A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations

Abstract

Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gaskinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is * Corresponding author mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it is still necessary to develop more accurate flow solver in the rarefied regime, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the flow study in this regime. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well.

Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations

“…Other approaches include the exponential integrator based methods [5,21], or micro-macro (MM) decomposition [2]. We also mention relevant works [33,23,28]. One should note that [2,33] only dealt with the BGK model, rather than the full Boltzmann equation.…”

A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures: Homogeneous relaxation and transport coefficients

“…Also, the FSM can be used to assess the accuracy of quantum kinetic models [43,44,45]. Furthermore, the FSM can be incorporated into other multi-scale methods [46,47] that solve the Boltzmann equation accurately and 24 efficiently from the hydrodynamic to free-molecular flow regimes, which is frequently encountered in experiments where the quantum gas is trapped so that its density is maximum at the trap center (i.e. hydrodynamic regime) and vanishes near the trap edge (i.e.…”

Molecular-level study of compressible gaseous continua

“…As analyzed by Xu [23,24], all hydrodynamic equations such as NSF are based on the fluid element assumption. Due to the individual isolated element approximation, there is only pressure, viscous friction and heat exchange between the elements, but no mass or molecules penetration, such as the absence of mass diffusion term.…”