In this article we propose the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system that models thermal convection of two-phase flows in a fluid layer overlying a porous medium. Based on operator splitting and pressure stabilization we propose a family of fully decoupled numerical schemes such that the Navier–Stokes equations, the Darcy equations, the heat equation and the Cahn–Hilliard equation are solved independently at each time step, thus significantly reducing the computational cost. We show that the schemes preserve the underlying energy law and hence are unconditionally long-time stable. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.

在本文中，我们提出了Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq系统，该系统对覆盖多孔介质的流体层中两相流的热对流进行建模。基于操作员分裂和压力稳定，我们提出了一系列完全解耦的数值方案，以便在每个时间步都独立求解Navier–Stokes方程，Darcy方程，热方程和Cahn–Hilliard方程，从而显着减少了计算量成本。我们表明，这些方案保留了基本的能量定律，因此无条件长期稳定。数值结果表明了算法的准确性和稳定性。