Inverse design of mesoscopic models for compressible flow using the Chapman-Enskog analysis

Tao Chen,Shiyi Chen,Jun Lai,Lian-Ping Wang

doi:10.1186/s42774-020-00059-2

Abstract

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.

Highlights

The Boltzmann equation is of vital importance in the kinetic theory of dilute gases [1]
We focus on the inverse design of the source term in the model Boltzmann equation for compressible flows
12 Conclusions In this paper, an inverse design approach of mesoscopic models for compressible flows in continuum or near-continuum regime has been explored based on the Chapman-Enskog analysis

Summary

Introduction

The Boltzmann equation is of vital importance in the kinetic theory of dilute gases [1]. By applying the Chapman-Enskog expansion to the Boltzmann equation with the BGK collision operator, the NSF system can be recovered, but with a unit Prandtl number which does not obey the physical reality. Liu et al [25] claimed that the predictions based on the BGK model for highly nonequilibrium flows are only qualitatively correct in the transitional regime since the BGK model filters out the information of the detailed molecular-interaction processes They compare the Boltzmann equation and its model equations through some test cases where the distribution functions are far from equilibrium. By comparing the first-order moment of the model Boltzmann equation with the Navier-Stokes equation, it can be shown that the viscous stress tensor σ is determined by the non-equilibrium part of the particle distribution function as σ = − cc f − f eq dξ dζ ,.

Notations and conventions

An examination of three existing mesoscopic models in our design framework

12 Conclusions

C Details in the derivations of the Rykov model