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

由于摩擦损失和河床变化，用带源项的双曲偏微分方程描述了水跃的动力学。传统上双曲系统的分析主要集中在没有非线性源项的系统上，因为双曲算子本身的非线性是相当复杂的，并且在解的性质中起着重要的作用。对于非线性源项的作用，如浅水方程和反应性欧拉方程中出现的非线性源项，人们的关注明显较少。这些源项造成了各种各样的现象，包括爆轰激波锋面的复杂动力学特征。在这个项目中，我们研究了由源项引起的非线性波浪相互作用在浅水系统中的作用，这可能是多边形水力跳跃形成的原因。这个项目关注的是可能从圆形水力跃变中出现的特殊流动结构的动力学。当垂直射流撞击平坦的固体表面时，射流在逐渐变薄的薄膜中径向扩散，直到达到一个临界半径，在这个半径处薄膜厚度急剧增加，这就是所谓的“液压跃变”。在某些参数条件下，尽管源条件是轴对称的，但仍会出现引人注目的不对称流动，包括多边形水力跳跃，其解释仍然难以捉摸。我们的综合理论，数值和实验项目将集中于通过发展水力跳跃和爆震前沿之间的数学类比来合理化这些微妙的流动。将开创和探索数学分析的一个新领域。

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.