使用双曲浅水矩方程捕捉径向对称流中的垂直信息
Capturing vertical information in radially symmetric flow using hyperbolic shallow water moment equations
AI总结：
本文针对二维轴对称浅水矩方程缺乏全局双曲性的问题，通过修改系统矩阵提出了一种保证双曲性的新模型，数值实验证明其能减少不连续初始条件下的振荡并随阶数提升降低误差。
摘要：
浅水流模型通常假设侧向速度在水深方向上是常数。最近推导的浅水矩方程是这些标准浅水方程的扩展。该扩展模型允许侧向速度在垂直方向上发生变化，从而在水平速度随流体高度变化显著的情况下，使得系统更加准确。然而，一维模型已经缺乏全局双曲性，这是偏微分方程的一个重要性质，确保扰动具有有限的传播速度。本文表明，双曲性的丧失也出现在二维轴对称系统中。首先，从柱坐标不可压缩 Navier-Stokes 方程出发，得到了一个柱坐标矩模型。通过在柱坐标模型中施加轴对称性，推导出二维轴对称浅水矩方程。通过直接评估传播速度并绘制双曲性区域，观察到双曲性的丧失。随后，类比于一维情况下已经观察到并解决了双曲性问题的方法，通过修改系统矩阵，获得了一个双曲轴对称矩模型。新模型以解析形式写出，我们证明了可以保证双曲性。为测试新模型，分别使用不连续和连续初始数据进行了数值模拟。结果表明，在不连续初始高度剖面的情况下，双曲模型比非双曲模型产生的振荡更少。在特定场景下，对于较短的时间尺度，双曲模型优于非双曲模型。此外，在具有平滑初始数据的测试案例中，我们观察到随着模型阶数的增加，误差会减小。
英文摘要：
Models for shallow water flow often assume that the lateral velocity is constant over the water height. The recently derived shallow water moment equations are an extension of these standard shallow water equations. The extended models allow for vertical changes in the lateral velocity, resulting in a system that is more accurate in situations where the horizontal velocity varies considerably over the height of the fluid. Unfortunately, already the one-dimensional models lack global hyperbolicity, an important property of partial differential equations that ensures that disturbances have a finite propagation speed. In this paper we show that the loss of hyperbolicity also occurs in two-dimensional axisymmetric systems. First, a cylindrical moment model is obtained by starting from the cylindrical incompressible Navier-Stokes equations. We derive two-dimensional axisymmetric Shallow Water Moment Equations by imposing axisymmetry in the cylindrical model. The loss of hyperbolicity is observed by directly evaluating the propagation speeds and plotting the hyperbolicity region. A hyperbolic axisymmetric moment model is then obtained by modifying the system matrix in analogy to the one-dimensional case, for which the hyperbolicity problem has already been observed and overcome. The new model is written in analytical form and we prove that hyperbolicity can be guaranteed. To test the new model, numerical simulations with both discontinuous and continuous initial data are performed. We show that the hyperbolic model leads to less oscillations than the non-hyperbolic model in the case of a discontinuous initial height profile. The hyperbolic model is preferred over the non-hyperbolic model in this specific scenario for shorter times. Further, we observe that the error decreases when the order of the model is increased in a test case with smooth initial data.
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