Shock Waves in Gas Dynamics

气体动力学中的冲击波

基本信息

批准号：
EP/W001888/1
负责人：
Matthew Schrecker
金额：
$ 38.97万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW001888%2F1
关键词：
Shock Waves Gas Dynamics

项目摘要

The problem of understanding the nature and behaviour of shock waves in fluids has been a puzzle to mathematicians and physicists for centuries. The term fluids encompasses both liquids and gases and the flow of such fluids is described by the famous Euler equations, first written down by Leonhard Euler in the 1750s. In the 1840s, Stokes discovered a new phenomenon in the solutions of these equations: the shock wave. A shock wave occurs when there is a sudden, very rapid change in the density or velocity of a fluid over a very short space of time, described mathematically as a discontinuity in the solution. Since then, shocks have been discovered to be a ubiquitous phenomenon throughout the theory of gases, occurring naturally in situations ranging from the flow of gas in an exhaust pipe or a trumpet to sonic booms, supernovas and explosions. This Fellowship proposal aims to investigate two key problems within the broader theory of shocks: the shock reflection problem and the stability of blast waves. Shock reflection occurs when a shock wave meets a solid object and is reflected from it, the reflected part of the shock interacting with those parts of the shock that have not yet met the object. Thus the very simple initial scenario leads to a potentially very complex pattern of gas flow. In fact, the shock reflection problem is an example of a Riemann problem, the fundamental building blocks of the solutions to the Euler equations. Blast waves, on the other hand, are spherical shock waves that surround an area of very high pressure, for example caused by a supernova or explosion, and expand into a surrounding gas of much lower pressure. In the 1940s, independent work of von Neumann, Taylor and Sedov established that these shock waves could be described by solutions of the Euler equations, but the fundamental question of the stability of these solutions to small perturbations of the surrounding gas remains a significant open question. In both of these problems, deep connections between mathematics, physics and geometry come into play, as the underlying symmetries of the equations and geometric structures of the shocks interact. Moreover, both of these problems are free boundary problems because the location of the shock, which is a boundary for the region in which the equations hold, depends on the solution of the equations themselves. The results of the Fellowship will be of interest not only in Mathematical Analysis, but in the areas of Partial Differential Equations, Mathematical Biology, and Geometry also, where free boundary problems occur naturally and frequently.

几个世纪以来，理解流体冲击波的性质和行为的问题一直是数学家和物理学家的难题。术语流体包括液体和气体，以及此类流体的流动，由著名的Euler方程描述，莱昂哈德·欧拉（Leonhard Euler）在1750年代首次写下来。在1840年代，斯托克斯在这些方程式的解决方案中发现了一种新现象：冲击波。当在很短的时间段内，流体的密度或速度突然变化时，就会发生冲击波，从数学上描述为溶液中的不连续性。从那时起，在整个气体理论中，人们发现冲击是一种无处不在的现象，在气体理论中自然发生，从排气管中的气体或小号的气流到声音繁荣，超新星和爆炸。该奖学金提案旨在调查更广泛的冲击理论中的两个关键问题：冲击反射问题和爆炸波的稳定性。当冲击波遇到一个实体物体并反射出来时，发生冲击反射会发生，冲击的反射部分与尚未符合对象的冲击部分相互作用。因此，非常简单的初始场景导致了气流的潜在非常复杂的模式。实际上，冲击反射问题是Riemann问题的一个例子，即Euler方程解决方案的基本构建基础。另一方面，爆炸波是围绕非常高压区域的球形冲击波，例如由超新星或爆炸引起，并扩展到周围压力较低的气体中。在1940年代，冯·诺伊曼（Von Neumann），泰勒（Taylor）和塞多夫（Sedov）的独立工作确定，这些冲击波可以用欧拉方程的解决方案来描述，但是这些解决方案对周围气体小扰动的稳定性的基本问题仍然是一个重要的开放问题。在这两个问题中，随着电击的方程式和几何结构的基础对称性相互作用，数学，物理和几何形状之间的深厚连接开始起作用。此外，这两个问题都是自由边界问题，因为冲击的位置是方程所在的区域的边界，取决于方程本身的解决方案。奖学金的结果不仅在数学分析中引起了人们的关注，而且在部分微分方程，数学生物学和几何形状的领域中，自由边界问题自然而然地发生。