In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

翻译：本文提出一个完整的框架，用于使用一维两相流模型对通道内分层不可压缩流动进行能量稳定模拟。在早期针对基础两相流模型能量守恒研究的基础上，我们的新框架包含了扩散、摩擦和表面张力。我们证明了表面张力可以以能量守恒的方式加入，而扩散和摩擦对能量具有严格的耗散效应。随后，我们针对这些项提出空间离散化方案，使得半离散模型具备与连续模型相同的守恒特性。此外，我们提出一种新的能量稳定平流通量方案，该方案在流动光滑区域呈能量守恒特性，而在尖锐梯度出现处呈严格耗散特性。这是通过结合使用通量限制器，将先前开发的能量守恒平流通量与一种被证明呈严格耗散的新型一阶迎风格式而实现的。包含扩散、表面张力和有界能量的完整框架对短波长扰动呈线性稳定，并在激波附近表现出非线性阻尼。即使是在基础两相流模型不适定条件下，该模型也能产生平滑收敛的数值解。借助耗散率的显式表达式，我们能够将非线性阻尼归因于不同的耗散机制，并比较它们的效果。