Regularization of the Chapman-Enskog Expansion and the Shock Structure Calculation

In order to extend the gas-kinetic BGK scheme for numerical solution of the compressible Navier-Stokes equations to the continuum transition flow regime, we propose to generalize the local particle collision time to depend not only on the local macroscopic flow variables, but also their gradients. Based on the gas-kinetic BGK model, the relation between the conventional collision time and the general one is obtained. With the truncation of the Chapman-Enskog expansion to the Navier-Stokes order, this generalized model is applied to the study of argon gas shock structure. There is good agreement between the predicted shock structure and experimental results for a wide range of Mach numbers. Also, some numerical results related to the application of the BGK scheme in the continuum flow regime will be presented. INTRODUCTION In the past years, the gas-kinetic BGK scheme has been successfully developed for the compressible Navier-Stokes equations, see [1] and references therein. The scheme not only captures accurately the Navier-Stokes solution when the mesh resolution is fine enough to resolve the physical structure, such as the computation of the NS shock structure and boundary layer, but also has excellent shock-capturing ability in case of under-resolveness of the physical structure with a coarse mesh. The kinematic and dynamic dissipative mechanism in the numerical schemes for the compressible flow calculations is analyzed in [1]. Even with the success of the gas-kinetic BGK scheme in the capturing of the Navier-Stokes solution, it is well recognized that the Navier-Stokes equations of the classical hydrodynamics are incapable of accurately describing shock wave phenomena and also for the flow phenomena in the rarefied regime. For example, the NS equations give a shock thickness which is thinner than that observed in the experiment [1]. In order to improve the NavierStokes solutions, much effort has been paid on the construction of higher-order hydrodynamic equations based on the Chapman-Enskog expansion. But the Burnett and super Burnett equations give unstable shock structures in high Mach number cases. For example, no shock structure can be obtained for the Burnett equations after a critical Mach number Mc 2 69 [2]. Even though the argumented Burnett of Zhong et. al. and BGK-Burnett equations of Agarwal et. al. can significantly improve the Navier-Stokes solutions in the continuum transition regime [3, 4, 5], it is unclear that the shock stability of these schemes are coming from the numerical dissipations, such as the use of Steger Warming flux splitting scheme for the inviscid part of the equations [6], or the selected higher-order terms. In order to get a stable solution, based on the mathematical analysis many authors also suggest new equations. As analyzed in [7], the failure of the Burnett equations for the shock structure calculation is not too surprising because the applicability of the Chapman-Enskog expansion itself is valid to the small Knudsen numbers. The possible generation of spurious solutions from the higher-order terms in the Chapman-Enskog expansion is another point for criticism [8]. This work is motivated originally by extending the gas-kinetic BGK Navier-Stokes solver [1] to the continuum transition regime. The direct adoption of the Chapman-Enskog expansion with the terms proportional to Knudsen number K2 n and K 3 n in the gas distribution function encounters great difficulty in the shock structure calculations. The critical Mach number for the shock structure based on the BGK-Burnett expansion is found to be around M c 4 5, and the number becomes even small, i.e. Mc 2 0, with the inclusion of super Burnett term. Our numerical experiments show clearly that the successive Chapman-Enskog expansions give divergent results as the Knudsen number increases. However, up to the Navier-Stokes order, there is no any limitation on the Mach number for the existence of the stable shock structure [2]. Therefore, it is reasonable to truncate the Chapman-Enskog expansion to the Navier-Stokes order only and include the possible non-equilibrium effect on the modification of the viscosity and heat conduction coefficients. Traditionally, the particle collision time τ is regarded as a function of macroscopic variables. For example, based on the BGK model [9], we have the collision time τ μ p, where μ is the dynamical viscosity coefficient, such as the Sutherland’s law, and p is the pressure. Those viscosity coefficients are basically obtained either experimentally or theoretically in the continuum flow regime [10]. There is no reason to guarantee that this relation is still applicable for the rarefied gas. In this paper, we are going to derive a general particle collision time τ , which is applicable in both continuum and continuum transition regime. This derivation is based on the closure of the Chapman-Enskog expansion on the Navier-Stokes order and the BGK equation. CLOSURE OF THE CHAPMAN-ENSKOG EXPANSION FOR THE BGK MODEL The BGK model in the x-direction can be written as [9]