Three asymptotic limits exist for the Euler equations at low Mach number - purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit. In this paper, we review the low-Mach behaviour using the modified equations - the continuous Euler equations augmented with artificial diffusion terms - which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single- and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting. As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows.

翻译：摘要：欧拉方程在低马赫数下存在三种渐近极限——纯对流、纯声学、以及对流-声学混合极限。标准基于密度同位网格的可压缩流数值格式因人工扩散的渐近标度不正确而无法处理低马赫数流动。先前关于此类格式的研究揭示了不同极限下的多种行为，并提出了低马赫格式的设计准则，但这些研究主要聚焦于特定离散化和/或仅考虑对流极限。本文利用修正方程（即含人工扩散项的连续欧拉方程）回顾低马赫数行为，该方法可代表此类格式的广泛类别。通过同时考虑对流效应与声学效应，我们证明自然存在三种扩散标度。对这些标度进行单尺度与多尺度渐近分析表明，此类格式许多重要的低马赫数特征可在连续设定下直接复现。以现有多种低马赫数Roe型有限体积格式为例，我们证明它们恰好符合这三种标度之一。分析结果验证了先前对这些格式的分析，并通过纳入对流与声学效应完善了低马赫格式的设计准则。针对对流、声学及混合流动，离散分析与数值算例展示了采用三种标度的最小化Roe型格式的行为特征。