This paper proposes an unconditionally energy stable method for incompressible heat conductive fluids under the phase–field framework. We combine the complicated system by the Navier–Stokes equation, Cahn–Hilliard equation, and heat transfer equation. A Crank–Nicolson type scheme is employed to discretize the governing equation with the second-order temporal accuracy. The unconditional energy stability of the proposed scheme is proved, which means that a significantly larger time step can be used. The Crank–Nicolson type discrete framework is applied to obtain the second-order temporal accuracy. We perform the biconjugate gradient method and Fourier transform method to solve the discrete system. Several computational tests are performed to show the efficiency and robustness of the proposed method.

本文提出了一种相场框架下不可压缩导热流体的无条件能量稳定方法。我们通过 Navier-Stokes 方程、Cahn-Hilliard 方程和传热方程组合了复杂的系统。采用 Crank-Nicolson 型方案以二阶时间精度离散控制方程。证明了所提出方案的无条件能量稳定性，这意味着可以使用明显更大的时间步长。应用 Crank-Nicolson 型离散框架来获得二阶时间精度。我们执行双共轭梯度法和傅里叶变换法来求解离散系统。进行了几个计算测试以显示所提出方法的效率和鲁棒性。