Shock formation and interface motion in fluids

自然界和技术上的许多流动模式都是通过存在尖锐的界面来区分的。这种模式的例子包括表面水波、分离具有不同密度、速度或温度的流体的不稳定锋面，以及流体在其上经历压力、温度和速度显著变化的冲击波前沿。这个项目将集中在流体动力学的三个基本领域：(1)精确描述导致激波形成的机制和由此产生的激波奇异性，(2)具有随时间变化的和可变形的角度波峰的水波(如海浪)的演变，以及对这些波中尖峰的形成的解释，以及(3)开发用于两种气体之间的化学反应的物质界面运动的快速计算模型，并描述这些气体的结果混合区。这些模型的运行速度大约是传统数值格式的100倍，可以用来模拟火球、野火、云层和各种其他反应流动。该项目还将为研究生提供研究机会。本项目将开发分析和数值方法来研究涉及尖锐界面的几个重要流体力学问题：(1)详细研究三维可压缩欧拉方程的激波形成和传播。这将通过建立调制自相似变量中一般激波轮廓的渐近稳定性，通过以下方式控制波族的相互作用来实现：(I)沿拉格朗日轨迹的逐点边界，(Ii)几何涡度结构，以及(Iii)Sobolev空间中的高阶能量估计。(2)研究了二维自由表面旋转不可压缩欧拉方程中的峰波及其在有限时间内尖点奇点的形成。(3)建立了燃烧条件下经典Rayleigh-Taylor(RT)、Kelvin-Helmholtz(KH)和Richtmyer-Meshkov(RM)不稳定性的快速计算模型。这些不稳定性非常不稳定，使用传统计算工具模拟非常昂贵。对于RT，材料界面的微小扰动最初根据线性理论增长，然后系统进入非线性区域，在该区域中，轻流体气泡进入重流体，而重流体进入轻流体。由此产生的流动的速度在这个界面上是不连续的，这引发了KH。这会导致界面卷曲成复杂的旋涡结构，并最终导致湍流混合。在燃烧存在的情况下，会形成一个巨大的不稳定混合区。这些不稳定性中的每一个都出现在许多重要的应用中，包括在天体物理学、惯性约束聚变和海洋混合中。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many flow patterns in nature and technology are distinguished by the presence of sharp interfaces. Examples of such patterns include surface water waves, unstable fronts separating fluids with different densities, velocities, or temperatures, and shock wave fronts across which the fluid experiences significant changes in pressure, temperature, and velocity. This project will focus on three fundamental areas of fluid dynamics: (1) a precise description of the mechanisms that lead to shock wave formation and a detailed description of the resulting shock singularity, (2) the evolution of water waves (such as ocean waves) with time-dependent and deformable angled crests and an explanation for the formation of cusps in these waves, and (3) the development of fast-running computational models for the motion of material interfaces between two gases that chemically react and a description of the resulting mixing zone of these gases. These models will run about 100 times faster than traditional numerical schemes, and can be used to model fireballs, wildfires, clouds, and variety of other reacting flows. The project will also provide research opportunities for graduate students.This project will develop analytical and numerical methods to study several important fluid mechanics problems involving sharp interfaces: (1) A detailed study of the formation and propagation of shock waves for the 3D compressible Euler equations. This shall be achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces. (2) Study of crested waves in the 2D free-surface rotational incompressible Euler equations and formation of cusp singularities in finite time. (3) Development of fast-running computational models of the classical Rayleigh-Taylor (RT), Kelvin-Helmholtz (KH), and Richtmyer-Meshkov (RM) instabilities in the presence of combustion. These instabilities are highly unstable and very expensive to simulate using traditional computational tools. For RT, small perturbations of the material interface initially grow according to the linear theory, before the system enters the nonlinear regime, in which the light fluid bubbles into the heavy fluid, while the heavy fluid spikes into the light fluid. The velocity of the resulting flow is discontinuous at this interface, which initiates KH. This causes the interface to roll up into complex vortical structures and eventually leads to turbulent mixing. In the presence of combustion, a large and unstable mixing zone is created. Each of these instabilities arises in numerous important applications, including in astrophysics, inertial confinement fusion, and ocean mixing.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.