Numerical comparison of Bhatnagar–Gross–Krook models with proper Prandtl number

Abstract

While the standard Bhatnagar–Gross–Krook (BGK) model leads to the wrong Prandtl number, the BGK model with velocity dependent collision frequency as well as the ellipsoidal statistical BGK (ES-BGK) model can be adjusted to give its proper value of 2/3. In this paper, the BGK model with velocity dependent collision frequency is considered in some detail. The corresponding thermal conductivity and viscosity are computed from the Chapman–Enskog method, and several velocity-dependent collision frequencies are introduced which all give the proper Prandtl number. The models are tested for Couette flow, and the results are compared to solutions obtained with the ES-BGK model, and the direct simulation Monte Carlo method. The simulations rely on a numerical scheme that ensures positivity of solutions, conservation of moments, and dissipation of entropy. The advantages and disadvantages of the various BGK models are discussed.