Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations

Abstract

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the $Mach$ number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

Introduction

With the rapid development of the computer techniques, the numerical simulation is becoming a more important method in scientific research. Apart from the conventional numerical methods including the finite difference method [1], [2], the finite volume method [3], [4], [5] and the finite element method [6], [7], the mesoscopic numerical methods have a beautiful performance in many hydrodynamic problems. For examples, the lattice Boltzmann (LB) method [8], [9], [10], [11] and the unified gas kinetic scheme (UGKS) [12], [13], they are constructed at the mesoscopic scale and obtain the macroscopic variables by the moment conditions. Recently, the discrete unified gas kinetic scheme (DUGKS) was proposed by Guo et al. [14]. The scheme not only can adopt the flexible meshes and maintain the asymptotic preserving (AP) properties like UGKS, but also have a simplified flux calculation with the idea of LB method.

The incompressible Navier-Stokes (N-S) equations are widely used in the field of computational fluid dynamics (CFD) [15], such as the multiphase flow [16], the double-diffusive convection [17] and the thermal turbulence [18]. At the mesoscopic scale, many researchers have made a lot of contributions. He and Luo proposed a LB model for the incompressible N-S equations by explicitly eliminating the terms of $O(M{a}^2)$ [19]. However, when this model is used, the average pressure must be specified in advance. Then, Guo et al. proposed a lattice Bhatnagar-Gross-Krook (LBGK) model for incompressible N-S equations [20]. In this model, the pressure is independent of the density, and the incompressible N-S equations can be recovered by the LBGK model exactly. To improve the stability of the incompressible LBGK model, Du et al. developed an incompressible multi-relaxation-time lattice Boltzmann (IMRT-LB) model [21]. They found that the incompressible LBGK model would be unstable when the Reynolds number (Re) of lid-driven cavity flow increases up to 52900, but the IMRT-LB model can give a stable numerical result when $Re=100000$. It should be noted that, all the above LB models have a shortcoming, they can not be implemented on the non-uniform meshes. To overcome the shortcoming, Guo et al. [14] proposed the DUGKS where the non-uniform meshes can be adopted. However, the compressible effect still exists in the original DUGKS. To eliminate the compressible effect and include the force term, Wu et al. [22] developed a DUGKS for incompressible fluid flows which was inspired by the idea of He and Luo [19]. But they didn't demonstrate whether the DUGKS can recover the incompressible N-S equations, and the average pressure must be specified in advance. In addition, the non-equilibrium extrapolation (NEE) scheme used in their work can not satisfy the Dirichlet boundary conditions exactly.

To overcome the shortcomings left in the previous works, a new DUGKS is developed for the incompressible N-S equations. The Chapman-Enskog (CE) expansion is used to analyze the present DUGKS. Besides, a NEE scheme different from the previous work [22] is adopted in our simulations. Then, four different problems are used to test the capacity of the present DUGKS, and the numerical results show that the present DUGKS has a good performance in stability, accuracy and efficiency.

The rest of the paper is organized as follows. In Section 2, the DUGKS for incompressible N-S equations is presented. In Section 3, a steady state problem, an unsteady state problem and two classical benchmark problems are adopted to test the capacity of the present DUGKS. Finally, Section 4 summarizes the results and concludes the paper.