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.

Cahn-Hilliard-Navier-Stokes (CHNS) 方程代表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，需要数值求解。传统上，使用耦合项的数值外推法。然而，这种蛮力外推通常会破坏该 CHNS 系统的内在热力学结构。本文提出了一种将 CHNS 系统重新构造为约束梯度流公式的新策略，其中清楚地揭示了可逆和不可逆结构。这指导我们提出具有几个有利特性的算子拆分方案。首先，提议的方案导致在每个时间行进步骤中解决几个较小尺寸的解耦系统。这显着降低了计算成本。其次，所提出的方案仍然在离散水平上保证了 CHNS 系统的热力学规律。此外，与最近使用辅助变量填充的 IEQ 或 SAV 方法不同，我们生成的能量定律是在原始变量中制定的。这是一项重大改进，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。我们提出的框架为设计用于流体动力相场模型的解耦和能量稳定数值算法奠定了基础。此外，给定不同的分裂步骤，可以获得各种数值算法，使得这个框架相当通用。实现了所提出的数值算法。它们的二阶时间和空间精度已通过数值验证。计算了一些数值例子和基准问题，以验证所提出方案的有效性。