A high order spectral difference-based phase field lattice Boltzmann method for incompressible two-phase flows

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.