application of the lattice Boltzmann method in fluid mechanics, we expect that the lattice Boltzmann method for fractional flow in porous media can deal with the problems in anomalous diffusion and abnormal flows. The fractional lattice Boltzmann model retains the advantages of classical lattice Boltzmann model. We choose two objects: (1) lattice Boltzmann model and algorithm for various types of fractional partial differential equations, (2) lattice Boltzmann model and algorithm for the fractional flow in porous media. We should analyses the intrinsic relations of the fractional lattice Boltzmann model and fractional fluid flows. We are to study the computational effectiveness of this method, and to find particularity of fractional lattice Boltzmann model. By comparing with results of classical numerical algorithms, we discover the advantages and disadvantages of this method.

本课题研究的是多孔介质中分数阶流动的格子Boltzmann模型与算法. 这个课题的切入点是: 异常的输运现象导致Darcy定律的理论预测与实验结果存在偏差, 而分数阶导数的应用能较好地描述一些异常现象..鉴于格子Boltzmann方法在流体力学中的成功应用, 可以预期, 将其用于多孔介质中的渗流可以处理这些异常扩散和流动问题. 该分数阶格子Boltzmann模型保留了经典格子Boltzmann模型的优越性. 我们选择两个研究对象: (1) 不同类型的分数阶偏微分方程的格子Boltzmann 模型及算法, (2) 多孔介质中分数阶渗流的格子Boltzmann模型以及算法. 一方面分析分数阶格子Boltzmann模型与分数阶流动的内在联系; 另一方面研究该算法的计算有效性和特殊性. 通过与经典数值方法的计算结果比较, 进而发现该方法的优势和不足.

本课题研究的是用格子Boltzmann方法实现分数阶偏微分方程的数值模拟和计算。首先，对于时间导数的分数阶、空间导数的分数阶、以及混合情况，分别给出了分数阶导数在格子Boltzmann模型中的实现方式，进而给出格子Boltzmann模型中的各阶矩及平衡态分布函数，最后恢复到相应的宏观方程。其次，把成功用于一般偏微分方程的分数阶处理方式，用于描述流体运动的分数阶方程，给出了相应的格子Boltzmann模型，并做了一些数值模拟工作。最后，对于多孔介质中的分数阶渗流，进行了理论推导以及一些初步的数值模拟工作。由于分数阶数值计算的计算量很大，做了一些并行计算的处理，并将其用于流体力学方程的计算。