In this paper, a high order spectral difference-based phase field lattice Boltzmann method (SD-PFLBM) is proposed for simulating incompressible two-phase flows. The spectral difference method (SDM) is used to discretize the convection term and the gradient term of the discrete Boltzmann equation for obtaining the flow field. Moreover, the SDM is also adopted to discretize the convection term and the high order partial derivative term of the Cahn–Hilliard equation for interface tracking. The proposed method can overcome the drawback of the standard LBM such as tie-up between the time step and the mesh spacing. Meanwhile, the present method still holds the locality of the standard LBM because each cell only needs its own information to complete the discretization. Numerical validations of the proposed method are implemented by simulating rigid-body rotation of Zalesak’s disk, layered Poiseuille flows, bubble deformation in shear flow, Rayleigh–Taylor instability, and bubble merging. More satisfactory interface shapes and flow properties can be achieved as compared with the published data in the literature. In addition, the convergence studies are also given, which prove that the current SD-PFLBM can achieve high order accuracy by increasing the order of cell local polynomials.

中文翻译：

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基于高阶谱差的不可压缩两相流相场格玻尔兹曼方法

本文提出了一种基于高阶谱差的相场格子玻尔兹曼方法（SD-PFLBM）来模拟不可压缩的两相流。使用谱差法（SDM）离散离散Boltzmann方程的对流项和梯度项，以获得流场。此外，还采用SDM离散化Cahn-Hilliard方程的对流项和高阶偏导数，以进行界面跟踪。所提出的方法可以克服标准LBM的缺点，例如时间步长和网格间距之间的联系。同时，由于每个小区仅需要其自身的信息来完成离散化，因此本方法仍然保持标准LBM的局部性。通过模拟Zalesak圆盘的刚体旋转，分层的Poiseuille流，剪切流中的气泡变形，Rayleigh-Taylor失稳以及气泡合并，对所提出方法进行了数值验证。与文献中公开的数据相比，可以获得更令人满意的界面形状和流动特性。另外，还进行了收敛性研究，证明当前的SD-PFLBM通过增加单元局部多项式的阶数可以达到高阶精度。