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.