Comparison of forcing models in the discrete unified gas kinetic scheme for compressible turbulence

External body forces influence the evolution of the distribution function in mesoscopic methods through the particle–velocity–space gradient term in the Boltzmann equation. Simplified forcing models for continuum flows can be categorized into moment-expansion-based and Chapman–Enskog-expansion-based models, while their performance in compressible turbulence remains insufficiently explored. Using the partial internal energy double-distribution-function (DDF) framework, this study addresses that gap. We systematically derive the moment constraints for any forcing model to recover the Navier–Stokes–Fourier equations with arbitrary bulk viscosity for the first time. Within the partial internal energy DDF framework, we novelly propose a moment-expansion-based forcing model for compressible flows, which accounts for the higher-order structure of the distribution function without requiring increased Gauss–Hermite quadrature accuracy. For comparison, we also consider two classical Chapman–Enskog-based models, the He model [He et al., J. Comput. Phys. 146, 282 (1998)], which requires higher-order quadrature, and the Kupershtokh model [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)], which neglects higher-order structural effects, even though both models satisfy the same velocity moment constraints. All three models yield stable simulations of forced compressible turbulence using partial internal energy DDF-based discrete unified gas kinetic scheme. However, nonlinear interactions lead to observable, though small, differences in turbulence statistics under identical initial conditions. Overall, the results indicate that each model is capable of providing reasonable turbulence statistics.