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

水跃稳定性：分析、计算和实验

• 批准号: 0907955
• 负责人: John Bush
• 金额: $ 36万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907955&HistoricalAwards=false
• 关键词: stability; hydraulic; jumps; analysis; computation

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.

由于摩擦损失和河床变化，水跃动力学可用带源项的双曲型偏微分方程组来描述。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。对于诸如浅水方程和反应欧拉方程中出现的那些非线性源项的作用，人们关注的明显较少。这些源项导致了各种各样的现象，包括爆轰激波锋面复杂的动力学特征。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，这可能是多边形水跃形成的原因。当垂直射流撞击平坦的固体表面时，射流流体在稀释膜中径向扩散，直到达到临界半径，膜厚急剧增加，这就是所谓的“水力跳跃”。在某些参数条件下，尽管源是轴对称的，但出现了惊人的不对称流动，包括多边形水跃，其原因仍然难以捉摸。我们的理论、数值和实验相结合的项目将致力于通过发展水力跳跃和爆轰激波锋面之间的数学类比来使这些微妙的流动合理化。将开创和探索数学分析的一个新领域。