基于格子玻尔兹曼方法的局部网格加密研究—粗细网格间分布函数转换及网格加密界面非物理震荡分析

格子Boltzmann方法作为一种高效的介观计算流体力学方法在过去二十多年里得到快速发展，其相对较高的计算效率和灵活性使其可以适用于各种复杂流动的模拟。然而标准的格子Boltzmann方法受限于自身的限制使其只能使用均匀的直角网格，这种网格排布方式并不利于复杂流动的计算。为此，基于格子Boltzmann方法的局部网格加密算法被提出。该算法需要在局部加密的界面处将粗细网格间的分布函数转换后交换。目前分布函数的转换方式大多是在没有源项的基础下推导的，而且现存考虑源项时转换公式的推导也都是基于Champan-Enskog展开寻找与分辨率无关的分布函数表达式。其推导过程相对复杂，且需要对分布函数的非平衡态部分做一阶Champan-Enskog近似，这有可能会限制局部网格加密算法在高阶格子Boltzmann方法中的应用。 为了进一步完善和扩展局部网格加密算法的适用范围，本文在忽略离散误差的前提下以保证粗细网格间时空离散前分布函数以及弛豫时间一致为基础，首次构建了一套规范且简洁的粗细网格间在考虑任意源项时分布函数转换关系的推导过程，该方法不依赖于Champan-Enskog展开以及Champan-Enskog近似，且该方法既可以适用于单松弛碰撞模型也可以适用于多松弛碰撞模型。此外，本文还首次从理论上证明了保证粗细网格间非平衡态部分的一阶 Champan-Enskog 近似一致，便可以保证整个非平衡态部分的一致，这也给之前学者对非平衡态部分做一阶 Champan-Enskog 近似提供了理论依据。 源项对于扩展格子Boltzmann方法的适用范围有着重要的意义。本文首次给出了考虑任意源项时递归正则化模型中非平衡态分布函数各阶Hermite矩间的递推关系并给出了考虑任意源项后该碰撞模型的具体实施方案，同时本文首次给出了混合递归正则化模型在考虑任意源项时需要做出的额外修正，这将有助于扩展两种碰撞模型的适用范围。针对分布函数的初始化问题，本文首次通过引入新的Hermite转换矩阵构建了考虑任意源项时多松弛碰撞模型下时空离散前后分布函数各阶Hermite矩间的关系，该方法的提出除了可以用于初始化分布函数外还有助于通过Hermite矩为桥梁将单松弛碰撞模型和多松弛碰撞模型有机统一。 为了验证分布函数转换公式对复杂源项的适应性，我们使用了局部网格加密算法模拟了具有复杂源项的强迫泰勒-格林涡流动以及二维平板泊肃叶流中的扩散问题并取得了较好的数值结果，同时我们以二维平板泊肃叶流和顶盖驱动方腔流为算例展示了局部网格加密算法对于计算效率的提升，其中我们在顶盖方腔驱动流中通过将整个顶盖附近加密使得计算时间节省了 2 倍以上。此外，我们以顶盖方腔驱动流为算例探索性的对比了不同碰撞模型对网格加密界面非物理震荡的抑制效果，实验结果表明在使用局部网格加密技术时除了合理选择加密区域的范围外，往往还需要配合使用具有良好数值稳定性的碰撞模型。在分析分布函数转换关系的推导过程后，我们认为网格加密界面附近非物理震荡的产生是由于粗细网格间离散误差不一致从而导致在加密界面处形成的“离散误差间断”引起的。 综上，本文的主要工作在于：完善了粗细网格间分布函数的转换关系，探索性的研究了局部加密网格界面处的非物理震荡并对震荡产生的本质给出了猜测。同时提出了不同碰撞模型下源项的处理方法，引入了Hermite转换矩阵用于多松弛碰撞模型的初始化。

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