Hongwei Liu

A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows

We compare the lattice Boltzmann equation (LBE) and the gas-kinetic scheme (GKS) applied to 2D incompressible laminar flows. Although both methods are derived from the Boltzmann equation thus share a common kinetic origin, numerically they are rather different. The LBE is a finite difference method, while the GKS is a finite-volume one. In addition, the LBE is valid for near incompressible flows with low-Mach number restriction Ma<0.3, while the GKS is valid for fully compressible flows. In this study, we use the generalized lattice Boltzmann equation (GLBE) with multiple-relaxation-time (MRT) collision model, which overcomes all the apparent defects in the popular lattice BGK equation. We use both the LBE and GKS methods to simulate the flow past a square block symmetrically placed in a 2D channel with the Reynolds number Re between 10 and 300. The LBE and GKS results are validated against the well-resolved results obtained using finite-volume method. Our results show that both the LBE and GKS yield quantitatively similar results for laminar flow simulations, and agree well with existing ones, provided that sufficient grid resolution is given. For 2D problems, the LBE is about 10 and 3 times faster than the GKS for steady and unsteady flow calculations, respectively, while the GKS uses less memory. We also observe that the GKS method is much more robust and stable for under-resolved cases due to its upwinding nature and interpolations used in calculating fluxes.

A Runge-Kutta discontinuous Galerkin method for viscous flow equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gas-kinetic BGK model and the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method.

Multiple-temperature kinetic model for continuum and near continuum flows

A gas-kinetic model with multiple translational temperature for the continuum and near continuum flow simulations is proposed. The main purpose for this work is to derive the generalized Navier-Stokes equations with multiple temperature. It is well recognized that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies may be partially due to the single temperature assumption in the standard Navier-Stokes equations. Here, based on an extended Bhatnagar-Gross-Krook (BGK) model with multiple translational temperature, the numerical scheme for its corresponding Navier-Stokes equations is also constructed. In the current approach, the energy exchange between x, y, and z directions is modeled through the particle collision, and individual energy equation in different direction is obtained. The kinetic model, newly constructed is an enlarged system in comparison with Holway’s ellipsoid statistical BGK model (ES-BGK)....

A gas-kinetic discontinuous Galerkin method for viscous flow equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at the cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gaskinetic BGK model and the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method.

Multiple Temperature Model for Near Continuum Flows

In the near continuum flow regime, the flow may have different translational temperatures in different directions, it is well known that for increasingly rarefied flow fields, the predictions from continuum formulation, such as the Navier-Stokes equations, lose accuracy. These inaccuracies may be partially due to the single temperature assumption in the Navier-Stokes equations. Here, based on the gas-kinetic Bhatnagar-Gross-Krook (BGK) equation, a multitranslational temperature model is proposed and used in the flow calculations. In order to fix all three translational temperatures, two constraints are additionally proposed to model the energy exchange in different directions. Based on the multiple temperature assumption, the Navier-Stokes relation between the stress and strain is replaced by the temperature relaxation term, and the Navier-Stokes assumption is recovered only in the limiting case when the flow is close to the equilibrium with the same temperature in different directions. In order to validate the current model, both the Couette and Poiseuille flows are studied in the transition flow regime.