Shock formation and interface motion in fluids

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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.