Zhang Qingdian, Jiang Mengyuan, Zhuo Congshan, Zhong Chengwen, Liu Sha
School of Aeronautics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China.
School of Aeronautics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China and National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China.
Phys Rev E. 2023 Nov;108(5-2):055309. doi: 10.1103/PhysRevE.108.055309.
In the multiphase flow simulations based on the lattice Boltzmann equation (LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation (WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.
在基于格子玻尔兹曼方程（LBE）的多相流模拟中，经常会观察到界面附近的虚假速度和不一致的密度特性。本文针对两相流开发了一种具有三阶埃尔米特展开的平衡正则化格子玻尔兹曼（WB-RLB）模型。为此，在考虑具有梯形积分的LBE时，将Guo [《物理流体》33, 031709 (2021)1070 - 663110.1063/5.0041446] 提出的平衡分布函数和修正力项直接引入到变换后的分布函数的正则化中。首先，为了详细比较平衡格子玻尔兹曼方程（WB-LBE）、WB-RLB和Lee与Fischer [《物理评论E》74, 046709 (2006)1539 - 375510.1103/PhysRevE.74.046709] 提出的二阶混合差分格式（SOMDS），对具有两种不同梯度算子（各向同性中心格式（ICS）和SOMDS）的LBE的力平衡进行了理论分析，并对静止液滴进行了数值模拟。力分析表明，对于力平衡，SOMDS比ICS能达到更高的精度，这在静止液滴情况的模拟中得到了验证。对于静止液滴情况，所有三个模型（WB-LBE、WB-RLB和SOMDS）即使在密度比为1000的大情况下也能捕捉到物理平衡状态。此外，对具有三阶展开的WB-RLB模型（WB-RLB3）的数值研究表明，调整三阶矩的松弛参数可以进一步提高WB-RLB模型的精度和稳定性。然后，考虑不同界面厚度的影响，对液滴聚并和相分离情况进行了研究，结果表明WB-RLB在两相动态问题上的性能仍然相当好，并且与WB-LBE相比表现出更好的数值稳定性。此外，用本文的WB-RLB模型研究了接触角问题；数值结果表明，预测的接触角值与解析解吻合良好，但平衡特性未得到验证，特别是在三相交界处附近。总体而言，本文的WB-RLB模型在静态和动态界面问题上均表现出优异的数值精度和稳定性。
