Approximate mathematical models in high-speed hydrodynamics

Approximate solutions of some problems in high-speed hydrodynamics are given, the solutions being based upon well-known approaches, such as the principle of independence of cavity expansion (Logvinovich), formulation of the problem of the immersion of a solid contour into liquid (Wagner), various models of cavity closure in its tail, etc. Theoretical studies of the dynamic properties of slender ventilated cavities are performed. The mathematical model of a cavity is obtained in the form of a system of nonlinear time-delay differential equations. The linear theory of cavity stability and oscillations is developed for various cavity types. The mechanism of nonlinear cavity oscillations accounting for gas-bubble detachment is considered, and the results of extensive numerical experimentation are presented. A theoretical model of cavity closure is proposed that develops the well-known Efros approach with a re-entrant jet. An approximate analysis of the model has been performed. A planar problem of the impact and immersion of an expanding cylinder into liquid with a cylindrical free surface of variable radius is solved in Wagner’s formulation.