Discrete Boltzmann method for non-equilibrium flows: based on Shakhov model

A general framework for constructing discrete Boltzmann model for non-equilibrium flows based on the Shakhov model is presented. The Hermite polynomial expansion and a set of discrete velocity with isotropy are adopted to solve the kinetic moments of discrete equilibrium distribution function. Such a model possesses both an adjustable specific heat ratio and Prandtl number, and can be applied to a wide range of flow regimes including continuous, slip, and transition flows. To recover results for actual situations, the nondimensionalization process is demonstrated. To verify and validate the new model, several typical non-equilibrium flows including the Couette flow, Fourier flow, unsteady boundary heating problem, cavity flow, and Kelvin-Helmholtz instability are simulated. Comparisons are made between the results of discrete Boltzmann model and those of previous models including analytic solution in slip flow, Lattice ES-BGK, and DSMC based on both BGK and hard-sphere models. The results show that the new model can accurately capture the velocity slip and temperature jump near the wall, and show excellent performance in predicting the non-equilibrium flow even in transition flow regime. In addition, the measurement of non-equilibrium effects is further developed and the non-equilibrium strength $D^*_n$ in the n-th order moment space is defined. The nonequilibrium characteristics and the advantage of using $D^*_n$ in Kelvin-Helmholtz instability are discussed. It concludes that the non-equilibrium strength $D^*_n$ is more appropriate to describe the interfaces than the individual components of $Delta^*_n$. Besides, the $D^*_3$ and $D^*_{3,1}$ can provide higher resolution interfaces in the simulation of Kelvin-Helmholtz instability.