[论文解读] An Improved Roe Scheme for All Mach-Number Flows Simultaneously Curing Known Problems

Xuesong Li, Xiaodong Ren+1 more|arXiv (Cornell University)|Nov 25, 2017

Computational Fluid Dynamics and Aerodynamics参考文献 23被引用 3

一句话总结

本文提出Roe-AM，一种改进的Roe格式，通过引入基于马赫数的可压缩性检测和压力-密度变化检测器，同时解决了所有马赫数流动中的主要问题——如非物理行为、棋盘振荡、激波不稳定性及膨胀激波问题。该方法在保持极低数值耗散的前提下，通过预处理和改进的熵修正，显著提升了从不可压到高超音速流动范围内的鲁棒性与精度。

ABSTRACT

Roe scheme is known for its good performance in moderate-Mach-number flows. However, this scheme and its extended versions suffers from many disastrous problems, such as non-physical behavior, global cut-off, and checkerboard problems, for incompressible flows; and shock instability, expansion shock, and positively non-conservative problems for hypersonic flows. In this paper, non-physical behavior problem, checkerboard problem, and main reason of shock instability problem are due to that the Roe scheme cannot identify multi-dimensional incompressible and compressible flows when normal Mach number on the cell face tends to zero, and then leads to incorrect cross modifications. Positively non-conservative problem is also identified as another important reason for shock instability. Therefore, Mach number and an assistant pressure-density-varying detector are introduced into the Roe scheme to judge compressibility, positivity condition is satisfied by a simple modification with minimal numerical dissipation increases and even with possible decreases in numerical dissipation, the mechanism of the preconditioned Roe scheme is introduced to suppress checkerboard problem, and modified entropy fix and the rotated Riemann solver is combined with complementary advantages as an assistant improvement for better robust. Based on above improvements and previous developments for global cut-off and expansion shock problems, an improvement Roe scheme for all Mach-number flow (Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacks of the classical Roe scheme. The Roe-AM scheme is simple, easy to implement, computationally low-cost, robust, good extensibility, and free of empirical parameters essentially, with increasing minimal numerical dissipation.

研究动机与目标

• 解决经典Roe格式在所有马赫数流动中长期存在的缺陷，包括非物理行为与激波不稳定性。
• 通过使用压力-密度变化检测器修改交叉导数修正项，解决不可压流动中的棋盘问题。
• 通过识别并修正根本原因——当法向马赫数趋近于零时发生的错误交叉修正——消除激波不稳定性。
• 通过简单而有效的修改，在最小化数值耗散的同时确保守恒性与正性。
• 将全局截断与膨胀激波问题的改进统一为单一、可扩展且无参数的格式。

提出的方法

• 引入基于马赫数的检测器，以识别可压缩性，并在低马赫数区域防止错误的交叉导数修正。
• 结合压力-密度变化检测器，提升在不可压与过渡流区域的检测精度。
• 采用预处理Roe格式机制，在不增加耗散的前提下抑制棋盘振荡。
• 结合改进的熵修正与旋转Riemann求解器，提升格式的鲁棒性与稳定性。
• 实施最小数值耗散调整，在保持或降低耗散的同时确保正性与守恒性。
• 将先前针对全局截断与膨胀激波问题的解决方案整合进统一框架。

实验结果

研究问题

• RQ1 经典Roe格式为何在不可压与低马赫数流动中失效？其非物理行为的根本原因是什么？
• RQ2 Roe格式中激波不稳定的根源是什么？它如何与交叉导数修正相关联？
• RQ3 如何在不增加数值耗散的前提下抑制棋盘问题？
• RQ4 能否通过单一格式有效处理所有马赫数流动，同时保持守恒性与正性？
• RQ5 是否可能在保持极低算法复杂度且无需经验参数的前提下，提升鲁棒性与精度？

主要发现

• Roe-AM格式通过马赫数与压力-密度检测，成功修正了不可压与低马赫数流动中因错误交叉导数修正导致的非物理行为。
• 通过预处理Roe机制，棋盘振荡得到有效抑制，同时保持低耗散与解的稳定性。
• 通过识别并修正低法向马赫数下单元面上的错误交叉修正，成功解决了激波不稳定性问题。
• 通过最小化、物理解释一致的修正，确保了正性与守恒性，且未增加数值耗散。
• 该方法在所有马赫数范围内（包括高超音速流动）均表现出优异的鲁棒性，且无需经验参数。
• 改进后的格式展现出良好的可扩展性与计算效率，适用于实际CFD应用。