Keon-Young Yun

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

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 conventio...

BGK-Burnett Equations for Flows in the Continuum-Transition Regime

To extend the range of applicability of continuum formulations into the continuum-transition regime, an extended set of fluid dynamic equations has been derived. These equations, termed as the Bhatnagar-Gross-Krook (BGK)-Burnett equations, have been derived by taking moments of the Boltzmann equation by using the BGK model for the collision integral, The second-order distribution function that forms the basis of this derivation is formulated by considering the first three terms of the Chapman-Enskog expansion. It is shown that the BGK-Burnett equations have been used to compute the hypersonic shock structure and the hypersonic flow past a blunt body. The results of these computations are compared with the augmented Burnett and Navier-Stokes solutions. The second-order distribution function does not violate Boltzmanns H-theorem; as a consequence the BGK-Burnett equations are entropy consistent for the range of Knudsen numbers for which computations have been performed

Numerical Simulation of Three-Dimensional Augmented Burnett Equations for Hypersonic Flow

For computation of hypersonic e owe elds about space vehicles in low Earth orbits, where the local Knudsen numbers Kn lie in the continuum-transition regime, a set of extended three-dimensional hydrodynamic equations is required that is more accurate than the Navier ‐Stokes equations and computationally more efe cient than the direct simulation Monte Carlo (DSMC) computations. The three-dimensional augmented Burnett equations are derived from the Chapman ‐Enskog expansion of the Boltzmann equation to O(Kn 2) and adding the augmented terms (linear third-order super Burnett terms with coefe cients determined from linearized stability analysis to ensure stability of the augmented Burnett equations to small wavelength disturbances ). The three-dimensional augmentedBurnettequationsareappliedtocomputethehypersonicblunt-bodye owsforvariousrangeofKnudsen numbers (0:0884 < ‐ Kn < ‐ 0:227) and Mach numbers (10 < ‐ M < ‐ 25:3). The computational results are compared with the Navier ‐Stokes solutions, the existing augmented Burnett solutions, and the available DSMC results. The comparisons show that the difference between the Navier ‐Stokes and the augmented Burnett solutions is very small (less than 3% for the shock layer thickness ) at Knudsen numbers less than 0.01; the difference becomes signie cant as the Knudsen number increases. The comparisons also show that the augmented Burnett solutions are signie cantly closer to the DSMC results for the temperature distributions in the continuum-transition regime than the Navier ‐Stokes calculations.

BURNETT SIMULATIONS OF FLOWS IN MICROGEOMETRIES

In recent years, there has been considerable interest in computing gas flows at high Knudsen numbers in microgeometries. At low Knudsen numbers, models based on the solution of compressible Navier-Stokes equations with slip boundary conditions are adequate. At high Knudsen numbers, either higher-order (beyond Navier- Stokes) continuum equations or the particle methods such as Direct Simulation Monte Carlo (DSMC) are employed to compute the flows. Higher-order continuum approximations are based on the Chapman-Enskog expansion of Boltzmann equation (leading to Burnett and super-Burnett equations), or moment methods based on taking the moments of the Boltzmann equation with flow variables (leading to Grads 13 moments equations or Levermores moments equations for example). In this paper, the augmented Burnett calculations are presented for Couette flow, subsonic channel flow, and lid-driven cavity flow in microscale. These computations are compared with Navier-Stokes solutions with slip boundary conditions. The computations provide some assessment of Burnett equations for computing microscale flows at high Knudsen numbers. Nomenclature