封闭水管中瞬变流中的空气夹带：两层方法
Air entrainment in transient flows in closed water pipes: a two-layer approach
摘要：
在本文中，我们首先建立一个自由表面流动模型，考虑到空气卷吸的四个偏微分方程的系统。我们通过取欧拉方程中跨表面流动的气体和流体速度的平均值（流体不可压缩，气体可压缩）来推导它。所得到的系统是条件双曲的。然后，我们提出了一个数学动力学解释这个系统，最终构建一个两层的动力学计划，其中的“失踪”的边界条件进行了特殊的处理。几个封闭的水管进行数值试验，并讨论和说明双曲线的损失的影响。最后，我们对动力学方法在非双曲系统中的阶进行了数值研究。这在空间网格步长变为零时提供了有用的稳定性结果。
英文摘要：
In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the "missing" boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero.
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