Supercavitating flow in multiply-connected domains

这项研究与流体力学和计算复分析的当代趋势密切相关，重点是自由边界区域内水翼超空泡流的数学模拟。中心思想是使用黎曼曲面对空腔后部的物理流动进行建模，并用于解决支配边值问题。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、Schwarz反射原理、对称肖特基群理论、黎曼曲面上的Riemann-Hilbert边值问题理论和对称自同构函数。流体动力空化的定义是液体在低压下的破裂。空化会导致壁面侵蚀，改变水翼的性能(例如，降低水翼的升力和增加水翼的阻力)，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引起人们的兴趣，而且还因为它在化学处理(电解沉积和乳化液的生产)、物理(粒子在液体中的分散)和医学(设备表面的细菌杀灭、治疗按摩和由于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区域中的自由边界问题发展一种统一的数学方法，以便解决绕水翼堆叠的超空泡流的非线性模型问题。这些解决方案将有助于了解空腔后部的流动机制、空腔如何相互作用以及升力、阻力和空腔形状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.