Derivation of stable Burnett equations for rarefied gas flows

Onsager-regularized lattice Boltzmann method: A nonequilibrium thermodynamics-based regularized lattice Boltzmann method

The regularized class of lattice Boltzmann methods (LBMs) leverage the potency of the standard lattice-Bhatnagar-Gross-Krook method by filtering out spurious nonhydrodynamic moments from the moment space; this is achieved through evaluating regularized populations via a multiscale or a Hermite polynomial expansion approach. In this paper, we propose an alternative approach for evaluating the lattice populations. This approach is based on a kinetic theory that is consistent with nonequilibrium thermodynamics and obeys the Onsager-reciprocity principle. The proposed method is verified and validated for a number of canonical problems such as the athermal shock tube, the double periodic shear layer, the lid driven cavity, flow past square cylinder, and Poiseuille flow at nonvanishing Knudsen numbers. Additionally, the proposed method is compared to existing regularized LBM schemes and is shown to yield significant improvement in the stability and accuracy of the simulations.

Strong shock as a stringent test for Onsager-Burnett equations

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)2470-004510.1103/PhysRevE.96.013106] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

Exact solutions for shock waves in dilute gases

In 1922 Becker found an exact solution for shock waves in gases using the Navier-Stokes-Fourier constitutive equations for a Prandtl number of value 3/4 with constant transport coefficients. His analysis has been extended to study some cases where an implicit solution can be found in an exact way. In this work we consider this problem for the so-called soft-spheres model in which the viscosity and thermal conductivity are proportional to a power of the temperature η,κ∝T^{σ}. In particular, we give implicit exact solutions for the Maxwell model (σ=1), hard spheres (σ=1/2), and when σ (the viscosity index) is a natural number.