Shock formation and interface motion in fluids

自然界和技术中的许多流动模式都以尖锐界面的存在为特征。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波前，以及流体在其上经历压力、温度和速度的显著变化的冲击波前。该项目将侧重于流体动力学的三个基本领域：（1）导致冲击波形成的机制的精确描述和由此产生的冲击奇异性的详细描述，（2）水波的演化（如海浪）与时间相关的和可变形的角度波峰和解释的形成尖点，在这些波，以及（3）发展快速运行的计算模型，用于化学反应的两种气体之间的物质界面的运动，并描述这些气体的最终混合区。这些模型的运行速度将比传统的数值方案快100倍，可用于模拟火球，野火，云和各种其他反应流。本计画亦将提供研究机会给研究生，本计画将发展解析与数值方法来研究几个重要的流体力学问题，包括锐界面：（1）详细研究三维可压缩欧拉方程的冲击波形成与传播。这应通过建立调制自相似变量中一般激波剖面的渐近稳定性来实现，通过以下方式控制波族的相互作用：（i）沿沿着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)二维自由面旋转不可压欧拉方程中的波峰波和有限时间内尖点奇点的形成。(3)在存在燃烧时，发展经典瑞利-泰勒（RT）、开尔文-亥姆霍兹（KH）和里希特迈尔-梅什科夫（RM）不稳定性的快速运行计算模型。这些不稳定性是高度不稳定的，并且使用传统的计算工具进行模拟非常昂贵。对于RT，在系统进入非线性区域之前，材料界面的小扰动最初根据线性理论增长，在非线性区域中，轻流体起泡进入重流体，而重流体尖峰进入轻流体。由此产生的流动的速度在该界面处是不连续的，这引发KH。这导致界面卷起形成复杂的旋涡结构，并最终导致湍流混合。在存在燃烧的情况下，产生大的且不稳定的混合区。这些不稳定性中的每一种都出现在许多重要的应用中，包括天体物理学、惯性约束聚变和海洋混合。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.