Augmented Burnett and Bhatnagar-Gross-Krook-Burnett Equations for Hypersonic Flow

Abstract

Two different forms of Burnett equations are studied that have been designated as augmented Burnett equations and Bhatnagar ‐Gross‐Krook‐Burnett (BGK‐Burnett) equations. The augmented Burnett equations were developed to stabilize the solution of the conventional Burnett equations that were derived from the Boltzmann equation using the second-order Chapman ‐Enskog expansion. In this formulation, the conventional Burnett equations are augmented by adding ad hoc third-order derivatives to stress and heat transfer terms so that the augmented equations are stable to small wavelength disturbances. The BGK‐Burnett equations have been recently derived from the Boltzmann equation using the BGK approximation for the collision integral. These equations have been shown to be entropy consistent and satisfy the Boltzmann H-theorem in contrast to the conventional Burnett equations that violate the second law of thermodynamics. Both sets of Burnett equations are applied to compute a two-dimensional hypersonic e ow over a circular cylinder at Knudsen numbers 0.001 ‐0.1. Comparison is made between the augmented and BGK ‐Burnett equations solutions and with the Navier ‐Stokes calculations. Comparison of the solutions of the augmented Burnett equations with the Navier ‐Stokes solutions shows that the difference is signie cant at high Knudsen number (Kn = 0.1). The solutions from the BGK ‐Burnett equations compare reasonably well with those from the Navier ‐Stokes equations and the augmented Burnett equations.