Second-order Decoupled Energy-stable Schemes for Cahn-Hilliard-Navier-Stokes equations
Second-order Decoupled Energy-stable Schemes for Cahn-Hilliard-Navier-Stokes equations
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DOI：
10.1016/j.jcp.2021.110536
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Jia Zhao
中科院分区：
文献类型：
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作者：
Jia Zhao
Cahn-Hilliard-Navier-Stokes（CHNS）方程代表了多相流体流动动力学的流体动力学相场模型的基本构建块。由于Navier-Stokes方程和Cahn-Hilliard方程之间的耦合，CHNS系统是非平凡的数值求解。传统上，使用耦合项的数值外推。然而，这样的蛮力外推通常会破坏该CHNS系统的内在热力学结构。本文提出了一种新的策略，将CHNS系统转化为约束梯度流形式，其中可逆和不可逆结构被清楚地揭示出来。这指导我们提出运营商分裂计划，有几个有利的属性。首先，所提出的计划导致几个解耦的系统在较小的尺寸要解决在每个时间推进步骤。这大大降低了计算成本。其次，所提出的方案仍然保证CHNS系统在离散水平上的热力学定律。此外，与最近流行的IEQ或SAV方法使用辅助变量不同，我们得到的能量定律在原始变量中制定。这是一个重大的改进，因为带有辅助变量的修正能量定律有时会偏离原始能量定律。该框架为流体力学相场模型的解耦和能量稳定的数值算法设计奠定了基础。此外，各种数值算法可以得到不同的分裂步骤，使这个框架相当普遍。所提出的数值算法的实施。它们的二阶时间和空间精度的数值验证。数值算例和基准问题的计算结果验证了所提格式的有效性。
The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force extrapolation usually destroys the intrinsic thermodynamic structures of this CHNS system. This paper proposes a new strategy to reformulate the CHNS system into a constraint gradient flow formulation, where the reversible and irreversible structures are clearly revealed. This guides us to propose operator splitting schemes that have several advantageous properties. First of all, the proposed schemes lead to several decoupled systems in smaller sizes to be solved at each time marching step. This significantly reduces computational costs. Secondly, the proposed schemes still guarantee the thermodynamic laws of the CHNS system at the discrete level. In addition, unlike the recently populated IEQ or SAV approaches using auxiliary variables, our resulting energy laws are formulated in the original variables. This is a significant improvement, as the modified energy laws with auxiliary variables sometimes deviate from the original energy law. Our proposed framework lays a foundation for designing decoupled and energy stable numerical algorithms for hydrodynamic phase-field models. Furthermore, various numerical algorithms can be obtained given different splitting steps, making this framework rather general. The proposed numerical algorithms are implemented. Their second-order temporal and spatial accuracy are verified numerically. Some numerical examples and benchmark problems are calculated to verify the effectiveness of the proposed schemes.