In spite of the increasing applications of the lattice Boltzmann method (LBM) in simulating various flow and transport systems in recent years, complex boundary conditions for the convection-diffusion and heat transfer processes in LBM have not been well addressed. In this paper, we propose an improved bounce-back method by using the midpoint concentration value to modify the bounced-back density distribution for LBM simulations of the concentration field. An accurate finite-difference scheme in the normal boundary direction has also been introduced for gradient boundary conditions. Compared with existing boundary methods, our method has a simple algorithm and can easily deal with boundaries with general geometries, motions, and surface conditions (the Dirichlet, Neumann, and mixed conditions). Carefully designed simulations are performed to examine the capacity and accuracy of this proposed boundary method. Simulation results are compared with those from theory and a representative boundary method, and an improved performance is observed. We have also simulated the effect of reference velocity on global accuracy to examine the performance of our model in preserving the fundamental Galilean invariance. These boundary treatments for concentration boundary conditions can be readily applied to other processes such as heat transfer systems.

Effects of Reynolds and Prandtl Numbers on Heat Transfer Around a Circular Cylinder by the Simplified Thermal Lattice Boltzmann Model - Volume 17 Issue 4 - Qing Chen, Xiaobing Zhang, Junfeng Zhang

In this paper, a coupled discrete unified gas kinetic scheme (CDUGKS) with a flexible Prandtl number is developed for the thermal compressible flows in all Knudsen number regimes. Different from the existing thermal discrete unified gas kinetic scheme based on the Shakhov model, the proposed CDUGKS based on the total energy double-distribution-function model can well preserve the nonnegative property of the distribution function, especially for the strong shock in the continuum regime. In the CDUGKS, the velocity distribution function (VDF) is used to recover the compressible continuity and momentum equations, while the energy distribution function (EDF) is used to recover the energy equation. The VDF and EDF are evaluated in a similar way and then coupled via the thermal equation of state. With the un-splitting treatment of the particle transport and collision in the distribution function evolution and the flux evaluation, the time step in CDUGKS is not limited by the particle collision time. Furthermore, the CDUGKS is an asymptotic preserving scheme, in which the Navier-Stokes solution in the hydrodynamic regime and the free transport mechanism in the kinetic regime can be precisely recovered with the second-order accuracy in both space and time. Finally, several numerical experiments, including the weak shock tube and the strong one in the whole Knudsen number flows, as well as the two-dimensional Riemann problem and the Rayleigh-Taylor instability in both hydrodynamic regime and kinetic regimes, are performed to validate the method. Numerical results agree fairly well with other benchmark results in different flow regimes, which demonstrates the current CDUGKS is a reliable and efficient method for multiscale flow problems.

In this paper, we present a finite-volume direct kinetic method, the so-called discrete unified gas kinetic scheme (DUGKS), for electrostatic plasma. One key feature of this method is the semi-implicit unsplitting treatment of particle transport and collision, and thus the time step in current DUGKS is not limited by the particle collision time. In addition, a fourth-order compact MUSCL scheme with a positivity preserving limiter is implemented in the interface reconstruction, which enables present DUGKS to preserve the favorable conservative property and positivity of distribution function. Combined with this MUSCL method, the semi-Lagrangian scheme is used for the particle transport in velocity space to remove Courant-Friedricks-Lewy restriction induced by the large electric force. As a result, the proposed DUGKS becomes an efficient and stable multiscale scheme. Several numerical experiments, including plasma sheath, linear Landau damping, collisional nonlinear Landau damping, and plasma ion acceleration, are performed to validate current DUGKS. A comparative study of current DUGKS with a general particle in cell (PIC) method which could handle particle collision in a conservative way is also presented. Theory and numerical experiments demonstrate that DUGKS is preferred for plasma flows involving small electrostatic perturbation and high collision regimes, while the PIC method is desired for the field- dominated plasma flows involving a wide range of velocities.