In this paper, based on simplified Boltzmann equation, we explore the inverse-design of mesoscopic models for compressible flow using the Chapman-Enskog analysis. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored using the DNS simulation data based on the proposed model.

中文翻译：

---

使用Chapman-Enskog分析的可压缩流介观模型的逆设计

在本文中，基于简化的玻尔兹曼方程，我们使用Chapman-Enskog分析探索了可压缩流动的介观模型的逆设计。从具有一个附加源项的单弛豫时间Boltzmann方程开始，获得了两个用于两个简化分布函数的模型Boltzmann方程，每个模型又具有一个附加的不确定源项。在此通用框架下，以Navier-Stokes-Fourier（NSF）方程为约束，通过前导Chapman-Enskog分析获得分布函数的结构。接下来，获得了两个源项设计的五个基本约束，以便在连续范围内恢复NSF系统。这些限制条件允许调节体积-剪切粘度比，Prandtl数以及热能源。可以通过适当的物理考虑和数字实现要求来确定两个源术语的具体形式。通过采用截断的Hermite展开，为两个源项提出了一种设计。此外，文献中的三个众所周知的介观模型显示与这五个约束条件兼容。此外，还通过使用NSF顺序的Chapman-Enskog扩展来探索边界条件的一致实现。最后，基于分布函数的高阶Chapman-Enskog展开，我们导出了粘性应力张量和热通量的完整解析表达式。基于提议的模型，可以使用DNS仿真数据进一步探索一些基础物理学。