Improved axisymmetric lattice Boltzmann scheme

Q. Li, Y. L. He, G. H. Tang, and W. Q. Tao National Key Laboratory of Multiphase Flow in Power Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

This paper proposes an improved lattice Boltzmann scheme for incompressible axisymmetric flows. The scheme has the following features. First, it is still within the framework of the standard lattice Boltzmann method using the single-particle density distribution function and consistent with the philosophy of the lattice Boltzmann method. Second, the source term of the scheme is simple and contains no velocity gradient terms. Owing to this feature, the scheme is easy to implement. In addition, the singularity problem at the axis can be appropriately handled without affecting an important advantage of the lattice Boltzmann method: the easy treatment of boundary conditions. The scheme is tested by simulating Hagen-Poiseuille flow, three-dimensional Womersley flow, Wheeler benchmark problem in crystal growth, and lid-driven rotational flow in cylindrical cavities. It is found that the numerical results agree well with the analytical solutions and/or the results reported in previous studies.

PACS: 47.11.-j

Ⅰ. INTRODUCTION Because of its kinetic nature and distinctive computational features, the lattice-Boltzmann (LB) method, which originates from the lattice-gas automata (LGA) method [1], has been developed into a

1 very attractive alternative to conventional numerical methods. In the LB method, instead of solving the macroscopic governing equations, the discrete Boltzmann equation with certain collision models, such as the matrix model [2, 3], Bhatnagar-Gross-Krook (BGK) model [4-7], multiple-relaxation-time (MRT) model [8-13], and the two-relaxation-time (TRT) model [14-16], is solved to simulate fluid flows and model physics in fluids. In the literature, the main advantages of the LB method are summarized as follows [17]: (i) non-linearity (collision process) is local and non-locality (streaming process) is linear, while in the Navier-Stokes equation the convective term u u ∇ is non-linear and non-local at a time; (ii) streaming is exact; (iii) complex boundary conditions can be easily formulated in terms of elementary mechanics rules; (iv) fluid pressure and the strain tensor are available locally; (v) nearly ideal amenability to parallel computing (low communication/computation ratio). Owing to these advantages, in the past two decades the LB method has been successfully applied to various flow problems in science and engineering [18-24] In recent years, the LB method for axisymmetric flows has attracted much attention. It is known that LB simulations of axisymmetric flows can be handled with a standard three-dimensional (3D) LB model. However, such a treatment does not take the advantage of the axisymmetric property of the flow: 3D axisymmetric flows are two-dimensional (2D) problems in a cylindrical coordinate system. To make use of this property, much research has been conducted. The first attempt was made by Halliday et al. [25]. The basic idea of Halliday et al.’s method is to incorporate spatial and velocity dependent source terms into the microscopic evolution equation to mimic the additional axisymmetric contributions in cylindrical coordinates. Following Halliday et al.’s work, Peng et al. [26] proposed a hybrid LB model for incompressible axisymmetric thermal flows by solving the azimuthal velocity and

2 the temperature with a second-order center-difference scheme. Nevertheless, it was later found that Halliday et al.’s model fails to reproduce the correct hydrodynamic momentum equation due to some missing terms. After considering these terms, Lee et al. [27] developed a more accurate axisymmetric LB model. Reis and Phillips [28] have also presented a modified version of Halliday et al.’s model by deriving the source terms in a different manner. The modified model was subsequently validated with several numerical tests [29]. In the above models, some complex differential terms were introduced into the second-order source term due to the discrete effects on the first-order source term (identical to a forcing term). These complex terms may introduce some additional errors and do harm to the numerical stability. He et al. [30, 31] have pointed out the trapezium rule is necessary for the integration of a forcing term to avoid the spurious effects in the recovered macroscopic equations. By using a new distribution function to eliminate the implicitness resulting from the trapezium rule, it can be found that a factor dependent on the relaxation time will be included in the forcing term and the macroscopic variables should be redefined [32]. Following this strategy, Premnath and Abraham [33] devised a LB scheme for axisymmetric multiphase flows. The scheme was extended to axisymmetric two-phase

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