Supercavitating flow in multiply-connected domains

拟议的研究是密切相关的流体力学和计算复杂性分析的当代趋势，侧重于在自由边界域的超空泡绕流的水翼的数学建模。 其中心思想是利用黎曼曲面来模拟空腔后部的物理流动，并解决控制边值问题。这项技术需要发展和应用各种现代纯数学和应用数学工具，即保角映射法、施瓦茨反射原理、对称肖特基群理论、黎曼曲面上的黎曼-希尔伯特边值问题理论和对称自守函数。流体动力空化被定义为液体在低压下的破裂。 气穴会导致壁面腐蚀，改变水翼性能（例如，减少升力并增加水翼的阻力），并产生噪音和振动。 长期以来，空化不仅在造船和水力机械领域受到关注，而且还因为其在化学加工（电解沉积和乳液的产生）、物理学（颗粒在流体中的分散）和医学（设备表面的细菌破坏、治疗按摩和血管中的声空化引起的超声波的潜在生物效应）中的积极作用。该项目的总体目标是为多连通域中的自由边界问题开发一种统一的数学方法，以解决围绕一堆水翼的超空泡流的非线性模型问题。这些解将有助于理解空腔后部的流动机制，空腔如何相互作用，以及升力、阻力和空腔构型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.