Ellipsoidal statistical Bhatnagar–Gross–Krook model with velocity-dependent collision frequency

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In this paper, an ellipsoidal statistical (ES) Bhatnagar–Gross–Krook (BGK)-type kinetic model with velocity-dependent collision frequency is proposed and further numerically tested for one-dimensional shock waves and planar Couette flow at steady state for hard sphere molecules. In this new kinetic model, a physically meaningful expression for the velocity-dependent collision frequency derived from the Boltzmann equation is used, while the important properties for a kinetic model are retained at the same time. This kinetic model can be simplified to the classical ES-BGK model and the BGK model with velocity-dependent collision frequency for suitable choices of parameters. The H theorem for this new kinetic model has so far been proven only for small Knudsen numbers. The numerical method used here for kinetic models is based on Mieussens’s discrete velocity model [L. Mieussens, J. Comput. Phys. 162, 429 (2000)]. Computational results from the kinetic models (including the BGK model, the ES-BGK model, the BGK model with velocity-dependent collision frequency, and this new kinetic model) are compared to results obtained from the direct simulation Monte Carlo (DSMC) method. It is found that results obtained from this new kinetic model lie in between results from the ES-BGK model and results from the BGK model with velocity-dependent collision frequency. For one-dimensional shock waves, results from this new kinetic model fit best with results from the DSMC, while for planar Couette flow, the classical ES-BGK model is suggested.