The stability of hydraulic jumps: analysis, computation, and experiment

水跃的动力学是由双曲型偏微分方程描述的，由于摩擦损失和河床变化的源项。双曲方程组的分析传统上集中在没有非线性源项的方程组上，因为双曲算子本身的非线性相当复杂，并且在解的性质中发挥着重要作用。对于浅水方程和反应欧拉方程中的非线性源项的作用，人们的关注明显较少。这样的源项是负责的各种各样的现象，包括复杂的动力学特征的爆轰激波阵面。在本计画中，我们将探讨浅水系统中源项所引起的非线性波浪相互作用的作用，这可能是多边形水跃形成的原因。本计画关注圆形水跃可能产生的特殊水流结构的动力学。当垂直射流冲击在一个平坦的固体表面上时，射流流体在薄膜中径向扩散，直到达到临界半径，在该临界半径处，薄膜厚度急剧增加，这被称为“水力跃变”。在某些参数制度，尽管轴对称的源条件下，显着的不对称流出现，包括多边形的水力跳跃，其解释仍然难以捉摸。我们的理论，数值和实验相结合的项目将集中在合理化这些微妙的流动，通过发展之间的数学类比水力跳跃和爆震冲击波阵面。数学分析的一个新领域将被启动和探索。

The dynamics of hydraulic jumps is described by hyperbolic partial differential equations with source terms due to frictional losses and channel-bed variations. The analysis of hyperbolic systems has traditionally been focused on systems without non-linear source terms, as the nonlinearities in the hyperbolic operators themselves are rather intricate, and play an important role in the nature of the solutions. Significantly less attention has been paid to the role of nonlinear source terms such as those arising in the shallow-water equations and in the reactive Euler equations. Such source terms are responsible for a rich variety of phenomena, including the complex dynamical features of detonation shock fronts. In this project, we investigate the role of the nonlinear wave interactions arising from source terms in the shallow-water system, which may be responsible for the formation of polygonal hydraulic jumps.This project concerns the dynamics of peculiar flow structures that may emerge from the circular hydraulic jump. When a vertical jet impinges on a flat solid surface, the jet fluid spreads radially in a thinning film until reaching a critical radius at which the film thickness increases dramatically in what is termed a `hydraulic jump'. In certain parameter regimes, despite the axisymmetric source conditions, striking asymmetric flows emerge, including polygonal hydraulic jumps, the explanation for which remains elusive. Our combined theoretical, numerical and experimental project will be focused towards rationalizing these subtle flows by developing the mathematical analogy between hydraulic jumps and detonation shock fronts. A new area of mathematical analysis will be initiated and explored.