Lifting relations for a generalized total-energy double-distribution-function kinetic model and their impact on compressible turbulence simulation

Recently, Qi et al. (2022) and Guo et al. (2023) proposed two alternative designs of an efficient mesoscopic method using the total-energy double-distribution-function (DDF) formulation, hereafter referred to as the Qi model and the Guo model. The two models share the same advantage of using only 40 discrete particle velocities to fully reproduce the Navier–Stokes-Fourier (NSF) system. However, the Guo model is based on a more rigorous kinetic consideration, while the Qi model relies on a more general design of the source term to allow for adjustable bulk-to-shear viscosity ratio. In this paper, we derive lifting relations for the Qi model based on two alternative approaches, namely, the Hermite expansion and the Chapman–Enskog expansion, which can be used to construct the boundary and initial conditions for the mesoscopic method. For three-dimensional compressible turbulence simulations, including compressible decaying homogeneous isotropic turbulence and Taylor–Green vortex flows, the derived two sets of lifting relations are applied to the initialization distribution function to study their impacts. Interestingly, for the Qi model, the two sets of lifting relations yield the same results without numerical artifacts, whereas for the Guo model, an appropriate lifting relation must be specified to avoid numerical artifacts resulting from the flow initialization (Qi et al., 2023).