Insight into enthalpy-based lattice Boltzmann method for solid-liquid phase change without numerical diffusion

Enthalpy-based lattice Boltzmann (LB) method is an effective approach for solid-liquid phase change problems. Meanwhile, numerical diffusion across the phase interface will degrade the numerical accuracy of the single-relaxation-time (SRT)-LB method. To address this deficiency, different multiple-relaxation-time (MRT) models and lattice structures with several free relaxation times are employed in the enthalpy-based LB method, which affects the numerical performance. In this work, systematic analyses of the enthalpy-based MRT-LB method are conducted and its connection with two-relaxation-time (TRT)-LB formulation is constructed. Derivation and validation demonstrate that to eliminate numerical diffusion, all the second-order relaxation factors should be equal and satisfy that the magic parameter, determined through the first- and second-order relaxation factors, equals 1/4. It is the basis that the MRT model can be reduced to the TRT counterpart. Under this condition, the simplified lattice structure-based MRT-LB method is equivalent to the TRT-LB method. For the MRT-LB method employing the standard lattice structure, besides the prerequisite of relaxation times within the first two orders, the high-order relaxation factors will affect the numerical accuracy. In the conventional and widely employed numerical implementation with standard lattice structure [R. Huang and H. Wu, J. Comput. Phys., 2015, 294: 346–362.], high-order odd and even relaxation factors are equal to first- and second-order factors, respectively. It can also be reduced to the TRT-LB method. Furthermore, based on the multiscale analysis within the TRT-LB framework, a unified enthalpy-based TRT-LB method without deviation terms is proposed by employing a linear distribution function and adopting an additional source term. Numerical diffusion across the phase interface can be reduced with the magic parameter equal to 1/4. The computational efficiency and numerical accuracy related to the numerical diffusion elimination and deviation terms reduction are validated through several benchmarks. The proposed unified TRT-LB method is recommended for solid-liquid phase transition.