Supercavitating flow in multiply-connected domains

本研究与当代流体力学和计算复杂分析的发展趋势密切相关，重点研究自由边界域中水翼周围超空泡流动的数学建模。中心思想是利用黎曼曲面对腔体后部的物理流动进行建模，并解决控制边值问题。该技术需要发展和实现各种纯数学和应用数学的现代工具，即共形映射方法，施瓦茨反射原理，对称肖特基群理论，以及黎曼曲面和对称自同构函数上的黎曼-希尔伯特边值问题理论。流体动力空化被定义为液体在低压下的破裂。空化引起壁面侵蚀，改变水翼的性能（例如，降低升力并增加水翼的阻力），并产生噪音和振动。空化不仅在造船和液压机械领域引起了人们的兴趣，而且还因为它在化学加工（电解沉积和乳液的生产）、物理（流体中颗粒的分散）和医学（设备表面的细菌破坏、治疗性按摩和血管中声波空化引起的超声波的潜在生物效应）方面的积极作用。本项目的总体目标是建立一个统一的数学方法来解决多连通域的自由边界问题，以解决水翼堆周围超空泡流动的非线性模型问题。这些解决方案将有助于了解空腔后部的流动机制，空腔之间如何相互作用，以及升力、阻力和空腔结构如何取决于壁面和自由表面的接近程度。

The proposed research is closely related to contemporary trends in fluid mechanics and computational complex analysis, focusing on mathematical modeling of supercavitating flow around hydrofoils in free boundary domains. The central idea is the use of Riemann surfaces for the modeling of physical flow in the rear part of a cavity and for the solution of the governing boundary-value problems. The technique requires the development and implementation of diverse modern tools of pure and applied mathematics, namely, the conformal mapping method, the Schwarz reflection principle, the theory of symmetric Schottky groups, and the theory of the Riemann-Hilbert boundary-value problem on Riemann surfaces and for symmetric automorphic functions.Hydrodynamic cavitation is defined as the breakdown of a liquid under low pressure. Cavitation causes wall erosion, alters hydrofoil performance (reduces lift and increases drag of a hydrofoil for example) and produces noise and vibration. Cavitation has long been of interest not only in the field of shipbuilding and hydraulic machinery, but also because of its positive effects in chemical processing (electrolytic deposition and the production of emulsions), physics (dispersion of particles in a fluid) and medicine (bacteria destruction from the surfaces of equipment, therapeutic massage and potential bioeffects of ultrasound caused by acoustic cavitation in blood vessels). The overall goal of this project is to develop a unified mathematical approach for free boundary problems in multiply-connected domains in order to solve non-linear model problems of supercavitating flow around a stack of hydrofoils. These solutions will help obtain an understanding of the mechanism of flow in the rear parts of cavities, of how the cavities interact with each other, and of how the lift, drag and the cavity configuration depend on the proximity of the walls and the free surfaces.