格子Boltzmann 方法中反弹格式隐藏误差的 理论分析及数值模拟研究

过去三十多年，格子Boltzmann方法（LBM）已经发展成为计算流体力学领域中一种高效且实用的介观数值模拟方法。由于LBM具有边界条件容易实施、并行性较好、数值耗散相对较低和易于编程等优势，已经被成功应用到微流动、壁湍流、多相流动和多孔介质流动等涉及流固边界复杂流动问题的数值模拟中。反弹格式是目前LBM方法中最常用的边界条件处理方法，其通过重构边界点上的未知分布函数来实现宏观固体壁面上的无滑移边界条件。众所周知，应用于内部流体点上的格子Boltzmann方程，通过一阶Chapman-Enskog展开，可以恢复宏观Navier-Stokes方程。但是，在边界点上由反弹格式所重构的未知分布函数可能与Chapman-Enskog展开结构不完全一致，这一问题及其可能带来的各种负面效应并没有被广泛认知和深入研究。

因此，本文引入“隐藏误差”这一概念，来描述由反弹格式实施所引入的、与格子Boltzmann方程Chapman-Enskog展开不一致的误差。为了能定量描述这些隐藏误差，首先需要深入理解LBM方法中的介观分布函数与宏观Navier-Stokes方程的物理变量之间的本质关联。本文通过一种更直接的Chapman-Enskog展开方法，将介观分布函数表达成碰撞模型的松弛参数、宏观物理变量及其对应空间导数的函数，揭示了分布函数的内在结构。基于这种多尺度展开分析，推导了BGK、TRT、MRT三种不同碰撞模型下对应的介观分布函数内在结构。

本文具体研究了两种常用反弹格式：离壁反弹格式和壁面反弹格式。通过将由反弹格式重构的未知分布函数与理论推导的介观分布函数结构进行对比，从而建立了一个能够识别并描述隐藏误差的理论分析方法。基于隐藏误差只出现在边界节点，导致方法误差在流域内不连续，本文提出误差分析需要考虑二阶Chapman-Enskog展开的一致性，并由此定义隐藏误差，极大的改善了类似分析的完整性和解释LBM各种内在问题的能力。为了进一步验证该理论分析方法的适用性和可靠性，对BGK、TRT、MRT三种不同碰撞模型下反弹格式所引入的隐藏误差进行了深度分析，并得到了对应的隐藏误差具体表达式。进一步地，从理论上预测了在均匀泊肃叶流动和非均匀流动中隐藏误差所导致的非物理滑移速度。从理论分析结果可以看出，碰撞模型中的松弛参数对隐藏误差的大小起着决定性作用，通过优化松弛参数的设置，可以有效地抑制或消除隐藏误差导致的负面效应。这一优化的碰撞参数设置虽然是在单向泊肃叶流动问题中推导出来的，但是在一定的物理限定条件下，本文证明它也可以被推广应用到非均匀流动中；推广得到新的理论结果显示局部滑移速度依赖壁面处切向速度的法向二阶导数，网格分辨率和流动雷诺数。

针对隐藏误差对粘性流动数值模拟的影响，本文对一系列经典二维粘性流动问题进行了数值模拟。在泊肃叶流动的数值模拟中，通过数值模拟得到的滑移速度与理论预测结果完全吻合。在顶盖驱动方腔流动的数值模拟中，隐藏误差导致了压力场和涡量场的非物理振荡，这些非物理振荡的幅值和波及范围与碰撞模型中的碰撞参数设置密切相关。在方柱绕流的数值模拟中，隐藏误差导致了方柱后尾流区内尾涡脱落的物理频率模态无法准确捕捉。根据上述这些数值模拟结果可以发现，BGK模型只能通过细化网格来减小隐藏误差的影响，而TRT模型则可以通过设置自由碰撞参数来消除隐藏误差。

最后，通过比较两种反弹格式下得到的宏观统计量、Navier-Stokes方程的局部平衡检测和数值稳定性，讨论了隐藏误差对三维槽道湍流LBM数值模拟结果的影响。数值模拟结果证实了隐藏误差可以导致Navier-Stokes方程在边界节点上无法完全满足，这一问题严重影响壁湍流数值模拟的精度和数值稳定性，尤其在粗网格上使用离壁反弹格式时最为明显。在相同数值模拟条件下，采用壁面反弹格式能够获得一组相对稳定且符合物理的数值结果，这是因为壁面反弹格式所引入的隐藏误差相对较小。

综上所述，本文针对反弹格式所引入的隐藏误差问题及其多种负面效应，开展了全新的理论分析和巧妙的数值模拟研究，系统性的解释了LBM各类内在问题如非物理滑移速度、网格尺度上的压力场和涡量场噪声、Navier-Stokes方程局部不成立和整体数值精度及稳定性下降。本项工作清晰展示了隐藏误差的正确定义方法，提出降低和消除隐藏误差的理论路径，为LBM方法准确模拟复杂流动提供重要的理论分析方法，同时也为LBM方法中曾经遇到的一系列令人困惑的问题找到了根本原因。本文提出了将LBM方法中两个组成部分––格子Boltzmann方程和反弹格式––整合分析的理论思路，为今后进一步完善LBM方法提供指引。

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