[论文解读] Performance evaluation of the general characteristics based off-lattice Boltzmann and DUGKS methods for low speed continuum flows: A comparative study

Lianhua Zhu, Peng Wang+1 more|arXiv (Cornell University)|Nov 1, 2015

Lattice Boltzmann Simulation Studies参考文献 14被引用 4

一句话总结

本研究对比了用于低速连续流的非结构网格BKG与DUGKS方法，结果表明DUGKS通过使用梯形求积法对碰撞项进行积分，实现了更高的精度与稳定性，而BKG则采用矩形求积法。DUGKS在CFL < 1时保持稳定，且在解析边界层和高雷诺数流场方面优于BKG，尽管其计算速度约为BKG的一半。

ABSTRACT

The general characteristics based off-lattice Boltzmann scheme (BKG) proposed by Bardow et~al.(2006), and the discrete unified gas kinetic scheme (DUGKS) are two methods that successfully overcome the time step restriction by the collision time, which is commonly seen in many other kinetic schemes. Basically, the BKG scheme is a time splitting scheme, while the DUGKS is an un-split finite volume scheme. In this work, we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations. It is found that the effects of collision term are considered in the reconstructions of the cell-interface distribution function in both schemes, which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time. The difference between the two schemes lies in the treatment of the integral of the collision term, in which the Bardow's scheme uses the rectangular rule while the DUGKS uses the trapezoidal rule. The performance of the two schemes, i.e., accuracy, stability, and efficiency are then compared by simulating several two dimensional flows, including the unsteady Taylor-Green vortex flow, the steady lid-driven cavity flow, and the laminar boundary layer problem. It is observed that, the DUGKS can give more accurate results than the BKG scheme. Furthermore, the numerical stability of the BKG scheme decreases as the Courant-Friedrichs-Lewy (CFL) number approaches to 1, while the stability of DUGKS is not affected by the CFL number apparently as long as CFL&lt;1. It is also observed that the BKG scheme is about one time faster than the DUGKS scheme with the same computational mesh and time step.

研究动机与目标

• 评估并比较通用特征基非结构网格BKG与DUGKS格式在低速连续流中的性能。
• 分析BKG与DUGKS在有限体积框架下的理论差异，重点关注数值通量的构建方式。
• 研究时间步长、网格分辨率与CFL数对两种格式精度与稳定性的影响。
• 确定哪种方法在模拟复杂连续流（如边界层与腔流）方面具有更好的鲁棒性与效率。

提出的方法

• BKG格式采用时间分裂方法，对碰撞项进行变量变换，并使用Lax-Wendroff格式处理平流步骤。
• DUGKS采用非分裂的有限体积格式，直接从离散的Boltzmann-BGK方程重构数值通量。
• 两种方法均沿特征线积分碰撞项，但BKG使用单点（矩形）求积，而DUGKS使用梯形求积。
• 通过D2Q9网格模型在二维流动中对两种方法进行数值对比：非定常Taylor-Green涡流、定常顶盖驱动腔流及层流边界层。
• 在有限体积框架中评估数值通量，以实现对碰撞积分处理方式的直接比较。
• 通过改变时间步长、网格分辨率与CFL数进行模拟，以评估精度、稳定性与效率。

实验结果

研究问题

• RQ1 BKG与DUGKS格式在有限体积框架中对碰撞项积分的处理方式有何不同？
• RQ2 求积规则（矩形与梯形）对非结构网格保角格式的精度与稳定性有何影响？
• RQ3 CFL数如何影响BKG与DUGKS在低速连续流中的数值稳定性？
• RQ4 网格分辨率与时间步长如何影响两种格式在解析边界层与涡流结构方面的精度？
• RQ5 在高雷诺数连续流中，哪种方法能提供更优的效率-精度权衡？

主要发现

• DUGKS方法在精度上优于BKG，尤其在使用粗网格解析层流边界层时，BKG即使在最细网格下仍会失效。
• DUGKS在所有测试的CFL值低于1的情况下均保持数值稳定，而BKG在CFL超过0.5后稳定性显著下降。
• BKG在相同网格与时间步长下速度约为DUGKS的1倍，但此速度优势被其较差的精度与鲁棒性所抵消。
• DUGKS对网格分辨率的敏感性较低，在Δy_min = 0.1时仍能产生准确结果，而BKG在Δy_min = 0.02时也表现出显著偏差。
• 采用小时间步长（CFL = 0.01）可提升BKG的精度，但仍与Blasius解存在明显偏差，表明其存在固有局限性。
• 理论分析确认，两种格式的差异仅源于碰撞积分求积规则的不同：BKG使用矩形求积，DUGKS使用梯形求积，这解释了观察到的性能差距。