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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.

 

The lattice Boltzmann method is employed to conduct direct numerical simulations of turbulent open channel flows with the presence of finite-size spherical sediment particles. The uniform particles have a diameter of approximately 18 wall units and a density of ρp=2.65ρf, where ρp and ρf are the particle and fluid densities, respectively. Three low particle volume fractions ϕ=0.11%, 0.22%, and 0.44% are used to investigate the particle-turbulence interactions. Simulation results indicate that particles are found to result in a more isotropic distribution of fluid turbulent kinetic energy (TKE) among different velocity components, and a more homogeneous distribution of the fluid TKE in the wall-normal direction. Particles tend to accumulate in the near-wall region due to the settling effect and they preferentially reside in low-speed streaks. The vertical particle volume fraction profiles are self-similar when normalized by the total particle volume fractions. Moreover, several typical transport modes of the sediment particles, such as resuspension, saltation, and rolling, are captured by tracking the trajectories of particles. Finally, the vertical profiles of particle concentration are shown to be consistent with a kinetic model.