Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime

Abstract

In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the “augmented Burnett equations” and the “BGK–Burnett equations.” This paper traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and the Grad’s 13 moment equations is elucidated by employing the Maxwell–Truesdell–Green iteration. Numerical solutions are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum–transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier–Stokes calculations, Burnett solutions of other investigators, and DSMC solutions as much as possible.