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.

Hydrodynamic shock wave studies within a kinetic Monte Carlo approach

“…In fluid dynamics, when the characteristic length scale of a flow is smaller than the mean free path of its components, the continuum approximation which is assumed by the Navier-Stokes (NS) equations breaks down and the particle nature of matter must be taken into account [1]. Under these conditions, flows are said to be rarefied.…”

A continuum model of gas flows with localized density variations

“…Heat transfer processes in a highly rarefied gas, in particular at high temperature gradients [5][6][7], and some thermally-driven flows are also not well understood [8][9][10]. Meanwhile, various continuum models based on approximate solutions to the Boltzmann equation, and which are expected to cover flows beyond the Navier-Stokes level, are the subject of different controversies and still under investigation [9,[11][12][13][14][15].…”

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

“…Until recently, DSMC simulations of low Mach number flows were complicated by excessive statistical noise (Hadjiconstantinou et al 2003;Baker & Hadjiconstantinou 2005;Chun & Koch 2005). This issue was addressed by Homolle & Hadjiconstantinou (2007), who developed the low variance DSMC (LVDSMC) method to solve the hard-sphere Boltzmann equation.…”

“…These include applications in plasma physics (Bittencourt 2004), cosmology and astrophysics (Dodelson 2003;Camenzind 2007), molecular biology (Dubois, Ouanounou & where λ is the mean free path and L is a characteristic geometric length scale of the flow. Importantly, solutions of the Boltzmann equation provide an accurate model of the (true) flow across the full range of Knudsen number for a dilute gas (Agarwal, Yun & Balakrishnan 2001;Hadjiconstantinou 2005b). For consistency, in this article we adopt the convention that the constituent molecules/particles of the dilute gas are simply referred to as particles.…”