We consider the initial-boundary value Euler equations with the aim to derive boundary conditions that yield an entropy bound for the physical (Navier-Stokes) entropy. We begin by reviewing the entropy bound obtained for standard no-penetration wall boundary conditions and propose a numerical implementation. The main results is the derivation of full-state boundary conditions (far-field, inlet, outlet) and the accompanying entropy stable implementations. We also show that boundary conditions obtained from linear theory are unable to bound the entropy and that non-linear bounds require additional boundary conditions. We corroborate our theoretical findings with numerical experiments.

我们考虑初始边界值Euler方程，目的是得出产生物理（Navier-Stokes）熵的熵界的边界条件。我们首先回顾为标准无穿透墙边界条件获得的熵界，并提出一个数值实现。主要结果是推导全状态边界条件（远场，入口，出口）和伴随的熵稳定实现。我们还表明，从线性理论获得的边界条件无法约束熵，并且非线性边界需要附加的边界条件。我们通过数值实验证实了我们的理论发现。